LMFDB - Embedded newform 731.2.s.a.214.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [731,2,Mod(214,731)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(731, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([3, 8]))

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

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

chi := DirichletCharacter("731.214");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 731 = 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	731.s (of order $16$, degree $8$, 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:	$5.83706438776$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{16})$
Coefficient field:	16.0.3289935900927224469054816256.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10319x^{8} + 214358881 $ x^16 + 10319*x^8 + 214358881
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			214.2
Root			$1.71665 - 2.83780i$ of defining polynomial
Character	$\chi$	$=$	731.214
Dual form			731.2.s.a.386.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +O(q^{10})$	$q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +(1.25563 + 1.87919i) q^{11} +(1.77872 - 1.77872i) q^{13} +4.00000i q^{16} +(1.05499 - 3.98585i) q^{17} +(1.81025 - 1.20957i) q^{23} +(4.61940 - 1.91342i) q^{25} +(4.54859 - 6.80745i) q^{31} +(2.29610 - 5.54328i) q^{36} +(0.646300 + 0.128557i) q^{41} +(-2.50942 - 6.05828i) q^{43} +(0.881839 - 4.43330i) q^{44} +(9.65010 - 9.65010i) q^{47} +(6.46716 + 2.67878i) q^{49} -5.03098 q^{52} +(-5.55651 + 13.4146i) q^{53} +(-2.19162 + 0.907798i) q^{59} +(5.65685 - 5.65685i) q^{64} +0.318032i q^{67} +(-7.12882 + 4.14487i) q^{68} +(9.76218 + 14.6101i) q^{79} +(-6.36396 + 6.36396i) q^{81} +(6.05828 + 2.50942i) q^{83} +(-4.27067 - 0.849489i) q^{92} +(-3.29947 - 16.5875i) q^{97} +(-3.76690 + 5.63756i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{13} + 56 q^{23} - 64 q^{59} + 96 q^{79}+O(q^{100}) $ 16 * q - 24 * q^13 + 56 * q^23 - 64 * q^59 + 96 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$e\left(\frac{3}{16}\right)$	$-1$

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 $

$2$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$2$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$3$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$3$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$4$	−1.41421	−	1.41421i	−0.707107	−	0.707107i
$5$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$5$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$6$		0			0
$7$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$7$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$8$		0			0
$9$	1.14805	+	2.77164i	0.382683	+	0.923880i
$10$		0			0
$11$	1.25563	+	1.87919i	0.378587	+	0.566596i	0.971012	−	0.239030i	$-0.0768293\pi$
$11$	1.25563	+	1.87919i	0.378587	+	0.566596i	−0.592425	+	0.805626i	$0.701829\pi$
$12$		0			0
$13$	1.77872	−	1.77872i	0.493328	−	0.493328i	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	1.77872	−	1.77872i	0.493328	−	0.493328i	0.909353	+	0.416025i	$0.136577\pi$
$14$		0			0
$15$		0			0
$16$			4.00000i			1.00000i
$17$	1.05499	−	3.98585i	0.255872	−	0.966711i
$17$	1.05499	−	3.98585i	0.255872	−	0.966711i
$18$		0			0
$19$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$19$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.81025	−	1.20957i	0.377463	−	0.252213i	−0.352337	−	0.935873i	$-0.614613\pi$
$23$	1.81025	−	1.20957i	0.377463	−	0.252213i	0.729800	+	0.683660i	$0.239613\pi$
$24$		0			0
$25$	4.61940	−	1.91342i	0.923880	−	0.382683i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$29$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$30$		0			0
$31$	4.54859	−	6.80745i	0.816952	−	1.22265i	−0.155103	−	0.987898i	$-0.549571\pi$
$31$	4.54859	−	6.80745i	0.816952	−	1.22265i	0.972054	−	0.234756i	$-0.0754291\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.29610	−	5.54328i	0.382683	−	0.923880i
$37$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$37$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.646300	+	0.128557i	0.100935	+	0.0200772i	0.245299	−	0.969447i	$-0.421114\pi$
$41$	0.646300	+	0.128557i	0.100935	+	0.0200772i	−0.144364	+	0.989525i	$0.546114\pi$
$42$		0			0
$43$	−2.50942	−	6.05828i	−0.382683	−	0.923880i
$43$	−2.50942	−	6.05828i	−0.382683	−	0.923880i
