LMFDB - Embedded newform 405.2.e.l.271.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not 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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.2.e.l.136.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36603 + 2.36603i) q^{2} +(-2.73205 + 4.73205i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(0.633975 + 1.09808i) q^{7} -9.46410 q^{8} -2.73205 q^{10} +(1.13397 + 1.96410i) q^{11} +(2.73205 - 4.73205i) q^{13} +(-1.73205 + 3.00000i) q^{14} +(-7.46410 - 12.9282i) q^{16} +0.732051 q^{17} -2.46410 q^{19} +(-2.73205 - 4.73205i) q^{20} +(-3.09808 + 5.36603i) q^{22} +(-1.73205 + 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +14.9282 q^{26} -6.92820 q^{28} +(3.59808 + 6.23205i) q^{29} +(1.50000 - 2.59808i) q^{31} +(10.9282 - 18.9282i) q^{32} +(1.00000 + 1.73205i) q^{34} -1.26795 q^{35} +0.732051 q^{37} +(-3.36603 - 5.83013i) q^{38} +(4.73205 - 8.19615i) q^{40} +(-1.59808 + 2.76795i) q^{41} +(5.09808 + 8.83013i) q^{43} -12.3923 q^{44} -9.46410 q^{46} +(2.63397 + 4.56218i) q^{47} +(2.69615 - 4.66987i) q^{49} +(1.36603 - 2.36603i) q^{50} +(14.9282 + 25.8564i) q^{52} +3.26795 q^{53} -2.26795 q^{55} +(-6.00000 - 10.3923i) q^{56} +(-9.83013 + 17.0263i) q^{58} +(5.86603 - 10.1603i) q^{59} +(-2.00000 - 3.46410i) q^{61} +8.19615 q^{62} +29.8564 q^{64} +(2.73205 + 4.73205i) q^{65} +(1.73205 - 3.00000i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(-1.73205 - 3.00000i) q^{70} -0.267949 q^{71} +9.66025 q^{73} +(1.00000 + 1.73205i) q^{74} +(6.73205 - 11.6603i) q^{76} +(-1.43782 + 2.49038i) q^{77} +(-4.26795 - 7.39230i) q^{79} +14.9282 q^{80} -8.73205 q^{82} +(4.09808 + 7.09808i) q^{83} +(-0.366025 + 0.633975i) q^{85} +(-13.9282 + 24.1244i) q^{86} +(-10.7321 - 18.5885i) q^{88} -5.19615 q^{89} +6.92820 q^{91} +(-9.46410 - 16.3923i) q^{92} +(-7.19615 + 12.4641i) q^{94} +(1.23205 - 2.13397i) q^{95} +(-3.83013 - 6.63397i) q^{97} +14.7321 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34}+ \cdots + 52 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 - 4 * q^10 + 8 * q^11 + 4 * q^13 - 16 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^25 + 32 * q^26 + 4 * q^29 + 6 * q^31 + 16 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 - 10 * q^38 + 12 * q^40 + 4 * q^41 + 10 * q^43 - 8 * q^44 - 24 * q^46 + 14 * q^47 - 10 * q^49 + 2 * q^50 + 32 * q^52 + 20 * q^53 - 16 * q^55 - 24 * q^56 - 22 * q^58 + 20 * q^59 - 8 * q^61 + 12 * q^62 + 64 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^71 + 4 * q^73 + 4 * q^74 + 20 * q^76 - 30 * q^77 - 24 * q^79 + 32 * q^80 - 28 * q^82 + 6 * q^83 + 2 * q^85 - 28 * q^86 - 36 * q^88 - 24 * q^92 - 8 * q^94 - 2 * q^95 + 2 * q^97 + 52 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.36603	+	2.36603i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$2$	1.36603	+	2.36603i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$3$		0			0
$3$		0			0
$4$	−2.73205	+	4.73205i	−1.36603	+	2.36603i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	0.633975	+	1.09808i	0.239620	+	0.415034i	0.960605	−	0.277916i	$-0.0896439\pi$
$7$	0.633975	+	1.09808i	0.239620	+	0.415034i	−0.720985	+	0.692950i	$0.756311\pi$
$8$	−9.46410			−3.34607
$9$		0			0
$10$	−2.73205			−0.863950
$11$	1.13397	+	1.96410i	0.341906	+	0.592199i	0.984787	−	0.173768i	$-0.0555941\pi$
$11$	1.13397	+	1.96410i	0.341906	+	0.592199i	−0.642880	+	0.765967i	$0.722261\pi$
$12$		0			0
$13$	2.73205	−	4.73205i	0.757735	−	1.31243i	−0.186269	−	0.982499i	$-0.559640\pi$
$13$	2.73205	−	4.73205i	0.757735	−	1.31243i	0.944003	−	0.329936i	$-0.107027\pi$
$14$	−1.73205	+	3.00000i	−0.462910	+	0.801784i
$15$		0			0
$16$	−7.46410	−	12.9282i	−1.86603	−	3.23205i
$17$	0.732051			0.177548			0.0887742	−	0.996052i	$-0.471705\pi$
$17$	0.732051			0.177548			0.0887742	+	0.996052i	$0.471705\pi$
$18$		0			0
$19$	−2.46410			−0.565304			−0.282652	−	0.959223i	$-0.591214\pi$
$19$	−2.46410			−0.565304			−0.282652	+	0.959223i	$0.591214\pi$
$20$	−2.73205	−	4.73205i	−0.610905	−	1.05812i
$21$		0			0
$22$	−3.09808	+	5.36603i	−0.660512	+	1.14404i
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	−0.988152	−	0.153481i	$-0.950952\pi$
$23$	−1.73205	+	3.00000i	−0.361158	+	0.625543i	0.626994	+	0.779024i	$0.284285\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	14.9282			2.92766
$27$		0			0
$28$	−6.92820			−1.30931
$29$	3.59808	+	6.23205i	0.668146	+	1.15726i	0.978422	+	0.206616i	$0.0662452\pi$
$29$	3.59808	+	6.23205i	0.668146	+	1.15726i	−0.310276	+	0.950646i	$0.600421\pi$
$30$		0			0
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	−0.699301	−	0.714827i	$-0.746505\pi$
$31$	1.50000	−	2.59808i	0.269408	−	0.466628i	0.968709	+	0.248199i	$0.0798387\pi$
$32$	10.9282	−	18.9282i	1.93185	−	3.34607i
$33$		0			0
$34$	1.00000	+	1.73205i	0.171499	+	0.297044i
$35$	−1.26795			−0.214323
$36$		0			0
$37$	0.732051			0.120348			0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051			0.120348			0.0601742	+	0.998188i	$0.480834\pi$
$38$	−3.36603	−	5.83013i	−0.546041	−	0.945771i
$39$		0			0
$40$	4.73205	−	8.19615i	0.748203	−	1.29593i
$41$	−1.59808	+	2.76795i	−0.249578	+	0.432281i	−0.963409	−	0.268037i	$-0.913625\pi$
$41$	−1.59808	+	2.76795i	−0.249578	+	0.432281i	0.713831	+	0.700318i	$0.246958\pi$
$42$		0			0
$43$	5.09808	+	8.83013i	0.777449	+	1.34658i	0.933408	+	0.358818i	$0.116820\pi$
$43$	5.09808	+	8.83013i	0.777449	+	1.34658i	−0.155958	+	0.987764i	$0.549847\pi$
$44$	−12.3923			−1.86821
$45$		0			0
$46$	−9.46410			−1.39541
$47$	2.63397	+	4.56218i	0.384205	+	0.665462i	0.991659	−	0.128893i	$-0.0411424\pi$
$47$	2.63397	+	4.56218i	0.384205	+	0.665462i	−0.607454	+	0.794355i	$0.707809\pi$
$48$		0			0
$49$	2.69615	−	4.66987i	0.385165	−	0.667125i
$50$	1.36603	−	2.36603i	0.193185	−	0.334607i
$51$		0			0
$52$	14.9282	+	25.8564i	2.07017	+	3.58564i
$53$	3.26795			0.448887			0.224444	−	0.974487i	$-0.427944\pi$
$53$	3.26795			0.448887			0.224444	+	0.974487i	$0.427944\pi$
$54$		0			0
$55$	−2.26795			−0.305810
$56$	−6.00000	−	10.3923i	−0.801784	−	1.38873i
$57$		0			0
$58$	−9.83013	+	17.0263i	−1.29076	+	2.23566i
$59$	5.86603	−	10.1603i	0.763691	−	1.32275i	−0.177244	−	0.984167i	$-0.556718\pi$
$59$	5.86603	−	10.1603i	0.763691	−	1.32275i	0.940936	−	0.338585i	$-0.109948\pi$
$60$		0			0
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	0.709113	−	0.705095i	$-0.249096\pi$
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	−0.965187	+	0.261562i	$0.915762\pi$
$62$	8.19615			1.04091
$63$		0			0
$64$	29.8564			3.73205
$65$	2.73205	+	4.73205i	0.338869	+	0.586939i
$66$		0			0
$67$	1.73205	−	3.00000i	0.211604	−	0.366508i	−0.740613	−	0.671932i	$-0.765465\pi$
$67$	1.73205	−	3.00000i	0.211604	−	0.366508i	0.952217	+	0.305424i	$0.0987981\pi$
$68$	−2.00000	+	3.46410i	−0.242536	+	0.420084i
$69$		0			0
$70$	−1.73205	−	3.00000i	−0.207020	−	0.358569i
$71$	−0.267949			−0.0317997			−0.0158999	−	0.999874i	$-0.505061\pi$
$71$	−0.267949			−0.0317997			−0.0158999	+	0.999874i	$0.505061\pi$
$72$		0			0
$73$	9.66025			1.13065			0.565324	−	0.824869i	$-0.308751\pi$
