LMFDB - Embedded newform 731.2.s.a.343.2

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			343.2
Root			$2.83780 - 1.71665i$ of defining polynomial
Character	$\chi$	$=$	731.343
Dual form			731.2.s.a.601.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +O(q^{10})$	$q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +(1.17208 + 5.89244i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(-3.98585 + 1.05499i) q^{17} +(7.98829 - 1.58897i) q^{23} +(-1.91342 + 4.61940i) q^{25} +(2.17064 - 10.9126i) q^{31} +(5.54328 - 2.29610i) q^{36} +(4.76537 - 7.13187i) q^{41} +(-6.05828 - 2.50942i) q^{43} +(9.99074 + 6.67560i) q^{44} +(0.935734 + 0.935734i) q^{47} +(-2.67878 - 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} +16.3676i q^{67} +(-4.14487 + 7.12882i) q^{68} +(-2.05270 - 10.3196i) q^{79} +(6.36396 + 6.36396i) q^{81} +(-2.50942 - 6.05828i) q^{83} +(9.05001 - 13.5443i) q^{92} +(15.8804 - 10.6110i) q^{97} +(-3.51624 + 17.6773i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

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

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		0.923880	−	0.382683i	$-0.125000\pi$
$2$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$3$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$4$	1.41421	−	1.41421i	0.707107	−	0.707107i
$5$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$5$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$6$		0			0
$7$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$7$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$8$		0			0
$9$	2.77164	+	1.14805i	0.923880	+	0.382683i
$10$		0			0
$11$	1.17208	+	5.89244i	0.353395	+	1.77664i	0.592425	+	0.805626i	$0.298171\pi$
$11$	1.17208	+	5.89244i	0.353395	+	1.77664i	−0.239030	+	0.971012i	$0.576829\pi$
$12$		0			0
$13$	1.77872	+	1.77872i	0.493328	+	0.493328i	0.909353	−	0.416025i	$-0.136577\pi$
$13$	1.77872	+	1.77872i	0.493328	+	0.493328i	−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$		0			0
$16$		−	4.00000i		−	1.00000i
$17$	−3.98585	+	1.05499i	−0.966711	+	0.255872i
$17$	−3.98585	+	1.05499i	−0.966711	+	0.255872i
$18$		0			0
$19$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.98829	−	1.58897i	1.66567	−	0.331323i	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.98829	−	1.58897i	1.66567	−	0.331323i	0.935873	+	0.352337i	$0.114613\pi$
$24$		0			0
$25$	−1.91342	+	4.61940i	−0.382683	+	0.923880i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$29$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$30$		0			0
$31$	2.17064	−	10.9126i	0.389859	−	1.95995i	0.155103	−	0.987898i	$-0.450429\pi$
$31$	2.17064	−	10.9126i	0.389859	−	1.95995i	0.234756	−	0.972054i	$-0.424571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.54328	−	2.29610i	0.923880	−	0.382683i
$37$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$37$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.76537	−	7.13187i	0.744225	−	1.11381i	−0.245299	−	0.969447i	$-0.578886\pi$
$41$	4.76537	−	7.13187i	0.744225	−	1.11381i	0.989525	−	0.144364i	$-0.0461137\pi$
$42$		0			0
$43$	−6.05828	−	2.50942i	−0.923880	−	0.382683i
$43$	−6.05828	−	2.50942i	−0.923880	−	0.382683i
$44$	9.99074	+	6.67560i	1.50616	+	1.00638i
$45$		0			0
$46$		0			0
$47$	0.935734	+	0.935734i	0.136491	+	0.136491i	0.772051	−	0.635560i	$-0.219231\pi$
$47$	0.935734	+	0.935734i	0.136491	+	0.136491i	−0.635560	+	0.772051i	$0.719231\pi$
$48$		0			0
$49$	−2.67878	−	6.46716i	−0.382683	−	0.923880i
$50$		0			0
$51$		0			0
$52$	5.03098			0.697671
$53$	−1.00093	+	0.414599i	−0.137488	+	0.0569495i	−0.450367	−	0.892844i	$-0.648707\pi$
$53$	−1.00093	+	0.414599i	−0.137488	+	0.0569495i	0.312878	+	0.949793i	$0.398707\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.80838	+	14.0227i	−0.756187	+	1.82560i	−0.235431	+	0.971891i	$0.575650\pi$
$59$	−5.80838	+	14.0227i	−0.756187	+	1.82560i	−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$61$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$62$		0			0
$63$		0			0
