LMFDB - Embedded newform 315.10.b.b.251.43

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(251,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("315.251");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.b (of order $2$, degree $1$, 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:	$162.236288392$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.43
Character	$\chi$	$=$	315.251
Dual form			315.10.b.b.251.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+38.0944i q^{2} -939.186 q^{4} +625.000 q^{5} +(-1841.41 + 6079.71i) q^{7} -16273.4i q^{8} +23809.0i q^{10} -84202.7i q^{11} +42662.6i q^{13} +(-231603. - 70147.3i) q^{14} +139063. q^{16} +337476. q^{17} +852828. i q^{19} -586991. q^{20} +3.20765e6 q^{22} -1.66390e6i q^{23} +390625. q^{25} -1.62521e6 q^{26} +(1.72942e6 - 5.70997e6i) q^{28} -1.11114e6i q^{29} +1.50291e6i q^{31} -3.03446e6i q^{32} +1.28559e7i q^{34} +(-1.15088e6 + 3.79982e6i) q^{35} +1.30751e7 q^{37} -3.24880e7 q^{38} -1.01709e7i q^{40} +1.50066e7 q^{41} -1.55380e7 q^{43} +7.90820e7i q^{44} +6.33852e7 q^{46} +2.26851e7 q^{47} +(-3.35721e7 - 2.23904e7i) q^{49} +1.48806e7i q^{50} -4.00681e7i q^{52} +3.52198e7i q^{53} -5.26267e7i q^{55} +(9.89375e7 + 2.99659e7i) q^{56} +4.23284e7 q^{58} +1.20666e8 q^{59} +1.46514e8i q^{61} -5.72524e7 q^{62} +1.86796e8 q^{64} +2.66641e7i q^{65} -4.59585e7 q^{67} -3.16952e8 q^{68} +(-1.44752e8 - 4.38421e7i) q^{70} +2.61665e8i q^{71} +1.84075e7i q^{73} +4.98090e8i q^{74} -8.00964e8i q^{76} +(5.11928e8 + 1.55051e8i) q^{77} +5.41476e8 q^{79} +8.69142e7 q^{80} +5.71670e8i q^{82} +1.05895e8 q^{83} +2.10922e8 q^{85} -5.91910e8i q^{86} -1.37026e9 q^{88} -4.37433e8 q^{89} +(-2.59376e8 - 7.85591e7i) q^{91} +1.56271e9i q^{92} +8.64175e8i q^{94} +5.33017e8i q^{95} -2.49633e7i q^{97} +(8.52950e8 - 1.27891e9i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12288 q^{4} + 30000 q^{5} + 1824 q^{7} - 442908 q^{14} + 3258948 q^{16} - 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} + 5432976 q^{26} - 3685092 q^{28} + 1140000 q^{35} + 7750344 q^{37} - 17423136 q^{38}+ \cdots + 2546372484 q^{98}+O(q^{100}) $ 48 * q - 12288 * q^4 + 30000 * q^5 + 1824 * q^7 - 442908 * q^14 + 3258948 * q^16 - 7680000 * q^20 - 2860668 * q^22 + 18750000 * q^25 + 5432976 * q^26 - 3685092 * q^28 + 1140000 * q^35 + 7750344 * q^37 - 17423136 * q^38 + 70626600 * q^41 + 16073832 * q^43 - 15083820 * q^46 + 33200856 * q^47 - 39367500 * q^49 + 318486852 * q^56 + 293819940 * q^58 - 342990864 * q^59 - 896801400 * q^62 - 1639789824 * q^64 - 980229936 * q^67 + 438669576 * q^68 - 276817500 * q^70 + 900236484 * q^77 + 325977144 * q^79 + 2036842500 * q^80 - 1415370096 * q^83 + 3319557300 * q^88 + 1132967016 * q^89 - 1162515060 * q^91 + 2546372484 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			38.0944i			1.68355i	0.539827	+	0.841776i	$0.318490\pi$
$2$			38.0944i			1.68355i	−0.539827	+	0.841776i	$0.681510\pi$
$3$		0			0
$3$		0			0
$4$	−939.186			−1.83435
$5$	625.000			0.447214
$5$	625.000			0.447214
$6$		0			0
$7$	−1841.41	+	6079.71i	−0.289874	+	0.957065i
$7$	−1841.41	+	6079.71i	−0.289874	+	0.957065i
$8$		−	16273.4i		−	1.40467i
$9$		0			0
$10$			23809.0i			0.752907i
$11$		−	84202.7i		−	1.73404i	−0.498274	−	0.867020i	$-0.666033\pi$
$11$		−	84202.7i		−	1.73404i	0.498274	−	0.867020i	$-0.333967\pi$
$12$		0			0
$13$			42662.6i			0.414287i	0.978311	+	0.207144i	$0.0664168\pi$
$13$			42662.6i			0.414287i	−0.978311	+	0.207144i	$0.933583\pi$
$14$	−231603.	−	70147.3i	−1.61127	−	0.488017i
$15$		0			0
$16$	139063.			0.530482
$17$	337476.			0.979992			0.489996	−	0.871725i	$-0.336998\pi$
$17$	337476.			0.979992			0.489996	+	0.871725i	$0.336998\pi$
$18$		0			0
$19$			852828.i			1.50131i	0.660695	+	0.750655i	$0.270262\pi$
$19$			852828.i			1.50131i	−0.660695	+	0.750655i	$0.729738\pi$
$20$	−586991.			−0.820345
$21$		0			0
$22$	3.20765e6			2.91935
$23$		−	1.66390e6i		−	1.23980i	−0.784682	−	0.619899i	$-0.787174\pi$
$23$		−	1.66390e6i		−	1.23980i	0.784682	−	0.619899i	$-0.212826\pi$
$24$		0			0
$25$	390625.			0.200000
$26$	−1.62521e6			−0.697474
$27$		0			0
$28$	1.72942e6	−	5.70997e6i	0.531729	−	1.75559i
$29$		−	1.11114e6i		−	0.291729i	−0.989305	−	0.145864i	$-0.953404\pi$
$29$		−	1.11114e6i		−	0.291729i	0.989305	−	0.145864i	$-0.0465963\pi$
$30$		0			0
$31$			1.50291e6i			0.292284i	0.989264	+	0.146142i	$0.0466856\pi$
$31$			1.50291e6i			0.292284i	−0.989264	+	0.146142i	$0.953314\pi$
$32$		−	3.03446e6i		−	0.511572i
$33$		0			0
$34$			1.28559e7i			1.64987i
$35$	−1.15088e6	+	3.79982e6i	−0.129635	+	0.428012i
$36$		0			0
$37$	1.30751e7			1.14693			0.573467	−	0.819229i	$-0.305598\pi$
$37$	1.30751e7			1.14693			0.573467	+	0.819229i	$0.305598\pi$
$38$	−3.24880e7			−2.52753
$39$		0			0
$40$		−	1.01709e7i		−	0.628186i
$41$	1.50066e7			0.829385			0.414693	−	0.909962i	$-0.363889\pi$
$41$	1.50066e7			0.829385			0.414693	+	0.909962i	$0.363889\pi$
$42$		0			0
$43$	−1.55380e7			−0.693085			−0.346542	−	0.938034i	$-0.612644\pi$
$43$	−1.55380e7			−0.693085			−0.346542	+	0.938034i	$0.612644\pi$
$44$			7.90820e7i			3.18083i
$45$		0			0
$46$	6.33852e7			2.08726
$47$	2.26851e7			0.678110			0.339055	−	0.940767i	$-0.389893\pi$
$47$	2.26851e7			0.678110			0.339055	+	0.940767i	$0.389893\pi$
$48$		0			0
$49$	−3.35721e7	−	2.23904e7i	−0.831947	−	0.554856i
$50$			1.48806e7i			0.336710i
$51$		0			0
$52$		−	4.00681e7i		−	0.759947i
$53$			3.52198e7i			0.613119i	0.951851	+	0.306560i	$0.0991779\pi$
$53$			3.52198e7i			0.613119i	−0.951851	+	0.306560i	$0.900822\pi$
$54$		0			0
$55$		−	5.26267e7i		−	0.775486i
$56$	9.89375e7	+	2.99659e7i	1.34436	+	0.407176i
$57$		0			0
$58$	4.23284e7			0.491140
$59$	1.20666e8			1.29643			0.648217	−	0.761456i	$-0.275515\pi$
$59$	1.20666e8			1.29643			0.648217	+	0.761456i	$0.275515\pi$
$60$		0			0
$61$			1.46514e8i			1.35486i	0.735586	+	0.677431i	$0.236907\pi$
$61$			1.46514e8i			1.35486i	−0.735586	+	0.677431i	$0.763093\pi$
$62$	−5.72524e7			−0.492075
$63$		0			0
$64$	1.86796e8			1.39174
$65$			2.66641e7i			0.185275i
$66$		0			0
$67$	−4.59585e7			−0.278631			−0.139315	−	0.990248i	$-0.544490\pi$
$67$	−4.59585e7			−0.278631			−0.139315	+	0.990248i	$0.544490\pi$
$68$	−3.16952e8			−1.79765
$69$		0			0
$70$	−1.44752e8	−	4.38421e7i	−0.720581	−	0.218248i