$44$	0.881839	−	4.43330i	0.132942	−	0.668346i
$45$		0			0
$46$		0			0
$47$	9.65010	−	9.65010i	1.40761	−	1.40761i	0.635560	−	0.772051i	$-0.280769\pi$
$47$	9.65010	−	9.65010i	1.40761	−	1.40761i	0.772051	−	0.635560i	$-0.219231\pi$
$48$		0			0
$49$	6.46716	+	2.67878i	0.923880	+	0.382683i
$50$		0			0
$51$		0			0
$52$	−5.03098			−0.697671
$53$	−5.55651	+	13.4146i	−0.763245	+	1.84264i	−0.312878	+	0.949793i	$0.601293\pi$
$53$	−5.55651	+	13.4146i	−0.763245	+	1.84264i	−0.450367	+	0.892844i	$0.648707\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.19162	+	0.907798i	−0.285324	+	0.118185i	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−2.19162	+	0.907798i	−0.285324	+	0.118185i	0.235431	+	0.971891i	$0.424350\pi$
$60$		0			0
$61$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$61$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	−	5.65685i	0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$			0.318032i			0.0388538i	0.999811	+	0.0194269i	$0.00618416\pi$
$67$			0.318032i			0.0388538i	−0.999811	+	0.0194269i	$0.993816\pi$
$68$	−7.12882	+	4.14487i	−0.864496	+	0.502639i
$69$		0			0
$70$		0			0
$71$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$71$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$72$		0			0
$73$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$73$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	9.76218	+	14.6101i	1.09833	+	1.64377i	0.675053	+	0.737769i	$0.264121\pi$
$79$	9.76218	+	14.6101i	1.09833	+	1.64377i	0.423279	+	0.906000i	$0.360879\pi$
$80$		0			0
$81$	−6.36396	+	6.36396i	−0.707107	+	0.707107i
$82$		0			0
$83$	6.05828	+	2.50942i	0.664983	+	0.275445i	0.689534	−	0.724254i	$-0.257816\pi$
$83$	6.05828	+	2.50942i	0.664983	+	0.275445i	−0.0245507	+	0.999699i	$0.507816\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$	−4.27067	−	0.849489i	−0.445248	−	0.0885654i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.29947	−	16.5875i	−0.335010	−	1.68421i	−0.670297	−	0.742093i	$-0.733833\pi$
$97$	−3.29947	−	16.5875i	−0.335010	−	1.68421i	0.335287	−	0.942116i	$-0.391167\pi$
$98$		0			0
$99$	−3.76690	+	5.63756i	−0.378587	+	0.566596i
$100$	−9.23880	−	3.82683i	−0.923880	−	0.382683i
$101$			13.3293i			1.32631i	0.748481	+	0.663156i	$0.230783\pi$
$101$			13.3293i			1.32631i	−0.748481	+	0.663156i	$0.769217\pi$
$102$		0			0
$103$	−12.1477			−1.19695			−0.598473	−	0.801143i	$-0.704225\pi$
$103$	−12.1477			−1.19695			−0.598473	+	0.801143i	$0.704225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.55425	+	1.90046i	−0.923645	+	0.183724i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	−9.55425	+	1.90046i	−0.923645	+	0.183724i	−0.289713	+	0.957114i	$0.593560\pi$
$108$		0			0
$109$	17.5842	+	3.49771i	1.68426	+	0.335020i	0.942133	−	0.335239i	$-0.108817\pi$
$109$	17.5842	+	3.49771i	1.68426	+	0.335020i	0.742127	+	0.670259i	$0.233817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$113$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	6.97203	+	2.88791i	0.644564	+	0.266987i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.25479	−	5.44354i	0.204981	−	0.494867i
$122$		0			0
$123$		0			0
$124$	−16.0599	+	3.19451i	−1.44222	+	0.286875i
$125$		0			0
$126$		0			0
$127$	−12.9023	+	5.34430i	−1.14489	+	0.474230i	−0.872818	−	0.488046i	$-0.837710\pi$
$127$	−12.9023	+	5.34430i	−1.14489	+	0.474230i	−0.272075	+	0.962276i	$0.587710\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$131$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−18.0749	−	12.0772i	−1.53309	−	1.02438i	−0.981767	−	0.190086i	$-0.939123\pi$
$139$	−18.0749	−	12.0772i	−1.53309	−	1.02438i	−0.551323	−	0.834292i	$-0.685877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.57596	+	1.10913i	0.466285	+	0.0927499i
$144$	−11.0866	+	4.59220i	−0.923880	+	0.382683i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$149$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$150$		0			0
$151$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$151$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$152$		0			0
$153$	12.2585	−	1.65191i	0.991042	−	0.133549i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$163$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$164$	−0.732199	−	1.09581i	−0.0571751	−	0.0855686i
$165$		0			0
$166$		0			0
$167$	0.863457	−	1.29225i	0.0668163	−	0.0999976i	−0.796557	−	0.604564i	$-0.793347\pi$
