LMFDB - Embedded newform 45.22.f.a.17.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,22,Mod(8,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.8");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$125.764804929$
Analytic rank:	$0$
Dimension:	$84$
Relative dimension:	$42$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.13
Character	$\chi$	$=$	45.17
Dual form			45.22.f.a.8.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-957.532 + 957.532i) q^{2} +263415. i q^{4} +(-1.22450e7 - 1.80803e7i) q^{5} +(-5.88402e8 - 5.88402e8i) q^{7} +(-2.26032e9 - 2.26032e9i) q^{8} +O(q^{10})$	\(q+(-957.532 + 957.532i) q^{2} +263415. i q^{4} +(-1.22450e7 - 1.80803e7i) q^{5} +(-5.88402e8 - 5.88402e8i) q^{7} +(-2.26032e9 - 2.26032e9i) q^{8} +(2.90375e10 + 5.58742e9i) q^{10} -3.48289e10i q^{11} +(2.12201e11 - 2.12201e11i) q^{13} +1.12683e12 q^{14} +3.77624e12 q^{16} +(-1.02005e13 + 1.02005e13i) q^{17} +2.75067e13i q^{19} +(4.76262e12 - 3.22553e12i) q^{20} +(3.33498e13 + 3.33498e13i) q^{22} +(2.25988e14 + 2.25988e14i) q^{23} +(-1.76955e14 + 4.42787e14i) q^{25} +4.06378e14i q^{26} +(1.54994e14 - 1.54994e14i) q^{28} -1.56108e15 q^{29} +4.20472e15 q^{31} +(1.12436e15 - 1.12436e15i) q^{32} -1.95346e16i q^{34} +(-3.43346e15 + 1.78435e16i) q^{35} +(1.33190e16 + 1.33190e16i) q^{37} +(-2.63385e16 - 2.63385e16i) q^{38} +(-1.31895e16 + 6.85449e16i) q^{40} +1.01890e17i q^{41} +(7.64240e16 - 7.64240e16i) q^{43} +9.17446e15 q^{44} -4.32782e17 q^{46} +(3.70927e17 - 3.70927e17i) q^{47} +1.33887e17i q^{49} +(-2.54543e17 - 5.93423e17i) q^{50} +(5.58969e16 + 5.58969e16i) q^{52} +(-1.12846e18 - 1.12846e18i) q^{53} +(-6.29715e17 + 4.26481e17i) q^{55} +2.65995e18i q^{56} +(1.49479e18 - 1.49479e18i) q^{58} -2.19964e17 q^{59} +6.62431e17 q^{61} +(-4.02616e18 + 4.02616e18i) q^{62} +1.00726e19i q^{64} +(-6.43505e18 - 1.23824e18i) q^{65} +(1.39064e19 + 1.39064e19i) q^{67} +(-2.68696e18 - 2.68696e18i) q^{68} +(-1.37980e19 - 2.03733e19i) q^{70} -2.64738e19i q^{71} +(-4.31833e19 + 4.31833e19i) q^{73} -2.55068e19 q^{74} -7.24568e18 q^{76} +(-2.04934e19 + 2.04934e19i) q^{77} +5.25073e19i q^{79} +(-4.62402e19 - 6.82754e19i) q^{80} +(-9.75627e19 - 9.75627e19i) q^{82} +(-1.92753e20 - 1.92753e20i) q^{83} +(3.09333e20 + 5.95221e19i) q^{85} +1.46357e20i q^{86} +(-7.87244e19 + 7.87244e19i) q^{88} -2.42769e20 q^{89} -2.49718e20 q^{91} +(-5.95288e19 + 5.95288e19i) q^{92} +7.10349e20i q^{94} +(4.97328e20 - 3.36820e20i) q^{95} +(-5.11344e19 - 5.11344e19i) q^{97} +(-1.28201e20 - 1.28201e20i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q + 687552792 q^{7}+O(q^{10}) $ 84 * q + 687552792 * q^7	$ 84 q + 687552792 q^{7} + 134292212112 q^{10} - 1733796401796 q^{13} - 65338919579016 q^{16} - 6984840176592 q^{22} - 11\!\cdots\!84 q^{25}+ \cdots + 37\!\cdots\!56 q^{97}+O(q^{100}) $ 84 * q + 687552792 * q^7 + 134292212112 * q^10 - 1733796401796 * q^13 - 65338919579016 * q^16 - 6984840176592 * q^22 - 1180149988019184 * q^25 - 1688782704439392 * q^28 - 142192095805536 * q^31 - 26634804805627596 * q^37 - 179561523957547992 * q^40 - 568410120821540832 * q^43 - 1043921016330318336 * q^46 + 4197387941225299704 * q^52 + 2218912905679837848 * q^55 + 11205074447113145712 * q^58 - 13309008883877058528 * q^61 - 487057429881367584 * q^67 - 73413022182740953296 * q^70 + 103761433025202728604 * q^73 - 71719061752861140000 * q^76 - 826264089579051988704 * q^82 + 414467640188195121048 * q^85 - 1465877561249256251424 * q^88 + 746298426009998704464 * q^91 + 3706758938239203368556 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−957.532	+	957.532i	−0.661209	+	0.661209i	−0.955665	−	0.294456i	$-0.904861\pi$
$2$	−957.532	+	957.532i	−0.661209	+	0.661209i	0.294456	+	0.955665i	$0.404861\pi$
$3$		0			0
$3$		0			0
$4$			263415.i			0.125606i
$5$	−1.22450e7	−	1.80803e7i	−0.560758	−	0.827980i
$5$	−1.22450e7	−	1.80803e7i	−0.560758	−	0.827980i
$6$		0			0
$7$	−5.88402e8	−	5.88402e8i	−0.787308	−	0.787308i	0.193744	−	0.981052i	$-0.437937\pi$
$7$	−5.88402e8	−	5.88402e8i	−0.787308	−	0.787308i	−0.981052	+	0.193744i	$0.937937\pi$
$8$	−2.26032e9	−	2.26032e9i	−0.744261	−	0.744261i
$9$		0			0
$10$	2.90375e10	+	5.58742e9i	0.918245	+	0.176690i
$11$		−	3.48289e10i		−	0.404870i	−0.979296	−	0.202435i	$-0.935114\pi$
$11$		−	3.48289e10i		−	0.404870i	0.979296	−	0.202435i	$-0.0648856\pi$
$12$		0			0
$13$	2.12201e11	−	2.12201e11i	0.426915	−	0.426915i	−0.460661	−	0.887576i	$-0.652388\pi$
$13$	2.12201e11	−	2.12201e11i	0.426915	−	0.426915i	0.887576	+	0.460661i	$0.152388\pi$
$14$	1.12683e12			1.04115
$15$		0			0
$16$	3.77624e12			0.858617
$17$	−1.02005e13	+	1.02005e13i	−1.22718	+	1.22718i	−0.262149	+	0.965028i	$0.584431\pi$
$17$	−1.02005e13	+	1.02005e13i	−1.22718	+	1.22718i	−0.965028	+	0.262149i	$0.915569\pi$
$18$		0			0
$19$			2.75067e13i			1.02926i	0.857412	+	0.514630i	$0.172071\pi$
$19$			2.75067e13i			1.02926i	−0.857412	+	0.514630i	$0.827929\pi$
$20$	4.76262e12	−	3.22553e12i	0.103999	−	0.0704347i
$21$		0			0
$22$	3.33498e13	+	3.33498e13i	0.267704	+	0.267704i
$23$	2.25988e14	+	2.25988e14i	1.13748	+	1.13748i	0.988901	+	0.148579i	$0.0474699\pi$
$23$	2.25988e14	+	2.25988e14i	1.13748	+	1.13748i	0.148579	+	0.988901i	$0.452530\pi$
$24$		0			0
$25$	−1.76955e14	+	4.42787e14i	−0.371101	+	0.928592i
$26$			4.06378e14i			0.564560i
$27$		0			0
$28$	1.54994e14	−	1.54994e14i	0.0988907	−	0.0988907i
$29$	−1.56108e15			−0.689044			−0.344522	−	0.938778i	$-0.611959\pi$
$29$	−1.56108e15			−0.689044			−0.344522	+	0.938778i	$0.611959\pi$
$30$		0			0
$31$	4.20472e15			0.921381			0.460690	−	0.887561i	$-0.347602\pi$
$31$	4.20472e15			0.921381			0.460690	+	0.887561i	$0.347602\pi$
$32$	1.12436e15	−	1.12436e15i	0.176536	−	0.176536i
$33$		0			0
$34$		−	1.95346e16i		−	1.62284i
$35$	−3.43346e15	+	1.78435e16i	−0.210386	+	1.09336i
$36$		0			0
$37$	1.33190e16	+	1.33190e16i	0.455359	+	0.455359i	0.897129	−	0.441769i	$-0.145649\pi$
$37$	1.33190e16	+	1.33190e16i	0.455359	+	0.455359i	−0.441769	+	0.897129i	$0.645649\pi$
$38$	−2.63385e16	−	2.63385e16i	−0.680556	−	0.680556i
$39$		0			0
$40$	−1.31895e16	+	6.85449e16i	−0.198883	+	1.03358i
$41$			1.01890e17i			1.18550i	0.805388	+	0.592748i	$0.201957\pi$
$41$			1.01890e17i			1.18550i	−0.805388	+	0.592748i	$0.798043\pi$
$42$		0			0
$43$	7.64240e16	−	7.64240e16i	0.539276	−	0.539276i	−0.384040	−	0.923316i	$-0.625468\pi$
$43$	7.64240e16	−	7.64240e16i	0.539276	−	0.539276i	0.923316	+	0.384040i	$0.125468\pi$
$44$	9.17446e15			0.0508542
$45$		0			0
$46$	−4.32782e17			−1.50422
$47$	3.70927e17	−	3.70927e17i	1.02863	−	1.02863i	0.0290553	−	0.999578i	$-0.490750\pi$
$47$	3.70927e17	−	3.70927e17i	1.02863	−	1.02863i	0.999578	−	0.0290553i	$-0.00924990\pi$
$48$		0			0
$49$			1.33887e17i			0.239707i
$50$	−2.54543e17	−	5.93423e17i	−0.368618	−	0.859369i
$51$		0			0
$52$	5.58969e16	+	5.58969e16i	0.0536232	+	0.0536232i
$53$	−1.12846e18	−	1.12846e18i	−0.886319	−	0.886319i	0.107849	−	0.994167i	$-0.465604\pi$
$53$	−1.12846e18	−	1.12846e18i	−0.886319	−	0.886319i	−0.994167	+	0.107849i	$0.965604\pi$
$54$		0			0
$55$	−6.29715e17	+	4.26481e17i	−0.335225	+	0.227034i
$56$			2.65995e18i			1.17192i
$57$		0			0
$58$	1.49479e18	−	1.49479e18i	0.455602	−	0.455602i
$59$	−2.19964e17			−0.0560280			−0.0280140	−	0.999608i	$-0.508918\pi$
$59$	−2.19964e17			−0.0560280			−0.0280140	+	0.999608i	$0.508918\pi$
$60$		0			0
$61$	6.62431e17			0.118899			0.0594494	−	0.998231i	$-0.481065\pi$
$61$	6.62431e17			0.118899			0.0594494	+	0.998231i	$0.481065\pi$
$62$	−4.02616e18	+	4.02616e18i	−0.609225	+	0.609225i
$63$		0			0
$64$			1.00726e19i			1.09207i
$65$	−6.43505e18	−	1.23824e18i	−0.592873	−	0.114081i