$64$	−5.65685	−	5.65685i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$			16.3676i			1.99962i	0.0194269	+	0.999811i	$0.493816\pi$
$67$			16.3676i			1.99962i	−0.0194269	+	0.999811i	$0.506184\pi$
$68$	−4.14487	+	7.12882i	−0.502639	+	0.864496i
$69$		0			0
$70$		0			0
$71$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$71$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$72$		0			0
$73$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$73$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.05270	−	10.3196i	−0.230947	−	1.16105i	−0.906000	−	0.423279i	$-0.860879\pi$
$79$	−2.05270	−	10.3196i	−0.230947	−	1.16105i	0.675053	−	0.737769i	$-0.264121\pi$
$80$		0			0
$81$	6.36396	+	6.36396i	0.707107	+	0.707107i
$82$		0			0
$83$	−2.50942	−	6.05828i	−0.275445	−	0.664983i	0.724254	−	0.689534i	$-0.242184\pi$
$83$	−2.50942	−	6.05828i	−0.275445	−	0.664983i	−0.999699	+	0.0245507i	$0.992184\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$	9.05001	−	13.5443i	0.943528	−	1.41209i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	15.8804	−	10.6110i	1.61241	−	1.07738i	0.670297	−	0.742093i	$-0.266167\pi$
$97$	15.8804	−	10.6110i	1.61241	−	1.07738i	0.942116	−	0.335287i	$-0.108833\pi$
$98$		0			0
$99$	−3.51624	+	17.6773i	−0.353395	+	1.77664i
$100$	3.82683	+	9.23880i	0.382683	+	0.923880i
$101$		−	15.0443i		−	1.49696i	−0.663156	−	0.748481i	$-0.730783\pi$
$101$		−	15.0443i		−	1.49696i	0.663156	−	0.748481i	$-0.269217\pi$
$102$		0			0
$103$	−16.2614			−1.60229			−0.801143	−	0.598473i	$-0.795775\pi$
$103$	−16.2614			−1.60229			−0.801143	+	0.598473i	$0.795775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.34302	+	5.00318i	0.323182	+	0.483676i	0.957114	−	0.289713i	$-0.0935599\pi$
$107$	3.34302	+	5.00318i	0.323182	+	0.483676i	−0.633932	+	0.773389i	$0.718560\pi$
$108$		0			0
$109$	2.83844	−	4.24803i	0.271874	−	0.406888i	−0.670259	−	0.742127i	$-0.733817\pi$
$109$	2.83844	−	4.24803i	0.271874	−	0.406888i	0.942133	+	0.335239i	$0.108817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$113$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.88791	+	6.97203i	0.266987	+	0.644564i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−23.1844	+	9.60330i	−2.10767	+	0.873027i
$122$		0			0
$123$		0			0
$124$	−12.3629	−	18.5024i	−1.11022	−	1.66157i
$125$		0			0
$126$		0			0
$127$	−6.77003	+	16.3443i	−0.600743	+	1.45032i	0.272075	+	0.962276i	$0.412290\pi$
$127$	−6.77003	+	16.3443i	−0.600743	+	1.45032i	−0.872818	+	0.488046i	$0.837710\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$131$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$132$		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
$139$	−8.74108	−	1.73871i	−0.741409	−	0.147475i	−0.190086	−	0.981767i	$-0.560877\pi$
$139$	−8.74108	−	1.73871i	−0.741409	−	0.147475i	−0.551323	+	0.834292i	$0.685877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.39620	+	12.5658i	−0.702125	+	1.05080i
$144$	4.59220	−	11.0866i	0.382683	−	0.923880i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$151$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$152$		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	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$163$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$164$	−3.34675	−	16.8252i	−0.261337	−	1.31383i
$165$		0			0
$166$		0			0
$167$	−3.34455	+	16.8142i	−0.258809	+	1.30112i	0.604564	+	0.796557i	$0.293347\pi$
$167$	−3.34455	+	16.8142i	−0.258809	+	1.30112i	−0.863373	+	0.504566i	$0.831653\pi$
$168$		0			0
$169$		−	6.67232i		−	0.513255i
$170$		0			0
$171$		0			0
$172$	−12.1166	+	5.01885i	−0.923880	+	0.382683i
$173$	−15.9235	−	3.16737i	−1.21064	−	0.240811i	−0.451817	−	0.892111i	$-0.649224\pi$
$173$	−15.9235	−	3.16737i	−1.21064	−	0.240811i	−0.758820	+	0.651300i	$0.774224\pi$
$174$		0			0
$175$		0			0
$176$	23.5698	−	4.68832i	1.77664	−	0.353395i
$177$		0			0
$178$		0			0
$179$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$179$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$180$		0			0