$167$	0.863457	−	1.29225i	0.0668163	−	0.0999976i	0.863373	+	0.504566i	$0.168347\pi$
$168$		0			0
$169$			6.67232i			0.513255i
$170$		0			0
$171$		0			0
$172$	−5.01885	+	12.1166i	−0.382683	+	0.923880i
$173$	21.7146	+	14.5092i	1.65093	+	1.10312i	0.892111	+	0.451817i	$0.149224\pi$
$173$	21.7146	+	14.5092i	1.65093	+	1.10312i	0.758820	+	0.651300i	$0.225776\pi$
$174$		0			0
$175$		0			0
$176$	−7.51674	+	5.02253i	−0.566596	+	0.378587i
$177$		0			0
$178$		0			0
$179$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$179$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$180$		0			0
$181$	−10.8677	−	16.2646i	−0.807788	−	1.20894i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$	−10.8677	−	16.2646i	−0.807788	−	1.20894i	0.167034	−	0.985951i	$-0.446581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.81483	−	3.02224i	0.644604	−	0.221008i
$188$	−27.2946			−1.99066
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$		0			0
$193$	−21.6954	+	14.4964i	−1.56167	+	1.04347i	−0.589915	+	0.807465i	$0.700839\pi$
$193$	−21.6954	+	14.4964i	−1.56167	+	1.04347i	−0.971751	+	0.236007i	$0.924161\pi$
$194$		0			0
$195$		0			0
$196$	−5.35757	−	12.9343i	−0.382683	−	0.923880i
$197$	3.17091	−	15.9412i	0.225918	−	1.13577i	−0.686696	−	0.726944i	$-0.740940\pi$
$197$	3.17091	−	15.9412i	0.225918	−	1.13577i	0.912614	−	0.408822i	$-0.134060\pi$
$198$		0			0
$199$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$199$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.43075	+	3.62871i	0.377463	+	0.252213i
$208$	7.11488	+	7.11488i	0.493328	+	0.493328i
$209$		0			0
$210$		0			0
$211$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$211$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$212$	26.8292	−	11.1130i	1.84264	−	0.763245i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.21318	−	8.96623i	−0.350677	−	0.603134i
$222$		0			0
$223$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$223$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$	10.6066	+	10.6066i	0.707107	+	0.707107i
$226$		0			0
$227$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$227$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$228$		0			0
$229$	−24.9231	+	10.3235i	−1.64696	+	0.682195i	−0.996973	−	0.0777462i	$-0.975228\pi$
$229$	−24.9231	+	10.3235i	−1.64696	+	0.682195i	−0.649992	+	0.759941i	$0.725228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$233$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$234$		0			0
$235$		0			0
$236$	4.38324	+	1.81560i	0.285324	+	0.118185i
$237$		0			0
$238$		0			0
$239$	−29.0726			−1.88055			−0.940275	−	0.340415i	$-0.889432\pi$
$239$	−29.0726			−1.88055			−0.940275	+	0.340415i	$0.889432\pi$
$240$		0			0
$241$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$241$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.7787	−	18.7787i	1.18530	−	1.18530i	0.206951	−	0.978351i	$-0.433646\pi$
$251$	18.7787	−	18.7787i	1.18530	−	1.18530i	0.978351	−	0.206951i	$-0.0663540\pi$
$252$		0			0
$253$	4.54602	+	1.88302i	0.285806	+	0.118385i
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$257$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$263$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.449765	−	0.449765i	0.0274738	−	0.0274738i
$269$	−9.65810	+	14.4544i	−0.588865	+	0.881298i	−0.999535	−	0.0304855i	$-0.990295\pi$
$269$	−9.65810	+	14.4544i	−0.588865	+	0.881298i	0.410671	+	0.911784i	$0.365295\pi$
$270$		0			0
$271$			9.77548i			0.593818i	0.954906	+	0.296909i	$0.0959558\pi$
$271$			9.77548i			0.593818i	−0.954906	+	0.296909i	$0.904044\pi$
$272$	15.9434	+	4.21995i	0.966711	+	0.255872i
$273$		0			0
$274$		0			0
$275$	9.39593	+	6.27816i	0.566596	+	0.378587i
$276$		0			0
$277$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$277$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$278$		0			0
$279$	24.0898	+	4.79176i	1.44222	+	0.286875i
$280$		0			0
$281$	10.5669	+	25.5107i	0.630368	+	1.52184i	0.839161	+	0.543884i	$0.183047\pi$
$281$	10.5669	+	25.5107i	0.630368	+	1.52184i	−0.208792	+	0.977960i	$0.566953\pi$
$282$		0			0
$283$	−17.2708	−	25.8476i	−1.02664	−	1.53648i	−0.831362	−	0.555732i	$-0.812438\pi$