$66$		0			0
$67$	1.39064e19	+	1.39064e19i	0.932028	+	0.932028i	0.997833	−	0.0658047i	$-0.0209614\pi$
$67$	1.39064e19	+	1.39064e19i	0.932028	+	0.932028i	−0.0658047	+	0.997833i	$0.520961\pi$
$68$	−2.68696e18	−	2.68696e18i	−0.154141	−	0.154141i
$69$		0			0
$70$	−1.37980e19	−	2.03733e19i	−0.583833	−	0.862051i
$71$		−	2.64738e19i		−	0.965169i	−0.875849	−	0.482585i	$-0.839698\pi$
$71$		−	2.64738e19i		−	0.965169i	0.875849	−	0.482585i	$-0.160302\pi$
$72$		0			0
$73$	−4.31833e19	+	4.31833e19i	−1.17605	+	1.17605i	−0.195307	+	0.980742i	$0.562570\pi$
$73$	−4.31833e19	+	4.31833e19i	−1.17605	+	1.17605i	−0.980742	+	0.195307i	$0.937430\pi$
$74$	−2.55068e19			−0.602175
$75$		0			0
$76$	−7.24568e18			−0.129281
$77$	−2.04934e19	+	2.04934e19i	−0.318758	+	0.318758i
$78$		0			0
$79$			5.25073e19i			0.623929i	0.950094	+	0.311965i	$0.100987\pi$
$79$			5.25073e19i			0.623929i	−0.950094	+	0.311965i	$0.899013\pi$
$80$	−4.62402e19	−	6.82754e19i	−0.481476	−	0.710918i
$81$		0			0
$82$	−9.75627e19	−	9.75627e19i	−0.783860	−	0.783860i
$83$	−1.92753e20	−	1.92753e20i	−1.36358	−	1.36358i	−0.869302	−	0.494282i	$-0.835431\pi$
$83$	−1.92753e20	−	1.92753e20i	−1.36358	−	1.36358i	−0.494282	−	0.869302i	$-0.664569\pi$
$84$		0			0
$85$	3.09333e20	+	5.95221e19i	1.70423	+	0.327929i
$86$			1.46357e20i			0.713148i
$87$		0			0
$88$	−7.87244e19	+	7.87244e19i	−0.301329	+	0.301329i
$89$	−2.42769e20			−0.825273			−0.412637	−	0.910896i	$-0.635392\pi$
$89$	−2.42769e20			−0.825273			−0.412637	+	0.910896i	$0.635392\pi$
$90$		0			0
$91$	−2.49718e20			−0.672227
$92$	−5.95288e19	+	5.95288e19i	−0.142874	+	0.142874i
$93$		0			0
$94$			7.10349e20i			1.36028i
$95$	4.97328e20	−	3.36820e20i	0.852207	−	0.577166i
$96$		0			0
$97$	−5.11344e19	−	5.11344e19i	−0.0704060	−	0.0704060i	0.671027	−	0.741433i	$-0.265853\pi$
$97$	−5.11344e19	−	5.11344e19i	−0.0704060	−	0.0704060i	−0.741433	+	0.671027i	$0.765853\pi$
$98$	−1.28201e20	−	1.28201e20i	−0.158496	−	0.158496i
$99$		0			0
$100$	−1.16637e20	−	4.66126e19i	−0.116637	−	0.0466126i
$101$			1.16354e21i			1.04811i	0.851685	+	0.524054i	$0.175581\pi$
$101$			1.16354e21i			1.04811i	−0.851685	+	0.524054i	$0.824419\pi$
$102$		0			0
$103$	1.23972e21	−	1.23972e21i	0.908934	−	0.908934i	−0.0872522	−	0.996186i	$-0.527809\pi$
$103$	1.23972e21	−	1.23972e21i	0.908934	−	0.908934i	0.996186	+	0.0872522i	$0.0278086\pi$
$104$	−9.59282e20			−0.635472
$105$		0			0
$106$	2.16108e21			1.17208
$107$	1.00269e21	−	1.00269e21i	0.492763	−	0.492763i	−0.416413	−	0.909176i	$-0.636713\pi$
$107$	1.00269e21	−	1.00269e21i	0.492763	−	0.492763i	0.909176	+	0.416413i	$0.136713\pi$
$108$		0			0
$109$			8.69875e20i			0.351948i	0.984395	+	0.175974i	$0.0563075\pi$
$109$			8.69875e20i			0.351948i	−0.984395	+	0.175974i	$0.943693\pi$
$110$	1.94603e20	−	1.01134e21i	0.0715364	−	0.371770i
$111$		0			0
$112$	−2.22194e21	−	2.22194e21i	−0.675996	−	0.675996i
$113$	−2.06353e21	−	2.06353e21i	−0.571857	−	0.571857i	0.360790	−	0.932647i	$-0.382507\pi$
$113$	−2.06353e21	−	2.06353e21i	−0.571857	−	0.571857i	−0.932647	+	0.360790i	$0.882507\pi$
$114$		0			0
$115$	1.31869e21	−	6.85316e21i	0.303960	−	1.57966i
$116$		−	4.11213e20i		−	0.0865482i
$117$		0			0
$118$	2.10622e20	−	2.10622e20i	0.0370462	−	0.0370462i
$119$	1.20040e22			1.93233
$120$		0			0
$121$	6.18720e21			0.836080
$122$	−6.34299e20	+	6.34299e20i	−0.0786169	+	0.0786169i
$123$		0			0
$124$			1.10759e21i			0.115731i
$125$	1.01725e22	−	2.22256e21i	0.976954	−	0.213451i
$126$		0			0
$127$	1.70704e21	+	1.70704e21i	0.138773	+	0.138773i	0.773081	−	0.634308i	$-0.218715\pi$
$127$	1.70704e21	+	1.70704e21i	0.138773	+	0.138773i	−0.634308	+	0.773081i	$0.718715\pi$
$128$	−7.28685e21	−	7.28685e21i	−0.545551	−	0.545551i
$129$		0			0
$130$	7.34742e21	−	4.97612e21i	0.467444	−	0.316581i
$131$			1.49759e22i			0.879112i	0.898215	+	0.439556i	$0.144864\pi$
$131$			1.49759e22i			0.879112i	−0.898215	+	0.439556i	$0.855136\pi$
$132$		0			0
$133$	1.61850e22	−	1.61850e22i	0.810345	−	0.810345i
$134$	−2.66316e22			−1.23253
$135$		0			0
$136$	4.61127e22			1.82668
$137$	1.44342e22	−	1.44342e22i	0.529452	−	0.529452i	−0.390957	−	0.920409i	$-0.627856\pi$
$137$	1.44342e22	−	1.44342e22i	0.529452	−	0.529452i	0.920409	+	0.390957i	$0.127856\pi$
$138$		0			0
$139$		−	5.80685e22i		−	1.82930i	−0.404245	−	0.914651i	$-0.632466\pi$
$139$		−	5.80685e22i		−	1.82930i	0.404245	−	0.914651i	$-0.367534\pi$
$140$	−4.70024e21	−	9.04425e20i	−0.137333	−	0.0264258i
$141$		0			0
$142$	2.53495e22	+	2.53495e22i	0.638178	+	0.638178i
$143$	−7.39071e21	−	7.39071e21i	−0.172845	−	0.172845i
$144$		0			0
$145$	1.91155e22	+	2.82248e22i	0.386387	+	0.570515i
$146$		−	8.26988e22i		−	1.55523i
$147$		0			0
$148$	−3.50843e21	+	3.50843e21i	−0.0571959	+	0.0571959i
$149$	−1.09487e23			−1.66306			−0.831530	−	0.555479i	$-0.812535\pi$
$149$	−1.09487e23			−1.66306			−0.831530	+	0.555479i	$0.812535\pi$
$150$		0			0
$151$	−1.78084e22			−0.235162			−0.117581	−	0.993063i	$-0.537514\pi$
$151$	−1.78084e22			−0.235162			−0.117581	+	0.993063i	$0.537514\pi$
$152$	6.21739e22	−	6.21739e22i	0.766038	−	0.766038i
$153$		0			0
$154$		−	3.92461e22i		−	0.421531i
$155$	−5.14870e22	−	7.60225e22i	−0.516672	−	0.762885i
$156$		0			0
$157$	6.95180e22	+	6.95180e22i	0.609749	+	0.609749i	0.942880	−	0.333131i	$-0.108105\pi$
$157$	6.95180e22	+	6.95180e22i	0.609749	+	0.609749i	−0.333131	+	0.942880i	$0.608105\pi$
$158$	−5.02775e22	−	5.02775e22i	−0.412547	−	0.412547i
$159$		0			0
$160$	−3.40967e22	−	6.56092e21i	−0.245162	−	0.0471742i
$161$		−	2.65944e23i		−	1.79109i
$162$		0			0
$163$	1.58264e23	−	1.58264e23i	0.936296	−	0.936296i	−0.0617927	−	0.998089i	$-0.519682\pi$
$163$	1.58264e23	−	1.58264e23i	0.936296	−	0.936296i	0.998089	+	0.0617927i	$0.0196818\pi$
$164$	−2.68393e22			−0.148906
$165$		0			0
$166$	3.69135e23			1.80323
$167$	−1.79270e23	+	1.79270e23i	−0.822211	+	0.822211i	−0.986425	−	0.164213i	$-0.947491\pi$
$167$	−1.79270e23	+	1.79270e23i	−0.822211	+	0.822211i	0.164213	+	0.986425i	$0.447491\pi$
$168$		0			0
$169$			1.57006e23i			0.635487i
$170$	−3.53190e23	+	2.39202e23i	−1.34368	+	0.910020i
$171$		0			0
$172$	2.01313e22	+	2.01313e22i	0.0677364	+	0.0677364i
$173$	−2.96424e23	−	2.96424e23i	−0.938488	−	0.938488i	0.0597268	−	0.998215i	$-0.480977\pi$
$173$	−2.96424e23	−	2.96424e23i	−0.938488	−	0.938488i	−0.998215	+	0.0597268i	$0.980977\pi$
$174$		0			0
$175$	3.64657e23	−	1.56416e23i	1.02326	−	0.438917i
$176$		−	1.31522e23i		−	0.347629i
$177$		0			0
$178$	2.32459e23	−	2.32459e23i	0.545678	−	0.545678i
$179$	5.09775e23			1.12829			0.564146	−	0.825675i	$-0.309205\pi$
$179$	5.09775e23			1.12829			0.564146	+	0.825675i	$0.309205\pi$
$180$		0			0
$181$	−1.38975e23			−0.273723			−0.136861	−	0.990590i	$-0.543702\pi$
$181$	−1.38975e23			−0.273723			−0.136861	+	0.990590i	$0.543702\pi$
$182$	2.39113e23	−	2.39113e23i	0.444482	−	0.444482i
$183$		0			0
$184$		−	1.02161e24i		−	1.69316i
$185$	7.77195e22	−	4.03903e23i	0.121682	−	0.632375i
$186$		0			0
$187$	3.55271e23	+	3.55271e23i	0.496847	+	0.496847i
$188$	9.77078e22	+	9.77078e22i	0.129203	+	0.129203i
$189$		0			0
$190$	−1.53691e23	+	7.98724e23i	−0.181860	+	0.945114i
$191$		−	4.98751e23i		−	0.558513i	−0.960217	−	0.279256i	$-0.909912\pi$
$191$		−	4.98751e23i		−	0.558513i	0.960217	−	0.279256i	$-0.0900879\pi$
$192$		0			0
$193$	5.03988e23	−	5.03988e23i	0.505905	−	0.505905i	−0.407362	−	0.913267i	$-0.633551\pi$
$193$	5.03988e23	−	5.03988e23i	0.505905	−	0.505905i	0.913267	+	0.407362i	$0.133551\pi$
$194$	9.79257e22			0.0931062
$195$		0			0
$196$	−3.52679e22			−0.0301086
$197$	−2.10363e23	+	2.10363e23i	−0.170245	+	0.170245i	−0.787087	−	0.616842i	$-0.788412\pi$