$181$	0.149740	+	0.752792i	0.0111301	+	0.0559546i	0.985951	−	0.167034i	$-0.0534188\pi$
$181$	0.149740	+	0.752792i	0.0111301	+	0.0559546i	−0.974821	+	0.222988i	$0.928419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.8882	−	22.2499i	−0.796223	−	1.62707i
$188$	2.64666			0.193027
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−24.7177	+	4.91665i	−1.77922	+	0.353908i	−0.971751	−	0.236007i	$-0.924161\pi$
$193$	−24.7177	+	4.91665i	−1.77922	+	0.353908i	−0.807465	+	0.589915i	$0.799161\pi$
$194$		0			0
$195$		0			0
$196$	−12.9343	−	5.35757i	−0.923880	−	0.382683i
$197$	−23.0123	−	15.3763i	−1.63956	−	1.09552i	−0.912614	−	0.408822i	$-0.865940\pi$
$197$	−23.0123	−	15.3763i	−1.63956	−	1.09552i	−0.726944	−	0.686696i	$-0.759060\pi$
$198$		0			0
$199$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$199$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	23.9649	+	4.76691i	1.66567	+	0.331323i
$208$	7.11488	−	7.11488i	0.493328	−	0.493328i
$209$		0			0
$210$		0			0
$211$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$211$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$212$	−0.829197	+	2.00186i	−0.0569495	+	0.137488i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.96623	−	5.21318i	−0.603134	−	0.350677i
$222$		0			0
$223$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$223$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$	−10.6066	+	10.6066i	−0.707107	+	0.707107i
$226$		0			0
$227$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$227$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$228$		0			0
$229$	5.25078	−	12.6765i	0.346982	−	0.837688i	−0.649992	−	0.759941i	$-0.725228\pi$
$229$	5.25078	−	12.6765i	0.346982	−	0.837688i	0.996973	−	0.0777462i	$-0.0247724\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$233$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$234$		0			0
$235$		0			0
$236$	11.6168	+	28.0453i	0.756187	+	1.82560i
$237$		0			0
$238$		0			0
$239$	29.0726			1.88055			0.940275	−	0.340415i	$-0.110568\pi$
$239$	29.0726			1.88055			0.940275	+	0.340415i	$0.110568\pi$
$240$		0			0
$241$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$241$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$242$		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	0.978351	+	0.206951i	$0.0663540\pi$
$251$	18.7787	+	18.7787i	1.18530	+	1.18530i	0.206951	+	0.978351i	$0.433646\pi$
$252$		0			0
$253$	18.7258	+	45.2081i	1.17728	+	2.84221i
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$257$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$263$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	23.1473	+	23.1473i	1.41395	+	1.41395i
$269$	−1.43923	+	7.23550i	−0.0877514	+	0.441156i	0.911784	+	0.410671i	$0.134705\pi$
$269$	−1.43923	+	7.23550i	−0.0877514	+	0.441156i	−0.999535	+	0.0304855i	$0.990295\pi$
$270$		0			0
$271$		−	31.4395i		−	1.90981i	−0.296909	−	0.954906i	$-0.595956\pi$
$271$		−	31.4395i		−	1.90981i	0.296909	−	0.954906i	$-0.404044\pi$
$272$	4.21995	+	15.9434i	0.255872	+	0.966711i
$273$		0			0
$274$		0			0
$275$	−29.4622	−	5.86040i	−1.77664	−	0.353395i
$276$		0			0
$277$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$277$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$278$		0			0
$279$	18.5444	−	27.7537i	1.11022	−	1.66157i
$280$		0			0
$281$	−17.5669	−	7.27645i	−1.04795	−	0.434076i	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−17.5669	−	7.27645i	−1.04795	−	0.434076i	−0.839161	+	0.543884i	$0.816953\pi$
$282$		0			0
$283$	−2.51306	−	12.6340i	−0.149386	−	0.751013i	−0.980747	−	0.195281i	$-0.937438\pi$
$283$	−2.51306	−	12.6340i	−0.149386	−	0.751013i	0.831362	−	0.555732i	$-0.187562\pi$
$284$		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	0.642627	−	0.766179i	$-0.277845\pi$
$293$	24.1149	−	24.1149i	1.40881	−	1.40881i	0.766179	−	0.642627i	$-0.222155\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	17.0353	+	11.3826i	0.985174	+	0.658272i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−29.2727			−1.67068			−0.835340	−	0.549734i	$-0.814729\pi$