$283$	−17.2708	−	25.8476i	−1.02664	−	1.53648i	−0.195281	−	0.980747i	$-0.562562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−14.7740	−	8.41004i	−0.869059	−	0.494708i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.1149	+	24.1149i	1.40881	+	1.40881i	0.766179	+	0.642627i	$0.222155\pi$
$293$	24.1149	+	24.1149i	1.40881	+	1.40881i	0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.06844	−	5.37141i	0.0617895	−	0.310637i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	19.2642			1.09947			0.549734	−	0.835340i	$-0.314729\pi$
$307$	19.2642			1.09947			0.549734	+	0.835340i	$0.314729\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−33.8214	+	6.72749i	−1.91783	+	0.381481i	−0.999884	−	0.0152162i	$-0.995156\pi$
$311$	−33.8214	+	6.72749i	−1.91783	+	0.381481i	−0.917950	+	0.396697i	$0.870156\pi$
$312$		0			0
$313$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$313$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$314$		0			0
$315$		0			0
$316$	6.85605	−	34.4677i	0.385683	−	1.93896i
$317$	19.7821	+	29.6060i	1.11107	+	1.66284i	0.564799	+	0.825228i	$0.308954\pi$
$317$	19.7821	+	29.6060i	1.11107	+	1.66284i	0.546272	+	0.837608i	$0.316046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	4.81318	−	11.6200i	0.266987	−	0.644564i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$331$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$332$	−5.01885	−	12.1166i	−0.275445	−	0.664983i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.7595	−	22.0892i	0.804003	−	1.20328i	−0.171909	−	0.985113i	$-0.554994\pi$
$337$	14.7595	−	22.0892i	0.804003	−	1.20328i	0.975912	−	0.218163i	$-0.0700065\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.5038			1.00204
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$347$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.2213	+	15.2213i	−0.810147	+	0.810147i	−0.984656	−	0.174509i	$-0.944166\pi$
$353$	−15.2213	+	15.2213i	−0.810147	+	0.810147i	0.174509	+	0.984656i	$0.444166\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0135	−	33.8316i	0.739605	−	1.78556i	0.132119	−	0.991234i	$-0.457822\pi$
$359$	14.0135	−	33.8316i	0.739605	−	1.78556i	0.607486	−	0.794330i	$-0.292178\pi$
$360$		0			0
$361$	−13.4350	−	13.4350i	−0.707107	−	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.38746	−	37.1393i	0.385622	−	1.93865i	0.0433266	−	0.999061i	$-0.486204\pi$
$367$	7.38746	−	37.1393i	0.385622	−	1.93865i	0.342296	−	0.939592i	$-0.388796\pi$
$368$	4.83828	+	7.24100i	0.252213	+	0.377463i
$369$	0.385671	+	1.93890i	0.0200772	+	0.100935i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−15.2949	+	3.04235i	−0.785648	+	0.156275i	−0.571583	−	0.820544i	$-0.693670\pi$
$379$	−15.2949	+	3.04235i	−0.785648	+	0.156275i	−0.214065	+	0.976819i	$0.568670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.9104	−	13.9104i	0.707107	−	0.707107i
$388$	−18.7922	+	28.1245i	−0.954028	+	1.42780i
$389$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$389$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$390$		0			0
$391$	−2.91138	−	8.49147i	−0.147235	−	0.429432i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	13.2999	−	2.64552i	0.668346	−	0.132942i
$397$	20.2100	−	13.5039i	1.01431	−	0.677742i	0.0669005	−	0.997760i	$-0.478689\pi$
$397$	20.2100	−	13.5039i	1.01431	−	0.677742i	0.947411	+	0.320018i	$0.103689\pi$
$398$		0			0
$399$		0			0
$400$	7.65367	+	18.4776i	0.382683	+	0.923880i
$401$	5.39817	−	27.1384i	0.269572	−	1.35523i	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	5.39817	−	27.1384i	0.269572	−	1.35523i	0.843854	−	0.536572i	$-0.180281\pi$
$402$		0			0
$403$	−4.01787	−	20.1992i	−0.200145	−	1.00619i
$404$	18.8504	−	18.8504i	0.937844	−	0.937844i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	17.1794	+	17.1794i	0.846368	+	0.846368i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$419$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$	37.8254	+	15.6678i	1.83913	+	0.761794i
$424$		0			0
$425$	−2.75319	−	20.4309i	−0.133549	−	0.991042i
$426$		0			0
$427$		0			0
$428$	16.1994	+	10.8241i	0.783028	+	0.523203i
$429$		0			0
$430$		0			0
$431$	25.3159	−	16.9155i	1.21942	−	0.814792i	0.231972	−	0.972722i	$-0.425482\pi$
$431$	25.3159	−	16.9155i	1.21942	−	0.814792i	0.987450	+	0.157930i	$0.0504821\pi$
$432$		0			0
$433$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$433$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$434$		0			0
$435$		0			0
$436$	−19.9213	−	29.8143i	−0.954057	−	1.42785i