$197$	−2.10363e23	+	2.10363e23i	−0.170245	+	0.170245i	0.616842	+	0.787087i	$0.288412\pi$
$198$		0			0
$199$			3.67186e23i			0.267257i	0.991031	+	0.133628i	$0.0426629\pi$
$199$			3.67186e23i			0.267257i	−0.991031	+	0.133628i	$0.957337\pi$
$200$	1.40082e24	−	6.00866e23i	0.967311	−	0.414918i
$201$		0			0
$202$	−1.11413e24	−	1.11413e24i	−0.693018	−	0.693018i
$203$	9.18544e23	+	9.18544e23i	0.542490	+	0.542490i
$204$		0			0
$205$	1.84219e24	−	1.24764e24i	0.981566	−	0.664776i
$206$			2.37414e24i			1.20199i
$207$		0			0
$208$	8.01320e23	−	8.01320e23i	0.366556	−	0.366556i
$209$	9.58026e23			0.416717
$210$		0			0
$211$	−1.46881e24			−0.578094			−0.289047	−	0.957315i	$-0.593338\pi$
$211$	−1.46881e24			−0.578094			−0.289047	+	0.957315i	$0.593338\pi$
$212$	2.97254e23	−	2.97254e23i	0.111327	−	0.111327i
$213$		0			0
$214$			1.92022e24i			0.651638i
$215$	−2.31758e24	−	4.45951e23i	−0.748913	−	0.144106i
$216$		0			0
$217$	−2.47406e24	−	2.47406e24i	−0.725410	−	0.725410i
$218$	−8.32933e23	−	8.32933e23i	−0.232711	−	0.232711i
$219$		0			0
$220$	−1.12342e23	−	1.65877e23i	−0.0285169	−	0.0421063i
$221$			4.32910e24i			1.04780i
$222$		0			0
$223$	−5.64239e24	+	5.64239e24i	−1.24240	+	1.24240i	−0.283400	+	0.959002i	$0.591462\pi$
$223$	−5.64239e24	+	5.64239e24i	−1.24240	+	1.24240i	−0.959002	+	0.283400i	$0.908538\pi$
$224$	−1.32316e24			−0.277976
$225$		0			0
$226$	3.95180e24			0.756233
$227$	2.09575e24	−	2.09575e24i	0.382884	−	0.382884i	−0.489256	−	0.872140i	$-0.662732\pi$
$227$	2.09575e24	−	2.09575e24i	0.382884	−	0.382884i	0.872140	+	0.489256i	$0.162732\pi$
$228$		0			0
$229$			4.07052e24i			0.678231i	0.940745	+	0.339115i	$0.110128\pi$
$229$			4.07052e24i			0.678231i	−0.940745	+	0.339115i	$0.889872\pi$
$230$	5.29944e24	+	7.82482e24i	0.843504	+	1.24547i
$231$		0			0
$232$	3.52855e24	+	3.52855e24i	0.512828	+	0.512828i
$233$	5.74900e23	+	5.74900e23i	0.0798647	+	0.0798647i	0.745911	−	0.666046i	$-0.232015\pi$
$233$	5.74900e23	+	5.74900e23i	0.0798647	+	0.0798647i	−0.666046	+	0.745911i	$0.732015\pi$
$234$		0			0
$235$	−1.12485e25	−	2.16444e24i	−1.42850	−	0.274874i
$236$		−	5.79418e22i		−	0.00703746i
$237$		0			0
$238$	−1.14942e25	+	1.14942e25i	−1.27767	+	1.27767i
$239$	−6.90560e24			−0.734554			−0.367277	−	0.930112i	$-0.619710\pi$
$239$	−6.90560e24			−0.734554			−0.367277	+	0.930112i	$0.619710\pi$
$240$		0			0
$241$	−1.85129e25			−1.80425			−0.902124	−	0.431477i	$-0.857993\pi$
$241$	−1.85129e25			−1.80425			−0.902124	+	0.431477i	$0.857993\pi$
$242$	−5.92444e24	+	5.92444e24i	−0.552823	+	0.552823i
$243$		0			0
$244$			1.74495e23i			0.0149344i
$245$	2.42072e24	−	1.63945e24i	0.198472	−	0.134417i
$246$		0			0
$247$	5.83693e24	+	5.83693e24i	0.439407	+	0.439407i
$248$	−9.50401e24	−	9.50401e24i	−0.685748	−	0.685748i
$249$		0			0
$250$	−7.61236e24	+	1.18687e25i	−0.504835	+	0.787106i
$251$			5.46480e24i			0.347536i	0.984787	+	0.173768i	$0.0555944\pi$
$251$			5.46480e24i			0.347536i	−0.984787	+	0.173768i	$0.944406\pi$
$252$		0			0
$253$	7.87091e24	−	7.87091e24i	0.460532	−	0.460532i
$254$	−3.26909e24			−0.183516
$255$		0			0
$256$	−7.16892e24			−0.370625
$257$	1.12155e25	−	1.12155e25i	0.556571	−	0.556571i	−0.371758	−	0.928330i	$-0.621245\pi$
$257$	1.12155e25	−	1.12155e25i	0.556571	−	0.556571i	0.928330	+	0.371758i	$0.121245\pi$
$258$		0			0
$259$		−	1.56739e25i		−	0.717016i
$260$	3.26171e23	−	1.69509e24i	0.0143293	−	0.0744685i
$261$		0			0
$262$	−1.43399e25	−	1.43399e25i	−0.581276	−	0.581276i
$263$	1.82457e25	+	1.82457e25i	0.710601	+	0.710601i	0.966661	−	0.256060i	$-0.0824243\pi$
$263$	1.82457e25	+	1.82457e25i	0.710601	+	0.710601i	−0.256060	+	0.966661i	$0.582424\pi$
$264$		0			0
$265$	−6.58482e24	+	3.42209e25i	−0.236844	+	1.23086i
$266$			3.09953e25i			1.07161i
$267$		0			0
$268$	−3.66315e24	+	3.66315e24i	−0.117068	+	0.117068i
$269$	1.47829e25			0.454320			0.227160	−	0.973857i	$-0.427056\pi$
$269$	1.47829e25			0.454320			0.227160	+	0.973857i	$0.427056\pi$
$270$		0			0
$271$	−9.75015e24			−0.277226			−0.138613	−	0.990347i	$-0.544264\pi$
$271$	−9.75015e24			−0.277226			−0.138613	+	0.990347i	$0.544264\pi$
$272$	−3.85194e25	+	3.85194e25i	−1.05367	+	1.05367i
$273$		0			0
$274$			2.76424e25i			0.700157i
$275$	1.54218e25	+	6.16314e24i	0.375960	+	0.150248i
$276$		0			0
$277$	9.91878e24	+	9.91878e24i	0.224089	+	0.224089i	0.810218	−	0.586129i	$-0.199349\pi$
$277$	9.91878e24	+	9.91878e24i	0.224089	+	0.224089i	−0.586129	+	0.810218i	$0.699349\pi$
$278$	5.56025e25	+	5.56025e25i	1.20955	+	1.20955i
$279$		0			0
$280$	4.80926e25	−	3.25712e25i	0.970330	−	0.657166i
$281$		−	4.45030e25i		−	0.864914i	−0.901655	−	0.432457i	$-0.857647\pi$
$281$		−	4.45030e25i		−	0.864914i	0.901655	−	0.432457i	$-0.142353\pi$
$282$		0			0
$283$	1.09235e25	−	1.09235e25i	0.197062	−	0.197062i	−0.601677	−	0.798739i	$-0.705501\pi$
$283$	1.09235e25	−	1.09235e25i	0.197062	−	0.197062i	0.798739	+	0.601677i	$0.205501\pi$
$284$	6.97360e24			0.121231
$285$		0			0
$286$	1.41537e25			0.228574
$287$	5.99521e25	−	5.99521e25i	0.933349	−	0.933349i
$288$		0			0
$289$		−	1.39008e26i		−	2.01192i
$290$	−4.53299e25	−	8.72242e24i	−0.632712	−	0.121747i
$291$		0			0
$292$	−1.13751e25	−	1.13751e25i	−0.147719	−	0.147719i
$293$	−7.61675e25	−	7.61675e25i	−0.954245	−	0.954245i	0.0447534	−	0.998998i	$-0.485750\pi$
$293$	−7.61675e25	−	7.61675e25i	−0.954245	−	0.954245i	−0.998998	+	0.0447534i	$0.985750\pi$
$294$		0			0
$295$	2.69347e24	+	3.97700e24i	0.0314181	+	0.0463901i
$296$		−	6.02105e25i		−	0.677812i
$297$		0			0
$298$	1.04838e26	−	1.04838e26i	1.09963	−	1.09963i
$299$	9.59097e25			0.971214
$300$		0			0
$301$	−8.99360e25			−0.849152
$302$	1.70522e25	−	1.70522e25i	0.155491	−	0.155491i
$303$		0			0
$304$			1.03872e26i			0.883740i
$305$	−8.11150e24	−	1.19769e25i	−0.0666734	−	0.0984458i
$306$		0			0
$307$	3.57720e25	+	3.57720e25i	0.274531	+	0.274531i	0.830921	−	0.556390i	$-0.187814\pi$
$307$	3.57720e25	+	3.57720e25i	0.274531	+	0.274531i	−0.556390	+	0.830921i	$0.687814\pi$
$308$	−5.39826e24	−	5.39826e24i	−0.0400379	−	0.0400379i
$309$		0			0
$310$	1.22094e26	+	2.34935e25i	0.846054	+	0.162798i
$311$			1.80776e25i			0.121103i	0.998165	+	0.0605517i	$0.0192860\pi$
$311$			1.80776e25i			0.121103i	−0.998165	+	0.0605517i	$0.980714\pi$
$312$		0			0
$313$	1.33865e26	−	1.33865e26i	0.838402	−	0.838402i	−0.150246	−	0.988649i	$-0.548007\pi$
$313$	1.33865e26	−	1.33865e26i	0.838402	−	0.838402i	0.988649	+	0.150246i	$0.0480066\pi$
$314$	−1.33132e26			−0.806343
$315$		0			0
$316$	−1.38312e25			−0.0783694
$317$	−9.10856e25	+	9.10856e25i	−0.499261	+	0.499261i	−0.911208	−	0.411947i	$-0.864849\pi$
$317$	−9.10856e25	+	9.10856e25i	−0.499261	+	0.499261i	0.411947	+	0.911208i	$0.364849\pi$
$318$		0			0
$319$			5.43707e25i			0.278974i
$320$	1.82115e26	−	1.23339e26i	0.904213	−	0.612387i
$321$		0			0
$322$	2.54650e26	+	2.54650e26i	1.18429	+	1.18429i
$323$	−2.80581e26	−	2.80581e26i	−1.26308	−	1.26308i
$324$		0			0
$325$	5.64098e25	+	1.31510e26i	0.238001	+	0.554859i
$326$			3.03087e26i			1.23817i
$327$		0			0
$328$	2.30303e26	−	2.30303e26i	0.882317	−	0.882317i
$329$	−4.36508e26			−1.61970
$330$		0			0
$331$	−6.04213e25			−0.210376			−0.105188	−	0.994452i	$-0.533544\pi$
$331$	−6.04213e25			−0.210376			−0.105188	+	0.994452i	$0.533544\pi$
$332$	5.07742e25	−	5.07742e25i	0.171275	−	0.171275i
$333$		0			0
$334$		−	3.43313e26i		−	1.08731i
$335$	8.11469e25	−	4.21715e26i	0.249058	−	1.29434i
$336$		0			0
$337$	−1.35119e26	−	1.35119e26i	−0.389585	−	0.389585i	0.484954	−	0.874539i	$-0.338836\pi$
$337$	−1.35119e26	−	1.35119e26i	−0.389585	−	0.389585i	−0.874539	+	0.484954i	$0.838836\pi$