$71$			2.61665e8i			1.22203i	0.791617	+	0.611017i	$0.209239\pi$
$71$			2.61665e8i			1.22203i	−0.791617	+	0.611017i	$0.790761\pi$
$72$		0			0
$73$			1.84075e7i			0.0758653i	0.999280	+	0.0379326i	$0.0120772\pi$
$73$			1.84075e7i			0.0758653i	−0.999280	+	0.0379326i	$0.987923\pi$
$74$			4.98090e8i			1.93092i
$75$		0			0
$76$		−	8.00964e8i		−	2.75392i
$77$	5.11928e8	+	1.55051e8i	1.65959	+	0.502652i
$78$		0			0
$79$	5.41476e8			1.56407			0.782037	−	0.623232i	$-0.214181\pi$
$79$	5.41476e8			1.56407			0.782037	+	0.623232i	$0.214181\pi$
$80$	8.69142e7			0.237239
$81$		0			0
$82$			5.71670e8i			1.39631i
$83$	1.05895e8			0.244920			0.122460	−	0.992473i	$-0.460922\pi$
$83$	1.05895e8			0.244920			0.122460	+	0.992473i	$0.460922\pi$
$84$		0			0
$85$	2.10922e8			0.438266
$86$		−	5.91910e8i		−	1.16684i
$87$		0			0
$88$	−1.37026e9			−2.43575
$89$	−4.37433e8			−0.739020			−0.369510	−	0.929227i	$-0.620475\pi$
$89$	−4.37433e8			−0.739020			−0.369510	+	0.929227i	$0.620475\pi$
$90$		0			0
$91$	−2.59376e8	−	7.85591e7i	−0.396500	−	0.120091i
$92$			1.56271e9i			2.27422i
$93$		0			0
$94$			8.64175e8i			1.14163i
$95$			5.33017e8i			0.671406i
$96$		0			0
$97$		−	2.49633e7i		−	0.0286305i	−0.999898	−	0.0143153i	$-0.995443\pi$
$97$		−	2.49633e7i		−	0.0286305i	0.999898	−	0.0143153i	$-0.00455685\pi$
$98$	8.52950e8	−	1.27891e9i	0.934128	−	1.40063i
$99$		0			0
$100$	−3.66869e8			−0.366869
$101$	5.12779e7			0.0490325			0.0245163	−	0.999699i	$-0.492195\pi$
$101$	5.12779e7			0.0490325			0.0245163	+	0.999699i	$0.492195\pi$
$102$		0			0
$103$			1.53085e9i			1.34019i	0.742277	+	0.670093i	$0.233746\pi$
$103$			1.53085e9i			1.34019i	−0.742277	+	0.670093i	$0.766254\pi$
$104$	6.94265e8			0.581936
$105$		0			0
$106$	−1.34168e9			−1.03222
$107$		−	4.51557e8i		−	0.333032i	−0.986039	−	0.166516i	$-0.946748\pi$
$107$		−	4.51557e8i		−	0.333032i	0.986039	−	0.166516i	$-0.0532517\pi$
$108$		0			0
$109$	−1.72072e9			−1.16759			−0.583797	−	0.811899i	$-0.698434\pi$
$109$	−1.72072e9			−1.16759			−0.583797	+	0.811899i	$0.698434\pi$
$110$	2.00478e9			1.30557
$111$		0			0
$112$	−2.56071e8	+	8.45461e8i	−0.153773	+	0.507706i
$113$			1.77082e9i			1.02170i	0.859671	+	0.510848i	$0.170669\pi$
$113$			1.77082e9i			1.02170i	−0.859671	+	0.510848i	$0.829331\pi$
$114$		0			0
$115$		−	1.03993e9i		−	0.554454i
$116$			1.04357e9i			0.535132i
$117$		0			0
$118$			4.59670e9i			2.18261i
$119$	−6.21430e8	+	2.05175e9i	−0.284074	+	0.937916i
$120$		0			0
$121$	−4.73215e9			−2.00689
$122$	−5.58137e9			−2.28098
$123$		0			0
$124$		−	1.41151e9i		−	0.536150i
$125$	2.44141e8			0.0894427
$126$		0			0
$127$	−4.13404e9			−1.41012			−0.705062	−	0.709145i	$-0.749081\pi$
$127$	−4.13404e9			−1.41012			−0.705062	+	0.709145i	$0.749081\pi$
$128$			5.56225e9i			1.83150i
$129$		0			0
$130$	−1.01575e9			−0.311920
$131$	−8.73121e8			−0.259032			−0.129516	−	0.991577i	$-0.541342\pi$
$131$	−8.73121e8			−0.259032			−0.129516	+	0.991577i	$0.541342\pi$
$132$		0			0
$133$	−5.18494e9	−	1.57040e9i	−1.43685	−	0.435190i
$134$		−	1.75076e9i		−	0.469089i
$135$		0			0
$136$		−	5.49188e9i		−	1.37656i
$137$		−	2.57450e9i		−	0.624382i	−0.950019	−	0.312191i	$-0.898937\pi$
$137$		−	2.57450e9i		−	0.624382i	0.950019	−	0.312191i	$-0.101063\pi$
$138$		0			0
$139$			6.52630e9i			1.48286i	0.671030	+	0.741430i	$0.265852\pi$
$139$			6.52630e9i			1.48286i	−0.671030	+	0.741430i	$0.734148\pi$
$140$	1.08089e9	−	3.56873e9i	0.237796	−	0.785123i
$141$		0			0
$142$	−9.96799e9			−2.05736
$143$	3.59230e9			0.718391
$144$		0			0
$145$		−	6.94465e8i		−	0.130465i
$146$	−7.01225e8			−0.127723
$147$		0			0
$148$	−1.22800e10			−2.10387
$149$			9.44902e8i			0.157054i	0.996912	+	0.0785269i	$0.0250216\pi$
$149$			9.44902e8i			0.157054i	−0.996912	+	0.0785269i	$0.974978\pi$
$150$		0			0
$151$	−1.32097e9			−0.206775			−0.103388	−	0.994641i	$-0.532968\pi$
$151$	−1.32097e9			−0.206775			−0.103388	+	0.994641i	$0.532968\pi$
$152$	1.38784e10			2.10884
$153$		0			0
$154$	−5.90660e9	+	1.95016e10i	−0.846241	+	2.79400i
$155$			9.39317e8i			0.130713i
$156$		0			0
$157$		−	1.12248e10i		−	1.47445i	−0.675645	−	0.737227i	$-0.736135\pi$
$157$		−	1.12248e10i		−	1.47445i	0.675645	−	0.737227i	$-0.263865\pi$
$158$			2.06272e10i			2.63320i
$159$		0			0
$160$		−	1.89654e9i		−	0.228782i
$161$	1.01160e10	+	3.06391e9i	1.18657	+	0.359384i
$162$		0			0
$163$	1.49926e10			1.66353			0.831767	−	0.555125i	$-0.187330\pi$
$163$	1.49926e10			1.66353			0.831767	+	0.555125i	$0.187330\pi$
$164$	−1.40940e10			−1.52138
$165$		0			0
$166$			4.03400e9i			0.412335i
$167$	3.60836e9			0.358993			0.179496	−	0.983759i	$-0.442553\pi$
$167$	3.60836e9			0.358993			0.179496	+	0.983759i	$0.442553\pi$
$168$		0			0
$169$	8.78441e9			0.828366
$170$			8.03497e9i			0.737843i
$171$		0			0
$172$	1.45930e10			1.27136
$173$	−1.98627e10			−1.68590			−0.842949	−	0.537993i	$-0.819183\pi$
$173$	−1.98627e10			−1.68590			−0.842949	+	0.537993i	$0.819183\pi$
$174$		0			0
$175$	−7.19299e8	+	2.37489e9i	−0.0579747	+	0.191413i
$176$		−	1.17095e10i		−	0.919877i
$177$		0			0
$178$		−	1.66638e10i		−	1.24418i
$179$		−	1.27350e10i		−	0.927174i	−0.886051	−	0.463587i	$-0.846562\pi$
$179$		−	1.27350e10i		−	0.927174i	0.886051	−	0.463587i	$-0.153438\pi$
$180$		0			0
$181$		−	7.85158e8i		−	0.0543756i	−0.999630	−	0.0271878i	$-0.991345\pi$
$181$		−	7.85158e8i		−	0.0543756i	0.999630	−	0.0271878i	$-0.00865521\pi$
$182$	2.99266e9	−	9.88077e9i	0.202179	−	0.667528i
$183$		0			0
$184$	−2.70772e10			−1.74150
$185$	8.17196e9			0.512924
$186$		0			0
$187$		−	2.84164e10i		−	1.69934i
$188$	−2.13055e10			−1.24389
$189$		0			0
$190$	−2.03050e10			−1.13035
$191$		−	2.37307e10i		−	1.29021i	−0.764093	−	0.645106i	$-0.776813\pi$
$191$		−	2.37307e10i		−	1.29021i	0.764093	−	0.645106i	$-0.223187\pi$
$192$		0			0
$193$	1.38650e9			0.0719306			0.0359653	−	0.999353i	$-0.488549\pi$
$193$	1.38650e9			0.0719306			0.0359653	+	0.999353i	$0.488549\pi$
$194$	9.50964e8			0.0482010
$195$		0			0
$196$	3.15304e10	+	2.10288e10i	1.52608	+	1.01780i
$197$			1.95840e10i			0.926411i	0.886251	+	0.463205i	$0.153301\pi$
$197$			1.95840e10i			0.926411i	−0.886251	+	0.463205i	$0.846699\pi$
$198$		0			0
$199$			1.54831e10i			0.699875i	0.936773	+	0.349937i	$0.113797\pi$
$199$			1.54831e10i			0.699875i	−0.936773	+	0.349937i	$0.886203\pi$