$437$		0			0
$438$		0			0
$439$	−22.2933	+	33.3643i	−1.06400	+	1.59239i	−0.292391	+	0.956299i	$0.594451\pi$
$439$	−22.2933	+	33.3643i	−1.06400	+	1.59239i	−0.771612	+	0.636093i	$0.780549\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$		0			0
$443$	19.0107			0.903223			0.451612	−	0.892215i	$-0.350849\pi$
$443$	19.0107			0.903223			0.451612	+	0.892215i	$0.350849\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$449$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$450$		0			0
$451$	0.569932	+	1.37594i	0.0268371	+	0.0647904i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$457$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.0377	+	24.2331i	−0.467502	+	1.12865i	0.497748	+	0.867322i	$0.334160\pi$
$461$	−10.0377	+	24.2331i	−0.467502	+	1.12865i	−0.965250	+	0.261328i	$0.915840\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$	−5.77582	−	13.9441i	−0.266987	−	0.644564i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.23373	−	12.3226i	0.378587	−	0.566596i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−43.5596			−1.99446
$478$		0			0
$479$	−18.3308	−	12.2483i	−0.837557	−	0.559638i	0.0611793	−	0.998127i	$-0.480514\pi$
$479$	−18.3308	−	12.2483i	−0.837557	−	0.559638i	−0.898737	+	0.438489i	$0.855514\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−10.8871	+	4.50958i	−0.494867	+	0.204981i
$485$		0			0
$486$		0			0
$487$	−23.6064	−	35.3295i	−1.06971	−	1.60093i	−0.759788	−	0.650171i	$-0.774697\pi$
$487$	−23.6064	−	35.3295i	−1.06971	−	1.60093i	−0.309921	−	0.950762i	$-0.600303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	27.2298	+	18.1944i	1.22265	+	0.816952i
$497$		0			0
$498$		0			0
$499$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$499$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$503$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	25.8046	+	10.6886i	1.14489	+	0.474230i
$509$			41.1874i			1.82560i	0.408408	+	0.912799i	$0.366084\pi$
$509$			41.1874i			1.82560i	−0.408408	+	0.912799i	$0.633916\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	30.2513	+	6.01736i	1.33045	+	0.264643i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$521$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.3348	−	25.3118i	−0.972918	−	1.10260i
$528$		0			0
$529$	−6.98777	+	16.8700i	−0.303816	+	0.733477i
$530$		0			0
$531$	−5.03218	−	5.03218i	−0.218378	−	0.218378i
$532$		0			0
$533$	1.37825	−	0.920919i	0.0596988	−	0.0398894i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.08644	+	15.5166i	0.132942	+	0.668346i
$540$		0			0
$541$	−25.3641	+	37.9600i	−1.09049	+	1.63203i	−0.385670	+	0.922637i	$0.626030\pi$
$541$	−25.3641	+	37.9600i	−1.09049	+	1.63203i	−0.704816	+	0.709390i	$0.748970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−27.5766	−	18.4261i	−1.17909	−	0.787842i	−0.197773	−	0.980248i	$-0.563371\pi$
$547$	−27.5766	−	18.4261i	−1.17909	−	0.787842i	−0.981315	+	0.192406i	$0.938371\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	8.48193	+	42.6415i	0.359714	+	1.80840i
$557$	−5.23491	+	5.23491i	−0.221810	+	0.221810i	−0.809260	−	0.587450i	$-0.800132\pi$
$557$	−5.23491	+	5.23491i	−0.221810	+	0.221810i	0.587450	+	0.809260i	$0.300132\pi$
$558$		0			0
$559$	−15.2395	−	6.31243i	−0.644564	−	0.266987i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.5773	+	42.4354i	−0.740796	+	1.78844i	−0.138182	+	0.990407i	$0.544126\pi$
$563$	−17.5773	+	42.4354i	−0.740796	+	1.78844i	−0.602614	+	0.798033i	$0.705874\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	38.4796	−	15.9388i	1.61315	−	0.668189i	0.619953	−	0.784639i	$-0.287151\pi$
$569$	38.4796	−	15.9388i	1.61315	−	0.668189i	0.993197	+	0.116450i	$0.0371515\pi$
$570$		0			0
$571$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$571$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$572$	−6.31706	−	9.45414i	−0.264129	−	0.395298i
$573$		0			0
$574$		0			0
$575$	6.04785	−	9.05125i	0.252213	−	0.377463i
$576$	22.1731	+	9.18440i	0.923880	+	0.382683i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−32.1855	+	6.40209i	−1.33299	+	0.265147i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$593$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	32.4577	+	32.4577i	1.32618	+	1.32618i	0.908671	+	0.417514i	$0.137098\pi$
$599$	32.4577	+	32.4577i	1.32618	+	1.32618i	0.417514	+	0.908671i	$0.362902\pi$