$338$	−1.50339e26	−	1.50339e26i	−0.420190	−	0.420190i
$339$		0			0
$340$	−1.56790e25	+	8.14830e25i	−0.0411899	+	0.214061i
$341$		−	1.46446e26i		−	0.373040i
$342$		0			0
$343$	−2.49870e26	+	2.49870e26i	−0.598585	+	0.598585i
$344$	−3.45485e26			−0.802724
$345$		0			0
$346$	5.67670e26			1.24107
$347$	4.34175e26	−	4.34175e26i	0.920885	−	0.920885i	−0.0762066	−	0.997092i	$-0.524281\pi$
$347$	4.34175e26	−	4.34175e26i	0.920885	−	0.920885i	0.997092	+	0.0762066i	$0.0242809\pi$
$348$		0			0
$349$		−	3.59635e26i		−	0.718116i	−0.933315	−	0.359058i	$-0.883098\pi$
$349$		−	3.59635e26i		−	0.718116i	0.933315	−	0.359058i	$-0.116902\pi$
$350$	−1.99398e26	+	4.98945e26i	−0.386372	+	0.966803i
$351$		0			0
$352$	−3.91603e25	−	3.91603e25i	−0.0714741	−	0.0714741i
$353$	−3.93232e26	−	3.93232e26i	−0.696649	−	0.696649i	0.267037	−	0.963686i	$-0.413955\pi$
$353$	−3.93232e26	−	3.93232e26i	−0.696649	−	0.696649i	−0.963686	+	0.267037i	$0.913955\pi$
$354$		0			0
$355$	−4.78653e26	+	3.24173e26i	−0.799141	+	0.541226i
$356$		−	6.39490e25i		−	0.103659i
$357$		0			0
$358$	−4.88126e26	+	4.88126e26i	−0.746037	+	0.746037i
$359$	1.61743e26			0.240068			0.120034	−	0.992770i	$-0.461700\pi$
$359$	1.61743e26			0.240068			0.120034	+	0.992770i	$0.461700\pi$
$360$		0			0
$361$	−4.24076e25			−0.0593769
$362$	1.33073e26	−	1.33073e26i	0.180988	−	0.180988i
$363$		0			0
$364$		−	6.57796e25i		−	0.0844359i
$365$	1.30955e27	+	2.51984e26i	1.63322	+	0.314266i
$366$		0			0
$367$	2.02638e26	+	2.02638e26i	0.238631	+	0.238631i	0.816283	−	0.577652i	$-0.196031\pi$
$367$	2.02638e26	+	2.02638e26i	0.238631	+	0.238631i	−0.577652	+	0.816283i	$0.696031\pi$
$368$	8.53385e26	+	8.53385e26i	0.976659	+	0.976659i
$369$		0			0
$370$	3.12332e26	+	4.61170e26i	0.337674	+	0.498589i
$371$			1.32798e27i			1.39561i
$372$		0			0
$373$	1.30798e27	−	1.30798e27i	1.29914	−	1.29914i	0.370188	−	0.928957i	$-0.379293\pi$
$373$	1.30798e27	−	1.30798e27i	1.29914	−	1.29914i	0.928957	−	0.370188i	$-0.120707\pi$
$374$	−6.80367e26			−0.657040
$375$		0			0
$376$	−1.67683e27			−1.53114
$377$	−3.31263e26	+	3.31263e26i	−0.294163	+	0.294163i
$378$		0			0
$379$		−	4.16142e26i		−	0.349567i	−0.984607	−	0.174783i	$-0.944078\pi$
$379$		−	4.16142e26i		−	0.349567i	0.984607	−	0.174783i	$-0.0559225\pi$
$380$	8.87237e25	+	1.31004e26i	0.0724956	+	0.107042i
$381$		0			0
$382$	4.77570e26	+	4.77570e26i	0.369293	+	0.369293i
$383$	2.17485e26	+	2.17485e26i	0.163622	+	0.163622i	0.784169	−	0.620547i	$-0.213089\pi$
$383$	2.17485e26	+	2.17485e26i	0.163622	+	0.163622i	−0.620547	+	0.784169i	$0.713089\pi$
$384$		0			0
$385$	6.21467e26	+	1.19583e26i	0.442671	+	0.0851791i
$386$			9.65170e26i			0.669017i
$387$		0			0
$388$	1.34696e25	−	1.34696e25i	0.00884344	−	0.00884344i
$389$	2.15040e27			1.37419			0.687096	−	0.726566i	$-0.258885\pi$
$389$	2.15040e27			1.37419			0.687096	+	0.726566i	$0.258885\pi$
$390$		0			0
$391$	−4.61038e27			−2.79178
$392$	3.02628e26	−	3.02628e26i	0.178404	−	0.178404i
$393$		0			0
$394$		−	4.02859e26i		−	0.225135i
$395$	9.49346e26	−	6.42954e26i	0.516601	−	0.349873i
$396$		0			0
$397$	−2.03518e26	−	2.03518e26i	−0.105028	−	0.105028i	0.652640	−	0.757668i	$-0.273661\pi$
$397$	−2.03518e26	−	2.03518e26i	−0.105028	−	0.105028i	−0.757668	+	0.652640i	$0.773661\pi$
$398$	−3.51593e26	−	3.51593e26i	−0.176713	−	0.176713i
$399$		0			0
$400$	−6.68224e26	+	1.67207e27i	−0.318634	+	0.797305i
$401$			2.73487e27i			1.27034i	0.772372	+	0.635170i	$0.219070\pi$
$401$			2.73487e27i			1.27034i	−0.772372	+	0.635170i	$0.780930\pi$
$402$		0			0
$403$	8.92244e26	−	8.92244e26i	0.393351	−	0.393351i
$404$	−3.06494e26			−0.131649
$405$		0			0
$406$	−1.75907e27			−0.717398
$407$	4.63886e26	−	4.63886e26i	0.184361	−	0.184361i
$408$		0			0
$409$		−	2.40855e27i		−	0.909203i	−0.890695	−	0.454601i	$-0.849782\pi$
$409$		−	2.40855e27i		−	0.909203i	0.890695	−	0.454601i	$-0.150218\pi$
$410$	−5.69300e26	+	2.95862e27i	−0.209465	+	1.08858i
$411$		0			0
$412$	3.26561e26	+	3.26561e26i	0.114168	+	0.114168i
$413$	1.29427e26	+	1.29427e26i	0.0441113	+	0.0441113i
$414$		0			0
$415$	−1.12476e27	+	5.84530e27i	−0.364380	+	1.89366i
$416$		−	4.77182e26i		−	0.150731i
$417$		0			0
$418$	−9.17341e26	+	9.17341e26i	−0.275537	+	0.275537i
$419$	−2.05719e27			−0.602596			−0.301298	−	0.953530i	$-0.597420\pi$
$419$	−2.05719e27			−0.602596			−0.301298	+	0.953530i	$0.597420\pi$
$420$		0			0
$421$	−3.41737e27			−0.952203			−0.476101	−	0.879390i	$-0.657950\pi$
$421$	−3.41737e27			−0.952203			−0.476101	+	0.879390i	$0.657950\pi$
$422$	1.40643e27	−	1.40643e27i	0.382241	−	0.382241i
$423$		0			0
$424$			5.10137e27i			1.31930i
$425$	−2.71162e27	−	6.32167e27i	−0.684139	−	1.59495i
$426$		0			0
$427$	−3.89776e26	−	3.89776e26i	−0.0936100	−	0.0936100i
$428$	2.64124e26	+	2.64124e26i	0.0618941	+	0.0618941i
$429$		0			0
$430$	2.64617e27	−	1.79215e27i	0.590472	−	0.399903i
$431$			4.97365e27i			1.08309i	0.840672	+	0.541544i	$0.182160\pi$
$431$			4.97365e27i			1.08309i	−0.840672	+	0.541544i	$0.817840\pi$
$432$		0			0
$433$	−2.60856e27	+	2.60856e27i	−0.541099	+	0.541099i	−0.923851	−	0.382752i	$-0.874976\pi$
$433$	−2.60856e27	+	2.60856e27i	−0.541099	+	0.541099i	0.382752	+	0.923851i	$0.374976\pi$
$434$	4.73799e27			0.959295
$435$		0			0
$436$	−2.29138e26			−0.0442069
$437$	−6.21618e27	+	6.21618e27i	−1.17076	+	1.17076i
$438$		0			0
$439$		−	4.62149e27i		−	0.829668i	−0.909897	−	0.414834i	$-0.863840\pi$
$439$		−	4.62149e27i		−	0.829668i	0.909897	−	0.414834i	$-0.136160\pi$
$440$	2.38734e27	+	4.59374e26i	0.418467	+	0.0805218i
$441$		0			0
$442$	−4.14525e27	−	4.14525e27i	−0.692814	−	0.692814i
$443$	−4.06230e27	−	4.06230e27i	−0.663029	−	0.663029i	0.293064	−	0.956093i	$-0.405325\pi$
$443$	−4.06230e27	−	4.06230e27i	−0.663029	−	0.663029i	−0.956093	+	0.293064i	$0.905325\pi$
$444$		0			0
$445$	2.97271e27	+	4.38932e27i	0.462778	+	0.683310i
$446$		−	1.08055e28i		−	1.64297i
$447$		0			0
$448$	5.92672e27	−	5.92672e27i	0.859796	−	0.859796i
$449$	−1.25782e28			−1.78251			−0.891253	−	0.453506i	$-0.850173\pi$
$449$	−1.25782e28			−1.78251			−0.891253	+	0.453506i	$0.850173\pi$
$450$		0			0
$451$	3.54870e27			0.479972
$452$	5.43566e26	−	5.43566e26i	0.0718288	−	0.0718288i
$453$		0			0
$454$			4.01349e27i			0.506332i
$455$	3.05781e27	+	4.51497e27i	0.376956	+	0.556590i
$456$		0			0
$457$	1.08104e28	+	1.08104e28i	1.27269	+	1.27269i	0.944673	+	0.328014i	$0.106379\pi$
$457$	1.08104e28	+	1.08104e28i	1.27269	+	1.27269i	0.328014	+	0.944673i	$0.393621\pi$
$458$	−3.89766e27	−	3.89766e27i	−0.448452	−	0.448452i
$459$		0			0
$460$	1.80523e27	+	3.47364e26i	0.198415	+	0.0381792i
$461$			1.61624e28i			1.73639i	0.496227	+	0.868193i	$0.334718\pi$
$461$			1.61624e28i			1.73639i	−0.496227	+	0.868193i	$0.665282\pi$
$462$		0			0
$463$	5.49258e27	−	5.49258e27i	0.563866	−	0.563866i	−0.366537	−	0.930403i	$-0.619457\pi$
$463$	5.49258e27	−	5.49258e27i	0.563866	−	0.563866i	0.930403	+	0.366537i	$0.119457\pi$
$464$	−5.89502e27			−0.591625
$465$		0			0
$466$	−1.10097e27			−0.105614
$467$	1.43777e28	−	1.43777e28i	1.34853	−	1.34853i	0.461278	−	0.887256i	$-0.347391\pi$
$467$	1.43777e28	−	1.43777e28i	1.34853	−	1.34853i	0.887256	−	0.461278i	$-0.152609\pi$
$468$		0			0
$469$		−	1.63651e28i		−	1.46759i
$470$	1.28433e28	−	8.69825e27i	1.12629	−	0.762789i
$471$		0			0
$472$	4.97189e26	+	4.97189e26i	0.0416994	+	0.0416994i
$473$	−2.66176e27	−	2.66176e27i	−0.218337	−	0.218337i
$474$		0			0
$475$	−1.21796e28	−	4.86744e27i	−0.955763	−	0.381960i
$476$			3.16203e27i			0.242713i
$477$		0			0
$478$	6.61234e27	−	6.61234e27i	0.485693	−	0.485693i
$479$	−5.27361e27			−0.378953			−0.189476	−	0.981885i	$-0.560679\pi$