$200$		−	6.35680e9i		−	0.280933i
$201$		0			0
$202$			1.95340e9i			0.0825488i
$203$	6.75543e9	+	2.04607e9i	0.279203	+	0.0845644i
$204$		0			0
$205$	9.37915e9			0.370912
$206$	−5.83169e10			−2.25627
$207$		0			0
$208$			5.93277e9i			0.219772i
$209$	7.18104e10			2.60333
$210$		0			0
$211$	4.15394e10			1.44274			0.721371	−	0.692549i	$-0.243512\pi$
$211$	4.15394e10			1.44274			0.721371	+	0.692549i	$0.243512\pi$
$212$		−	3.30779e10i		−	1.12467i
$213$		0			0
$214$	1.72018e10			0.560676
$215$	−9.71123e9			−0.309957
$216$		0			0
$217$	−9.13724e9	−	2.76746e9i	−0.279735	−	0.0847253i
$218$		−	6.55500e10i		−	1.96571i
$219$		0			0
$220$			4.94262e10i			1.42251i
$221$			1.43976e10i			0.405998i
$222$		0			0
$223$			4.08884e10i			1.10720i	0.832781	+	0.553602i	$0.186747\pi$
$223$			4.08884e10i			1.10720i	−0.832781	+	0.553602i	$0.813253\pi$
$224$	1.84486e10	+	5.58768e9i	0.489608	+	0.148291i
$225$		0			0
$226$	−6.74584e10			−1.72008
$227$	−3.06705e9			−0.0766664			−0.0383332	−	0.999265i	$-0.512205\pi$
$227$	−3.06705e9			−0.0766664			−0.0383332	+	0.999265i	$0.512205\pi$
$228$		0			0
$229$			7.70544e10i			1.85156i	0.378062	+	0.925780i	$0.376591\pi$
$229$			7.70544e10i			1.85156i	−0.378062	+	0.925780i	$0.623409\pi$
$230$	3.96157e10			0.933453
$231$		0			0
$232$	−1.80821e10			−0.409782
$233$			1.35023e10i			0.300127i	0.988676	+	0.150064i	$0.0479478\pi$
$233$			1.35023e10i			0.300127i	−0.988676	+	0.150064i	$0.952052\pi$
$234$		0			0
$235$	1.41782e10			0.303260
$236$	−1.13328e11			−2.37811
$237$		0			0
$238$	−7.81604e10	−	2.36730e10i	−1.57903	−	0.478253i
$239$		−	3.98522e10i		−	0.790064i	−0.918667	−	0.395032i	$-0.870734\pi$
$239$		−	3.98522e10i		−	0.790064i	0.918667	−	0.395032i	$-0.129266\pi$
$240$		0			0
$241$		−	1.02086e11i		−	1.94936i	−0.223612	−	0.974678i	$-0.571785\pi$
$241$		−	1.02086e11i		−	1.94936i	0.223612	−	0.974678i	$-0.428215\pi$
$242$		−	1.80268e11i		−	3.37871i
$243$		0			0
$244$		−	1.37604e11i		−	2.48529i
$245$	−2.09825e10	−	1.39940e10i	−0.372058	−	0.248139i
$246$		0			0
$247$	−3.63838e10			−0.621974
$248$	2.44574e10			0.410561
$249$		0			0
$250$			9.30040e9i			0.150581i
$251$	−6.84821e10			−1.08904			−0.544522	−	0.838747i	$-0.683289\pi$
$251$	−6.84821e10			−1.08904			−0.544522	+	0.838747i	$0.683289\pi$
$252$		0			0
$253$	−1.40105e11			−2.14986
$254$		−	1.57484e11i		−	2.37402i
$255$		0			0
$256$	−1.16251e11			−1.69168
$257$	−1.34746e11			−1.92671			−0.963354	−	0.268234i	$-0.913560\pi$
$257$	−1.34746e11			−1.92671			−0.963354	+	0.268234i	$0.913560\pi$
$258$		0			0
$259$	−2.40766e10	+	7.94929e10i	−0.332466	+	1.09769i
$260$		−	2.50425e10i		−	0.339859i
$261$		0			0
$262$		−	3.32610e10i		−	0.436094i
$263$		−	9.76583e10i		−	1.25866i	−0.777138	−	0.629330i	$-0.783329\pi$
$263$		−	9.76583e10i		−	1.25866i	0.777138	−	0.629330i	$-0.216671\pi$
$264$		0			0
$265$			2.20124e10i			0.274195i
$266$	5.98236e10	−	1.97517e11i	0.732665	−	2.41901i
$267$		0			0
$268$	4.31635e10			0.511105
$269$	1.40523e11			1.63629			0.818147	−	0.575009i	$-0.195001\pi$
$269$	1.40523e11			1.63629			0.818147	+	0.575009i	$0.195001\pi$
$270$		0			0
$271$			5.84926e8i			0.00658778i	0.999995	+	0.00329389i	$0.00104848\pi$
$271$			5.84926e8i			0.00658778i	−0.999995	+	0.00329389i	$0.998952\pi$
$272$	4.69303e10			0.519868
$273$		0			0
$274$	9.80740e10			1.05118
$275$		−	3.28917e10i		−	0.346808i
$276$		0			0
$277$	−4.68309e10			−0.477941			−0.238970	−	0.971027i	$-0.576810\pi$
$277$	−4.68309e10			−0.477941			−0.238970	+	0.971027i	$0.576810\pi$
$278$	−2.48616e11			−2.49647
$279$		0			0
$280$	6.18359e10	+	1.87287e10i	0.601215	+	0.182095i
$281$			5.07549e10i			0.485623i	0.970073	+	0.242812i	$0.0780697\pi$
$281$			5.07549e10i			0.485623i	−0.970073	+	0.242812i	$0.921930\pi$
$282$		0			0
$283$		−	6.30599e10i		−	0.584406i	−0.956356	−	0.292203i	$-0.905612\pi$
$283$		−	6.30599e10i		−	0.584406i	0.956356	−	0.292203i	$-0.0943883\pi$
$284$		−	2.45752e11i		−	2.24164i
$285$		0			0
$286$			1.36847e11i			1.20945i
$287$	−2.76333e10	+	9.12360e10i	−0.240417	+	0.793775i
$288$		0			0
$289$	−4.69796e9			−0.0396159
$290$	2.64552e10			0.219645
$291$		0			0
$292$		−	1.72881e10i		−	0.139163i
$293$	−1.90956e11			−1.51366			−0.756831	−	0.653611i	$-0.773253\pi$
$293$	−1.90956e11			−1.51366			−0.756831	+	0.653611i	$0.773253\pi$
$294$		0			0
$295$	7.54162e10			0.579783
$296$		−	2.12777e11i		−	1.61106i
$297$		0			0
$298$	−3.59955e10			−0.264408
$299$	7.09860e10			0.513633
$300$		0			0
$301$	2.86117e10	−	9.44663e10i	0.200907	−	0.663327i
$302$		−	5.03218e10i		−	0.348117i
$303$		0			0
$304$			1.18597e11i			0.796418i
$305$			9.15713e10i			0.605913i
$306$		0			0
$307$			2.40435e11i			1.54481i	0.635131	+	0.772404i	$0.280946\pi$
$307$			2.40435e11i			1.54481i	−0.635131	+	0.772404i	$0.719054\pi$
$308$	−4.80795e11	−	1.45622e11i	−3.04426	−	0.922038i
$309$		0			0
$310$	−3.57828e10			−0.220063
$311$	2.26474e11			1.37277			0.686384	−	0.727239i	$-0.259197\pi$
$311$	2.26474e11			1.37277			0.686384	+	0.727239i	$0.259197\pi$
$312$		0			0
$313$		−	7.18510e10i		−	0.423140i	−0.977363	−	0.211570i	$-0.932142\pi$
$313$		−	7.18510e10i		−	0.423140i	0.977363	−	0.211570i	$-0.0678576\pi$
$314$	4.27603e11			2.48232
$315$		0			0
$316$	−5.08547e11			−2.86906
$317$			2.51383e11i			1.39820i	0.715024	+	0.699100i	$0.246416\pi$
$317$			2.51383e11i			1.39820i	−0.715024	+	0.699100i	$0.753584\pi$
$318$		0			0
$319$	−9.35613e10			−0.505869
$320$	1.16748e11			0.622405
$321$		0			0
$322$	−1.16718e11	+	3.85363e11i	−0.605042	+	1.99765i
$323$			2.87809e11i			1.47127i
$324$		0			0
$325$			1.66651e10i			0.0828575i
$326$			5.71133e11i			2.80065i
$327$		0			0
$328$		−	2.44209e11i		−	1.16501i
$329$	−4.17725e10	+	1.37919e11i	−0.196566	+	0.648995i
$330$		0			0
$331$	3.02578e11			1.38552			0.692758	−	0.721170i	$-0.256396\pi$
$331$	3.02578e11			1.38552			0.692758	+	0.721170i	$0.256396\pi$
$332$	−9.94549e10			−0.449267
$333$		0			0
$334$			1.37458e11i			0.604383i
$335$	−2.87240e10			−0.124607
$336$		0			0
$337$	9.07356e9			0.0383216			0.0191608	−	0.999816i	$-0.493901\pi$
$337$	9.07356e9			0.0383216			0.0191608	+	0.999816i	$0.493901\pi$
$338$			3.34637e11i			1.39460i
$339$		0			0
$340$	−1.98095e11			−0.803931
$341$	1.26549e11			0.506832
$342$		0			0
$343$	1.97947e11	−	1.62878e11i	0.772192	−	0.635389i