$307$	−29.2727			−1.67068			−0.835340	+	0.549734i	$0.814729\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.4566	+	24.6290i	0.933166	+	1.39658i	0.917950	+	0.396697i	$0.129844\pi$
$311$	16.4566	+	24.6290i	0.933166	+	1.39658i	0.0152162	+	0.999884i	$0.495156\pi$
$312$		0			0
$313$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$313$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$314$		0			0
$315$		0			0
$316$	−17.4971	−	11.6912i	−0.984289	−	0.657681i
$317$	4.85722	+	24.4189i	0.272808	+	1.37150i	0.837608	+	0.546272i	$0.183954\pi$
$317$	4.85722	+	24.4189i	0.272808	+	1.37150i	−0.564799	+	0.825228i	$0.691046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	−11.6200	+	4.81318i	−0.644564	+	0.266987i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$331$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$332$	−12.1166	−	5.01885i	−0.664983	−	0.275445i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.168898	−	0.849110i	0.00920048	−	0.0462539i	−0.975912	−	0.218163i	$-0.929994\pi$
$337$	0.168898	−	0.849110i	0.00920048	−	0.0462539i	0.985113	+	0.171909i	$0.0549935\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	66.8458			3.61990
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$347$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.2213	−	15.2213i	−0.810147	−	0.810147i	0.174509	−	0.984656i	$-0.444166\pi$
$353$	−15.2213	−	15.2213i	−0.810147	−	0.810147i	−0.984656	+	0.174509i	$0.944166\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−33.8316	+	14.0135i	−1.78556	+	0.739605i	−0.794330	+	0.607486i	$0.792178\pi$
$359$	−33.8316	+	14.0135i	−1.78556	+	0.739605i	−0.991234	+	0.132119i	$0.957822\pi$
$360$		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
$367$	25.6967	+	17.1700i	1.34136	+	0.896266i	0.999061	−	0.0433266i	$-0.0137956\pi$
$367$	25.6967	+	17.1700i	1.34136	+	0.896266i	0.342296	+	0.939592i	$0.388796\pi$
$368$	−6.35588	−	31.9532i	−0.331323	−	1.66567i
$369$	21.3956	−	14.2961i	1.11381	−	0.744225i
$370$		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
$379$	20.1417	+	30.1442i	1.03461	+	1.54840i	0.820544	+	0.571583i	$0.193670\pi$
$379$	20.1417	+	30.1442i	1.03461	+	1.54840i	0.214065	+	0.976819i	$0.431330\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$383$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.9104	−	13.9104i	−0.707107	−	0.707107i
$388$	7.45215	−	37.4645i	0.378326	−	1.90197i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$	−30.1638	+	14.7609i	−1.52545	+	0.746493i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	20.0268	+	29.9722i	1.00638	+	1.50616i
$397$	5.04333	−	1.00318i	0.253118	−	0.0503482i	−0.0669005	−	0.997760i	$-0.521311\pi$
$397$	5.04333	−	1.00318i	0.253118	−	0.0503482i	0.320018	+	0.947411i	$0.396311\pi$
$398$		0			0
$399$		0			0
$400$	18.4776	+	7.65367i	0.923880	+	0.382683i
$401$	−0.755151	−	0.504576i	−0.0377105	−	0.0251973i	0.536572	−	0.843854i	$-0.319719\pi$
$401$	−0.755151	−	0.504576i	−0.0377105	−	0.0251973i	−0.574283	+	0.818657i	$0.694719\pi$
$402$		0			0
$403$	23.2713	−	15.5494i	1.15923	−	0.774571i
$404$	−21.2758	−	21.2758i	−1.05851	−	1.05851i
$405$		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
$412$	−22.9971	+	22.9971i	−1.13299	+	1.13299i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$419$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$	1.51925	+	3.66779i	0.0738683	+	0.178334i
$424$		0			0
$425$	2.75319	−	20.4309i	0.133549	−	0.991042i
$426$		0			0
$427$		0			0
$428$	11.8033	+	2.34783i	0.570535	+	0.113486i
$429$		0			0
$430$		0			0
$431$	40.6942	−	8.09459i	1.96017	−	0.389903i	0.972722	−	0.231972i	$-0.0745179\pi$
$431$	40.6942	−	8.09459i	1.96017	−	0.389903i	0.987450	−	0.157930i	$-0.0504821\pi$
$432$		0			0
$433$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$433$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$434$		0			0
$435$		0			0
$436$	−1.99346	−	10.0218i	−0.0954693	−	0.479957i
$437$		0			0
$438$		0			0
$439$	−7.20138	+	36.2038i	−0.343703	+	1.72791i	0.292391	+	0.956299i	$0.405549\pi$