$73$	9.66025			1.13065			0.565324	+	0.824869i	$0.308751\pi$
$74$	1.00000	+	1.73205i	0.116248	+	0.201347i
$75$		0			0
$76$	6.73205	−	11.6603i	0.772219	−	1.33752i
$77$	−1.43782	+	2.49038i	−0.163855	+	0.283805i
$78$		0			0
$79$	−4.26795	−	7.39230i	−0.480182	−	0.831699i	0.519560	−	0.854434i	$-0.326096\pi$
$79$	−4.26795	−	7.39230i	−0.480182	−	0.831699i	−0.999742	+	0.0227349i	$0.992763\pi$
$80$	14.9282			1.66902
$81$		0			0
$82$	−8.73205			−0.964294
$83$	4.09808	+	7.09808i	0.449822	+	0.779115i	0.998374	−	0.0570015i	$-0.0181540\pi$
$83$	4.09808	+	7.09808i	0.449822	+	0.779115i	−0.548552	+	0.836117i	$0.684821\pi$
$84$		0			0
$85$	−0.366025	+	0.633975i	−0.0397010	+	0.0687642i
$86$	−13.9282	+	24.1244i	−1.50192	+	2.60140i
$87$		0			0
$88$	−10.7321	−	18.5885i	−1.14404	−	1.98154i
$89$	−5.19615			−0.550791			−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615			−0.550791			−0.275396	+	0.961331i	$0.588809\pi$
$90$		0			0
$91$	6.92820			0.726273
$92$	−9.46410	−	16.3923i	−0.986701	−	1.70902i
$93$		0			0
$94$	−7.19615	+	12.4641i	−0.742226	+	1.28557i
$95$	1.23205	−	2.13397i	0.126406	−	0.218941i
$96$		0			0
$97$	−3.83013	−	6.63397i	−0.388890	−	0.673578i	0.603410	−	0.797431i	$-0.293808\pi$
$97$	−3.83013	−	6.63397i	−0.388890	−	0.673578i	−0.992301	+	0.123853i	$0.960475\pi$
$98$	14.7321			1.48816
$99$		0			0
$100$	5.46410			0.546410
$101$	−7.33013	−	12.6962i	−0.729375	−	1.26331i	−0.957148	−	0.289600i	$-0.906478\pi$
$101$	−7.33013	−	12.6962i	−0.729375	−	1.26331i	0.227773	−	0.973714i	$-0.426856\pi$
$102$		0			0
$103$	3.73205	−	6.46410i	0.367730	−	0.636927i	−0.621480	−	0.783430i	$-0.713468\pi$
$103$	3.73205	−	6.46410i	0.367730	−	0.636927i	0.989210	+	0.146503i	$0.0468018\pi$
$104$	−25.8564	+	44.7846i	−2.53543	+	4.39149i
$105$		0			0
$106$	4.46410	+	7.73205i	0.433592	+	0.751003i
$107$	−15.4641			−1.49497			−0.747486	−	0.664278i	$-0.768739\pi$
$107$	−15.4641			−1.49497			−0.747486	+	0.664278i	$0.768739\pi$
$108$		0			0
$109$	−19.9282			−1.90878			−0.954388	−	0.298570i	$-0.903490\pi$
$109$	−19.9282			−1.90878			−0.954388	+	0.298570i	$0.903490\pi$
$110$	−3.09808	−	5.36603i	−0.295390	−	0.511630i
$111$		0			0
$112$	9.46410	−	16.3923i	0.894274	−	1.54893i
$113$	−2.56218	+	4.43782i	−0.241029	+	0.417475i	−0.961008	−	0.276521i	$-0.910818\pi$
$113$	−2.56218	+	4.43782i	−0.241029	+	0.417475i	0.719978	+	0.693997i	$0.244152\pi$
$114$		0			0
$115$	−1.73205	−	3.00000i	−0.161515	−	0.279751i
$116$	−39.3205			−3.65082
$117$		0			0
$118$	32.0526			2.95068
$119$	0.464102	+	0.803848i	0.0425441	+	0.0736886i
$120$		0			0
$121$	2.92820	−	5.07180i	0.266200	−	0.461072i
$122$	5.46410	−	9.46410i	0.494697	−	0.856840i
$123$		0			0
$124$	8.19615	+	14.1962i	0.736036	+	1.27485i
$125$	1.00000			0.0894427
$126$		0			0
$127$	16.5885			1.47199			0.735994	−	0.676988i	$-0.236715\pi$
$127$	16.5885			1.47199			0.735994	+	0.676988i	$0.236715\pi$
$128$	18.9282	+	32.7846i	1.67303	+	2.89778i
$129$		0			0
$130$	−7.46410	+	12.9282i	−0.654645	+	1.13388i
$131$	7.79423	−	13.5000i	0.680985	−	1.17950i	−0.293696	−	0.955899i	$-0.594885\pi$
$131$	7.79423	−	13.5000i	0.680985	−	1.17950i	0.974681	−	0.223602i	$-0.0717814\pi$
$132$		0			0
$133$	−1.56218	−	2.70577i	−0.135458	−	0.234620i
$134$	9.46410			0.817574
$135$		0			0
$136$	−6.92820			−0.594089
$137$	−4.73205	−	8.19615i	−0.404286	−	0.700245i	0.589952	−	0.807439i	$-0.299147\pi$
$137$	−4.73205	−	8.19615i	−0.404286	−	0.700245i	−0.994238	+	0.107194i	$0.965813\pi$
$138$		0			0
$139$	−10.6962	+	18.5263i	−0.907236	+	1.57138i	−0.0893482	+	0.996000i	$0.528478\pi$
$139$	−10.6962	+	18.5263i	−0.907236	+	1.57138i	−0.817888	+	0.575378i	$0.804855\pi$
$140$	3.46410	−	6.00000i	0.292770	−	0.507093i
$141$		0			0
$142$	−0.366025	−	0.633975i	−0.0307162	−	0.0532020i
$143$	12.3923			1.03630
$144$		0			0
$145$	−7.19615			−0.597608
$146$	13.1962	+	22.8564i	1.09212	+	1.89161i
$147$		0			0
$148$	−2.00000	+	3.46410i	−0.164399	+	0.284747i
$149$	4.00000	−	6.92820i	0.327693	−	0.567581i	−0.654361	−	0.756182i	$-0.727062\pi$
$149$	4.00000	−	6.92820i	0.327693	−	0.567581i	0.982054	+	0.188602i	$0.0603956\pi$
$150$		0			0
$151$	7.69615	+	13.3301i	0.626304	+	1.08479i	0.988287	+	0.152606i	$0.0487665\pi$
$151$	7.69615	+	13.3301i	0.626304	+	1.08479i	−0.361983	+	0.932185i	$0.617900\pi$
$152$	23.3205			1.89154
$153$		0			0
$154$	−7.85641			−0.633087
$155$	1.50000	+	2.59808i	0.120483	+	0.208683i
$156$		0			0
$157$	−2.56218	+	4.43782i	−0.204484	+	0.354177i	−0.949968	−	0.312347i	$-0.898885\pi$
$157$	−2.56218	+	4.43782i	−0.204484	+	0.354177i	0.745484	+	0.666523i	$0.232218\pi$
$158$	11.6603	−	20.1962i	0.927640	−	1.60672i
$159$		0			0
$160$	10.9282	+	18.9282i	0.863950	+	1.49641i
$161$	−4.39230			−0.346162
$162$		0			0
$163$	9.26795			0.725922			0.362961	−	0.931804i	$-0.381766\pi$
$163$	9.26795			0.725922			0.362961	+	0.931804i	$0.381766\pi$
$164$	−8.73205	−	15.1244i	−0.681859	−	1.18101i
$165$		0			0
$166$	−11.1962	+	19.3923i	−0.868990	+	1.50513i
$167$	0.169873	−	0.294229i	0.0131452	−	0.0227681i	−0.859378	−	0.511341i	$-0.829149\pi$
$167$	0.169873	−	0.294229i	0.0131452	−	0.0227681i	0.872523	+	0.488573i	$0.162482\pi$
$168$		0			0
$169$	−8.42820	−	14.5981i	−0.648323	−	1.12293i
$170$	−2.00000			−0.153393
$171$		0			0
$172$	−55.7128			−4.24806
$173$	−7.73205	−	13.3923i	−0.587857	−	1.01820i	−0.994513	−	0.104617i	$-0.966638\pi$
$173$	−7.73205	−	13.3923i	−0.587857	−	1.01820i	0.406656	−	0.913581i	$-0.366695\pi$
$174$		0			0
$175$	0.633975	−	1.09808i	0.0479240	−	0.0830068i
$176$	16.9282	−	29.3205i	1.27601	−	2.21012i
$177$		0			0
$178$	−7.09808	−	12.2942i	−0.532023	−	0.921491i
$179$	−16.1244			−1.20519			−0.602595	−	0.798047i	$-0.705867\pi$
$179$	−16.1244			−1.20519			−0.602595	+	0.798047i	$0.705867\pi$
$180$		0			0
$181$	19.5359			1.45209			0.726046	−	0.687646i	$-0.241356\pi$
$181$	19.5359			1.45209			0.726046	+	0.687646i	$0.241356\pi$
$182$	9.46410	+	16.3923i	0.701526	+	1.21508i
$183$		0			0
$184$	16.3923	−	28.3923i	1.20846	−	2.09311i
$185$	−0.366025	+	0.633975i	−0.0269107	+	0.0466107i
$186$		0			0
$187$	0.830127	+	1.43782i	0.0607049	+	0.105144i
$188$	−28.7846			−2.09933
$189$		0			0
$190$	6.73205			0.488394
$191$	−8.06218	−	13.9641i	−0.583359	−	1.01041i	−0.995078	−	0.0990961i	$-0.968405\pi$
$191$	−8.06218	−	13.9641i	−0.583359	−	1.01041i	0.411719	−	0.911311i	$-0.364928\pi$
$192$		0			0
$193$	4.36603	−	7.56218i	0.314273	−	0.544337i	−0.665009	−	0.746835i	$-0.731572\pi$
$193$	4.36603	−	7.56218i	0.314273	−	0.544337i	0.979283	+	0.202498i	$0.0649058\pi$
$194$	10.4641	−	18.1244i	0.751279	−	1.30125i
$195$		0			0
$196$	14.7321	+	25.5167i	1.05229	+	1.82262i
$197$	13.8564			0.987228			0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564			0.987228			0.493614	+	0.869681i	$0.335676\pi$
$198$		0			0
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$	4.73205	+	8.19615i	0.334607	+	0.579555i
$201$		0			0
$202$	20.0263	−	34.6865i	1.40904	−	2.44054i
$203$	−4.56218	+	7.90192i	−0.320202	+	0.554606i
$204$		0			0
$205$	−1.59808	−	2.76795i	−0.111614	−	0.193322i
$206$	20.3923			1.42080
$207$		0			0
$208$	−81.5692			−5.65581
$209$	−2.79423	−	4.83975i	−0.193281	−	0.334772i