$479$	−5.27361e27			−0.378953			−0.189476	+	0.981885i	$0.560679\pi$
$480$		0			0
$481$	5.65261e27			0.388799
$482$	1.77267e28	−	1.77267e28i	1.19298	−	1.19298i
$483$		0			0
$484$			1.62980e27i			0.105017i
$485$	−2.98381e26	+	1.55067e27i	−0.0188141	+	0.0977755i
$486$		0			0
$487$	4.38865e27	+	4.38865e27i	0.265019	+	0.265019i	0.827089	−	0.562070i	$-0.189995\pi$
$487$	4.38865e27	+	4.38865e27i	0.265019	+	0.265019i	−0.562070	+	0.827089i	$0.689995\pi$
$488$	−1.49731e27	−	1.49731e27i	−0.0884917	−	0.0884917i
$489$		0			0
$490$	−7.48083e26	+	3.88774e27i	−0.0423537	+	0.220109i
$491$		−	1.92483e28i		−	1.06668i	−0.845900	−	0.533342i	$-0.820936\pi$
$491$		−	1.92483e28i		−	1.06668i	0.845900	−	0.533342i	$-0.179064\pi$
$492$		0			0
$493$	1.59238e28	−	1.59238e28i	0.845578	−	0.845578i
$494$	−1.11781e28			−0.581079
$495$		0			0
$496$	1.58780e28			0.791113
$497$	−1.55772e28	+	1.55772e28i	−0.759885	+	0.759885i
$498$		0			0
$499$		−	3.33022e28i		−	1.55746i	−0.627357	−	0.778732i	$-0.715863\pi$
$499$		−	3.33022e28i		−	1.55746i	0.627357	−	0.778732i	$-0.284137\pi$
$500$	5.85456e26	+	2.67960e27i	0.0268107	+	0.122711i
$501$		0			0
$502$	−5.23272e27	−	5.23272e27i	−0.229794	−	0.229794i
$503$	1.18041e27	+	1.18041e27i	0.0507655	+	0.0507655i	0.732034	−	0.681268i	$-0.238571\pi$
$503$	1.18041e27	+	1.18041e27i	0.0507655	+	0.0507655i	−0.681268	+	0.732034i	$0.738571\pi$
$504$		0			0
$505$	2.10371e28	−	1.42476e28i	0.867813	−	0.587735i
$506$			1.50733e28i			0.609015i
$507$		0			0
$508$	−4.49661e26	+	4.49661e26i	−0.0174307	+	0.0174307i
$509$	3.16476e28			1.20172			0.600861	−	0.799354i	$-0.294825\pi$
$509$	3.16476e28			1.20172			0.600861	+	0.799354i	$0.294825\pi$
$510$		0			0
$511$	5.08182e28			1.85182
$512$	2.21461e28	−	2.21461e28i	0.790611	−	0.790611i
$513$		0			0
$514$			2.14784e28i			0.736020i
$515$	−3.75949e28	−	7.23404e27i	−1.26227	−	0.242887i
$516$		0			0
$517$	−1.29190e28	−	1.29190e28i	−0.416463	−	0.416463i
$518$	1.50082e28	+	1.50082e28i	0.474097	+	0.474097i
$519$		0			0
$520$	1.17465e28	+	1.73441e28i	0.356346	+	0.526158i
$521$			3.01590e28i			0.896646i	0.893872	+	0.448323i	$0.147979\pi$
$521$			3.01590e28i			0.896646i	−0.893872	+	0.448323i	$0.852021\pi$
$522$		0			0
$523$	−2.41287e28	+	2.41287e28i	−0.689074	+	0.689074i	−0.962027	−	0.272953i	$-0.911999\pi$
$523$	−2.41287e28	+	2.41287e28i	−0.689074	+	0.689074i	0.272953	+	0.962027i	$0.411999\pi$
$524$	−3.94488e27			−0.110422
$525$		0			0
$526$	−3.49418e28			−0.939712
$527$	−4.28902e28	+	4.28902e28i	−1.13070	+	1.13070i
$528$		0			0
$529$			6.26698e28i			1.58772i
$530$	−2.64625e28	−	3.90728e28i	−0.657255	−	0.970461i
$531$		0			0
$532$	4.26337e27	+	4.26337e27i	0.101784	+	0.101784i
$533$	2.16211e28	+	2.16211e28i	0.506106	+	0.506106i
$534$		0			0
$535$	−3.04069e28	−	5.85093e27i	−0.684318	−	0.131677i
$536$		−	6.28658e28i		−	1.38734i
$537$		0			0
$538$	−1.41551e28	+	1.41551e28i	−0.300400	+	0.300400i
$539$	4.66314e27			0.0970501
$540$		0			0
$541$	−6.86087e28			−1.37344			−0.686718	−	0.726924i	$-0.740949\pi$
$541$	−6.86087e28			−1.37344			−0.686718	+	0.726924i	$0.740949\pi$
$542$	9.33608e27	−	9.33608e27i	0.183304	−	0.183304i
$543$		0			0
$544$			2.29381e28i			0.433281i
$545$	1.57276e28	−	1.06517e28i	0.291406	−	0.197358i
$546$		0			0
$547$	−5.14447e28	−	5.14447e28i	−0.917221	−	0.917221i	0.0796058	−	0.996826i	$-0.474634\pi$
$547$	−5.14447e28	−	5.14447e28i	−0.917221	−	0.917221i	−0.996826	+	0.0796058i	$0.974634\pi$
$548$	3.80219e27	+	3.80219e27i	0.0665025	+	0.0665025i
$549$		0			0
$550$	−2.06683e28	+	8.86545e27i	−0.347933	+	0.149242i
$551$		−	4.29402e28i		−	0.709206i
$552$		0			0
$553$	3.08954e28	−	3.08954e28i	0.491224	−	0.491224i
$554$	−1.89951e28			−0.296339
$555$		0			0
$556$	1.52961e28			0.229772
$557$	−2.66199e28	+	2.66199e28i	−0.392399	+	0.392399i	−0.875542	−	0.483143i	$-0.839495\pi$
$557$	−2.66199e28	+	2.66199e28i	−0.392399	+	0.392399i	0.483143	+	0.875542i	$0.339495\pi$
$558$		0			0
$559$		−	3.24344e28i		−	0.460450i
$560$	−1.29655e28	+	6.73811e28i	−0.180641	+	0.938781i
$561$		0			0
$562$	4.26131e28	+	4.26131e28i	0.571889	+	0.571889i
$563$	−3.42007e28	−	3.42007e28i	−0.450502	−	0.450502i	0.445019	−	0.895521i	$-0.353197\pi$
$563$	−3.42007e28	−	3.42007e28i	−0.450502	−	0.450502i	−0.895521	+	0.445019i	$0.853197\pi$
$564$		0			0
$565$	−1.20412e28	+	6.25772e28i	−0.152813	+	0.794159i
$566$			2.09192e28i			0.260598i
$567$		0			0
$568$	−5.98392e28	+	5.98392e28i	−0.718337	+	0.718337i
$569$	−2.93564e28			−0.345958			−0.172979	−	0.984925i	$-0.555339\pi$
$569$	−2.93564e28			−0.345958			−0.172979	+	0.984925i	$0.555339\pi$
$570$		0			0
$571$	−6.85441e28			−0.778558			−0.389279	−	0.921120i	$-0.627276\pi$
$571$	−6.85441e28			−0.778558			−0.389279	+	0.921120i	$0.627276\pi$
$572$	1.94682e27	−	1.94682e27i	0.0217104	−	0.0217104i
$573$		0			0
$574$			1.14812e29i			1.23428i
$575$	−1.40054e29	+	6.00750e28i	−1.47838	+	0.634134i
$576$		0			0
$577$	−1.29375e29	−	1.29375e29i	−1.31675	−	1.31675i	−0.916332	−	0.400420i	$-0.868864\pi$
$577$	−1.29375e29	−	1.29375e29i	−1.31675	−	1.31675i	−0.400420	−	0.916332i	$-0.631136\pi$
$578$	1.33104e29	+	1.33104e29i	1.33030	+	1.33030i
$579$		0			0
$580$	−7.43484e27	+	5.03532e27i	−0.0716602	+	0.0485326i
$581$			2.26833e29i			2.14712i
$582$		0			0
$583$	−3.93030e28	+	3.93030e28i	−0.358844	+	0.358844i
$584$	1.95216e29			1.75057
$585$		0			0
$586$	1.45866e29			1.26191
$587$	9.51184e28	−	9.51184e28i	0.808285	−	0.808285i	−0.176089	−	0.984374i	$-0.556345\pi$
$587$	9.51184e28	−	9.51184e28i	0.808285	−	0.808285i	0.984374	+	0.176089i	$0.0563447\pi$
$588$		0			0
$589$			1.15658e29i			0.948341i
$590$	−6.38719e27	−	1.22903e27i	−0.0514474	−	0.00989956i
$591$		0			0
$592$	5.02958e28	+	5.02958e28i	0.390979	+	0.390979i
$593$	9.87168e28	+	9.87168e28i	0.753905	+	0.753905i	0.975206	−	0.221301i	$-0.0710302\pi$
$593$	9.87168e28	+	9.87168e28i	0.753905	+	0.753905i	−0.221301	+	0.975206i	$0.571030\pi$
$594$		0			0
$595$	−1.46989e29	−	2.17035e29i	−1.08357	−	1.59993i
$596$		−	2.88406e28i		−	0.208891i
$597$		0			0
$598$	−9.18366e28	+	9.18366e28i	−0.642175	+	0.642175i
$599$	1.67174e27			0.0114865			0.00574323	−	0.999984i	$-0.498172\pi$
$599$	1.67174e27			0.0114865			0.00574323	+	0.999984i	$0.498172\pi$
$600$		0			0
$601$	2.32045e29			1.53954			0.769769	−	0.638322i	$-0.220371\pi$
$601$	2.32045e29			1.53954			0.769769	+	0.638322i	$0.220371\pi$
$602$	8.61167e28	−	8.61167e28i	0.561467	−	0.561467i
$603$		0			0
$604$		−	4.69102e27i		−	0.0295379i
$605$	−7.57625e28	−	1.11866e29i	−0.468838	−	0.692257i
$606$		0			0
$607$	−2.00869e29	−	2.00869e29i	−1.20069	−	1.20069i	−0.973956	−	0.226736i	$-0.927194\pi$
$607$	−2.00869e29	−	2.00869e29i	−1.20069	−	1.20069i	−0.226736	−	0.973956i	$-0.572806\pi$
$608$	3.09275e28	+	3.09275e28i	0.181701	+	0.181701i
$609$		0			0
$610$	1.92353e28	+	3.70128e27i	0.109178	+	0.0210082i
$611$		−	1.57422e29i		−	0.878278i
$612$		0			0
$613$	2.87544e28	−	2.87544e28i	0.155014	−	0.155014i	−0.625339	−	0.780353i	$-0.715039\pi$
$613$	2.87544e28	−	2.87544e28i	0.155014	−	0.155014i	0.780353	+	0.625339i	$0.215039\pi$
$614$	−6.85058e28			−0.363044
$615$		0			0
$616$	9.26431e28			0.474477
$617$	6.49326e27	−	6.49326e27i	0.0326940	−	0.0326940i	−0.690571	−	0.723265i	$-0.742641\pi$
$617$	6.49326e27	−	6.49326e27i	0.0326940	−	0.0326940i	0.723265	+	0.690571i	$0.242641\pi$
$618$		0			0
$619$		−	7.93542e28i		−	0.386205i	−0.981179	−	0.193103i	$-0.938145\pi$
$619$		−	7.93542e28i		−	0.386205i	0.981179	−	0.193103i	$-0.0618550\pi$
$620$	2.00255e28	−	1.35625e28i	0.0958231	−	0.0648972i
$621$		0			0
$622$	−1.73099e28	−	1.73099e28i	−0.0800746	−	0.0800746i
$623$	1.42846e29	+	1.42846e29i	0.649744	+	0.649744i