$439$	−7.20138	+	36.2038i	−0.343703	+	1.72791i	−0.636093	+	0.771612i	$0.719451\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$	−19.0107			−0.903223			−0.451612	−	0.892215i	$-0.649151\pi$
$443$	−19.0107			−0.903223			−0.451612	+	0.892215i	$0.649151\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$449$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$450$		0			0
$451$	47.6095	+	19.7205i	2.24185	+	0.928603i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$457$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.2331	+	10.0377i	−1.12865	+	0.467502i	−0.867322	−	0.497748i	$-0.834160\pi$
$461$	−24.2331	+	10.0377i	−1.12865	+	0.467502i	−0.261328	+	0.965250i	$0.584160\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$467$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$468$	13.9441	+	5.77582i	0.644564	+	0.266987i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.68584	−	38.6393i	0.353395	−	1.77664i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.25020			−0.148816
$478$		0			0
$479$	−10.9358	−	2.17526i	−0.499668	−	0.0993902i	−0.0611793	−	0.998127i	$-0.519486\pi$
$479$	−10.9358	−	2.17526i	−0.499668	−	0.0993902i	−0.438489	+	0.898737i	$0.644486\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−19.2066	+	46.3688i	−0.873027	+	2.10767i
$485$		0			0
$486$		0			0
$487$	−7.50866	−	37.7486i	−0.340250	−	1.71055i	−0.650171	−	0.759788i	$-0.725303\pi$
$487$	−7.50866	−	37.7486i	−0.340250	−	1.71055i	0.309921	−	0.950762i	$-0.399697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$491$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−43.6502	−	8.68257i	−1.95995	−	0.389859i
$497$		0			0
$498$		0			0
$499$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$499$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$503$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	13.5401	+	32.6886i	0.600743	+	1.45032i
$509$		−	18.4282i		−	0.816817i	−0.912799	−	0.408408i	$-0.866084\pi$
$509$		−	18.4282i		−	0.816817i	0.912799	−	0.408408i	$-0.133916\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.41700	+	6.61051i	−0.194260	+	0.290730i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$521$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.86075	+	45.7858i	0.124616	+	1.99446i
$528$		0			0
$529$	40.0387	−	16.5846i	1.74081	−	0.721069i
$530$		0			0
$531$	−32.1975	+	32.1975i	−1.39725	+	1.39725i
$532$		0			0
$533$	21.1619	−	4.20935i	0.916622	−	0.182327i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	34.9676	−	23.3646i	1.50616	−	1.00638i
$540$		0			0
$541$	5.06639	−	25.4705i	0.217821	−	1.09506i	−0.704816	−	0.709390i	$-0.748970\pi$
$541$	5.06639	−	25.4705i	0.217821	−	1.09506i	0.922637	−	0.385670i	$-0.126030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−45.8771	−	9.12552i	−1.96156	−	0.390179i	−0.981315	−	0.192406i	$-0.938371\pi$
$547$	−45.8771	−	9.12552i	−1.96156	−	0.390179i	−0.980248	−	0.197773i	$-0.936629\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−14.8207	+	9.90285i	−0.628536	+	0.419974i
$557$	32.9636	+	32.9636i	1.39671	+	1.39671i	0.809260	+	0.587450i	$0.199868\pi$
$557$	32.9636	+	32.9636i	1.39671	+	1.39671i	0.587450	+	0.809260i	$0.300132\pi$
$558$		0			0
$559$	−6.31243	−	15.2395i	−0.266987	−	0.644564i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.0199	−	4.56459i	0.464433	−	0.192374i	−0.138182	−	0.990407i	$-0.544126\pi$
$563$	11.0199	−	4.56459i	0.464433	−	0.192374i	0.602614	+	0.798033i	$0.294126\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.9388	−	38.4796i	0.668189	−	1.61315i	−0.116450	−	0.993197i	$-0.537151\pi$
$569$	15.9388	−	38.4796i	0.668189	−	1.61315i	0.784639	−	0.619953i	$-0.212849\pi$
$570$		0			0
$571$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$571$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$572$	5.89671	+	29.6447i	0.246554	+	1.23951i
$573$		0			0
$574$		0			0
$575$	−7.94485	+	39.9414i	−0.331323	+	1.66567i
$576$	−9.18440	−	22.1731i	−0.382683	−	0.923880i
$577$		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