$210$		0			0
$211$	−9.42820	+	16.3301i	−0.649064	+	1.12421i	0.334282	+	0.942473i	$0.391506\pi$
$211$	−9.42820	+	16.3301i	−0.649064	+	1.12421i	−0.983347	+	0.181739i	$0.941827\pi$
$212$	−8.92820	+	15.4641i	−0.613192	+	1.06208i
$213$		0			0
$214$	−21.1244	−	36.5885i	−1.44403	−	2.50114i
$215$	−10.1962			−0.695372
$216$		0			0
$217$	3.80385			0.258222
$218$	−27.2224	−	47.1506i	−1.84374	−	3.19344i
$219$		0			0
$220$	6.19615	−	10.7321i	0.417745	−	0.723555i
$221$	2.00000	−	3.46410i	0.134535	−	0.233021i
$222$		0			0
$223$	12.3923	+	21.4641i	0.829850	+	1.43734i	0.898155	+	0.439678i	$0.144907\pi$
$223$	12.3923	+	21.4641i	0.829850	+	1.43734i	−0.0683053	+	0.997664i	$0.521759\pi$
$224$	27.7128			1.85164
$225$		0			0
$226$	−14.0000			−0.931266
$227$	−10.0263	−	17.3660i	−0.665468	−	1.15262i	−0.979158	−	0.203098i	$-0.934899\pi$
$227$	−10.0263	−	17.3660i	−0.665468	−	1.15262i	0.313691	−	0.949525i	$-0.398434\pi$
$228$		0			0
$229$	6.00000	−	10.3923i	0.396491	−	0.686743i	−0.596799	−	0.802391i	$-0.703561\pi$
$229$	6.00000	−	10.3923i	0.396491	−	0.686743i	0.993290	+	0.115648i	$0.0368944\pi$
$230$	4.73205	−	8.19615i	0.312022	−	0.540438i
$231$		0			0
$232$	−34.0526	−	58.9808i	−2.23566	−	3.87228i
$233$	10.0526			0.658565			0.329283	−	0.944231i	$-0.393193\pi$
$233$	10.0526			0.658565			0.329283	+	0.944231i	$0.393193\pi$
$234$		0			0
$235$	−5.26795			−0.343643
$236$	32.0526	+	55.5167i	2.08644	+	3.61383i
$237$		0			0
$238$	−1.26795	+	2.19615i	−0.0821889	+	0.142355i
$239$	3.73205	−	6.46410i	0.241406	−	0.418128i	−0.719709	−	0.694276i	$-0.755725\pi$
$239$	3.73205	−	6.46410i	0.241406	−	0.418128i	0.961115	+	0.276148i	$0.0890580\pi$
$240$		0			0
$241$	−9.16025	−	15.8660i	−0.590064	−	1.02202i	−0.994223	−	0.107332i	$-0.965769\pi$
$241$	−9.16025	−	15.8660i	−0.590064	−	1.02202i	0.404159	−	0.914689i	$-0.367564\pi$
$242$	16.0000			1.02852
$243$		0			0
$244$	21.8564			1.39921
$245$	2.69615	+	4.66987i	0.172251	+	0.298347i
$246$		0			0
$247$	−6.73205	+	11.6603i	−0.428350	+	0.741924i
$248$	−14.1962	+	24.5885i	−0.901457	+	1.56137i
$249$		0			0
$250$	1.36603	+	2.36603i	0.0863950	+	0.149641i
$251$	10.3923			0.655956			0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923			0.655956			0.327978	+	0.944685i	$0.393633\pi$
$252$		0			0
$253$	−7.85641			−0.493928
$254$	22.6603	+	39.2487i	1.42183	+	2.46268i
$255$		0			0
$256$	−21.8564	+	37.8564i	−1.36603	+	2.36603i
$257$	5.19615	−	9.00000i	0.324127	−	0.561405i	−0.657208	−	0.753709i	$-0.728263\pi$
$257$	5.19615	−	9.00000i	0.324127	−	0.561405i	0.981335	+	0.192304i	$0.0615961\pi$
$258$		0			0
$259$	0.464102	+	0.803848i	0.0288379	+	0.0499487i
$260$	−29.8564			−1.85162
$261$		0			0
$262$	42.5885			2.63112
$263$	6.66025	+	11.5359i	0.410689	+	0.711334i	0.994965	−	0.100221i	$-0.0319549\pi$
$263$	6.66025	+	11.5359i	0.410689	+	0.711334i	−0.584276	+	0.811555i	$0.698622\pi$
$264$		0			0
$265$	−1.63397	+	2.83013i	−0.100374	+	0.173853i
$266$	4.26795	−	7.39230i	0.261685	−	0.453251i
$267$		0			0
$268$	9.46410	+	16.3923i	0.578112	+	1.00132i
$269$	−6.66025			−0.406083			−0.203041	−	0.979170i	$-0.565083\pi$
$269$	−6.66025			−0.406083			−0.203041	+	0.979170i	$0.565083\pi$
$270$		0			0
$271$	10.9282			0.663841			0.331921	−	0.943307i	$-0.392303\pi$
$271$	10.9282			0.663841			0.331921	+	0.943307i	$0.392303\pi$
$272$	−5.46410	−	9.46410i	−0.331310	−	0.573845i
$273$		0			0
$274$	12.9282	−	22.3923i	0.781021	−	1.35277i
$275$	1.13397	−	1.96410i	0.0683812	−	0.118440i
$276$		0			0
$277$	−7.09808	−	12.2942i	−0.426482	−	0.738689i	0.570075	−	0.821592i	$-0.306914\pi$
$277$	−7.09808	−	12.2942i	−0.426482	−	0.738689i	−0.996558	+	0.0829037i	$0.973581\pi$
$278$	−58.4449			−3.50529
$279$		0			0
$280$	12.0000			0.717137
$281$	−4.26795	−	7.39230i	−0.254605	−	0.440988i	0.710184	−	0.704017i	$-0.248612\pi$
$281$	−4.26795	−	7.39230i	−0.254605	−	0.440988i	−0.964788	+	0.263029i	$0.915279\pi$
$282$		0			0
$283$	−2.66025	+	4.60770i	−0.158136	+	0.273899i	−0.934196	−	0.356759i	$-0.883882\pi$
$283$	−2.66025	+	4.60770i	−0.158136	+	0.273899i	0.776061	+	0.630658i	$0.217215\pi$
$284$	0.732051	−	1.26795i	0.0434392	−	0.0752389i
$285$		0			0
$286$	16.9282	+	29.3205i	1.00099	+	1.73376i
$287$	−4.05256			−0.239215
$288$		0			0
$289$	−16.4641			−0.968477
$290$	−9.83013	−	17.0263i	−0.577245	−	0.999818i
$291$		0			0
$292$	−26.3923	+	45.7128i	−1.54449	+	2.67514i
$293$	−12.6340	+	21.8827i	−0.738085	+	1.27840i	0.215272	+	0.976554i	$0.430936\pi$
$293$	−12.6340	+	21.8827i	−0.738085	+	1.27840i	−0.953357	+	0.301846i	$0.902397\pi$
$294$		0			0
$295$	5.86603	+	10.1603i	0.341533	+	0.591553i
$296$	−6.92820			−0.402694
$297$		0			0
$298$	21.8564			1.26611
$299$	9.46410	+	16.3923i	0.547323	+	0.947991i
$300$		0			0
$301$	−6.46410	+	11.1962i	−0.372585	+	0.645335i
$302$	−21.0263	+	36.4186i	−1.20993	+	2.09565i
$303$		0			0
$304$	18.3923	+	31.8564i	1.05487	+	1.82709i
$305$	4.00000			0.229039
$306$		0			0
$307$	−24.0526			−1.37275			−0.686376	−	0.727247i	$-0.740800\pi$
$307$	−24.0526			−1.37275			−0.686376	+	0.727247i	$0.740800\pi$
$308$	−7.85641	−	13.6077i	−0.447660	−	0.775370i
$309$		0			0
$310$	−4.09808	+	7.09808i	−0.232755	+	0.403144i
$311$	−8.13397	+	14.0885i	−0.461235	+	0.798883i	−0.999023	−	0.0441973i	$-0.985927\pi$
$311$	−8.13397	+	14.0885i	−0.461235	+	0.798883i	0.537787	+	0.843080i	$0.319260\pi$
$312$		0			0
$313$	11.4641	+	19.8564i	0.647989	+	1.12235i	0.983602	+	0.180351i	$0.0577232\pi$
$313$	11.4641	+	19.8564i	0.647989	+	1.12235i	−0.335613	+	0.942000i	$0.608943\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	46.6410			2.62376
$317$	2.09808	+	3.63397i	0.117840	+	0.204104i	0.918911	−	0.394464i	$-0.129070\pi$
$317$	2.09808	+	3.63397i	0.117840	+	0.204104i	−0.801072	+	0.598568i	$0.795736\pi$
$318$		0			0
$319$	−8.16025	+	14.1340i	−0.456887	+	0.791351i
$320$	−14.9282	+	25.8564i	−0.834512	+	1.44542i
$321$		0			0
$322$	−6.00000	−	10.3923i	−0.334367	−	0.579141i
$323$	−1.80385			−0.100369
$324$		0			0
$325$	−5.46410			−0.303094
$326$	12.6603	+	21.9282i	0.701187	+	1.21449i
$327$		0			0
$328$	15.1244	−	26.1962i	0.835103	−	1.44644i
$329$	−3.33975	+	5.78461i	−0.184126	+	0.318916i
$330$		0			0
$331$	−3.23205	−	5.59808i	−0.177650	−	0.307698i	0.763425	−	0.645896i	$-0.223516\pi$
$331$	−3.23205	−	5.59808i	−0.177650	−	0.307698i	−0.941075	+	0.338198i	$0.890183\pi$
$332$	−44.7846			−2.45787
$333$		0			0
$334$	0.928203			0.0507890
$335$	1.73205	+	3.00000i	0.0946320	+	0.163908i
$336$		0			0
$337$	3.66025	−	6.33975i	0.199387	−	0.345348i	−0.748943	−	0.662634i	$-0.769438\pi$
$337$	3.66025	−	6.33975i	0.199387	−	0.345348i	0.948330	+	0.317286i	$0.102772\pi$
$338$	23.0263	−	39.8827i	1.25246	−	2.16933i
$339$		0			0
$340$	−2.00000	−	3.46410i	−0.108465	−	0.187867i
$341$	6.80385			0.368449
$342$		0			0
$343$	15.7128			0.848412
$344$	−48.2487	−	83.5692i	−2.60140	−	4.50575i
$345$		0			0
$346$	21.1244	−	36.5885i	1.13565	−	1.96701i
$347$	−1.29423	+	2.24167i	−0.0694778	+	0.120339i	−0.898672	−	0.438622i	$-0.855467\pi$
$347$	−1.29423	+	2.24167i	−0.0694778	+	0.120339i	0.829194	+	0.558961i	$0.188800\pi$
$348$		0			0