$600$		0			0
$601$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$601$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$602$		0			0
$603$	−0.881469	+	0.365116i	−0.0358962	+	0.0148687i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$607$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	34.3296i		−	1.38883i
$612$	−19.6723	−	15.0000i	−0.795206	−	0.606339i
$613$	−40.3058			−1.62794			−0.813969	−	0.580909i	$-0.802697\pi$
$613$	−40.3058			−1.62794			−0.813969	+	0.580909i	$0.802697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−40.5062	+	8.05718i	−1.63072	+	0.324370i	−0.923785	−	0.382911i	$-0.874922\pi$
$617$	−40.5062	+	8.05718i	−1.63072	+	0.324370i	−0.706932	+	0.707281i	$0.749922\pi$
$618$		0			0
$619$	−31.9974	−	6.36469i	−1.28609	−	0.255818i	−0.495735	−	0.868474i	$-0.665101\pi$
$619$	−31.9974	−	6.36469i	−1.28609	−	0.255818i	−0.790350	+	0.612655i	$0.790101\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.6777	−	17.6777i	0.707107	−	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	16.2681	−	6.73845i	0.644564	−	0.266987i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$641$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$642$		0			0
$643$	−8.29778	+	12.4185i	−0.327233	+	0.489738i	−0.958211	−	0.286062i	$-0.907654\pi$
$643$	−8.29778	+	12.4185i	−0.327233	+	0.489738i	0.630978	+	0.775800i	$0.282654\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−4.45779	−	2.97860i	−0.174983	−	0.116920i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$653$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$654$		0			0
$655$		0			0
$656$	−0.514228	+	2.58520i	−0.0200772	+	0.100935i
$657$		0			0
$658$		0			0
$659$	−27.4465	+	27.4465i	−1.06916	+	1.06916i	−0.0717399	+	0.997423i	$0.522855\pi$
$659$	−27.4465	+	27.4465i	−1.06916	+	1.06916i	−0.997423	+	0.0717399i	$0.977145\pi$
$660$		0			0
$661$	6.05828	+	2.50942i	0.235640	+	0.0976052i	0.497379	−	0.867533i	$-0.334296\pi$
$661$	6.05828	+	2.50942i	0.235640	+	0.0976052i	−0.261739	+	0.965139i	$0.584296\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−3.04864	+	0.606411i	−0.117955	+	0.0234628i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$673$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$674$		0			0
$675$		0			0
$676$	9.43608	−	9.43608i	0.362926	−	0.362926i
$677$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$677$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−43.2825	−	28.9204i	−1.65616	−	1.10661i	−0.878197	−	0.478299i	$-0.841253\pi$
$683$	−43.2825	−	28.9204i	−1.65616	−	1.10661i	−0.777962	−	0.628311i	$-0.783747\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	24.2331	−	10.0377i	0.923880	−	0.382683i
$689$	13.9773	+	33.7443i	0.532494	+	1.28555i
$690$		0			0
$691$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$691$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$692$	−10.1899	−	51.2283i	−0.387363	−	1.94741i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.19425	−	2.44043i	0.0452353	−	0.0924378i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.88512	−	9.88512i	−0.373356	−	0.373356i	0.495342	−	0.868698i	$-0.335043\pi$
$701$	−9.88512	−	9.88512i	−0.373356	−	0.373356i	−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$		0			0
$704$	17.7332	+	3.52735i	0.668346	+	0.132942i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.41918	−	17.1894i	−0.128410	−	0.645561i	−0.990354	−	0.138558i	$-0.955753\pi$
$709$	−3.41918	−	17.1894i	−0.128410	−	0.645561i	0.861944	−	0.507003i	$-0.169247\pi$
$710$		0			0
$711$	−29.2865	+	43.8304i	−1.09833	+	1.64377i
$712$		0			0
$713$		−	17.8250i		−	0.667553i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.5655	+	4.48856i	−0.841551	+	0.167395i	−0.597000	−	0.802242i	$-0.703641\pi$
$719$	−22.5655	+	4.48856i	−0.841551	+	0.167395i	−0.244551	+	0.969636i	$0.578641\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−7.63244	+	38.3708i	−0.283657	+	1.42604i
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−24.9447	−	10.3325i	−0.923880	−	0.382683i
$730$		0			0
$731$	−26.7948	+	3.61077i	−0.991042	+	0.133549i
$732$		0			0
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.597641	+	0.399331i	−0.0220144	+	0.0147095i
$738$		0			0