$344$			2.52856e11i			0.973553i
$345$		0			0
$346$		−	7.56659e11i		−	2.83830i
$347$			1.76345e11i			0.652953i	0.945205	+	0.326476i	$0.105861\pi$
$347$			1.76345e11i			0.652953i	−0.945205	+	0.326476i	$0.894139\pi$
$348$		0			0
$349$			4.88457e11i			1.76243i	0.472716	+	0.881215i	$0.343274\pi$
$349$			4.88457e11i			1.76243i	−0.472716	+	0.881215i	$0.656726\pi$
$350$	−9.04699e10	−	2.74013e10i	−0.322254	−	0.0976034i
$351$		0			0
$352$	−2.55510e11			−0.887086
$353$	1.41645e11			0.485530			0.242765	−	0.970085i	$-0.421946\pi$
$353$	1.41645e11			0.485530			0.242765	+	0.970085i	$0.421946\pi$
$354$		0			0
$355$			1.63541e11i			0.546511i
$356$	4.10831e11			1.35562
$357$		0			0
$358$	4.85134e11			1.56095
$359$			2.96641e11i			0.942555i	0.881985	+	0.471278i	$0.156207\pi$
$359$			2.96641e11i			0.942555i	−0.881985	+	0.471278i	$0.843793\pi$
$360$		0			0
$361$	−4.04628e11			−1.25393
$362$	2.99102e10			0.0915441
$363$		0			0
$364$	2.43602e11	+	7.37816e10i	0.727319	+	0.220288i
$365$			1.15047e10i			0.0339280i
$366$		0			0
$367$		−	4.07579e11i		−	1.17277i	−0.810031	−	0.586387i	$-0.800550\pi$
$367$		−	4.07579e11i		−	1.17277i	0.810031	−	0.586387i	$-0.199450\pi$
$368$		−	2.31386e11i		−	0.657691i
$369$		0			0
$370$			3.11306e11i			0.863535i
$371$	−2.14126e11	−	6.48539e10i	−0.586795	−	0.177727i
$372$		0			0
$373$	−2.59429e11			−0.693951			−0.346975	−	0.937874i	$-0.612791\pi$
$373$	−2.59429e11			−0.693951			−0.346975	+	0.937874i	$0.612791\pi$
$374$	1.08251e12			2.86093
$375$		0			0
$376$		−	3.69163e11i		−	0.952518i
$377$	4.74042e10			0.120860
$378$		0			0
$379$	−5.97278e11			−1.48696			−0.743481	−	0.668757i	$-0.766827\pi$
$379$	−5.97278e11			−1.48696			−0.743481	+	0.668757i	$0.766827\pi$
$380$		−	5.00602e11i		−	1.23159i
$381$		0			0
$382$	9.04009e11			2.17214
$383$	−1.37020e11			−0.325378			−0.162689	−	0.986677i	$-0.552017\pi$
$383$	−1.37020e11			−0.325378			−0.162689	+	0.986677i	$0.552017\pi$
$384$		0			0
$385$	3.19955e11	+	9.69071e10i	0.742190	+	0.224793i
$386$			5.28181e10i			0.121099i
$387$		0			0
$388$			2.34452e10i			0.0525184i
$389$		−	5.41587e11i		−	1.19921i	−0.800296	−	0.599605i	$-0.795324\pi$
$389$		−	5.41587e11i		−	1.19921i	0.800296	−	0.599605i	$-0.204676\pi$
$390$		0			0
$391$		−	5.61524e11i		−	1.21499i
$392$	−3.64368e11	+	5.46331e11i	−0.779387	+	1.16861i
$393$		0			0
$394$	−7.46042e11			−1.55966
$395$	3.38423e11			0.699475
$396$		0			0
$397$			6.47113e11i			1.30744i	0.756735	+	0.653722i	$0.226793\pi$
$397$			6.47113e11i			1.30744i	−0.756735	+	0.653722i	$0.773207\pi$
$398$	−5.89821e11			−1.17828
$399$		0			0
$400$	5.43214e10			0.106096
$401$			7.73138e11i			1.49316i	0.665294	+	0.746582i	$0.268306\pi$
$401$			7.73138e11i			1.49316i	−0.665294	+	0.746582i	$0.731694\pi$
$402$		0			0
$403$	−6.41179e10			−0.121089
$404$	−4.81595e10			−0.0899427
$405$		0			0
$406$	−7.79438e10	+	2.57344e11i	−0.142369	+	0.470053i
$407$		−	1.10096e12i		−	1.98883i
$408$		0			0
$409$		−	2.62991e11i		−	0.464715i	−0.972631	−	0.232357i	$-0.925356\pi$
$409$		−	2.62991e11i		−	0.464715i	0.972631	−	0.232357i	$-0.0746438\pi$
$410$			3.57294e11i			0.624450i
$411$		0			0
$412$		−	1.43775e12i		−	2.45837i
$413$	−2.22195e11	+	7.33613e11i	−0.375802	+	1.24077i
$414$		0			0
$415$	6.61843e10			0.109531
$416$	1.29458e11			0.211938
$417$		0			0
$418$			2.73558e12i			4.38284i
$419$	−1.26307e11			−0.200200			−0.100100	−	0.994977i	$-0.531916\pi$
$419$	−1.26307e11			−0.200200			−0.100100	+	0.994977i	$0.531916\pi$
$420$		0			0
$421$	−9.03070e11			−1.40104			−0.700522	−	0.713631i	$-0.747049\pi$
$421$	−9.03070e11			−1.40104			−0.700522	+	0.713631i	$0.747049\pi$
$422$			1.58242e12i			2.42893i
$423$		0			0
$424$	5.73145e11			0.861228
$425$	1.31826e11			0.195998
$426$		0			0
$427$	−8.90762e11	−	2.69792e11i	−1.29669	−	0.392739i
$428$			4.24096e11i			0.610896i
$429$		0			0
$430$		−	3.69944e11i		−	0.521828i
$431$			7.50217e11i			1.04722i	0.851957	+	0.523612i	$0.175416\pi$
$431$			7.50217e11i			1.04722i	−0.851957	+	0.523612i	$0.824584\pi$
$432$		0			0
$433$		−	1.05920e12i		−	1.44805i	−0.689775	−	0.724024i	$-0.742290\pi$
$433$		−	1.05920e12i		−	1.44805i	0.689775	−	0.724024i	$-0.257710\pi$
$434$	1.05425e11	−	3.48078e11i	0.142639	−	0.470948i
$435$		0			0
$436$	1.61608e12			2.14177
$437$	1.41902e12			1.86132
$438$		0			0
$439$			1.46305e12i			1.88004i	0.341117	+	0.940021i	$0.389195\pi$
$439$			1.46305e12i			1.88004i	−0.341117	+	0.940021i	$0.610805\pi$
$440$	−8.56415e11			−1.08930
$441$		0			0
$442$	−5.48468e11			−0.683519
$443$		−	5.76578e11i		−	0.711281i	−0.934623	−	0.355640i	$-0.884263\pi$
$443$		−	5.76578e11i		−	0.711281i	0.934623	−	0.355640i	$-0.115737\pi$
$444$		0			0
$445$	−2.73396e11			−0.330500
$446$	−1.55762e12			−1.86404
$447$		0			0
$448$	−3.43968e11	+	1.13567e12i	−0.403429	+	1.33199i
$449$			1.41523e12i			1.64330i	0.569990	+	0.821652i	$0.306947\pi$
$449$			1.41523e12i			1.64330i	−0.569990	+	0.821652i	$0.693053\pi$
$450$		0			0
$451$		−	1.26360e12i		−	1.43819i
$452$		−	1.66313e12i		−	1.87415i
$453$		0			0
$454$		−	1.16838e11i		−	0.129072i
$455$	−1.62110e11	−	4.90994e10i	−0.177320	−	0.0537063i
$456$		0			0
$457$	−8.57527e11			−0.919655			−0.459828	−	0.888008i	$-0.652089\pi$
$457$	−8.57527e11			−0.919655			−0.459828	+	0.888008i	$0.652089\pi$
$458$	−2.93535e12			−3.11720
$459$		0			0
$460$			9.76692e11i			1.01706i
$461$	1.00703e12			1.03846			0.519229	−	0.854635i	$-0.326219\pi$
$461$	1.00703e12			1.03846			0.519229	+	0.854635i	$0.326219\pi$
$462$		0			0
$463$	−5.94066e11			−0.600786			−0.300393	−	0.953815i	$-0.597118\pi$
$463$	−5.94066e11			−0.600786			−0.300393	+	0.953815i	$0.597118\pi$
$464$		−	1.54519e11i		−	0.154757i
$465$		0			0
$466$	−5.14361e11			−0.505279
$467$	1.31159e12			1.27607			0.638033	−	0.770009i	$-0.279749\pi$
$467$	1.31159e12			1.27607			0.638033	+	0.770009i	$0.279749\pi$
$468$		0			0
$469$	8.46282e10	−	2.79414e11i	0.0807676	−	0.266668i
$470$			5.40110e11i			0.510554i
$471$		0			0
$472$		−	1.96364e12i		−	1.82106i
$473$			1.30834e12i			1.20184i
$474$		0			0
$475$			3.33136e11i			0.300262i
$476$	5.83638e11	−	1.92698e12i	0.521090	−	1.72046i
$477$		0			0
$478$	1.51815e12			1.33011
$479$	−1.68046e12			−1.45854			−0.729272	−	0.684224i	$-0.760141\pi$