$583$	−3.61617	−	5.41198i	−0.149766	−	0.224141i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$587$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$593$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−32.4577	+	32.4577i	−1.32618	+	1.32618i	−0.417514	+	0.908671i	$0.637098\pi$
$599$	−32.4577	+	32.4577i	−1.32618	+	1.32618i	−0.908671	+	0.417514i	$0.862902\pi$
$600$		0			0
$601$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$601$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$602$		0			0
$603$	−18.7908	+	45.3651i	−0.765222	+	1.84741i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$607$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3.32882i			0.134670i
$612$	−19.6723	+	15.0000i	−0.795206	+	0.606339i
$613$	−28.7652			−1.16182			−0.580909	−	0.813969i	$-0.697303\pi$
$613$	−28.7652			−1.16182			−0.580909	+	0.813969i	$0.697303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.37785	+	8.04852i	0.216504	+	0.324021i	0.923785	−	0.382911i	$-0.125078\pi$
$617$	5.37785	+	8.04852i	0.216504	+	0.324021i	−0.707281	+	0.706932i	$0.750078\pi$
$618$		0			0
$619$	27.5764	−	41.2711i	1.10839	−	1.65882i	0.495735	−	0.868474i	$-0.334899\pi$
$619$	27.5764	−	41.2711i	1.10839	−	1.65882i	0.612655	−	0.790350i	$-0.290101\pi$
$620$		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		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.73845	−	16.2681i	0.266987	−	0.644564i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$641$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$642$		0			0
$643$	8.74620	−	43.9701i	0.344916	−	1.73401i	−0.286062	−	0.958211i	$-0.592346\pi$
$643$	8.74620	−	43.9701i	0.344916	−	1.73401i	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
$649$	−89.4357	−	17.7899i	−3.51066	−	0.698313i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$653$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$654$		0			0
$655$		0			0
$656$	−28.5275	−	19.0615i	−1.11381	−	0.744225i
$657$		0			0
$658$		0			0
$659$	23.7632	+	23.7632i	0.925684	+	0.925684i	0.997423	−	0.0717399i	$-0.0228551\pi$
$659$	23.7632	+	23.7632i	0.925684	+	0.925684i	−0.0717399	+	0.997423i	$0.522855\pi$
$660$		0			0
$661$	−2.50942	−	6.05828i	−0.0976052	−	0.235640i	0.867533	−	0.497379i	$-0.165704\pi$
$661$	−2.50942	−	6.05828i	−0.0976052	−	0.235640i	−0.965139	+	0.261739i	$0.915704\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	19.0490	+	28.5088i	0.737027	+	1.10304i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$673$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$674$		0			0
$675$		0			0
$676$	−9.43608	−	9.43608i	−0.362926	−	0.362926i
$677$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$677$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−39.3715	−	7.83147i	−1.50651	−	0.299663i	−0.628311	−	0.777962i	$-0.716253\pi$
$683$	−39.3715	−	7.83147i	−1.50651	−	0.299663i	−0.878197	+	0.478299i	$0.841253\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−10.0377	+	24.2331i	−0.382683	+	0.923880i
$689$	−2.51783	−	1.04292i	−0.0959216	−	0.0397320i
$690$		0			0
$691$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$691$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$692$	−26.9985	+	18.0398i	−1.02633	+	0.685771i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−11.4700	+	33.4540i	−0.434457	+	1.26716i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.88512	+	9.88512i	−0.373356	+	0.373356i	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−9.88512	+	9.88512i	−0.373356	+	0.373356i	0.495342	+	0.868698i	$0.335043\pi$
$702$		0			0
$703$		0			0
$704$	26.7024	−	39.9630i	1.00638	−	1.50616i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	19.2616	−	12.8702i	0.723386	−	0.483351i	−0.138558	−	0.990354i	$-0.544247\pi$
$709$	19.2616	−	12.8702i	0.723386	−	0.483351i	0.861944	+	0.507003i	$0.169247\pi$
$710$		0			0
$711$	6.15810	−	30.9589i	0.230947	−	1.16105i
$712$		0			0
$713$		−	90.6217i		−	3.39381i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.0689	−	42.0081i	−1.04679	−	1.56664i	−0.802242	−	0.597000i	$-0.796359\pi$