$349$	−4.42820	−	7.66987i	−0.237036	−	0.410559i	0.722826	−	0.691030i	$-0.242843\pi$
$349$	−4.42820	−	7.66987i	−0.237036	−	0.410559i	−0.959863	+	0.280471i	$0.909509\pi$
$350$	3.46410			0.185164
$351$		0			0
$352$	49.5692			2.64205
$353$	9.75833	+	16.9019i	0.519384	+	0.899599i	0.999746	+	0.0225287i	$0.00717171\pi$
$353$	9.75833	+	16.9019i	0.519384	+	0.899599i	−0.480363	+	0.877070i	$0.659495\pi$
$354$		0			0
$355$	0.133975	−	0.232051i	0.00711063	−	0.0123160i
$356$	14.1962	−	24.5885i	0.752395	−	1.30319i
$357$		0			0
$358$	−22.0263	−	38.1506i	−1.16413	−	2.01632i
$359$	18.1244			0.956567			0.478283	−	0.878206i	$-0.341259\pi$
$359$	18.1244			0.956567			0.478283	+	0.878206i	$0.341259\pi$
$360$		0			0
$361$	−12.9282			−0.680432
$362$	26.6865	+	46.2224i	1.40261	+	2.42940i
$363$		0			0
$364$	−18.9282	+	32.7846i	−0.992107	+	1.71838i
$365$	−4.83013	+	8.36603i	−0.252820	+	0.437898i
$366$		0			0
$367$	15.5885	+	27.0000i	0.813711	+	1.40939i	0.910250	+	0.414059i	$0.135889\pi$
$367$	15.5885	+	27.0000i	0.813711	+	1.40939i	−0.0965390	+	0.995329i	$0.530777\pi$
$368$	51.7128			2.69572
$369$		0			0
$370$	−2.00000			−0.103975
$371$	2.07180	+	3.58846i	0.107562	+	0.186303i
$372$		0			0
$373$	9.02628	−	15.6340i	0.467363	−	0.809497i	−0.531942	−	0.846781i	$-0.678537\pi$
$373$	9.02628	−	15.6340i	0.467363	−	0.809497i	0.999305	+	0.0372845i	$0.0118708\pi$
$374$	−2.26795	+	3.92820i	−0.117273	+	0.203123i
$375$		0			0
$376$	−24.9282	−	43.1769i	−1.28557	−	2.22668i
$377$	39.3205			2.02511
$378$		0			0
$379$	−18.3923			−0.944749			−0.472375	−	0.881398i	$-0.656603\pi$
$379$	−18.3923			−0.944749			−0.472375	+	0.881398i	$0.656603\pi$
$380$	6.73205	+	11.6603i	0.345347	+	0.598158i
$381$		0			0
$382$	22.0263	−	38.1506i	1.12696	−	1.95196i
$383$	−4.73205	+	8.19615i	−0.241797	+	0.418804i	−0.961226	−	0.275762i	$-0.911070\pi$
$383$	−4.73205	+	8.19615i	−0.241797	+	0.418804i	0.719430	+	0.694565i	$0.244403\pi$
$384$		0			0
$385$	−1.43782	−	2.49038i	−0.0732782	−	0.126922i
$386$	23.8564			1.21426
$387$		0			0
$388$	41.8564			2.12494
$389$	−10.2679	−	17.7846i	−0.520606	−	0.901716i	−0.999713	−	0.0239591i	$-0.992373\pi$
$389$	−10.2679	−	17.7846i	−0.520606	−	0.901716i	0.479107	−	0.877756i	$-0.340960\pi$
$390$		0			0
$391$	−1.26795	+	2.19615i	−0.0641229	+	0.111064i
$392$	−25.5167	+	44.1962i	−1.28879	+	2.23224i
$393$		0			0
$394$	18.9282	+	32.7846i	0.953589	+	1.65166i
$395$	8.53590			0.429488
$396$		0			0
$397$	−6.39230			−0.320821			−0.160410	−	0.987050i	$-0.551282\pi$
$397$	−6.39230			−0.320821			−0.160410	+	0.987050i	$0.551282\pi$
$398$	−2.73205	−	4.73205i	−0.136945	−	0.237196i
$399$		0			0
$400$	−7.46410	+	12.9282i	−0.373205	+	0.646410i
$401$	−5.53590	+	9.58846i	−0.276450	+	0.478825i	−0.970500	−	0.241102i	$-0.922491\pi$
$401$	−5.53590	+	9.58846i	−0.276450	+	0.478825i	0.694050	+	0.719927i	$0.255825\pi$
$402$		0			0
$403$	−8.19615	−	14.1962i	−0.408279	−	0.707161i
$404$	80.1051			3.98538
$405$		0			0
$406$	−24.9282			−1.23717
$407$	0.830127	+	1.43782i	0.0411479	+	0.0712702i
$408$		0			0
$409$	−4.92820	+	8.53590i	−0.243684	+	0.422073i	−0.961761	−	0.273891i	$-0.911689\pi$
$409$	−4.92820	+	8.53590i	−0.243684	+	0.422073i	0.718077	+	0.695964i	$0.245023\pi$
$410$	4.36603	−	7.56218i	0.215623	−	0.373469i
$411$		0			0
$412$	20.3923	+	35.3205i	1.00466	+	1.74012i
$413$	14.8756			0.731983
$414$		0			0
$415$	−8.19615			−0.402333
$416$	−59.7128	−	103.426i	−2.92766	−	5.07086i
$417$		0			0
$418$	7.63397	−	13.2224i	0.373390	−	0.646730i
$419$	0.196152	−	0.339746i	0.00958267	−	0.0165977i	−0.861194	−	0.508276i	$-0.830283\pi$
$419$	0.196152	−	0.339746i	0.00958267	−	0.0165977i	0.870777	+	0.491678i	$0.163616\pi$
$420$		0			0
$421$	3.89230	+	6.74167i	0.189699	+	0.328569i	0.945150	−	0.326637i	$-0.105915\pi$
$421$	3.89230	+	6.74167i	0.189699	+	0.328569i	−0.755451	+	0.655206i	$0.772582\pi$
$422$	−51.5167			−2.50779
$423$		0			0
$424$	−30.9282			−1.50201
$425$	−0.366025	−	0.633975i	−0.0177548	−	0.0307523i
$426$		0			0
$427$	2.53590	−	4.39230i	0.122721	−	0.212559i
$428$	42.2487	−	73.1769i	2.04217	−	3.53714i
$429$		0			0
$430$	−13.9282	−	24.1244i	−0.671678	−	1.16338i
$431$	−38.6603			−1.86220			−0.931099	−	0.364765i	$-0.881149\pi$
$431$	−38.6603			−1.86220			−0.931099	+	0.364765i	$0.881149\pi$
$432$		0			0
$433$	−28.5359			−1.37135			−0.685674	−	0.727909i	$-0.740492\pi$
$433$	−28.5359			−1.37135			−0.685674	+	0.727909i	$0.740492\pi$
$434$	5.19615	+	9.00000i	0.249423	+	0.432014i
$435$		0			0
$436$	54.4449	−	94.3013i	2.60744	−	4.51621i
$437$	4.26795	−	7.39230i	0.204164	−	0.353622i
$438$		0			0
$439$	7.69615	+	13.3301i	0.367317	+	0.636212i	0.989145	−	0.146942i	$-0.0469430\pi$
$439$	7.69615	+	13.3301i	0.367317	+	0.636212i	−0.621828	+	0.783154i	$0.713610\pi$
$440$	21.4641			1.02326
$441$		0			0
$442$	10.9282			0.519802
$443$	8.83013	+	15.2942i	0.419532	+	0.726651i	0.995892	−	0.0905449i	$-0.0288609\pi$
$443$	8.83013	+	15.2942i	0.419532	+	0.726651i	−0.576360	+	0.817196i	$0.695528\pi$
$444$		0			0
$445$	2.59808	−	4.50000i	0.123161	−	0.213320i
$446$	−33.8564	+	58.6410i	−1.60315	+	2.77673i
$447$		0			0
$448$	18.9282	+	32.7846i	0.894274	+	1.54893i
$449$	−16.1244			−0.760955			−0.380478	−	0.924790i	$-0.624240\pi$
$449$	−16.1244			−0.760955			−0.380478	+	0.924790i	$0.624240\pi$
$450$		0			0
$451$	−7.24871			−0.341328
$452$	−14.0000	−	24.2487i	−0.658505	−	1.14056i
$453$		0			0
$454$	27.3923	−	47.4449i	1.28558	−	2.22670i
$455$	−3.46410	+	6.00000i	−0.162400	+	0.281284i
$456$		0			0
$457$	0.366025	+	0.633975i	0.0171219	+	0.0296561i	0.874459	−	0.485099i	$-0.161216\pi$
$457$	0.366025	+	0.633975i	0.0171219	+	0.0296561i	−0.857337	+	0.514755i	$0.827883\pi$
$458$	32.7846			1.53192
$459$		0			0
$460$	18.9282			0.882532
$461$	−0.526279	−	0.911543i	−0.0245113	−	0.0424548i	0.853510	−	0.521077i	$-0.174470\pi$
$461$	−0.526279	−	0.911543i	−0.0245113	−	0.0424548i	−0.878021	+	0.478622i	$0.841136\pi$
$462$		0			0
$463$	−5.19615	+	9.00000i	−0.241486	+	0.418265i	−0.961138	−	0.276069i	$-0.910968\pi$
$463$	−5.19615	+	9.00000i	−0.241486	+	0.418265i	0.719652	+	0.694335i	$0.244301\pi$
$464$	53.7128	−	93.0333i	2.49355	−	4.31896i
$465$		0			0
$466$	13.7321	+	23.7846i	0.636125	+	1.10180i
$467$	−35.3731			−1.63687			−0.818435	−	0.574599i	$-0.805158\pi$
$467$	−35.3731			−1.63687			−0.818435	+	0.574599i	$0.805158\pi$
$468$		0			0
$469$	4.39230			0.202818
$470$	−7.19615	−	12.4641i	−0.331934	−	0.574926i
$471$		0			0
$472$	−55.5167	+	96.1577i	−2.55536	+	4.42602i
$473$	−11.5622	+	20.0263i	−0.531630	+	0.920809i
$474$		0			0
$475$	1.23205	+	2.13397i	0.0565304	+	0.0979135i
$476$	−5.07180			−0.232465
$477$		0			0
$478$	20.3923			0.932722
$479$	18.0622	+	31.2846i	0.825282	+	1.42943i	0.901704	+	0.432355i	$0.142317\pi$
$479$	18.0622	+	31.2846i	0.825282	+	1.42943i	−0.0764216	+	0.997076i	$0.524350\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$	25.0263	−	43.3468i	1.13992	−	1.97439i
$483$		0			0
$484$	16.0000	+	27.7128i	0.727273	+	1.25967i
$485$	7.66025			0.347834
$486$		0			0
$487$	3.60770			0.163480			0.0817401	−	0.996654i	$-0.473952\pi$
$487$	3.60770			0.163480			0.0817401	+	0.996654i	$0.473952\pi$