$479$	−1.68046e12			−1.45854			−0.729272	+	0.684224i	$0.760141\pi$
$480$		0			0
$481$			5.57818e11i			0.475160i
$482$	3.88892e12			3.28184
$483$		0			0
$484$	4.44437e12			3.68134
$485$		−	1.56021e10i		−	0.0128040i
$486$		0			0
$487$	1.37638e12			1.10881			0.554407	−	0.832246i	$-0.312945\pi$
$487$	1.37638e12			1.10881			0.554407	+	0.832246i	$0.312945\pi$
$488$	2.38428e12			1.90313
$489$		0			0
$490$	5.33094e11	−	7.99318e11i	0.417755	−	0.626379i
$491$			1.66072e12i			1.28953i	0.764381	+	0.644764i	$0.223044\pi$
$491$			1.66072e12i			1.28953i	−0.764381	+	0.644764i	$0.776956\pi$
$492$		0			0
$493$		−	3.74984e11i		−	0.285892i
$494$		−	1.38602e12i		−	1.04712i
$495$		0			0
$496$			2.08999e11i			0.155051i
$497$	−1.59085e12	−	4.81832e11i	−1.16957	−	0.354235i
$498$		0			0
$499$	1.79758e12			1.29788			0.648941	−	0.760839i	$-0.275212\pi$
$499$	1.79758e12			1.29788			0.648941	+	0.760839i	$0.275212\pi$
$500$	−2.29293e11			−0.164069
$501$		0			0
$502$		−	2.60879e12i		−	1.83346i
$503$	−3.81639e11			−0.265825			−0.132913	−	0.991128i	$-0.542433\pi$
$503$	−3.81639e11			−0.265825			−0.132913	+	0.991128i	$0.542433\pi$
$504$		0			0
$505$	3.20487e10			0.0219280
$506$		−	5.33720e12i		−	3.61940i
$507$		0			0
$508$	3.88263e12			2.58666
$509$	1.56092e11			0.103075			0.0515373	−	0.998671i	$-0.483588\pi$
$509$	1.56092e11			0.103075			0.0515373	+	0.998671i	$0.483588\pi$
$510$		0			0
$511$	−1.11912e11	−	3.38958e10i	−0.0726080	−	0.0219913i
$512$		−	1.58065e12i		−	1.01653i
$513$		0			0
$514$		−	5.13306e12i		−	3.24371i
$515$			9.56781e11i			0.599350i
$516$		0			0
$517$		−	1.91015e12i		−	1.17587i
$518$	−3.02824e12	−	9.17185e11i	−1.84802	−	0.559723i
$519$		0			0
$520$	4.33915e11			0.260250
$521$	2.23134e12			1.32677			0.663387	−	0.748277i	$-0.269118\pi$
$521$	2.23134e12			1.32677			0.663387	+	0.748277i	$0.269118\pi$
$522$		0			0
$523$			3.05055e12i			1.78288i	0.453143	+	0.891438i	$0.350303\pi$
$523$			3.05055e12i			1.78288i	−0.453143	+	0.891438i	$0.649697\pi$
$524$	8.20022e11			0.475155
$525$		0			0
$526$	3.72024e12			2.11902
$527$			5.07195e11i			0.286436i
$528$		0			0
$529$	−9.67396e11			−0.537098
$530$	−8.38548e11			−0.461622
$531$		0			0
$532$	4.86962e12	+	1.47490e12i	2.63568	+	0.798289i
$533$			6.40222e11i			0.343604i
$534$		0			0
$535$		−	2.82223e11i		−	0.148936i
$536$			7.47900e11i			0.391383i
$537$		0			0
$538$			5.35313e12i			2.75479i
$539$	−1.88533e12	+	2.82686e12i	−0.962141	+	1.44263i
$540$		0			0
$541$	−1.89145e11			−0.0949309			−0.0474655	−	0.998873i	$-0.515114\pi$
$541$	−1.89145e11			−0.0949309			−0.0474655	+	0.998873i	$0.515114\pi$
$542$	−2.22824e10			−0.0110909
$543$		0			0
$544$		−	1.02406e12i		−	0.501337i
$545$	−1.07545e12			−0.522164
$546$		0			0
$547$	1.08007e12			0.515832			0.257916	−	0.966167i	$-0.416964\pi$
$547$	1.08007e12			0.515832			0.257916	+	0.966167i	$0.416964\pi$
$548$			2.41793e12i			1.14533i
$549$		0			0
$550$	1.25299e12			0.583869
$551$	9.47614e11			0.437975
$552$		0			0
$553$	−9.97078e11	+	3.29202e12i	−0.453384	+	1.49692i
$554$		−	1.78400e12i		−	0.804638i
$555$		0			0
$556$		−	6.12941e12i		−	2.72008i
$557$			2.41084e11i			0.106125i	0.998591	+	0.0530627i	$0.0168983\pi$
$557$			2.41084e11i			0.106125i	−0.998591	+	0.0530627i	$0.983102\pi$
$558$		0			0
$559$		−	6.62889e11i		−	0.287136i
$560$	−1.60044e11	+	5.28413e11i	−0.0687693	+	0.227053i
$561$		0			0
$562$	−1.93348e12			−0.817572
$563$	1.54712e12			0.648986			0.324493	−	0.945888i	$-0.394806\pi$
$563$	1.54712e12			0.648986			0.324493	+	0.945888i	$0.394806\pi$
$564$		0			0
$565$			1.10676e12i			0.456916i
$566$	2.40223e12			0.983878
$567$		0			0
$568$	4.25818e12			1.71655
$569$			6.04337e11i			0.241698i	0.992671	+	0.120849i	$0.0385618\pi$
$569$			6.04337e11i			0.241698i	−0.992671	+	0.120849i	$0.961438\pi$
$570$		0			0
$571$	4.30931e11			0.169647			0.0848234	−	0.996396i	$-0.472967\pi$
$571$	4.30931e11			0.169647			0.0848234	+	0.996396i	$0.472967\pi$
$572$	−3.37384e12			−1.31778
$573$		0			0
$574$	−3.47558e12	−	1.05268e12i	−1.33636	−	0.404754i
$575$		−	6.49959e11i		−	0.247960i
$576$		0			0
$577$		−	1.27250e12i		−	0.477933i	−0.971028	−	0.238967i	$-0.923191\pi$
$577$		−	1.27250e12i		−	0.477933i	0.971028	−	0.238967i	$-0.0768086\pi$
$578$		−	1.78966e11i		−	0.0666954i
$579$		0			0
$580$			6.52231e11i			0.239318i
$581$	−1.94995e11	+	6.43810e11i	−0.0709957	+	0.234404i
$582$		0			0
$583$	2.96560e12			1.06317
$584$	2.99553e11			0.106565
$585$		0			0
$586$		−	7.27436e12i		−	2.54833i
$587$	9.00228e11			0.312954			0.156477	−	0.987682i	$-0.449986\pi$
$587$	9.00228e11			0.312954			0.156477	+	0.987682i	$0.449986\pi$
$588$		0			0
$589$	−1.28172e12			−0.438808
$590$			2.87294e12i			0.976094i
$591$		0			0
$592$	1.81826e12			0.608428
$593$	3.26508e12			1.08430			0.542148	−	0.840283i	$-0.317611\pi$
$593$	3.26508e12			1.08430			0.542148	+	0.840283i	$0.317611\pi$
$594$		0			0
$595$	−3.88394e11	+	1.28235e12i	−0.127042	+	0.419449i
$596$		−	8.87438e11i		−	0.288091i
$597$		0			0
$598$			2.70417e12i			0.864727i
$599$			3.36471e12i			1.06789i	0.845519	+	0.533945i	$0.179291\pi$
$599$			3.36471e12i			1.06789i	−0.845519	+	0.533945i	$0.820709\pi$
$600$		0			0
$601$			1.40213e11i			0.0438383i	0.999760	+	0.0219191i	$0.00697764\pi$
$601$			1.40213e11i			0.0438383i	−0.999760	+	0.0219191i	$0.993022\pi$
$602$	3.59864e12	+	1.08995e12i	1.11675	+	0.338237i
$603$		0			0
$604$	1.24064e12			0.379297
$605$	−2.95759e12			−0.897510
$606$		0			0
$607$			1.88297e12i			0.562983i	0.959564	+	0.281491i	$0.0908291\pi$
$607$			1.88297e12i			0.562983i	−0.959564	+	0.281491i	$0.909171\pi$
$608$	2.58787e12			0.768028
$609$		0			0
$610$	−3.48836e12			−1.02009
$611$			9.67804e11i			0.280932i
$612$		0			0
$613$	−1.00425e11			−0.0287255			−0.0143628	−	0.999897i	$-0.504572\pi$
$613$	−1.00425e11			−0.0287255			−0.0143628	+	0.999897i	$0.504572\pi$
$614$	−9.15923e12			−2.60077
$615$		0			0
$616$	2.52321e12	−	8.33080e12i	0.706059	−	2.33117i
$617$			1.73295e12i			0.481397i	0.970600	+	0.240698i	$0.0773765\pi$
$617$			1.73295e12i			0.481397i	−0.970600	+	0.240698i	$0.922624\pi$
$618$		0			0
$619$			3.70016e12i			1.01301i	0.862238	+	0.506503i	$0.169062\pi$
$619$			3.70016e12i			1.01301i	−0.862238	+	0.506503i	$0.830938\pi$
$620$		−	8.82194e11i		−	0.239774i
$621$		0			0
$622$			8.62741e12i			2.31113i