$719$	−28.0689	−	42.0081i	−1.04679	−	1.56664i	−0.244551	−	0.969636i	$-0.578641\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	1.27637	+	0.852845i	0.0474360	+	0.0316957i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	10.3325	+	24.9447i	0.382683	+	0.923880i
$730$		0			0
$731$	26.7948	+	3.61077i	0.991042	+	0.133549i
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−96.4452	+	19.1841i	−3.55261	+	0.706657i
$738$		0			0
$739$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$739$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$743$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	19.6723i		−	0.719772i
$748$	−46.8643	−	16.0678i	−1.71353	−	0.587498i
$749$		0			0
$750$		0			0
$751$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$751$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$752$	3.74294	−	3.74294i	0.136491	−	0.136491i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$757$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−35.2739	+	14.6109i	−1.27367	+	0.527570i
$768$		0			0
$769$	25.9545	−	25.9545i	0.935943	−	0.935943i	−0.0621249	−	0.998068i	$-0.519788\pi$
$769$	25.9545	−	25.9545i	0.935943	−	0.935943i	0.998068	+	0.0621249i	$0.0197877\pi$
$770$		0			0
$771$		0			0
$772$	−28.0029	+	41.9092i	−1.00785	+	1.50835i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$		0			0
$775$	46.2561	+	30.9073i	1.66157	+	1.11022i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−25.8686	+	10.7151i	−0.923880	+	0.382683i
$785$		0			0
$786$		0			0
$787$	30.5212	+	45.6782i	1.08796	+	1.62825i	0.712923	+	0.701242i	$0.247371\pi$
$787$	30.5212	+	45.6782i	1.08796	+	1.62825i	0.375040	+	0.927009i	$0.377629\pi$
$788$	−54.2897	+	10.7989i	−1.93399	+	0.384695i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	16.5959	+	40.0662i	0.587859	+	1.41922i	0.885545	+	0.464553i	$0.153785\pi$
$797$	16.5959	+	40.0662i	0.587859	+	1.41922i	−0.297687	+	0.954664i	$0.596215\pi$
$798$		0			0
$799$	−4.71688	−	2.74251i	−0.166871	−	0.0970230i
$800$		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
$809$	−26.4504	−	17.6736i	−0.929946	−	0.621370i	−0.00439415	−	0.999990i	$-0.501399\pi$
$809$	−26.4504	−	17.6736i	−0.929946	−	0.621370i	−0.925552	+	0.378620i	$0.876399\pi$
$810$		0			0
$811$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$811$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$812$		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
$821$	12.7714	+	19.1138i	0.445726	+	0.667076i	0.984502	−	0.175373i	$-0.0561132\pi$
$821$	12.7714	+	19.1138i	0.445726	+	0.667076i	−0.538776	+	0.842449i	$0.681113\pi$
$822$		0			0
$823$	−2.60491	+	3.89852i	−0.0908014	+	0.135894i	−0.874089	−	0.485765i	$-0.838541\pi$
$823$	−2.60491	+	3.89852i	−0.0908014	+	0.135894i	0.783288	+	0.621659i	$0.213541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.64080	−	23.3309i	−0.161376	−	0.811293i	−0.973655	−	0.228024i	$-0.926773\pi$
$827$	−4.64080	−	23.3309i	−0.161376	−	0.811293i	0.812279	−	0.583269i	$-0.198227\pi$
$828$	40.6329	−	27.1500i	1.41209	−	0.943528i
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$839$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$840$		0			0
$841$	11.0978	−	26.7925i	0.382683	−	0.923880i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	1.65839	+	4.00372i	0.0569495	+	0.137488i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	56.6708	+	11.2725i	1.94037	+	0.385964i	0.998790	+	0.0491804i	$0.0156609\pi$
$853$	56.6708	+	11.2725i	1.94037	+	0.385964i	0.941582	+	0.336784i	$0.109339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.2080	−	24.2570i	0.553655	−	0.828604i	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	16.2080	−	24.2570i	0.553655	−	0.828604i	0.997727	+	0.0673876i	$0.0214664\pi$
$858$		0			0
$859$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	58.4018	−	24.1908i	1.98115	−	0.820618i
$870$		0			0
$871$	−29.1134	+	29.1134i	−0.986470	+	0.986470i
$872$		0			0
$873$	56.1967	−	11.1782i	1.90197	−	0.378326i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.49577	+	3.67216i	0.185579	+	0.124000i	0.644889	−	0.764276i	$-0.276903\pi$