$624$		0			0
$625$	−1.64748e29	−	1.56707e29i	−0.724567	−	0.689204i
$626$			2.56361e29i			1.10872i
$627$		0			0
$628$	−1.83121e28	+	1.83121e28i	−0.0765883	+	0.0765883i
$629$	−2.71721e29			−1.11761
$630$		0			0
$631$	4.23522e29			1.68487			0.842437	−	0.538795i	$-0.181120\pi$
$631$	4.23522e29			1.68487			0.842437	+	0.538795i	$0.181120\pi$
$632$	1.18683e29	−	1.18683e29i	0.464366	−	0.464366i
$633$		0			0
$634$		−	1.74435e29i		−	0.660231i
$635$	9.96096e27	−	5.17665e28i	0.0370832	−	0.192719i
$636$		0			0
$637$	2.84109e28	+	2.84109e28i	0.102334	+	0.102334i
$638$	−5.20618e28	−	5.20618e28i	−0.184460	−	0.184460i
$639$		0			0
$640$	−4.25204e28	+	2.20976e29i	−0.145783	+	0.757627i
$641$		−	4.02891e29i		−	1.35887i	−0.733735	−	0.679435i	$-0.762225\pi$
$641$		−	4.02891e29i		−	1.35887i	0.733735	−	0.679435i	$-0.237775\pi$
$642$		0			0
$643$	−2.02727e29	+	2.02727e29i	−0.661753	+	0.661753i	−0.955793	−	0.294040i	$-0.905000\pi$
$643$	−2.02727e29	+	2.02727e29i	−0.661753	+	0.661753i	0.294040	+	0.955793i	$0.405000\pi$
$644$	7.00536e28			0.224972
$645$		0			0
$646$	5.37331e29			1.67032
$647$	5.94615e28	−	5.94615e28i	0.181861	−	0.181861i	−0.610305	−	0.792167i	$-0.708953\pi$
$647$	5.94615e28	−	5.94615e28i	0.181861	−	0.181861i	0.792167	+	0.610305i	$0.208953\pi$
$648$		0			0
$649$			7.66109e27i			0.0226841i
$650$	−1.79939e29	−	7.19106e28i	−0.524246	−	0.209509i
$651$		0			0
$652$	4.16893e28	+	4.16893e28i	0.117605	+	0.117605i
$653$	−1.91098e29	−	1.91098e29i	−0.530478	−	0.530478i	0.390237	−	0.920715i	$-0.372393\pi$
$653$	−1.91098e29	−	1.91098e29i	−0.530478	−	0.530478i	−0.920715	+	0.390237i	$0.872393\pi$
$654$		0			0
$655$	2.70768e29	−	1.83380e29i	0.727887	−	0.492969i
$656$			3.84760e29i			1.01789i
$657$		0			0
$658$	4.17970e29	−	4.17970e29i	1.07096	−	1.07096i
$659$	−3.55396e29			−0.896222			−0.448111	−	0.893978i	$-0.647903\pi$
$659$	−3.55396e29			−0.896222			−0.448111	+	0.893978i	$0.647903\pi$
$660$		0			0
$661$	−4.11569e29			−1.00537			−0.502687	−	0.864469i	$-0.667655\pi$
$661$	−4.11569e29			−1.00537			−0.502687	+	0.864469i	$0.667655\pi$
$662$	5.78554e28	−	5.78554e28i	0.139102	−	0.139102i
$663$		0			0
$664$			8.71368e29i			2.02972i
$665$	−4.90814e29	−	9.44429e28i	−1.12536	−	0.216542i
$666$		0			0
$667$	−3.52786e29	−	3.52786e29i	−0.783774	−	0.783774i
$668$	−4.72223e28	−	4.72223e28i	−0.103275	−	0.103275i
$669$		0			0
$670$	3.26105e29	+	4.81507e29i	0.691151	+	1.02051i
$671$		−	2.30717e28i		−	0.0481386i
$672$		0			0
$673$	6.66598e29	−	6.66598e29i	1.34805	−	1.34805i	0.460270	−	0.887779i	$-0.347753\pi$
$673$	6.66598e29	−	6.66598e29i	1.34805	−	1.34805i	0.887779	−	0.460270i	$-0.152247\pi$
$674$	2.58762e29			0.515194
$675$		0			0
$676$	−4.13579e28			−0.0798211
$677$	2.70711e29	−	2.70711e29i	0.514429	−	0.514429i	−0.401452	−	0.915880i	$-0.631494\pi$
$677$	2.70711e29	−	2.70711e29i	0.514429	−	0.514429i	0.915880	+	0.401452i	$0.131494\pi$
$678$		0			0
$679$			6.01751e28i			0.110862i
$680$	−5.64652e29	−	8.33730e29i	−1.02432	−	1.51245i
$681$		0			0
$682$	1.40226e29	+	1.40226e29i	0.246657	+	0.246657i
$683$	−8.65592e28	−	8.65592e28i	−0.149933	−	0.149933i	0.628155	−	0.778088i	$-0.283810\pi$
$683$	−8.65592e28	−	8.65592e28i	−0.149933	−	0.149933i	−0.778088	+	0.628155i	$0.783810\pi$
$684$		0			0
$685$	−4.37722e29	−	8.42268e28i	−0.735270	−	0.141481i
$686$		−	4.78517e29i		−	0.791579i
$687$		0			0
$688$	2.88595e29	−	2.88595e29i	0.463031	−	0.463031i
$689$	−4.78920e29			−0.756766
$690$		0			0
$691$	8.19365e29			1.25591			0.627954	−	0.778251i	$-0.283893\pi$
$691$	8.19365e29			1.25591			0.627954	+	0.778251i	$0.283893\pi$
$692$	7.80825e28	−	7.80825e28i	0.117880	−	0.117880i
$693$		0			0
$694$			8.31474e29i			1.21779i
$695$	−1.04989e30	+	7.11051e29i	−1.51462	+	1.02579i
$696$		0			0
$697$	−1.03932e30	−	1.03932e30i	−1.45481	−	1.45481i
$698$	3.44362e29	+	3.44362e29i	0.474825	+	0.474825i
$699$		0			0
$700$	4.12024e28	+	9.60563e28i	0.0551307	+	0.128528i
$701$			1.26986e29i			0.167385i	0.996492	+	0.0836923i	$0.0266713\pi$
$701$			1.26986e29i			0.167385i	−0.996492	+	0.0836923i	$0.973329\pi$
$702$		0			0
$703$	−3.66362e29	+	3.66362e29i	−0.468683	+	0.468683i
$704$	3.50816e29			0.442147
$705$		0			0
$706$	7.53064e29			0.921261
$707$	6.84628e29	−	6.84628e29i	0.825184	−	0.825184i
$708$		0			0
$709$		−	1.16150e30i		−	1.35904i	−0.733656	−	0.679521i	$-0.762188\pi$
$709$		−	1.16150e30i		−	1.35904i	0.733656	−	0.679521i	$-0.237812\pi$
$710$	1.47920e29	−	7.68732e29i	0.170535	−	0.886262i
$711$		0			0
$712$	5.48735e29	+	5.48735e29i	0.614218	+	0.614218i
$713$	9.50217e29	+	9.50217e29i	1.04805	+	1.04805i
$714$		0			0
$715$	−4.31264e28	+	2.24125e29i	−0.0461881	+	0.240037i
$716$			1.34283e29i			0.141721i
$717$		0			0
$718$	−1.54874e29	+	1.54874e29i	−0.158735	+	0.158735i
$719$	−8.03314e28			−0.0811394			−0.0405697	−	0.999177i	$-0.512917\pi$
$719$	−8.03314e28			−0.0811394			−0.0405697	+	0.999177i	$0.512917\pi$
$720$		0			0
$721$	−1.45891e30			−1.43122
$722$	4.06066e28	−	4.06066e28i	0.0392605	−	0.0392605i
$723$		0			0
$724$		−	3.66081e28i		−	0.0343813i
$725$	2.76241e29	−	6.91228e29i	0.255705	−	0.639841i
$726$		0			0
$727$	−8.69987e27	−	8.69987e27i	−0.00782351	−	0.00782351i	0.703184	−	0.711008i	$-0.251761\pi$
$727$	−8.69987e27	−	8.69987e27i	−0.00782351	−	0.00782351i	−0.711008	+	0.703184i	$0.751761\pi$
$728$	5.64443e29	+	5.64443e29i	0.500312	+	0.500312i
$729$		0			0
$730$	−1.49522e30	+	1.01265e30i	−1.28770	+	0.872106i
$731$			1.55912e30i			1.32357i
$732$		0			0
$733$	1.17633e30	−	1.17633e30i	0.970367	−	0.970367i	−0.0292068	−	0.999573i	$-0.509298\pi$
$733$	1.17633e30	−	1.17633e30i	0.970367	−	0.970367i	0.999573	+	0.0292068i	$0.00929812\pi$
$734$	−3.88065e29			−0.315570
$735$		0			0
$736$	5.08186e29			0.401611
$737$	4.84344e29	−	4.84344e29i	0.377351	−	0.377351i
$738$		0			0
$739$		−	4.34493e29i		−	0.329015i	−0.986376	−	0.164508i	$-0.947396\pi$
$739$		−	4.34493e29i		−	0.329015i	0.986376	−	0.164508i	$-0.0526035\pi$
$740$	1.06394e29	+	2.04725e28i	0.0794302	+	0.0152840i
$741$		0			0
$742$	−1.27158e30	−	1.27158e30i	−0.922790	−	0.922790i
$743$	1.46973e30	+	1.46973e30i	1.05161	+	1.05161i	0.998594	+	0.0530156i	$0.0168833\pi$
$743$	1.46973e30	+	1.46973e30i	1.05161	+	1.05161i	0.0530156	+	0.998594i	$0.483117\pi$
$744$		0			0
$745$	1.34068e30	+	1.97956e30i	0.932574	+	1.37698i
$746$			2.50486e30i			1.71801i
$747$		0			0
$748$	−9.35838e28	+	9.35838e28i	−0.0624071	+	0.0624071i
$749$	−1.17997e30			−0.775912
$750$		0			0
$751$	−1.07677e30			−0.688497			−0.344249	−	0.938878i	$-0.611866\pi$
$751$	−1.07677e30			−0.688497			−0.344249	+	0.938878i	$0.611866\pi$
$752$	1.40071e30	−	1.40071e30i	0.883202	−	0.883202i
$753$		0			0
$754$		−	6.34390e29i		−	0.389007i
$755$	2.18065e29	+	3.21981e29i	0.131869	+	0.194710i
$756$		0			0
$757$	5.35044e29	+	5.35044e29i	0.314690	+	0.314690i	0.846723	−	0.532033i	$-0.178572\pi$
$757$	5.35044e29	+	5.35044e29i	0.314690	+	0.314690i	−0.532033	+	0.846723i	$0.678572\pi$
$758$	3.98469e29	+	3.98469e29i	0.231136	+	0.231136i
$759$		0			0
$760$	−1.88544e30	−	3.62798e29i	−1.06383	−	0.204702i
$761$			3.47174e30i			1.93201i	0.258532	+	0.966003i	$0.416761\pi$
$761$			3.47174e30i			1.93201i	−0.258532	+	0.966003i	$0.583239\pi$
$762$		0			0
$763$	5.11836e29	−	5.11836e29i	0.277091	−	0.277091i
$764$	1.31379e29			0.0701527
$765$		0			0
$766$	−4.16498e29			−0.216377
$767$	−4.66765e28	+	4.66765e28i	−0.0239192	+	0.0239192i
$768$		0			0
$769$		−	1.23557e29i		−	0.0616084i	−0.999525	−	0.0308042i	$-0.990193\pi$
$769$		−	1.23557e29i		−	0.0616084i	0.999525	−	0.0308042i	$-0.00980683\pi$
$770$	−7.09580e29	+	4.80570e29i	−0.349019	+	0.236377i
$771$		0			0