$623$	8.05492e11	−	2.65946e12i	0.214222	−	0.707291i
$624$		0			0
$625$	1.52588e11			0.0400000
$626$	2.73712e12			0.712377
$627$		0			0
$628$			1.05422e13i			2.70466i
$629$	4.41254e12			1.12399
$630$		0			0
$631$	−2.29169e12			−0.575471			−0.287736	−	0.957710i	$-0.592902\pi$
$631$	−2.29169e12			−0.575471			−0.287736	+	0.957710i	$0.592902\pi$
$632$		−	8.81166e12i		−	2.19700i
$633$		0			0
$634$	−9.57629e12			−2.35394
$635$	−2.58377e12			−0.630627
$636$		0			0
$637$	9.55233e11	−	1.43227e12i	0.229870	−	0.344665i
$638$		−	3.56416e12i		−	0.851657i
$639$		0			0
$640$			3.47641e12i			0.819070i
$641$		−	4.88525e12i		−	1.14295i	−0.820621	−	0.571473i	$-0.806372\pi$
$641$		−	4.88525e12i		−	1.14295i	0.820621	−	0.571473i	$-0.193628\pi$
$642$		0			0
$643$			6.62542e12i			1.52849i	0.644923	+	0.764247i	$0.276889\pi$
$643$			6.62542e12i			1.52849i	−0.644923	+	0.764247i	$0.723111\pi$
$644$	−9.50080e12	−	2.87758e12i	−2.17658	−	0.659236i
$645$		0			0
$646$	−1.09639e13			−2.47696
$647$	−5.93113e12			−1.33066			−0.665331	−	0.746548i	$-0.731710\pi$
$647$	−5.93113e12			−1.33066			−0.665331	+	0.746548i	$0.731710\pi$
$648$		0			0
$649$		−	1.01604e13i		−	2.24807i
$650$	−6.34846e11			−0.139495
$651$		0			0
$652$	−1.40808e13			−3.05150
$653$			9.60892e11i			0.206807i	0.994639	+	0.103404i	$0.0329733\pi$
$653$			9.60892e11i			0.206807i	−0.994639	+	0.103404i	$0.967027\pi$
$654$		0			0
$655$	−5.45700e11			−0.115843
$656$	2.08687e12			0.439974
$657$		0			0
$658$	−5.25393e12	−	1.59130e12i	−1.09262	−	0.330929i
$659$			2.73844e12i			0.565611i	0.959177	+	0.282806i	$0.0912652\pi$
$659$			2.73844e12i			0.565611i	−0.959177	+	0.282806i	$0.908735\pi$
$660$		0			0
$661$		−	8.95430e12i		−	1.82442i	−0.409721	−	0.912211i	$-0.634374\pi$
$661$		−	8.95430e12i		−	1.82442i	0.409721	−	0.912211i	$-0.365626\pi$
$662$			1.15265e13i			2.33259i
$663$		0			0
$664$		−	1.72327e12i		−	0.344030i
$665$	−3.24059e12	−	9.81502e11i	−0.642579	−	0.194623i
$666$		0			0
$667$	−1.84883e12			−0.361685
$668$	−3.38892e12			−0.658517
$669$		0			0
$670$		−	1.09423e12i		−	0.209783i
$671$	1.23369e13			2.34938
$672$		0			0
$673$	2.60506e12			0.489497			0.244749	−	0.969587i	$-0.421295\pi$
$673$	2.60506e12			0.489497			0.244749	+	0.969587i	$0.421295\pi$
$674$			3.45652e11i			0.0645163i
$675$		0			0
$676$	−8.25019e12			−1.51951
$677$	2.75069e12			0.503260			0.251630	−	0.967824i	$-0.419033\pi$
$677$	2.75069e12			0.503260			0.251630	+	0.967824i	$0.419033\pi$
$678$		0			0
$679$	1.51770e11	+	4.59676e10i	0.0274013	+	0.00829924i
$680$		−	3.43242e12i		−	0.615617i
$681$		0			0
$682$			4.82081e12i			0.853277i
$683$			9.39157e12i			1.65137i	0.564130	+	0.825686i	$0.309212\pi$
$683$			9.39157e12i			1.65137i	−0.564130	+	0.825686i	$0.690788\pi$
$684$		0			0
$685$		−	1.60906e12i		−	0.279232i
$686$	6.20476e12	+	7.54068e12i	1.06971	+	1.30003i
$687$		0			0
$688$	−2.16075e12			−0.367669
$689$	−1.50256e12			−0.254008
$690$		0			0
$691$			4.34647e12i			0.725246i	0.931936	+	0.362623i	$0.118119\pi$
$691$			4.34647e12i			0.725246i	−0.931936	+	0.362623i	$0.881881\pi$
$692$	1.86548e13			3.09252
$693$		0			0
$694$	−6.71778e12			−1.09928
$695$			4.07894e12i			0.663155i
$696$		0			0
$697$	5.06438e12			0.812791
$698$	−1.86075e13			−2.96714
$699$		0			0
$700$	6.75556e11	−	2.23046e12i	0.106346	−	0.351118i
$701$		−	4.71651e12i		−	0.737716i	−0.929486	−	0.368858i	$-0.879749\pi$
$701$		−	4.71651e12i		−	0.737716i	0.929486	−	0.368858i	$-0.120251\pi$
$702$		0			0
$703$			1.11508e13i			1.72190i
$704$		−	1.57288e13i		−	2.41333i
$705$		0			0
$706$			5.39590e12i			0.817415i
$707$	−9.44235e10	+	3.11755e11i	−0.0142132	+	0.0469273i
$708$		0			0
$709$	3.84343e12			0.571230			0.285615	−	0.958344i	$-0.407802\pi$
$709$	3.84343e12			0.571230			0.285615	+	0.958344i	$0.407802\pi$
$710$	−6.22999e12			−0.920079
$711$		0			0
$712$			7.11852e12i			1.03808i
$713$	2.50068e12			0.362373
$714$		0			0
$715$	2.24519e12			0.321274
$716$			1.19606e13i			1.70076i
$717$		0			0
$718$	−1.13004e13			−1.58684
$719$	1.09093e13			1.52235			0.761176	−	0.648545i	$-0.224622\pi$
$719$	1.09093e13			1.52235			0.761176	+	0.648545i	$0.224622\pi$
$720$		0			0
$721$	−9.30712e12	−	2.81892e12i	−1.28265	−	0.388485i
$722$		−	1.54141e13i		−	2.11106i
$723$		0			0
$724$			7.37410e11i			0.0997437i
$725$		−	4.34040e11i		−	0.0583457i
$726$		0			0
$727$		−	7.43636e12i		−	0.987315i	−0.869656	−	0.493657i	$-0.835660\pi$
$727$		−	7.43636e12i		−	0.987315i	0.869656	−	0.493657i	$-0.164340\pi$
$728$	−1.27842e12	+	4.22093e12i	−0.168688	+	0.556950i
$729$		0			0
$730$	−4.38266e11			−0.0571195
$731$	−5.24369e12			−0.679217
$732$		0			0
$733$		−	1.51330e12i		−	0.193623i	−0.995303	−	0.0968115i	$-0.969136\pi$
$733$		−	1.51330e12i		−	0.193623i	0.995303	−	0.0968115i	$-0.0308644\pi$
$734$	1.55265e13			1.97443
$735$		0			0
$736$	−5.04903e12			−0.634246
$737$			3.86983e12i			0.483156i
$738$		0			0
$739$	−9.36352e12			−1.15489			−0.577443	−	0.816431i	$-0.695949\pi$
$739$	−9.36352e12			−1.15489			−0.577443	+	0.816431i	$0.695949\pi$
$740$	−7.67498e12			−0.940881
$741$		0			0
$742$	2.47057e12	−	8.15700e12i	0.299213	−	0.987900i
$743$			1.53465e13i			1.84739i	0.383126	+	0.923696i	$0.374847\pi$
$743$			1.53465e13i			1.84739i	−0.383126	+	0.923696i	$0.625153\pi$
$744$		0			0
$745$			5.90563e11i			0.0702366i
$746$		−	9.88280e12i		−	1.16830i
$747$		0			0
$748$			2.66883e13i			3.11719i
$749$	2.74533e12	+	8.31500e11i	0.318733	+	0.0965370i
$750$		0			0
$751$	−1.15451e13			−1.32439			−0.662197	−	0.749330i	$-0.730376\pi$
$751$	−1.15451e13			−1.32439			−0.662197	+	0.749330i	$0.730376\pi$
$752$	3.15465e12			0.359725
$753$		0			0
$754$			1.80584e12i			0.203473i
$755$	−8.25609e11			−0.0924726
$756$		0			0
$757$	−1.73462e12			−0.191987			−0.0959937	−	0.995382i	$-0.530603\pi$
$757$	−1.73462e12			−0.191987			−0.0959937	+	0.995382i	$0.530603\pi$
$758$		−	2.27530e13i		−	2.50338i
$759$		0			0
$760$	8.67400e12			0.943102
$761$	3.49684e12			0.377959			0.188980	−	0.981981i	$-0.439482\pi$
$761$	3.49684e12			0.377959			0.188980	+	0.981981i	$0.439482\pi$
$762$		0			0
$763$	3.16855e12	−	1.04615e13i	0.338455	−	1.11746i
$764$			2.22876e13i			2.36670i
$765$		0			0
$766$		−	5.21969e12i		−	0.547792i
$767$			5.14791e12i			0.537096i
$768$		0			0