$488$	18.9282	+	32.7846i	0.856840	+	1.48409i
$489$		0			0
$490$	−7.36603	+	12.7583i	−0.332763	+	0.576363i
$491$	19.0622	−	33.0167i	0.860264	−	1.49002i	−0.0114100	−	0.999935i	$-0.503632\pi$
$491$	19.0622	−	33.0167i	0.860264	−	1.49002i	0.871674	−	0.490086i	$-0.163035\pi$
$492$		0			0
$493$	2.63397	+	4.56218i	0.118628	+	0.205470i
$494$	−36.7846			−1.65502
$495$		0			0
$496$	−44.7846			−2.01089
$497$	−0.169873	−	0.294229i	−0.00761984	−	0.0131980i
$498$		0			0
$499$	5.16025	−	8.93782i	0.231005	−	0.400112i	−0.727099	−	0.686532i	$-0.759132\pi$
$499$	5.16025	−	8.93782i	0.231005	−	0.400112i	0.958104	+	0.286420i	$0.0924654\pi$
$500$	−2.73205	+	4.73205i	−0.122181	+	0.211624i
$501$		0			0
$502$	14.1962	+	24.5885i	0.633605	+	1.09744i
$503$	27.3205			1.21816			0.609081	−	0.793108i	$-0.291539\pi$
$503$	27.3205			1.21816			0.609081	+	0.793108i	$0.291539\pi$
$504$		0			0
$505$	14.6603			0.652373
$506$	−10.7321	−	18.5885i	−0.477098	−	0.826358i
$507$		0			0
$508$	−45.3205	+	78.4974i	−2.01077	+	3.48276i
$509$	−17.3923	+	30.1244i	−0.770900	+	1.33524i	0.166170	+	0.986097i	$0.446860\pi$
$509$	−17.3923	+	30.1244i	−0.770900	+	1.33524i	−0.937070	+	0.349141i	$0.886473\pi$
$510$		0			0
$511$	6.12436	+	10.6077i	0.270926	+	0.469257i
$512$	−43.7128			−1.93185
$513$		0			0
$514$	28.3923			1.25233
$515$	3.73205	+	6.46410i	0.164454	+	0.284842i
$516$		0			0
$517$	−5.97372	+	10.3468i	−0.262724	+	0.455051i
$518$	−1.26795	+	2.19615i	−0.0557105	+	0.0964934i
$519$		0			0
$520$	−25.8564	−	44.7846i	−1.13388	−	1.96394i
$521$	12.5359			0.549208			0.274604	−	0.961557i	$-0.411453\pi$
$521$	12.5359			0.549208			0.274604	+	0.961557i	$0.411453\pi$
$522$		0			0
$523$	−26.2487			−1.14778			−0.573888	−	0.818934i	$-0.694566\pi$
$523$	−26.2487			−1.14778			−0.573888	+	0.818934i	$0.694566\pi$
$524$	42.5885	+	73.7654i	1.86049	+	3.22246i
$525$		0			0
$526$	−18.1962	+	31.5167i	−0.793390	+	1.37419i
$527$	1.09808	−	1.90192i	0.0478330	−	0.0828491i
$528$		0			0
$529$	5.50000	+	9.52628i	0.239130	+	0.414186i
$530$	−8.92820			−0.387816
$531$		0			0
$532$	17.0718			0.740156
$533$	8.73205	+	15.1244i	0.378227	+	0.655109i
$534$		0			0
$535$	7.73205	−	13.3923i	0.334286	−	0.579000i
$536$	−16.3923	+	28.3923i	−0.708040	+	1.22636i
$537$		0			0
$538$	−9.09808	−	15.7583i	−0.392246	−	0.679390i
$539$	12.2295			0.526761
$540$		0			0
$541$	−17.5359			−0.753927			−0.376964	−	0.926228i	$-0.623032\pi$
$541$	−17.5359			−0.753927			−0.376964	+	0.926228i	$0.623032\pi$
$542$	14.9282	+	25.8564i	0.641221	+	1.11063i
$543$		0			0
$544$	8.00000	−	13.8564i	0.342997	−	0.594089i
$545$	9.96410	−	17.2583i	0.426815	−	0.739266i
$546$		0			0
$547$	3.07180	+	5.32051i	0.131341	+	0.227488i	0.924194	−	0.381924i	$-0.124739\pi$
$547$	3.07180	+	5.32051i	0.131341	+	0.227488i	−0.792853	+	0.609413i	$0.791405\pi$
$548$	51.7128			2.20906
$549$		0			0
$550$	6.19615			0.264205
$551$	−8.86603	−	15.3564i	−0.377705	−	0.654205i
$552$		0			0
$553$	5.41154	−	9.37307i	0.230122	−	0.398583i
$554$	19.3923	−	33.5885i	0.823900	−	1.42704i
$555$		0			0
$556$	−58.4449	−	101.229i	−2.47861	−	4.29309i
$557$	−9.46410			−0.401007			−0.200503	−	0.979693i	$-0.564258\pi$
$557$	−9.46410			−0.401007			−0.200503	+	0.979693i	$0.564258\pi$
$558$		0			0
$559$	55.7128			2.35640
$560$	9.46410	+	16.3923i	0.399931	+	0.692701i
$561$		0			0
$562$	11.6603	−	20.1962i	0.491858	−	0.851923i
$563$	6.63397	−	11.4904i	0.279589	−	0.484262i	−0.691694	−	0.722191i	$-0.743135\pi$
$563$	6.63397	−	11.4904i	0.279589	−	0.484262i	0.971283	+	0.237929i	$0.0764686\pi$
$564$		0			0
$565$	−2.56218	−	4.43782i	−0.107792	−	0.186701i
$566$	−14.5359			−0.610989
$567$		0			0
$568$	2.53590			0.106404
$569$	−16.4545	−	28.5000i	−0.689808	−	1.19478i	−0.971900	−	0.235395i	$-0.924362\pi$
$569$	−16.4545	−	28.5000i	−0.689808	−	1.19478i	0.282092	−	0.959387i	$-0.408972\pi$
$570$		0			0
$571$	−0.892305	+	1.54552i	−0.0373418	+	0.0646779i	−0.884092	−	0.467312i	$-0.845222\pi$
$571$	−0.892305	+	1.54552i	−0.0373418	+	0.0646779i	0.846750	+	0.531990i	$0.178556\pi$
$572$	−33.8564	+	58.6410i	−1.41561	+	2.45190i
$573$		0			0
$574$	−5.53590	−	9.58846i	−0.231064	−	0.400214i
$575$	3.46410			0.144463
$576$		0			0
$577$	18.7321			0.779825			0.389913	−	0.920852i	$-0.372505\pi$
$577$	18.7321			0.779825			0.389913	+	0.920852i	$0.372505\pi$
$578$	−22.4904	−	38.9545i	−0.935477	−	1.62029i
$579$		0			0
$580$	19.6603	−	34.0526i	0.816348	−	1.41396i
$581$	−5.19615	+	9.00000i	−0.215573	+	0.373383i
$582$		0			0
$583$	3.70577	+	6.41858i	0.153477	+	0.265831i
$584$	−91.4256			−3.78322
$585$		0			0
$586$	−69.0333			−2.85174
$587$	−2.83013	−	4.90192i	−0.116812	−	0.202324i	0.801691	−	0.597739i	$-0.203934\pi$
$587$	−2.83013	−	4.90192i	−0.116812	−	0.202324i	−0.918503	+	0.395415i	$0.870601\pi$
$588$		0			0
$589$	−3.69615	+	6.40192i	−0.152297	+	0.263787i
$590$	−16.0263	+	27.7583i	−0.659791	+	1.14279i
$591$		0			0
$592$	−5.46410	−	9.46410i	−0.224573	−	0.388972i
$593$	−27.8564			−1.14393			−0.571963	−	0.820280i	$-0.693818\pi$
$593$	−27.8564			−1.14393			−0.571963	+	0.820280i	$0.693818\pi$
$594$		0			0
$595$	−0.928203			−0.0380526
$596$	21.8564	+	37.8564i	0.895273	+	1.55066i
$597$		0			0
$598$	−25.8564	+	44.7846i	−1.05735	+	1.83138i
$599$	−8.40192	+	14.5526i	−0.343293	+	0.594601i	−0.985042	−	0.172313i	$-0.944876\pi$
$599$	−8.40192	+	14.5526i	−0.343293	+	0.594601i	0.641749	+	0.766915i	$0.278209\pi$
$600$		0			0
$601$	−8.62436	−	14.9378i	−0.351795	−	0.609326i	0.634769	−	0.772702i	$-0.281095\pi$
$601$	−8.62436	−	14.9378i	−0.351795	−	0.609326i	−0.986564	+	0.163375i	$0.947762\pi$
$602$	−35.3205			−1.43956
$603$		0			0
$604$	−84.1051			−3.42219
$605$	2.92820	+	5.07180i	0.119048	+	0.206198i
$606$		0			0
$607$	2.90192	−	5.02628i	0.117785	−	0.204010i	−0.801104	−	0.598525i	$-0.795754\pi$
$607$	2.90192	−	5.02628i	0.117785	−	0.204010i	0.918890	+	0.394514i	$0.129087\pi$
$608$	−26.9282	+	46.6410i	−1.09208	+	1.89154i
$609$		0			0
$610$	5.46410	+	9.46410i	0.221235	+	0.383190i
$611$	28.7846			1.16450
$612$		0			0
$613$	−5.46410			−0.220693			−0.110346	−	0.993893i	$-0.535196\pi$
$613$	−5.46410			−0.220693			−0.110346	+	0.993893i	$0.535196\pi$
$614$	−32.8564	−	56.9090i	−1.32598	−	2.29666i
$615$		0			0
$616$	13.6077	−	23.5692i	0.548270	−	0.949631i
$617$	−3.46410	+	6.00000i	−0.139459	+	0.241551i	−0.927292	−	0.374338i	$-0.877870\pi$
$617$	−3.46410	+	6.00000i	−0.139459	+	0.241551i	0.787833	+	0.615889i	$0.211203\pi$
$618$		0			0
$619$	−7.92820	−	13.7321i	−0.318661	−	0.551938i	0.661548	−	0.749903i	$-0.269900\pi$
$619$	−7.92820	−	13.7321i	−0.318661	−	0.551938i	−0.980209	+	0.197965i	$0.936567\pi$
$620$	−16.3923			−0.658331
$621$		0			0
$622$	−44.4449			−1.78208
$623$	−3.29423	−	5.70577i	−0.131980	−	0.228597i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	−31.3205	+	54.2487i	−1.25182	+	2.16821i
$627$		0			0
$628$	−14.0000	−	24.2487i	−0.558661	−	0.967629i
$629$	0.535898			0.0213677
$630$		0			0
$631$	22.7128			0.904183			0.452091	−	0.891972i	$-0.350678\pi$
$631$	22.7128			0.904183			0.452091	+	0.891972i	$0.350678\pi$
$632$	40.3923	+	69.9615i	1.60672	+	2.78292i