$772$	1.32758e29	+	1.32758e29i	0.0635447	+	0.0635447i
$773$	−2.03622e30	−	2.03622e30i	−0.961480	−	0.961480i	0.0378049	−	0.999285i	$-0.487963\pi$
$773$	−2.03622e30	−	2.03622e30i	−0.961480	−	0.961480i	−0.999285	+	0.0378049i	$0.987963\pi$
$774$		0			0
$775$	−7.44046e29	+	1.86180e30i	−0.341926	+	0.855587i
$776$			2.31160e29i			0.104801i
$777$		0			0
$778$	−2.05907e30	+	2.05907e30i	−0.908628	+	0.908628i
$779$	−2.80265e30			−1.22018
$780$		0			0
$781$	−9.22052e29			−0.390768
$782$	4.41459e30	−	4.41459e30i	1.84595	−	1.84595i
$783$		0			0
$784$			5.05590e29i			0.205816i
$785$	4.05653e29	−	2.10816e30i	0.162938	−	0.846782i
$786$		0			0
$787$	−3.05416e30	−	3.05416e30i	−1.19442	−	1.19442i	−0.975813	−	0.218608i	$-0.929848\pi$
$787$	−3.05416e30	−	3.05416e30i	−1.19442	−	1.19442i	−0.218608	−	0.975813i	$-0.570152\pi$
$788$	−5.54128e28	−	5.54128e28i	−0.0213838	−	0.0213838i
$789$		0			0
$790$	−2.93380e29	+	1.52468e30i	−0.110242	+	0.572920i
$791$			2.42837e30i			0.900455i
$792$		0			0
$793$	1.40568e29	−	1.40568e29i	0.0507597	−	0.0507597i
$794$	3.89751e29			0.138890
$795$		0			0
$796$	−9.67225e28			−0.0335691
$797$	1.55837e30	−	1.55837e30i	0.533776	−	0.533776i	−0.387918	−	0.921694i	$-0.626806\pi$
$797$	1.55837e30	−	1.55837e30i	0.533776	−	0.533776i	0.921694	+	0.387918i	$0.126806\pi$
$798$		0			0
$799$			7.56726e30i			2.52463i
$800$	2.98893e29	+	6.96816e29i	0.0984170	+	0.229442i
$801$		0			0
$802$	−2.61872e30	−	2.61872e30i	−0.839960	−	0.839960i
$803$	1.50402e30	+	1.50402e30i	0.476148	+	0.476148i
$804$		0			0
$805$	−4.80833e30	+	3.25649e30i	−1.48299	+	1.00437i
$806$			1.70871e30i			0.520175i
$807$		0			0
$808$	2.62997e30	−	2.62997e30i	0.780066	−	0.780066i
$809$	−6.12048e30			−1.79195			−0.895975	−	0.444105i	$-0.853522\pi$
$809$	−6.12048e30			−1.79195			−0.895975	+	0.444105i	$0.853522\pi$
$810$		0			0
$811$	4.15917e30			1.18656			0.593278	−	0.804998i	$-0.297834\pi$
$811$	4.15917e30			1.18656			0.593278	+	0.804998i	$0.297834\pi$
$812$	−2.41959e29	+	2.41959e29i	−0.0681401	+	0.0681401i
$813$		0			0
$814$			8.88372e29i			0.243803i
$815$	−4.79942e30	−	9.23508e29i	−1.30027	−	0.250199i
$816$		0			0
$817$	2.10217e30	+	2.10217e30i	0.555055	+	0.555055i
$818$	2.30626e30	+	2.30626e30i	0.601173	+	0.601173i
$819$		0			0
$820$	3.28648e29	+	4.85262e29i	0.0834999	+	0.123291i
$821$			3.76196e30i			0.943650i	0.881692	+	0.471825i	$0.156405\pi$
$821$			3.76196e30i			0.943650i	−0.881692	+	0.471825i	$0.843595\pi$
$822$		0			0
$823$	4.00221e30	−	4.00221e30i	0.978592	−	0.978592i	−0.0211840	−	0.999776i	$-0.506744\pi$
$823$	4.00221e30	−	4.00221e30i	0.978592	−	0.978592i	0.999776	+	0.0211840i	$0.00674357\pi$
$824$	−5.60432e30			−1.35297
$825$		0			0
$826$	−2.47861e29			−0.0583335
$827$	−1.09772e30	+	1.09772e30i	−0.255083	+	0.255083i	−0.823051	−	0.567968i	$-0.807730\pi$
$827$	−1.09772e30	+	1.09772e30i	−0.255083	+	0.255083i	0.567968	+	0.823051i	$0.307730\pi$
$828$		0			0
$829$		−	7.60657e30i		−	1.72332i	−0.507484	−	0.861661i	$-0.669424\pi$
$829$		−	7.60657e30i		−	1.72332i	0.507484	−	0.861661i	$-0.330576\pi$
$830$	−4.52007e30	−	6.67406e30i	−1.01117	−	1.49304i
$831$		0			0
$832$	2.13741e30	+	2.13741e30i	0.466221	+	0.466221i
$833$	−1.36571e30	−	1.36571e30i	−0.294162	−	0.294162i
$834$		0			0
$835$	5.43640e30	+	1.04608e30i	1.14184	+	0.219713i
$836$			2.52359e29i			0.0523423i
$837$		0			0
$838$	1.96982e30	−	1.96982e30i	0.398442	−	0.398442i
$839$	7.81664e30			1.56142			0.780710	−	0.624894i	$-0.214858\pi$
$839$	7.81664e30			1.56142			0.780710	+	0.624894i	$0.214858\pi$
$840$		0			0
$841$	−2.69586e30			−0.525218
$842$	3.27224e30	−	3.27224e30i	0.629605	−	0.629605i
$843$		0			0
$844$		−	3.86906e29i		−	0.0726122i
$845$	2.83872e30	−	1.92255e30i	0.526171	−	0.356354i
$846$		0			0
$847$	−3.64056e30	−	3.64056e30i	−0.658252	−	0.658252i
$848$	−4.26134e30	−	4.26134e30i	−0.761008	−	0.761008i
$849$		0			0
$850$	8.64966e30	+	3.45674e30i	1.50696	+	0.602238i
$851$			6.01988e30i			1.03592i
$852$		0			0
$853$	4.14777e30	−	4.14777e30i	0.696385	−	0.696385i	−0.267244	−	0.963629i	$-0.586113\pi$
$853$	4.14777e30	−	4.14777e30i	0.696385	−	0.696385i	0.963629	+	0.267244i	$0.0861129\pi$
$854$	7.46446e29			0.123791
$855$		0			0
$856$	−4.53281e30			−0.733488
$857$	4.90754e30	−	4.90754e30i	0.784449	−	0.784449i	−0.196129	−	0.980578i	$-0.562837\pi$
$857$	4.90754e30	−	4.90754e30i	0.784449	−	0.784449i	0.980578	+	0.196129i	$0.0628371\pi$
$858$		0			0
$859$			2.45720e30i			0.383276i	0.981466	+	0.191638i	$0.0613800\pi$
$859$			2.45720e30i			0.383276i	−0.981466	+	0.191638i	$0.938620\pi$
$860$	1.17470e29	−	6.10487e29i	0.0181007	−	0.0940681i
$861$		0			0
$862$	−4.76243e30	−	4.76243e30i	−0.716148	−	0.716148i
$863$	1.80803e30	+	1.80803e30i	0.268592	+	0.268592i	0.828533	−	0.559941i	$-0.189176\pi$
$863$	1.80803e30	+	1.80803e30i	0.268592	+	0.268592i	−0.559941	+	0.828533i	$0.689176\pi$
$864$		0			0
$865$	−1.72970e30	+	8.98914e30i	−0.250785	+	1.30331i
$866$		−	4.99555e30i		−	0.715559i
$867$		0			0
$868$	6.51706e29	−	6.51706e29i	0.0911160	−	0.0911160i
$869$	1.82877e30			0.252611
$870$		0			0
$871$	5.90189e30			0.795793
$872$	1.96620e30	−	1.96620e30i	0.261941	−	0.261941i
$873$		0			0
$874$		−	1.19044e31i		−	1.54824i
$875$	−7.29329e30	−	4.67778e30i	−0.937215	−	0.601112i
$876$		0			0
$877$	1.99001e30	+	1.99001e30i	0.249666	+	0.249666i	0.820834	−	0.571167i	$-0.193509\pi$
$877$	1.99001e30	+	1.99001e30i	0.249666	+	0.249666i	−0.571167	+	0.820834i	$0.693509\pi$
$878$	4.42522e30	+	4.42522e30i	0.548583	+	0.548583i
$879$		0			0
$880$	−2.37795e30	+	1.61049e30i	−0.287830	+	0.194935i
$881$		−	3.09947e30i		−	0.370715i	−0.982671	−	0.185358i	$-0.940656\pi$
$881$		−	3.09947e30i		−	0.370715i	0.982671	−	0.185358i	$-0.0593444\pi$
$882$		0			0
$883$	6.11836e30	−	6.11836e30i	0.714575	−	0.714575i	−0.252914	−	0.967489i	$-0.581389\pi$
$883$	6.11836e30	−	6.11836e30i	0.714575	−	0.714575i	0.967489	+	0.252914i	$0.0813890\pi$
$884$	−1.14035e30			−0.131610
$885$		0			0
$886$	7.77956e30			0.876801
$887$	1.06912e31	−	1.06912e31i	1.19077	−	1.19077i	0.213917	−	0.976852i	$-0.431378\pi$
$887$	1.06912e31	−	1.06912e31i	1.19077	−	1.19077i	0.976852	−	0.213917i	$-0.0686224\pi$
$888$		0			0
$889$		−	2.00885e30i		−	0.218514i
$890$	−7.04939e30	−	1.35645e30i	−0.757803	−	0.145817i
$891$		0			0
$892$	−1.48629e30	−	1.48629e30i	−0.156053	−	0.156053i
$893$	1.02030e31	+	1.02030e31i	1.05873	+	1.05873i
$894$		0			0
$895$	−6.24222e30	−	9.21687e30i	−0.632699	−	0.934204i
$896$			8.57519e30i			0.859033i
$897$		0			0
$898$	1.20440e31	−	1.20440e31i	1.17861	−	1.17861i
$899$	−6.56392e30			−0.634872
$900$		0			0
$901$	2.30217e31			2.17534
$902$	−3.39800e30	+	3.39800e30i	−0.317362	+	0.317362i
$903$		0			0
$904$			9.32848e30i			0.851221i
$905$	1.70175e30	+	2.51270e30i	0.153492	+	0.226637i
$906$		0			0
$907$	8.06881e30	+	8.06881e30i	0.711104	+	0.711104i	0.966766	−	0.255662i	$-0.0822935\pi$
$907$	8.06881e30	+	8.06881e30i	0.711104	+	0.711104i	−0.255662	+	0.966766i	$0.582293\pi$
$908$	5.52051e29	+	5.52051e29i	0.0480926	+	0.0480926i
$909$		0			0
$910$	−7.25119e30	−	1.39528e30i	−0.617269	−	0.118776i
$911$		−	1.06614e31i		−	0.897164i	−0.893742	−	0.448582i	$-0.851929\pi$
$911$		−	1.06614e31i		−	0.897164i	0.893742	−	0.448582i	$-0.148071\pi$
$912$		0			0
$913$	−6.71338e30	+	6.71338e30i	−0.552075	+	0.552075i
$914$	−2.07026e31			−1.68302
$915$		0			0
$916$	−1.07224e30			−0.0851900
$917$	8.81184e30	−	8.81184e30i	0.692131	−	0.692131i
$918$		0			0
$919$		−	3.83734e30i		−	0.294590i	−0.989093	−	0.147295i	$-0.952943\pi$
$919$		−	3.83734e30i		−	0.294590i	0.989093	−	0.147295i	$-0.0470566\pi$
$920$	−1.84710e31	+	1.25097e31i	−1.40190	+	0.949454i