$739$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$739$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$743$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			19.6723i			0.719772i
$748$	−16.7402	−	8.19195i	−0.612081	−	0.299528i
$749$		0			0
$750$		0			0
$751$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$751$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$752$	38.6004	+	38.6004i	1.40761	+	1.40761i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.28356	+	5.51299i	−0.0824544	+	0.199063i
$768$		0			0
$769$	29.4001	+	29.4001i	1.06019	+	1.06019i	0.998068	+	0.0621249i	$0.0197877\pi$
$769$	29.4001	+	29.4001i	1.06019	+	1.06019i	0.0621249	+	0.998068i	$0.480212\pi$
$770$		0			0
$771$		0			0
$772$	51.1829	+	10.1809i	1.84211	+	0.366419i
$773$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$773$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$774$		0			0
$775$	7.98627	−	40.1497i	0.286875	−	1.44222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−10.7151	+	25.8686i	−0.382683	+	0.923880i
$785$		0			0
$786$		0			0
$787$	46.0059	−	9.15113i	1.63993	−	0.326203i	0.712923	−	0.701242i	$-0.247371\pi$
$787$	46.0059	−	9.15113i	1.63993	−	0.326203i	0.927009	+	0.375040i	$0.122371\pi$
$788$	−27.0287	+	18.0600i	−0.962856	+	0.643360i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	33.4041	+	13.8364i	1.18323	+	0.490111i	0.885545	−	0.464553i	$-0.153785\pi$
$797$	33.4041	+	13.8364i	1.18323	+	0.490111i	0.297687	+	0.954664i	$0.403785\pi$
$798$		0			0
$799$	−28.2831	−	48.6446i	−1.00058	−	1.72092i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.11724	+	10.6441i	−0.0744382	+	0.374226i	−0.999990	−	0.00439415i	$-0.998601\pi$
$809$	−2.11724	+	10.6441i	−0.0744382	+	0.374226i	0.925552	+	0.378620i	$0.123601\pi$
$810$		0			0
$811$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$811$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.3478	+	10.4126i	−1.82695	+	0.363403i	−0.984502	−	0.175373i	$-0.943887\pi$
$821$	−52.3478	+	10.4126i	−1.82695	+	0.363403i	−0.842449	+	0.538776i	$0.818887\pi$
$822$		0			0
$823$	−36.4066	−	7.24172i	−1.26905	−	0.252430i	−0.485765	−	0.874089i	$-0.661459\pi$
$823$	−36.4066	−	7.24172i	−1.26905	−	0.252430i	−0.783288	+	0.621659i	$0.786459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.2266	−	16.8018i	−0.390386	−	0.584255i	0.583269	−	0.812279i	$-0.301773\pi$
$827$	−11.2266	−	16.8018i	−0.390386	−	0.584255i	−0.973655	+	0.228024i	$0.926773\pi$
$828$	−2.54847	−	12.8120i	−0.0885654	−	0.445248i
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		−	20.1239i		−	0.697671i
$833$	17.5000	−	22.9510i	0.606339	−	0.795206i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$839$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$840$		0			0
$841$	−26.7925	+	11.0978i	−0.923880	+	0.382683i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−53.6584	−	22.2260i	−1.84264	−	0.763245i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	28.9364	+	19.3347i	0.990762	+	0.662006i	0.941582	−	0.336784i	$-0.109339\pi$
$853$	28.9364	+	19.3347i	0.990762	+	0.662006i	0.0491804	+	0.998790i	$0.484339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.9727	−	2.97826i	−0.511459	−	0.101736i	−0.0673876	−	0.997727i	$-0.521466\pi$
$857$	−14.9727	−	2.97826i	−0.511459	−	0.101736i	−0.444072	+	0.895991i	$0.646466\pi$
$858$		0			0
$859$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$859$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.1975	+	36.6899i	−0.515539	+	1.24462i
$870$		0			0
$871$	0.565689	+	0.565689i	0.0191676	+	0.0191676i
$872$		0			0
$873$	42.1867	−	28.1882i	1.42780	−	0.954028i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.20769	−	36.2355i	0.243386	−	1.22359i	−0.644889	−	0.764276i	$-0.723097\pi$
$877$	7.20769	−	36.2355i	0.243386	−	1.22359i	0.888276	−	0.459310i	$-0.151903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.82168	+	4.22294i	−0.0950648	+	0.142275i	−0.875953	−	0.482396i	$-0.839767\pi$
$881$	−2.82168	+	4.22294i	−0.0950648	+	0.142275i	0.780889	+	0.624670i	$0.214767\pi$
$882$		0			0
$883$			46.6810i			1.57094i	0.618899	+	0.785471i	$0.287579\pi$
$883$			46.6810i			1.57094i	−0.618899	+	0.785471i	$0.712421\pi$
$884$	−5.30762	+	20.0527i	−0.178514	+	0.674446i
$885$		0			0
$886$		0			0
$887$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$887$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.9499	−	3.96827i	−0.668346	−	0.132942i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		−	30.0000i		−	1.00000i