$769$		−	1.03414e12i		−	0.106638i	−0.998578	−	0.0533188i	$-0.983020\pi$
$769$		−	1.03414e12i		−	0.106638i	0.998578	−	0.0533188i	$-0.0169800\pi$
$770$	−3.69162e12	+	1.21885e13i	−0.378450	+	1.24952i
$771$		0			0
$772$	−1.30219e12			−0.131946
$773$	−1.51631e13			−1.52750			−0.763749	−	0.645514i	$-0.776643\pi$
$773$	−1.51631e13			−1.52750			−0.763749	+	0.645514i	$0.776643\pi$
$774$		0			0
$775$			5.87073e11i			0.0584568i
$776$	−4.06238e11			−0.0402164
$777$		0			0
$778$	2.06315e13			2.01893
$779$			1.27981e13i			1.24516i
$780$		0			0
$781$	2.20329e13			2.11906
$782$	2.13910e13			2.04550
$783$		0			0
$784$	−4.66862e12	−	3.11367e12i	−0.441333	−	0.294341i
$785$		−	7.01551e12i		−	0.659396i
$786$		0			0
$787$			5.57880e12i			0.518387i	0.965825	+	0.259194i	$0.0834568\pi$
$787$			5.57880e12i			0.518387i	−0.965825	+	0.259194i	$0.916543\pi$
$788$		−	1.83930e13i		−	1.69936i
$789$		0			0
$790$			1.28920e13i			1.17760i
$791$	−1.07661e13	−	3.26080e12i	−0.977830	−	0.296163i
$792$		0			0
$793$	−6.25066e12			−0.561302
$794$	−2.46514e13			−2.20115
$795$		0			0
$796$		−	1.45415e13i		−	1.28381i
$797$	−2.60232e12			−0.228453			−0.114227	−	0.993455i	$-0.536439\pi$
$797$	−2.60232e12			−0.228453			−0.114227	+	0.993455i	$0.536439\pi$
$798$		0			0
$799$	7.65567e12			0.664542
$800$		−	1.18534e12i		−	0.102314i
$801$		0			0
$802$	−2.94522e13			−2.51382
$803$	1.54997e12			0.131553
$804$		0			0
$805$	6.32250e12	+	1.91494e12i	0.530649	+	0.160722i
$806$		−	2.44253e12i		−	0.203860i
$807$		0			0
$808$		−	8.34466e11i		−	0.0688744i
$809$			3.57628e12i			0.293538i	0.989171	+	0.146769i	$0.0468873\pi$
$809$			3.57628e12i			0.293538i	−0.989171	+	0.146769i	$0.953113\pi$
$810$		0			0
$811$		−	5.59959e11i		−	0.0454530i	−0.999742	−	0.0227265i	$-0.992765\pi$
$811$		−	5.59959e11i		−	0.0454530i	0.999742	−	0.0227265i	$-0.00723469\pi$
$812$	−6.34460e12	−	1.92164e12i	−0.512156	−	0.155121i
$813$		0			0
$814$	4.19405e13			3.34829
$815$	9.37035e12			0.743955
$816$		0			0
$817$		−	1.32512e13i		−	1.04053i
$818$	1.00185e13			0.782371
$819$		0			0
$820$	−8.80877e12			−0.680382
$821$			2.03191e13i			1.56085i	0.625251	+	0.780424i	$0.284997\pi$
$821$			2.03191e13i			1.56085i	−0.625251	+	0.780424i	$0.715003\pi$
$822$		0			0
$823$	2.34007e13			1.77799			0.888995	−	0.457917i	$-0.151404\pi$
$823$	2.34007e13			1.77799			0.888995	+	0.457917i	$0.151404\pi$
$824$	2.49121e13			1.88252
$825$		0			0
$826$	−2.79466e13	−	8.46439e12i	−2.08890	−	0.632682i
$827$		−	1.30144e13i		−	0.967496i	−0.875207	−	0.483748i	$-0.839275\pi$
$827$		−	1.30144e13i		−	0.967496i	0.875207	−	0.483748i	$-0.160725\pi$
$828$		0			0
$829$			2.58006e12i			0.189730i	0.995490	+	0.0948648i	$0.0302419\pi$
$829$			2.58006e12i			0.189730i	−0.995490	+	0.0948648i	$0.969758\pi$
$830$			2.52125e12i			0.184402i
$831$		0			0
$832$			7.96921e12i			0.576581i
$833$	−1.13298e13	−	7.55623e12i	−0.815301	−	0.543754i
$834$		0			0
$835$	2.25522e12			0.160546
$836$	−6.74433e13			−4.77541
$837$		0			0
$838$		−	4.81158e12i		−	0.337047i
$839$	9.19034e12			0.640329			0.320164	−	0.947362i	$-0.396262\pi$
$839$	9.19034e12			0.640329			0.320164	+	0.947362i	$0.396262\pi$
$840$		0			0
$841$	1.32725e13			0.914894
$842$		−	3.44019e13i		−	2.35873i
$843$		0			0
$844$	−3.90132e13			−2.64649
$845$	5.49025e12			0.370457
$846$		0			0
$847$	8.71381e12	−	2.87701e13i	0.581745	−	1.92073i
$848$			4.89776e12i			0.325249i
$849$		0			0
$850$			5.02185e12i			0.329973i
$851$		−	2.17556e13i		−	1.42197i
$852$		0			0
$853$			2.27465e13i			1.47111i	0.677467	+	0.735553i	$0.263078\pi$
$853$			2.27465e13i			1.47111i	−0.677467	+	0.735553i	$0.736922\pi$
$854$	1.02776e13	−	3.39331e13i	0.661196	−	2.18305i
$855$		0			0
$856$	−7.34836e12			−0.467798
$857$	−1.19907e13			−0.759327			−0.379664	−	0.925125i	$-0.623960\pi$
$857$	−1.19907e13			−0.759327			−0.379664	+	0.925125i	$0.623960\pi$
$858$		0			0
$859$			1.01502e13i			0.636072i	0.948079	+	0.318036i	$0.103023\pi$
$859$			1.01502e13i			0.636072i	−0.948079	+	0.318036i	$0.896977\pi$
$860$	9.12065e12			0.568568
$861$		0			0
$862$	−2.85791e13			−1.76305
$863$		−	2.40731e13i		−	1.47735i	−0.674063	−	0.738674i	$-0.735452\pi$
$863$		−	2.40731e13i		−	1.47735i	0.674063	−	0.738674i	$-0.264548\pi$
$864$		0			0
$865$	−1.24142e13			−0.753957
$866$	4.03497e13			2.43786
$867$		0			0
$868$	8.58156e12	+	2.59916e12i	0.513130	+	0.155416i
$869$		−	4.55938e13i		−	2.71217i
$870$		0			0
$871$		−	1.96070e12i		−	0.115433i
$872$			2.80020e13i			1.64008i
$873$		0			0
$874$			5.40566e13i			3.13363i
$875$	−4.49562e11	+	1.48430e12i	−0.0259271	+	0.0856025i
$876$		0			0
$877$	3.30746e13			1.88798			0.943988	−	0.329980i	$-0.107042\pi$
$877$	3.30746e13			1.88798			0.943988	+	0.329980i	$0.107042\pi$
$878$	−5.57339e13			−3.16515
$879$		0			0
$880$		−	7.31841e12i		−	0.411382i
$881$	−2.27328e13			−1.27134			−0.635669	−	0.771962i	$-0.719276\pi$
$881$	−2.27328e13			−1.27134			−0.635669	+	0.771962i	$0.719276\pi$
$882$		0			0
$883$	2.52535e12			0.139797			0.0698987	−	0.997554i	$-0.477732\pi$
$883$	2.52535e12			0.139797			0.0698987	+	0.997554i	$0.477732\pi$
$884$		−	1.35220e13i		−	0.744742i
$885$		0			0
$886$	2.19644e13			1.19748
$887$	3.55886e13			1.93043			0.965216	−	0.261452i	$-0.0842014\pi$
$887$	3.55886e13			1.93043			0.965216	+	0.261452i	$0.0842014\pi$
$888$		0			0
$889$	7.61244e12	−	2.51337e13i	0.408758	−	1.34958i
$890$		−	1.04148e13i		−	0.556414i
$891$		0			0
$892$		−	3.84018e13i		−	2.03100i
$893$			1.93465e13i			1.01805i
$894$		0			0
$895$		−	7.95939e12i		−	0.414645i
$896$	−3.38169e13	−	1.02424e13i	−1.75286	−	0.530902i
$897$		0			0
$898$	−5.39123e13			−2.76659
$899$	1.66995e12			0.0852676
$900$		0			0
$901$			1.18858e13i			0.600852i
$902$	4.81361e13			2.42126
$903$		0			0
$904$	2.88173e13			1.43514
$905$		−	4.90724e11i		−	0.0243175i
$906$		0			0
$907$	2.90847e13			1.42703			0.713513	−	0.700642i	$-0.247103\pi$
$907$	2.90847e13			1.42703			0.713513	+	0.700642i	$0.247103\pi$
$908$	2.88053e12			0.140633
$909$		0			0
$910$	1.87042e12	−	6.17548e12i	0.0904174	−	0.298528i
$911$			6.16034e12i			0.296328i	0.988963	+	0.148164i	$0.0473363\pi$
$911$			6.16034e12i			0.296328i	−0.988963	+	0.148164i	$0.952664\pi$
$912$		0			0
$913$		−	8.91663e12i		−	0.424700i
$914$		−	3.26670e13i		−	1.54829i
$915$		0			0
$916$		−	7.23684e13i		−	3.39641i