$633$		0			0
$634$	−5.73205	+	9.92820i	−0.227649	+	0.394299i
$635$	−8.29423	+	14.3660i	−0.329146	+	0.570098i
$636$		0			0
$637$	−14.7321	−	25.5167i	−0.583705	−	1.01101i
$638$	−44.5885			−1.76527
$639$		0			0
$640$	−37.8564			−1.49641
$641$	−13.3301	−	23.0885i	−0.526508	−	0.911939i	−0.999523	−	0.0308846i	$-0.990168\pi$
$641$	−13.3301	−	23.0885i	−0.526508	−	0.911939i	0.473015	−	0.881055i	$-0.343166\pi$
$642$		0			0
$643$	−12.1699	+	21.0788i	−0.479933	+	0.831268i	−0.999735	−	0.0230185i	$-0.992672\pi$
$643$	−12.1699	+	21.0788i	−0.479933	+	0.831268i	0.519802	+	0.854287i	$0.326006\pi$
$644$	12.0000	−	20.7846i	0.472866	−	0.819028i
$645$		0			0
$646$	−2.46410	−	4.26795i	−0.0969488	−	0.167920i
$647$	4.53590			0.178325			0.0891623	−	0.996017i	$-0.471581\pi$
$647$	4.53590			0.178325			0.0891623	+	0.996017i	$0.471581\pi$
$648$		0			0
$649$	26.6077			1.04444
$650$	−7.46410	−	12.9282i	−0.292766	−	0.507086i
$651$		0			0
$652$	−25.3205	+	43.8564i	−0.991628	+	1.71755i
$653$	5.26795	−	9.12436i	0.206151	−	0.357064i	−0.744348	−	0.667792i	$-0.767240\pi$
$653$	5.26795	−	9.12436i	0.206151	−	0.357064i	0.950499	+	0.310728i	$0.100573\pi$
$654$		0			0
$655$	7.79423	+	13.5000i	0.304546	+	0.527489i
$656$	47.7128			1.86287
$657$		0			0
$658$	−18.2487			−0.711409
$659$	−2.73205	−	4.73205i	−0.106426	−	0.184335i	0.807894	−	0.589328i	$-0.200607\pi$
$659$	−2.73205	−	4.73205i	−0.106426	−	0.184335i	−0.914320	+	0.404993i	$0.867274\pi$
$660$		0			0
$661$	8.16025	−	14.1340i	0.317397	−	0.549748i	−0.662547	−	0.749020i	$-0.730525\pi$
$661$	8.16025	−	14.1340i	0.317397	−	0.549748i	0.979944	+	0.199272i	$0.0638579\pi$
$662$	8.83013	−	15.2942i	0.343193	−	0.594427i
$663$		0			0
$664$	−38.7846	−	67.1769i	−1.50513	−	2.60697i
$665$	3.12436			0.121157
$666$		0			0
$667$	−24.9282			−0.965224
$668$	0.928203	+	1.60770i	0.0359133	+	0.0622036i
$669$		0			0
$670$	−4.73205	+	8.19615i	−0.182815	+	0.316645i
$671$	4.53590	−	7.85641i	0.175106	−	0.303293i
$672$		0			0
$673$	−5.19615	−	9.00000i	−0.200297	−	0.346925i	0.748327	−	0.663330i	$-0.230857\pi$
$673$	−5.19615	−	9.00000i	−0.200297	−	0.346925i	−0.948624	+	0.316405i	$0.897524\pi$
$674$	20.0000			0.770371
$675$		0			0
$676$	92.1051			3.54250
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	0.985517	−	0.169580i	$-0.0542410\pi$
$677$	9.00000	+	15.5885i	0.345898	+	0.599113i	−0.639618	+	0.768693i	$0.720908\pi$
$678$		0			0
$679$	4.85641	−	8.41154i	0.186372	−	0.322805i
$680$	3.46410	−	6.00000i	0.132842	−	0.230089i
$681$		0			0
$682$	9.29423	+	16.0981i	0.355894	+	0.616427i
$683$	−40.3923			−1.54557			−0.772784	−	0.634669i	$-0.781137\pi$
$683$	−40.3923			−1.54557			−0.772784	+	0.634669i	$0.781137\pi$
$684$		0			0
$685$	9.46410			0.361605
$686$	21.4641	+	37.1769i	0.819503	+	1.41942i
$687$		0			0
$688$	76.1051	−	131.818i	2.90148	−	5.02551i
$689$	8.92820	−	15.4641i	0.340137	−	0.589135i
$690$		0			0
$691$	−18.8564	−	32.6603i	−0.717332	−	1.24245i	−0.962053	−	0.272861i	$-0.912030\pi$
$691$	−18.8564	−	32.6603i	−0.717332	−	1.24245i	0.244722	−	0.969593i	$-0.421303\pi$
$692$	84.4974			3.21211
$693$		0			0
$694$	−7.07180			−0.268442
$695$	−10.6962	−	18.5263i	−0.405728	−	0.702742i
$696$		0			0
$697$	−1.16987	+	2.02628i	−0.0443121	+	0.0767508i
$698$	12.0981	−	20.9545i	0.457919	−	0.793139i
$699$		0			0
$700$	3.46410	+	6.00000i	0.130931	+	0.226779i
$701$	−31.1962			−1.17826			−0.589131	−	0.808037i	$-0.700530\pi$
$701$	−31.1962			−1.17826			−0.589131	+	0.808037i	$0.700530\pi$
$702$		0			0
$703$	−1.80385			−0.0680334
$704$	33.8564	+	58.6410i	1.27601	+	2.21012i
$705$		0			0
$706$	−26.6603	+	46.1769i	−1.00337	+	1.73789i
$707$	9.29423	−	16.0981i	0.349545	−	0.605430i
$708$		0			0
$709$	14.7321	+	25.5167i	0.553274	+	0.958298i	0.998036	+	0.0626494i	$0.0199550\pi$
$709$	14.7321	+	25.5167i	0.553274	+	0.958298i	−0.444762	+	0.895649i	$0.646712\pi$
$710$	0.732051			0.0274734
$711$		0			0
$712$	49.1769			1.84298
$713$	5.19615	+	9.00000i	0.194597	+	0.337053i
$714$		0			0
$715$	−6.19615	+	10.7321i	−0.231723	+	0.401356i
$716$	44.0526	−	76.3013i	1.64632	−	2.85151i
$717$		0			0
$718$	24.7583	+	42.8827i	0.923973	+	1.60037i
$719$	39.5885			1.47640			0.738200	−	0.674582i	$-0.235676\pi$
$719$	39.5885			1.47640			0.738200	+	0.674582i	$0.235676\pi$
$720$		0			0
$721$	9.46410			0.352462
$722$	−17.6603	−	30.5885i	−0.657247	−	1.13838i
$723$		0			0
$724$	−53.3731	+	92.4449i	−1.98359	+	3.43569i
$725$	3.59808	−	6.23205i	0.133629	−	0.231453i
$726$		0			0
$727$	−6.19615	−	10.7321i	−0.229803	−	0.398030i	0.727947	−	0.685633i	$-0.240475\pi$
$727$	−6.19615	−	10.7321i	−0.229803	−	0.398030i	−0.957749	+	0.287604i	$0.907141\pi$
$728$	−65.5692			−2.43016
$729$		0			0
$730$	−26.3923			−0.976823
$731$	3.73205	+	6.46410i	0.138035	+	0.239083i
$732$		0			0
$733$	3.39230	−	5.87564i	0.125298	−	0.217022i	−0.796552	−	0.604571i	$-0.793345\pi$
$733$	3.39230	−	5.87564i	0.125298	−	0.217022i	0.921849	+	0.387549i	$0.126678\pi$
$734$	−42.5885	+	73.7654i	−1.57197	+	2.72273i
$735$		0			0
$736$	37.8564	+	65.5692i	1.39541	+	2.41691i
$737$	7.85641			0.289394
$738$		0			0
$739$	15.5359			0.571497			0.285749	−	0.958305i	$-0.407758\pi$
$739$	15.5359			0.571497			0.285749	+	0.958305i	$0.407758\pi$
$740$	−2.00000	−	3.46410i	−0.0735215	−	0.127343i
$741$		0			0
$742$	−5.66025	+	9.80385i	−0.207794	+	0.359911i
$743$	22.9545	−	39.7583i	0.842118	−	1.45859i	−0.0459822	−	0.998942i	$-0.514642\pi$
$743$	22.9545	−	39.7583i	0.842118	−	1.45859i	0.888100	−	0.459649i	$-0.152025\pi$
$744$		0			0
$745$	4.00000	+	6.92820i	0.146549	+	0.253830i
$746$	49.3205			1.80575
$747$		0			0
$748$	−9.07180			−0.331698
$749$	−9.80385	−	16.9808i	−0.358225	−	0.620464i
$750$		0			0
$751$	−3.60770	+	6.24871i	−0.131647	+	0.228019i	−0.924311	−	0.381639i	$-0.875360\pi$
$751$	−3.60770	+	6.24871i	−0.131647	+	0.228019i	0.792665	+	0.609658i	$0.208693\pi$
$752$	39.3205	−	68.1051i	1.43387	−	2.48354i
$753$		0			0
$754$	53.7128	+	93.0333i	1.95611	+	3.38807i
$755$	−15.3923			−0.560183
$756$		0			0
$757$	53.1769			1.93275			0.966374	−	0.257141i	$-0.0827805\pi$
$757$	53.1769			1.93275			0.966374	+	0.257141i	$0.0827805\pi$
$758$	−25.1244	−	43.5167i	−0.912558	−	1.58060i
$759$		0			0
$760$	−11.6603	+	20.1962i	−0.422962	+	0.732591i
$761$	17.7224	−	30.6962i	0.642438	−	1.11273i	−0.342449	−	0.939536i	$-0.611256\pi$
$761$	17.7224	−	30.6962i	0.642438	−	1.11273i	0.984887	−	0.173198i	$-0.0554102\pi$
$762$		0			0
$763$	−12.6340	−	21.8827i	−0.457381	−	0.792206i
$764$	88.1051			3.18753
$765$		0			0
$766$	−25.8564			−0.934230
$767$	−32.0526	−	55.5167i	−1.15735	−	2.00459i
$768$		0			0
$769$	8.23205	−	14.2583i	0.296855	−	0.514169i	−0.678559	−	0.734545i	$-0.737395\pi$
$769$	8.23205	−	14.2583i	0.296855	−	0.514169i	0.975415	+	0.220377i	$0.0707287\pi$
$770$	3.92820	−	6.80385i	0.141563	−	0.245194i
$771$		0			0
$772$	23.8564	+	41.3205i	0.858611	+	1.48716i
$773$	−43.5167			−1.56519			−0.782593	−	0.622534i	$-0.786103\pi$
$773$	−43.5167			−1.56519			−0.782593	+	0.622534i	$0.786103\pi$
$774$		0			0
$775$	−3.00000			−0.107763
$776$	36.2487	+	62.7846i	1.30125	+	2.25384i
$777$		0			0
$778$	28.0526	−	48.5885i	1.00573	−	1.74198i