$921$		0			0
$922$	−1.54760e31	−	1.54760e31i	−1.14811	−	1.14811i
$923$	−5.61775e30	−	5.61775e30i	−0.412045	−	0.412045i
$924$		0			0
$925$	−8.25436e30	+	3.54063e30i	−0.591828	+	0.253859i
$926$			1.05186e31i			0.745666i
$927$		0			0
$928$	−1.75523e30	+	1.75523e30i	−0.121641	+	0.121641i
$929$	1.42887e31			0.979102			0.489551	−	0.871975i	$-0.337161\pi$
$929$	1.42887e31			0.979102			0.489551	+	0.871975i	$0.337161\pi$
$930$		0			0
$931$	−3.68279e30			−0.246720
$932$	−1.51437e29	+	1.51437e29i	−0.0100315	+	0.0100315i
$933$		0			0
$934$			2.75342e31i			1.78332i
$935$	2.07309e30	−	1.07737e31i	0.132769	−	0.689991i
$936$		0			0
$937$	6.34566e30	+	6.34566e30i	0.397385	+	0.397385i	0.877310	−	0.479925i	$-0.159336\pi$
$937$	6.34566e30	+	6.34566e30i	0.397385	+	0.397385i	−0.479925	+	0.877310i	$0.659336\pi$
$938$	1.56701e31	+	1.56701e31i	0.970380	+	0.970380i
$939$		0			0
$940$	5.70147e29	−	2.96302e30i	0.0345258	−	0.179429i
$941$			5.30410e30i			0.317629i	0.987308	+	0.158815i	$0.0507672\pi$
$941$			5.30410e30i			0.317629i	−0.987308	+	0.158815i	$0.949233\pi$
$942$		0			0
$943$	−2.30259e31	+	2.30259e31i	−1.34848	+	1.34848i
$944$	−8.30635e29			−0.0481066
$945$		0			0
$946$	5.09745e30			0.288732
$947$	6.68629e29	−	6.68629e29i	0.0374551	−	0.0374551i	−0.688131	−	0.725586i	$-0.741569\pi$
$947$	6.68629e29	−	6.68629e29i	0.0374551	−	0.0374551i	0.725586	+	0.688131i	$0.241569\pi$
$948$		0			0
$949$			1.83270e31i			1.00415i
$950$	1.63231e31	−	7.00163e30i	0.884514	−	0.379404i
$951$		0			0
$952$	−2.71328e31	−	2.71328e31i	−1.43816	−	1.43816i
$953$	−5.29617e30	−	5.29617e30i	−0.277643	−	0.277643i	0.554525	−	0.832167i	$-0.312900\pi$
$953$	−5.29617e30	−	5.29617e30i	−0.277643	−	0.277643i	−0.832167	+	0.554525i	$0.812900\pi$
$954$		0			0
$955$	−9.01754e30	+	6.10722e30i	−0.462437	+	0.313190i
$956$		−	1.81904e30i		−	0.0922645i
$957$		0			0
$958$	5.04965e30	−	5.04965e30i	0.250567	−	0.250567i
$959$	−1.69862e31			−0.833683
$960$		0			0
$961$	−3.14584e30			−0.151057
$962$	−5.41256e30	+	5.41256e30i	−0.257078	+	0.257078i
$963$		0			0
$964$		−	4.87659e30i		−	0.226625i
$965$	−1.52836e31	−	2.94088e30i	−0.702569	−	0.135189i
$966$		0			0
$967$	1.32785e30	+	1.32785e30i	0.0597272	+	0.0597272i	0.736339	−	0.676612i	$-0.236553\pi$
$967$	1.32785e30	+	1.32785e30i	0.0597272	+	0.0597272i	−0.676612	+	0.736339i	$0.736553\pi$
$968$	−1.39851e31	−	1.39851e31i	−0.622261	−	0.622261i
$969$		0			0
$970$	−1.19910e30	−	1.77052e30i	−0.0522100	−	0.0770900i
$971$			1.58635e30i			0.0683276i	0.999416	+	0.0341638i	$0.0108768\pi$
$971$			1.58635e30i			0.0683276i	−0.999416	+	0.0341638i	$0.989123\pi$
$972$		0			0
$973$	−3.41676e31	+	3.41676e31i	−1.44022	+	1.44022i
$974$	−8.40456e30			−0.350466
$975$		0			0
$976$	2.50150e30			0.102089
$977$	−1.57029e31	+	1.57029e31i	−0.633995	+	0.633995i	−0.949068	−	0.315073i	$-0.897971\pi$
$977$	−1.57029e31	+	1.57029e31i	−0.633995	+	0.633995i	0.315073	+	0.949068i	$0.397971\pi$
$978$		0			0
$979$			8.45536e30i			0.334129i
$980$	4.31857e29	+	6.37653e29i	0.0168837	+	0.0249293i
$981$		0			0
$982$	1.84308e31	+	1.84308e31i	0.705301	+	0.705301i
$983$	9.53643e30	+	9.53643e30i	0.361055	+	0.361055i	0.864201	−	0.503146i	$-0.167824\pi$
$983$	9.53643e30	+	9.53643e30i	0.361055	+	0.361055i	−0.503146	+	0.864201i	$0.667824\pi$
$984$		0			0
$985$	6.37932e30	+	1.22751e30i	0.236425	+	0.0454932i
$986$			3.04951e31i			1.11821i
$987$		0			0
$988$	−1.53754e30	+	1.53754e30i	−0.0551922	+	0.0551922i
$989$	3.45419e31			1.22683
$990$		0			0
$991$	−2.69150e30			−0.0935881			−0.0467941	−	0.998905i	$-0.514900\pi$
$991$	−2.69150e30			−0.0935881			−0.0467941	+	0.998905i	$0.514900\pi$
$992$	4.72764e30	−	4.72764e30i	0.162657	−	0.162657i
$993$		0			0
$994$		−	2.98314e31i		−	1.00489i
$995$	6.63882e30	−	4.49621e30i	0.221283	−	0.149866i
$996$		0			0
$997$	−3.96709e31	−	3.96709e31i	−1.29471	−	1.29471i	−0.931838	−	0.362874i	$-0.881796\pi$
$997$	−3.96709e31	−	3.96709e31i	−1.29471	−	1.29471i	−0.362874	−	0.931838i	$-0.618204\pi$
$998$	3.18880e31	+	3.18880e31i	1.02981	+	1.02981i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.22.f.a.17.13	yes	84
3.2	odd	2	inner	45.22.f.a.17.30	yes	84
5.3	odd	4	inner	45.22.f.a.8.30	yes	84
15.8	even	4	inner	45.22.f.a.8.13	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.22.f.a.8.13	✓	84	15.8	even	4	inner
45.22.f.a.8.30	yes	84	5.3	odd	4	inner
45.22.f.a.17.13	yes	84	1.1	even	1	trivial
45.22.f.a.17.30	yes	84	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-957.532 + 957.532i) q^{2} +263415. i q^{4} +(-1.22450e7 - 1.80803e7i) q^{5} +(-5.88402e8 - 5.88402e8i) q^{7} +(-2.26032e9 - 2.26032e9i) q^{8} +O(q^{10})\)	\(q+(-957.532 + 957.532i) q^{2} +263415. i q^{4} +(-1.22450e7 - 1.80803e7i) q^{5} +(-5.88402e8 - 5.88402e8i) q^{7} +(-2.26032e9 - 2.26032e9i) q^{8} +(2.90375e10 + 5.58742e9i) q^{10} -3.48289e10i q^{11} +(2.12201e11 - 2.12201e11i) q^{13} +1.12683e12 q^{14} +3.77624e12 q^{16} +(-1.02005e13 + 1.02005e13i) q^{17} +2.75067e13i q^{19} +(4.76262e12 - 3.22553e12i) q^{20} +(3.33498e13 + 3.33498e13i) q^{22} +(2.25988e14 + 2.25988e14i) q^{23} +(-1.76955e14 + 4.42787e14i) q^{25} +4.06378e14i q^{26} +(1.54994e14 - 1.54994e14i) q^{28} -1.56108e15 q^{29} +4.20472e15 q^{31} +(1.12436e15 - 1.12436e15i) q^{32} -1.95346e16i q^{34} +(-3.43346e15 + 1.78435e16i) q^{35} +(1.33190e16 + 1.33190e16i) q^{37} +(-2.63385e16 - 2.63385e16i) q^{38} +(-1.31895e16 + 6.85449e16i) q^{40} +1.01890e17i q^{41} +(7.64240e16 - 7.64240e16i) q^{43} +9.17446e15 q^{44} -4.32782e17 q^{46} +(3.70927e17 - 3.70927e17i) q^{47} +1.33887e17i q^{49} +(-2.54543e17 - 5.93423e17i) q^{50} +(5.58969e16 + 5.58969e16i) q^{52} +(-1.12846e18 - 1.12846e18i) q^{53} +(-6.29715e17 + 4.26481e17i) q^{55} +2.65995e18i q^{56} +(1.49479e18 - 1.49479e18i) q^{58} -2.19964e17 q^{59} +6.62431e17 q^{61} +(-4.02616e18 + 4.02616e18i) q^{62} +1.00726e19i q^{64} +(-6.43505e18 - 1.23824e18i) q^{65} +(1.39064e19 + 1.39064e19i) q^{67} +(-2.68696e18 - 2.68696e18i) q^{68} +(-1.37980e19 - 2.03733e19i) q^{70} -2.64738e19i q^{71} +(-4.31833e19 + 4.31833e19i) q^{73} -2.55068e19 q^{74} -7.24568e18 q^{76} +(-2.04934e19 + 2.04934e19i) q^{77} +5.25073e19i q^{79} +(-4.62402e19 - 6.82754e19i) q^{80} +(-9.75627e19 - 9.75627e19i) q^{82} +(-1.92753e20 - 1.92753e20i) q^{83} +(3.09333e20 + 5.95221e19i) q^{85} +1.46357e20i q^{86} +(-7.87244e19 + 7.87244e19i) q^{88} -2.42769e20 q^{89} -2.49718e20 q^{91} +(-5.95288e19 + 5.95288e19i) q^{92} +7.10349e20i q^{94} +(4.97328e20 - 3.36820e20i) q^{95} +(-5.11344e19 - 5.11344e19i) q^{97} +(-1.28201e20 - 1.28201e20i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q + 687552792 q^{7}+O(q^{10}) \) 84 * q + 687552792 * q^7	\( 84 q + 687552792 q^{7} + 134292212112 q^{10} - 1733796401796 q^{13} - 65338919579016 q^{16} - 6984840176592 q^{22} - 11\!\cdots\!84 q^{25}+ \cdots + 37\!\cdots\!56 q^{97}+O(q^{100}) \) 84 * q + 687552792 * q^7 + 134292212112 * q^10 - 1733796401796 * q^13 - 65338919579016 * q^16 - 6984840176592 * q^22 - 1180149988019184 * q^25 - 1688782704439392 * q^28 - 142192095805536 * q^31 - 26634804805627596 * q^37 - 179561523957547992 * q^40 - 568410120821540832 * q^43 - 1043921016330318336 * q^46 + 4197387941225299704 * q^52 + 2218912905679837848 * q^55 + 11205074447113145712 * q^58 - 13309008883877058528 * q^61 - 487057429881367584 * q^67 - 73413022182740953296 * q^70 + 103761433025202728604 * q^73 - 71719061752861140000 * q^76 - 826264089579051988704 * q^82 + 414467640188195121048 * q^85 - 1465877561249256251424 * q^88 + 746298426009998704464 * q^91 + 3706758938239203368556 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more