$917$	1.60777e12	−	5.30832e12i	0.0750865	−	0.247910i
$918$		0			0
$919$	−4.45657e12			−0.206101			−0.103051	−	0.994676i	$-0.532860\pi$
$919$	−4.45657e12			−0.206101			−0.103051	+	0.994676i	$0.532860\pi$
$920$	−1.69233e13			−0.778824
$921$		0			0
$922$			3.83623e13i			1.74830i
$923$	−1.11633e13			−0.506274
$924$		0			0
$925$	5.10747e12			0.229387
$926$		−	2.26306e13i		−	1.01146i
$927$		0			0
$928$	−3.37172e12			−0.149240
$929$	−1.03844e13			−0.457413			−0.228707	−	0.973495i	$-0.573450\pi$
$929$	−1.03844e13			−0.457413			−0.228707	+	0.973495i	$0.573450\pi$
$930$		0			0
$931$	1.90952e13	−	2.86312e13i	0.833010	−	1.24901i
$932$		−	1.26811e13i		−	0.550537i
$933$		0			0
$934$			4.99644e13i			2.14832i
$935$		−	1.77602e13i		−	0.759970i
$936$		0			0
$937$			3.54858e13i			1.50393i	0.659205	+	0.751963i	$0.270893\pi$
$937$			3.54858e13i			1.50393i	−0.659205	+	0.751963i	$0.729107\pi$
$938$	1.06441e13	+	3.22386e12i	0.448949	+	0.135976i
$939$		0			0
$940$	−1.33159e13			−0.556284
$941$	9.59076e12			0.398749			0.199375	−	0.979923i	$-0.436109\pi$
$941$	9.59076e12			0.398749			0.199375	+	0.979923i	$0.436109\pi$
$942$		0			0
$943$		−	2.49695e13i		−	1.02827i
$944$	1.67801e13			0.687735
$945$		0			0
$946$	−4.98404e13			−2.02335
$947$		−	2.11868e13i		−	0.856032i	−0.903771	−	0.428016i	$-0.859213\pi$
$947$		−	2.11868e13i		−	0.856032i	0.903771	−	0.428016i	$-0.140787\pi$
$948$		0			0
$949$	−7.85313e11			−0.0314300
$950$	−1.26906e13			−0.505506
$951$		0			0
$952$	3.33890e13	+	1.01128e13i	1.31746	+	0.399029i
$953$			2.05972e12i			0.0808891i	0.999182	+	0.0404445i	$0.0128774\pi$
$953$			2.05972e12i			0.0808891i	−0.999182	+	0.0404445i	$0.987123\pi$
$954$		0			0
$955$		−	1.48317e13i		−	0.577001i
$956$			3.74287e13i			1.44925i
$957$		0			0
$958$		−	6.40163e13i		−	2.45553i
$959$	1.56522e13	+	4.74070e12i	0.597574	+	0.180992i
$960$		0			0
$961$	2.41809e13			0.914570
$962$	−2.12498e13			−0.799957
$963$		0			0
$964$			9.58781e13i			3.57580i
$965$	8.66565e11			0.0321683
$966$		0			0
$967$	−3.70576e12			−0.136288			−0.0681442	−	0.997675i	$-0.521708\pi$
$967$	−3.70576e12			−0.136288			−0.0681442	+	0.997675i	$0.521708\pi$
$968$			7.70081e13i			2.81902i
$969$		0			0
$970$	5.94352e11			0.0215561
$971$	−2.46210e13			−0.888833			−0.444416	−	0.895820i	$-0.646589\pi$
$971$	−2.46210e13			−0.888833			−0.444416	+	0.895820i	$0.646589\pi$
$972$		0			0
$973$	−3.96780e13	−	1.20176e13i	−1.41919	−	0.429842i
$974$			5.24325e13i			1.86674i
$975$		0			0
$976$			2.03746e13i			0.718730i
$977$		−	1.82838e13i		−	0.642009i	−0.947078	−	0.321005i	$-0.895980\pi$
$977$		−	1.82838e13i		−	0.642009i	0.947078	−	0.321005i	$-0.104020\pi$
$978$		0			0
$979$			3.68330e13i			1.28149i
$980$	1.97065e13	+	1.31430e13i	0.682483	+	0.455173i
$981$		0			0
$982$	−6.32644e13			−2.17099
$983$	−3.16369e13			−1.08070			−0.540348	−	0.841442i	$-0.681707\pi$
$983$	−3.16369e13			−1.08070			−0.540348	+	0.841442i	$0.681707\pi$
$984$		0			0
$985$			1.22400e13i			0.414304i
$986$	1.42848e13			0.481314
$987$		0			0
$988$	3.41712e13			1.14092
$989$			2.58536e13i			0.859285i
$990$		0			0
$991$	1.23580e13			0.407020			0.203510	−	0.979073i	$-0.434765\pi$
$991$	1.23580e13			0.407020			0.203510	+	0.979073i	$0.434765\pi$
$992$	4.56052e12			0.149524
$993$		0			0
$994$	1.83551e13	−	6.06025e13i	0.596374	−	1.96903i
$995$			9.67696e12i			0.312993i
$996$		0			0
$997$			4.00822e13i			1.28476i	0.766385	+	0.642382i	$0.222054\pi$
$997$			4.00822e13i			1.28476i	−0.766385	+	0.642382i	$0.777946\pi$
$998$			6.84777e13i			2.18505i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.b.b.251.43	yes	48
3.2	odd	2		315.10.b.a.251.6	✓	48
7.6	odd	2		315.10.b.a.251.43	yes	48
21.20	even	2	inner	315.10.b.b.251.6	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.10.b.a.251.6	✓	48	3.2	odd	2
315.10.b.a.251.43	yes	48	7.6	odd	2
315.10.b.b.251.6	yes	48	21.20	even	2	inner
315.10.b.b.251.43	yes	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+38.0944i q^{2} -939.186 q^{4} +625.000 q^{5} +(-1841.41 + 6079.71i) q^{7} -16273.4i q^{8} +23809.0i q^{10} -84202.7i q^{11} +42662.6i q^{13} +(-231603. - 70147.3i) q^{14} +139063. q^{16} +337476. q^{17} +852828. i q^{19} -586991. q^{20} +3.20765e6 q^{22} -1.66390e6i q^{23} +390625. q^{25} -1.62521e6 q^{26} +(1.72942e6 - 5.70997e6i) q^{28} -1.11114e6i q^{29} +1.50291e6i q^{31} -3.03446e6i q^{32} +1.28559e7i q^{34} +(-1.15088e6 + 3.79982e6i) q^{35} +1.30751e7 q^{37} -3.24880e7 q^{38} -1.01709e7i q^{40} +1.50066e7 q^{41} -1.55380e7 q^{43} +7.90820e7i q^{44} +6.33852e7 q^{46} +2.26851e7 q^{47} +(-3.35721e7 - 2.23904e7i) q^{49} +1.48806e7i q^{50} -4.00681e7i q^{52} +3.52198e7i q^{53} -5.26267e7i q^{55} +(9.89375e7 + 2.99659e7i) q^{56} +4.23284e7 q^{58} +1.20666e8 q^{59} +1.46514e8i q^{61} -5.72524e7 q^{62} +1.86796e8 q^{64} +2.66641e7i q^{65} -4.59585e7 q^{67} -3.16952e8 q^{68} +(-1.44752e8 - 4.38421e7i) q^{70} +2.61665e8i q^{71} +1.84075e7i q^{73} +4.98090e8i q^{74} -8.00964e8i q^{76} +(5.11928e8 + 1.55051e8i) q^{77} +5.41476e8 q^{79} +8.69142e7 q^{80} +5.71670e8i q^{82} +1.05895e8 q^{83} +2.10922e8 q^{85} -5.91910e8i q^{86} -1.37026e9 q^{88} -4.37433e8 q^{89} +(-2.59376e8 - 7.85591e7i) q^{91} +1.56271e9i q^{92} +8.64175e8i q^{94} +5.33017e8i q^{95} -2.49633e7i q^{97} +(8.52950e8 - 1.27891e9i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12288 q^{4} + 30000 q^{5} + 1824 q^{7} - 442908 q^{14} + 3258948 q^{16} - 7680000 q^{20} - 2860668 q^{22} + 18750000 q^{25} + 5432976 q^{26} - 3685092 q^{28} + 1140000 q^{35} + 7750344 q^{37} - 17423136 q^{38}+ \cdots + 2546372484 q^{98}+O(q^{100}) \) 48 * q - 12288 * q^4 + 30000 * q^5 + 1824 * q^7 - 442908 * q^14 + 3258948 * q^16 - 7680000 * q^20 - 2860668 * q^22 + 18750000 * q^25 + 5432976 * q^26 - 3685092 * q^28 + 1140000 * q^35 + 7750344 * q^37 - 17423136 * q^38 + 70626600 * q^41 + 16073832 * q^43 - 15083820 * q^46 + 33200856 * q^47 - 39367500 * q^49 + 318486852 * q^56 + 293819940 * q^58 - 342990864 * q^59 - 896801400 * q^62 - 1639789824 * q^64 - 980229936 * q^67 + 438669576 * q^68 - 276817500 * q^70 + 900236484 * q^77 + 325977144 * q^79 + 2036842500 * q^80 - 1415370096 * q^83 + 3319557300 * q^88 + 1132967016 * q^89 - 1162515060 * q^91 + 2546372484 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more