$779$	3.93782	−	6.82051i	0.141087	−	0.244370i
$780$		0			0
$781$	−0.303848	−	0.526279i	−0.0108725	−	0.0188318i
$782$	−6.92820			−0.247752
$783$		0			0
$784$	−80.4974			−2.87491
$785$	−2.56218	−	4.43782i	−0.0914480	−	0.158393i
$786$		0			0
$787$	4.97372	−	8.61474i	0.177294	−	0.307082i	−0.763659	−	0.645620i	$-0.776599\pi$
$787$	4.97372	−	8.61474i	0.177294	−	0.307082i	0.940953	+	0.338538i	$0.109932\pi$
$788$	−37.8564	+	65.5692i	−1.34858	+	2.33581i
$789$		0			0
$790$	11.6603	+	20.1962i	0.414853	+	0.718547i
$791$	−6.49742			−0.231022
$792$		0			0
$793$	−21.8564			−0.776144
$794$	−8.73205	−	15.1244i	−0.309889	−	0.536743i
$795$		0			0
$796$	5.46410	−	9.46410i	0.193670	−	0.335446i
$797$	18.1962	−	31.5167i	0.644541	−	1.11638i	−0.339867	−	0.940474i	$-0.610382\pi$
$797$	18.1962	−	31.5167i	0.644541	−	1.11638i	0.984407	−	0.175904i	$-0.0562847\pi$
$798$		0			0
$799$	1.92820	+	3.33975i	0.0682149	+	0.118152i
$800$	−21.8564			−0.772741
$801$		0			0
$802$	−30.2487			−1.06812
$803$	10.9545	+	18.9737i	0.386575	+	0.669568i
$804$		0			0
$805$	2.19615	−	3.80385i	0.0774042	−	0.134068i
$806$	22.3923	−	38.7846i	0.788735	−	1.36613i
$807$		0			0
$808$	69.3731	+	120.158i	2.44054	+	4.22713i
$809$	−13.4449			−0.472696			−0.236348	−	0.971668i	$-0.575951\pi$
$809$	−13.4449			−0.472696			−0.236348	+	0.971668i	$0.575951\pi$
$810$		0			0
$811$	−11.5359			−0.405080			−0.202540	−	0.979274i	$-0.564920\pi$
$811$	−11.5359			−0.405080			−0.202540	+	0.979274i	$0.564920\pi$
$812$	−24.9282	−	43.1769i	−0.874808	−	1.51521i
$813$		0			0
$814$	−2.26795	+	3.92820i	−0.0794916	+	0.137683i
$815$	−4.63397	+	8.02628i	−0.162321	+	0.281148i
$816$		0			0
$817$	−12.5622	−	21.7583i	−0.439495	−	0.761228i
$818$	−26.9282			−0.941523
$819$		0			0
$820$	17.4641			0.609873
$821$	17.1340	+	29.6769i	0.597980	+	1.03573i	0.993119	+	0.117110i	$0.0373631\pi$
$821$	17.1340	+	29.6769i	0.597980	+	1.03573i	−0.395139	+	0.918621i	$0.629304\pi$
$822$		0			0
$823$	11.9282	−	20.6603i	0.415791	−	0.720171i	−0.579720	−	0.814816i	$-0.696838\pi$
$823$	11.9282	−	20.6603i	0.415791	−	0.720171i	0.995511	+	0.0946445i	$0.0301714\pi$
$824$	−35.3205	+	61.1769i	−1.23045	+	2.13120i
$825$		0			0
$826$	20.3205	+	35.1962i	0.707041	+	1.22463i
$827$	−32.3923			−1.12639			−0.563195	−	0.826324i	$-0.690428\pi$
$827$	−32.3923			−1.12639			−0.563195	+	0.826324i	$0.690428\pi$
$828$		0			0
$829$	17.7846			0.617685			0.308843	−	0.951113i	$-0.400058\pi$
$829$	17.7846			0.617685			0.308843	+	0.951113i	$0.400058\pi$
$830$	−11.1962	−	19.3923i	−0.388624	−	0.673117i
$831$		0			0
$832$	81.5692	−	141.282i	2.82790	−	4.89807i
$833$	1.97372	−	3.41858i	0.0683854	−	0.118447i
$834$		0			0
$835$	0.169873	+	0.294229i	0.00587870	+	0.0101822i
$836$	30.5359			1.05611
$837$		0			0
$838$	1.07180			0.0370246
$839$	−11.4019	−	19.7487i	−0.393638	−	0.681801i	0.599288	−	0.800533i	$-0.295450\pi$
$839$	−11.4019	−	19.7487i	−0.393638	−	0.681801i	−0.992926	+	0.118732i	$0.962117\pi$
$840$		0			0
$841$	−11.3923	+	19.7321i	−0.392838	+	0.680416i
$842$	−10.6340	+	18.4186i	−0.366471	+	0.634746i
$843$		0			0
$844$	−51.5167	−	89.2295i	−1.77328	−	3.07141i
$845$	16.8564			0.579878
$846$		0			0
$847$	7.42563			0.255148
$848$	−24.3923	−	42.2487i	−0.837635	−	1.45083i
$849$		0			0
$850$	1.00000	−	1.73205i	0.0342997	−	0.0594089i
$851$	−1.26795	+	2.19615i	−0.0434647	+	0.0752831i
$852$		0			0
$853$	−27.7583	−	48.0788i	−0.950427	−	1.64619i	−0.744502	−	0.667620i	$-0.767313\pi$
$853$	−27.7583	−	48.0788i	−0.950427	−	1.64619i	−0.205925	−	0.978568i	$-0.566020\pi$
$854$	13.8564			0.474156
$855$		0			0
$856$	146.354			5.00227
$857$	26.2487	+	45.4641i	0.896639	+	1.55302i	0.831763	+	0.555131i	$0.187332\pi$
$857$	26.2487	+	45.4641i	0.896639	+	1.55302i	0.0648758	+	0.997893i	$0.479335\pi$
$858$		0			0
$859$	5.50000	−	9.52628i	0.187658	−	0.325032i	−0.756811	−	0.653633i	$-0.773244\pi$
$859$	5.50000	−	9.52628i	0.187658	−	0.325032i	0.944469	+	0.328601i	$0.106577\pi$
$860$	27.8564	−	48.2487i	0.949896	−	1.64527i
$861$		0			0
$862$	−52.8109	−	91.4711i	−1.79875	−	3.11552i
$863$	1.12436			0.0382735			0.0191368	−	0.999817i	$-0.493908\pi$
$863$	1.12436			0.0382735			0.0191368	+	0.999817i	$0.493908\pi$
$864$		0			0
$865$	15.4641			0.525795
$866$	−38.9808	−	67.5167i	−1.32462	−	2.29431i
$867$		0			0
$868$	−10.3923	+	18.0000i	−0.352738	+	0.610960i
$869$	9.67949	−	16.7654i	0.328354	−	0.568726i
$870$		0			0
$871$	−9.46410	−	16.3923i	−0.320679	−	0.555432i
$872$	188.603			6.38689
$873$		0			0
$874$	23.3205			0.788828
$875$	0.633975	+	1.09808i	0.0214323	+	0.0371218i
$876$		0			0
$877$	−13.2679	+	22.9808i	−0.448027	+	0.776005i	−0.998258	−	0.0590075i	$-0.981206\pi$
$877$	−13.2679	+	22.9808i	−0.448027	+	0.776005i	0.550231	+	0.835013i	$0.314540\pi$
$878$	−21.0263	+	36.4186i	−0.709603	+	1.22907i
$879$		0			0
$880$	16.9282	+	29.3205i	0.570650	+	0.988394i
$881$	−8.94744			−0.301447			−0.150723	−	0.988576i	$-0.548160\pi$
$881$	−8.94744			−0.301447			−0.150723	+	0.988576i	$0.548160\pi$
$882$		0			0
$883$	−19.8038			−0.666453			−0.333226	−	0.942847i	$-0.608137\pi$
$883$	−19.8038			−0.666453			−0.333226	+	0.942847i	$0.608137\pi$
$884$	10.9282	+	18.9282i	0.367555	+	0.636624i
$885$		0			0
$886$	−24.1244	+	41.7846i	−0.810474	+	1.40378i
$887$	−22.0981	+	38.2750i	−0.741981	+	1.28515i	0.209611	+	0.977785i	$0.432780\pi$
$887$	−22.0981	+	38.2750i	−0.741981	+	1.28515i	−0.951592	+	0.307364i	$0.900553\pi$
$888$		0			0
$889$	10.5167	+	18.2154i	0.352717	+	0.610925i
$890$	14.1962			0.475856
$891$		0			0
$892$	−135.426			−4.53439
$893$	−6.49038	−	11.2417i	−0.217192	−	0.376188i
$894$		0			0
$895$	8.06218	−	13.9641i	0.269489	−	0.466768i
$896$	−24.0000	+	41.5692i	−0.801784	+	1.38873i
$897$		0			0
$898$	−22.0263	−	38.1506i	−0.735026	−	1.27310i
$899$	21.5885			0.720015
$900$		0			0
$901$	2.39230			0.0796992
$902$	−9.90192	−	17.1506i	−0.329698	−	0.571054i
$903$		0			0
$904$	24.2487	−	42.0000i	0.806500	−	1.39690i
$905$	−9.76795	+	16.9186i	−0.324698	+	0.562393i
$906$		0			0
$907$	15.0000	+	25.9808i	0.498067	+	0.862677i	0.999998	−	0.00223080i	$-0.000710087\pi$
$907$	15.0000	+	25.9808i	0.498067	+	0.862677i	−0.501931	+	0.864908i	$0.667377\pi$
$908$	109.569			3.63618
$909$		0			0
$910$	−18.9282			−0.627464
$911$	3.79423	+	6.57180i	0.125708	+	0.217733i	0.922010	−	0.387167i	$-0.126546\pi$
$911$	3.79423	+	6.57180i	0.125708	+	0.217733i	−0.796301	+	0.604900i	$0.793213\pi$
$912$		0			0
$913$	−9.29423	+	16.0981i	−0.307594	+	0.532769i
$914$	−1.00000	+	1.73205i	−0.0330771	+	0.0572911i
$915$		0			0
$916$	32.7846	+	56.7846i	1.08323	+	1.87622i
$917$	19.7654			0.652710
$918$		0			0
$919$	46.9615			1.54912			0.774559	−	0.632502i	$-0.217972\pi$
$919$	46.9615			1.54912			0.774559	+	0.632502i	$0.217972\pi$
$920$	16.3923	+	28.3923i	0.540438	+	0.936067i
$921$		0			0
$922$	1.43782	−	2.49038i	0.0473522	−	0.0820163i
$923$	−0.732051	+	1.26795i	−0.0240957	+	0.0417351i
$924$		0			0
$925$	−0.366025	−	0.633975i	−0.0120348	−	0.0208450i
$926$	−28.3923			−0.933029
$927$		0			0
$928$	157.282			5.16304
$929$	−14.1865	−	24.5718i	−0.465445	−	0.806175i	0.533776	−	0.845626i	$-0.320772\pi$