$901$	47.6065	+	36.2996i	1.58600	+	1.20932i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	31.1066	−	20.7847i	1.03288	−	0.690146i	0.0810272	−	0.996712i	$-0.474180\pi$
$907$	31.1066	−	20.7847i	1.03288	−	0.690146i	0.951849	+	0.306566i	$0.0991799\pi$
$908$		0			0
$909$	−36.9439	+	15.3027i	−1.22535	+	0.507558i
$910$		0			0
$911$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$911$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$912$		0			0
$913$	2.89130	+	14.5356i	0.0956881	+	0.481057i
$914$		0			0
$915$		0			0
$916$	49.8462	+	20.6470i	1.64696	+	0.682195i
$917$		0			0
$918$		0			0
$919$	32.7872			1.08155			0.540775	−	0.841167i	$-0.318131\pi$
$919$	32.7872			1.08155			0.540775	+	0.841167i	$0.318131\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−13.9461	−	33.6689i	−0.458051	−	1.10583i
$928$		0			0
$929$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$929$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$937$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−45.2010	+	30.2023i	−1.47351	+	0.984568i	−0.479238	+	0.877685i	$0.659087\pi$
$941$	−45.2010	+	30.2023i	−1.47351	+	0.984568i	−0.994272	+	0.106883i	$0.965913\pi$
$942$		0			0
$943$	1.32546	−	0.549025i	0.0431630	−	0.0178787i
$944$	−3.63119	−	8.76647i	−0.118185	−	0.285324i
$945$		0			0
$946$		0			0
$947$	9.59205	+	48.2225i	0.311700	+	1.56702i	0.745808	+	0.666160i	$0.232063\pi$
$947$	9.59205	+	48.2225i	0.311700	+	1.56702i	−0.434109	+	0.900861i	$0.642937\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	41.1149	+	41.1149i	1.32975	+	1.32975i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−13.7885	−	33.2884i	−0.444790	−	1.07382i
$962$		0			0
$963$	−16.2362	−	24.2991i	−0.523203	−	0.783028i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	51.3952	+	21.2886i	1.65276	+	0.684594i	0.997490	−	0.0708031i	$-0.0225562\pi$
$967$	51.3952	+	21.2886i	1.65276	+	0.684594i	0.655267	+	0.755397i	$0.272556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	23.2138	−	56.0432i	0.744968	−	1.79851i	0.160605	−	0.987019i	$-0.448655\pi$
$971$	23.2138	−	56.0432i	0.744968	−	1.79851i	0.584363	−	0.811493i	$-0.301345\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	48.4663	−	20.0754i	1.55057	−	0.642268i	0.567153	−	0.823612i	$-0.308045\pi$
$977$	48.4663	−	20.0754i	1.55057	−	0.642268i	0.983420	+	0.181344i	$0.0580446\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	10.4931	+	52.7526i	0.335020	+	1.68426i
$982$		0			0
$983$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$983$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.8706	−	7.93169i	−0.377463	−	0.252213i
$990$		0			0
$991$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$991$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$997$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.a.214.2	✓	16
17.12	odd	16	inner	731.2.s.a.386.2	yes	16
43.42	odd	2	CM	731.2.s.a.214.2	✓	16
731.386	even	16	inner	731.2.s.a.386.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.214.2	✓	16	1.1	even	1	trivial
731.2.s.a.214.2	✓	16	43.42	odd	2	CM
731.2.s.a.386.2	yes	16	17.12	odd	16	inner
731.2.s.a.386.2	yes	16	731.386	even	16	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +O(q^{10})\)	\(q+(-1.41421 - 1.41421i) q^{4} +(1.14805 + 2.77164i) q^{9} +(1.25563 + 1.87919i) q^{11} +(1.77872 - 1.77872i) q^{13} +4.00000i q^{16} +(1.05499 - 3.98585i) q^{17} +(1.81025 - 1.20957i) q^{23} +(4.61940 - 1.91342i) q^{25} +(4.54859 - 6.80745i) q^{31} +(2.29610 - 5.54328i) q^{36} +(0.646300 + 0.128557i) q^{41} +(-2.50942 - 6.05828i) q^{43} +(0.881839 - 4.43330i) q^{44} +(9.65010 - 9.65010i) q^{47} +(6.46716 + 2.67878i) q^{49} -5.03098 q^{52} +(-5.55651 + 13.4146i) q^{53} +(-2.19162 + 0.907798i) q^{59} +(5.65685 - 5.65685i) q^{64} +0.318032i q^{67} +(-7.12882 + 4.14487i) q^{68} +(9.76218 + 14.6101i) q^{79} +(-6.36396 + 6.36396i) q^{81} +(6.05828 + 2.50942i) q^{83} +(-4.27067 - 0.849489i) q^{92} +(-3.29947 - 16.5875i) q^{97} +(-3.76690 + 5.63756i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{13} + 56 q^{23} - 64 q^{59} + 96 q^{79}+O(q^{100}) \) 16 * q - 24 * q^13 + 56 * q^23 - 64 * q^59 + 96 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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