$929$	−14.1865	−	24.5718i	−0.465445	−	0.806175i	−0.999222	+	0.0394511i	$0.987439\pi$
$930$		0			0
$931$	−6.64359	+	11.5070i	−0.217735	+	0.377128i
$932$	−27.4641	+	47.5692i	−0.899617	+	1.55818i
$933$		0			0
$934$	−48.3205	−	83.6936i	−1.58110	−	2.73854i
$935$	−1.66025			−0.0542961
$936$		0			0
$937$	−31.8564			−1.04070			−0.520352	−	0.853952i	$-0.674199\pi$
$937$	−31.8564			−1.04070			−0.520352	+	0.853952i	$0.674199\pi$
$938$	6.00000	+	10.3923i	0.195907	+	0.339321i
$939$		0			0
$940$	14.3923	−	24.9282i	0.469425	−	0.813068i
$941$	13.5885	−	23.5359i	0.442971	−	0.767248i	−0.554937	−	0.831892i	$-0.687258\pi$
$941$	13.5885	−	23.5359i	0.442971	−	0.767248i	0.997908	+	0.0646438i	$0.0205911\pi$
$942$		0			0
$943$	−5.53590	−	9.58846i	−0.180274	−	0.312243i
$944$	−175.138			−5.70027
$945$		0			0
$946$	−63.1769			−2.05406
$947$	1.14359	+	1.98076i	0.0371618	+	0.0643661i	0.884008	−	0.467472i	$-0.154835\pi$
$947$	1.14359	+	1.98076i	0.0371618	+	0.0643661i	−0.846846	+	0.531838i	$0.821502\pi$
$948$		0			0
$949$	26.3923	−	45.7128i	0.856730	−	1.48390i
$950$	−3.36603	+	5.83013i	−0.109208	+	0.189154i
$951$		0			0
$952$	−4.39230	−	7.60770i	−0.142355	−	0.246567i
$953$	15.6077			0.505583			0.252791	−	0.967521i	$-0.418651\pi$
$953$	15.6077			0.505583			0.252791	+	0.967521i	$0.418651\pi$
$954$		0			0
$955$	16.1244			0.521772
$956$	20.3923	+	35.3205i	0.659534	+	1.14235i
$957$		0			0
$958$	−49.3468	+	85.4711i	−1.59432	+	2.76145i
$959$	6.00000	−	10.3923i	0.193750	−	0.335585i
$960$		0			0
$961$	11.0000	+	19.0526i	0.354839	+	0.614599i
$962$	10.9282			0.352339
$963$		0			0
$964$	100.105			3.22417
$965$	4.36603	+	7.56218i	0.140547	+	0.243435i
$966$		0			0
$967$	18.0981	−	31.3468i	0.581995	−	1.00804i	−0.413248	−	0.910619i	$-0.635606\pi$
$967$	18.0981	−	31.3468i	0.581995	−	1.00804i	0.995243	−	0.0974261i	$-0.0310610\pi$
$968$	−27.7128	+	48.0000i	−0.890724	+	1.54278i
$969$		0			0
$970$	10.4641	+	18.1244i	0.335982	+	0.581938i
$971$	−41.4449			−1.33003			−0.665014	−	0.746830i	$-0.731575\pi$
$971$	−41.4449			−1.33003			−0.665014	+	0.746830i	$0.731575\pi$
$972$		0			0
$973$	−27.1244			−0.869567
$974$	4.92820	+	8.53590i	0.157910	+	0.273508i
$975$		0			0
$976$	−29.8564	+	51.7128i	−0.955680	+	1.65529i
$977$	0.732051	−	1.26795i	0.0234204	−	0.0405653i	−0.854078	−	0.520145i	$-0.825878\pi$
$977$	0.732051	−	1.26795i	0.0234204	−	0.0405653i	0.877498	+	0.479580i	$0.159211\pi$
$978$		0			0
$979$	−5.89230	−	10.2058i	−0.188319	−	0.326178i
$980$	−29.4641			−0.941196
$981$		0			0
$982$	104.158			3.32381
$983$	−8.70577	−	15.0788i	−0.277671	−	0.480940i	0.693135	−	0.720808i	$-0.256229\pi$
$983$	−8.70577	−	15.0788i	−0.277671	−	0.480940i	−0.970806	+	0.239868i	$0.922896\pi$
$984$		0			0
$985$	−6.92820	+	12.0000i	−0.220751	+	0.382352i
$986$	−7.19615	+	12.4641i	−0.229172	+	0.396938i
$987$		0			0
$988$	−36.7846	−	63.7128i	−1.17027	−	2.02697i
$989$	−35.3205			−1.12313
$990$		0			0
$991$	3.14359			0.0998595			0.0499298	−	0.998753i	$-0.484100\pi$
$991$	3.14359			0.0998595			0.0499298	+	0.998753i	$0.484100\pi$
$992$	−32.7846	−	56.7846i	−1.04091	−	1.80291i
$993$		0			0
$994$	0.464102	−	0.803848i	0.0147204	−	0.0254965i
$995$	1.00000	−	1.73205i	0.0317021	−	0.0549097i
$996$		0			0
$997$	9.75833	+	16.9019i	0.309049	+	0.535289i	0.978155	−	0.207878i	$-0.0666558\pi$
$997$	9.75833	+	16.9019i	0.309049	+	0.535289i	−0.669105	+	0.743168i	$0.733322\pi$
$998$	28.1962			0.892534
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.l.271.2		4
3.2	odd	2		405.2.e.i.271.1		4
9.2	odd	6		405.2.e.i.136.1		4
9.4	even	3		405.2.a.g.1.1	✓	2
9.5	odd	6		405.2.a.h.1.2	yes	2
9.7	even	3	inner	405.2.e.l.136.2		4
36.23	even	6		6480.2.a.bi.1.1		2
36.31	odd	6		6480.2.a.br.1.1		2
45.4	even	6		2025.2.a.m.1.2		2
45.13	odd	12		2025.2.b.g.649.4		4
45.14	odd	6		2025.2.a.g.1.1		2
45.22	odd	12		2025.2.b.g.649.1		4
45.23	even	12		2025.2.b.h.649.1		4
45.32	even	12		2025.2.b.h.649.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.1	✓	2	9.4	even	3
405.2.a.h.1.2	yes	2	9.5	odd	6
405.2.e.i.136.1		4	9.2	odd	6
405.2.e.i.271.1		4	3.2	odd	2
405.2.e.l.136.2		4	9.7	even	3	inner
405.2.e.l.271.2		4	1.1	even	1	trivial
2025.2.a.g.1.1		2	45.14	odd	6
2025.2.a.m.1.2		2	45.4	even	6
2025.2.b.g.649.1		4	45.22	odd	12
2025.2.b.g.649.4		4	45.13	odd	12
2025.2.b.h.649.1		4	45.23	even	12
2025.2.b.h.649.4		4	45.32	even	12
6480.2.a.bi.1.1		2	36.23	even	6
6480.2.a.br.1.1		2	36.31	odd	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 + 2.36603i) q^{2} +(-2.73205 + 4.73205i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(0.633975 + 1.09808i) q^{7} -9.46410 q^{8} -2.73205 q^{10} +(1.13397 + 1.96410i) q^{11} +(2.73205 - 4.73205i) q^{13} +(-1.73205 + 3.00000i) q^{14} +(-7.46410 - 12.9282i) q^{16} +0.732051 q^{17} -2.46410 q^{19} +(-2.73205 - 4.73205i) q^{20} +(-3.09808 + 5.36603i) q^{22} +(-1.73205 + 3.00000i) q^{23} +(-0.500000 - 0.866025i) q^{25} +14.9282 q^{26} -6.92820 q^{28} +(3.59808 + 6.23205i) q^{29} +(1.50000 - 2.59808i) q^{31} +(10.9282 - 18.9282i) q^{32} +(1.00000 + 1.73205i) q^{34} -1.26795 q^{35} +0.732051 q^{37} +(-3.36603 - 5.83013i) q^{38} +(4.73205 - 8.19615i) q^{40} +(-1.59808 + 2.76795i) q^{41} +(5.09808 + 8.83013i) q^{43} -12.3923 q^{44} -9.46410 q^{46} +(2.63397 + 4.56218i) q^{47} +(2.69615 - 4.66987i) q^{49} +(1.36603 - 2.36603i) q^{50} +(14.9282 + 25.8564i) q^{52} +3.26795 q^{53} -2.26795 q^{55} +(-6.00000 - 10.3923i) q^{56} +(-9.83013 + 17.0263i) q^{58} +(5.86603 - 10.1603i) q^{59} +(-2.00000 - 3.46410i) q^{61} +8.19615 q^{62} +29.8564 q^{64} +(2.73205 + 4.73205i) q^{65} +(1.73205 - 3.00000i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(-1.73205 - 3.00000i) q^{70} -0.267949 q^{71} +9.66025 q^{73} +(1.00000 + 1.73205i) q^{74} +(6.73205 - 11.6603i) q^{76} +(-1.43782 + 2.49038i) q^{77} +(-4.26795 - 7.39230i) q^{79} +14.9282 q^{80} -8.73205 q^{82} +(4.09808 + 7.09808i) q^{83} +(-0.366025 + 0.633975i) q^{85} +(-13.9282 + 24.1244i) q^{86} +(-10.7321 - 18.5885i) q^{88} -5.19615 q^{89} +6.92820 q^{91} +(-9.46410 - 16.3923i) q^{92} +(-7.19615 + 12.4641i) q^{94} +(1.23205 - 2.13397i) q^{95} +(-3.83013 - 6.63397i) q^{97} +14.7321 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34}+ \cdots + 52 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^4 - 2 * q^5 + 6 * q^7 - 24 * q^8 - 4 * q^10 + 8 * q^11 + 4 * q^13 - 16 * q^16 - 4 * q^17 + 4 * q^19 - 4 * q^20 - 2 * q^22 - 2 * q^25 + 32 * q^26 + 4 * q^29 + 6 * q^31 + 16 * q^32 + 4 * q^34 - 12 * q^35 - 4 * q^37 - 10 * q^38 + 12 * q^40 + 4 * q^41 + 10 * q^43 - 8 * q^44 - 24 * q^46 + 14 * q^47 - 10 * q^49 + 2 * q^50 + 32 * q^52 + 20 * q^53 - 16 * q^55 - 24 * q^56 - 22 * q^58 + 20 * q^59 - 8 * q^61 + 12 * q^62 + 64 * q^64 + 4 * q^65 - 8 * q^68 - 8 * q^71 + 4 * q^73 + 4 * q^74 + 20 * q^76 - 30 * q^77 - 24 * q^79 + 32 * q^80 - 28 * q^82 + 6 * q^83 + 2 * q^85 - 28 * q^86 - 36 * q^88 - 24 * q^92 - 8 * q^94 - 2 * q^95 + 2 * q^97 + 52 * q^98

Introduction

L-functions

Modular forms

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Database

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