$877$	5.49577	+	3.67216i	0.185579	+	0.124000i	−0.459310	+	0.888276i	$0.651903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−8.85975	+	44.5410i	−0.298493	+	1.50062i	0.482396	+	0.875953i	$0.339767\pi$
$881$	−8.85975	+	44.5410i	−0.298493	+	1.50062i	−0.780889	+	0.624670i	$0.785233\pi$
$882$		0			0
$883$			46.6810i			1.57094i	0.618899	+	0.785471i	$0.287579\pi$
$883$			46.6810i			1.57094i	−0.618899	+	0.785471i	$0.712421\pi$
$884$	−20.0527	+	5.30762i	−0.674446	+	0.178514i
$885$		0			0
$886$		0			0
$887$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$887$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−30.0402	+	44.9583i	−1.00638	+	1.50616i
$892$		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
$901$	3.55216	−	2.70850i	0.118340	−	0.0902331i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.79242	−	1.35110i	0.225539	−	0.0448624i	−0.0810272	−	0.996712i	$-0.525820\pi$
$907$	6.79242	−	1.35110i	0.225539	−	0.0448624i	0.306566	+	0.951849i	$0.400820\pi$
$908$		0			0
$909$	17.2716	−	41.6973i	0.572863	−	1.38301i
$910$		0			0
$911$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$911$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$912$		0			0
$913$	32.7568	−	21.8874i	1.08409	−	0.724368i
$914$		0			0
$915$		0			0
$916$	−10.5016	−	25.3530i	−0.346982	−	0.837688i
$917$		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
$927$	−45.0708	−	18.6689i	−1.48032	−	0.613169i
$928$		0			0
$929$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$929$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$937$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−57.4236	+	11.4223i	−1.87196	+	0.372355i	−0.994272	−	0.106883i	$-0.965913\pi$
$941$	−57.4236	+	11.4223i	−1.87196	+	0.372355i	−0.877685	+	0.479238i	$0.840913\pi$
$942$		0			0
$943$	26.7348	−	64.5435i	0.870605	−	2.10183i
$944$	56.0907	+	23.2335i	1.82560	+	0.756187i
$945$		0			0
$946$		0			0
$947$	50.6735	−	33.8590i	1.64667	−	1.10027i	0.745808	−	0.666160i	$-0.232063\pi$
$947$	50.6735	−	33.8590i	1.64667	−	1.10027i	0.900861	−	0.434109i	$-0.142937\pi$
$948$		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
$961$	−85.7319	−	35.5113i	−2.76554	−	1.14553i
$962$		0			0
$963$	3.52174	+	17.7050i	0.113486	+	0.570535i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	21.2886	+	51.3952i	0.684594	+	1.65276i	0.755397	+	0.655267i	$0.227444\pi$
$967$	21.2886	+	51.3952i	0.684594	+	1.65276i	−0.0708031	+	0.997490i	$0.522556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−56.0432	+	23.2138i	−1.79851	+	0.744968i	−0.811493	+	0.584363i	$0.801345\pi$
$971$	−56.0432	+	23.2138i	−1.79851	+	0.744968i	−0.987019	+	0.160605i	$0.948655\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.0754	+	48.4663i	−0.642268	+	1.55057i	0.181344	+	0.983420i	$0.441955\pi$
$977$	−20.0754	+	48.4663i	−0.642268	+	1.55057i	−0.823612	+	0.567153i	$0.808045\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	12.7441	−	8.51533i	0.406888	−	0.271874i
$982$		0			0
$983$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$983$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−52.3827	−	10.4196i	−1.66567	−	0.331323i
$990$		0			0
$991$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$991$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$997$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$998$		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.343.2	✓	16
17.6	odd	16	inner	731.2.s.a.601.2	yes	16
43.42	odd	2	CM	731.2.s.a.343.2	✓	16
731.601	even	16	inner	731.2.s.a.601.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.343.2	✓	16	1.1	even	1	trivial
731.2.s.a.343.2	✓	16	43.42	odd	2	CM
731.2.s.a.601.2	yes	16	17.6	odd	16	inner
731.2.s.a.601.2	yes	16	731.601	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)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +(1.17208 + 5.89244i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(-3.98585 + 1.05499i) q^{17} +(7.98829 - 1.58897i) q^{23} +(-1.91342 + 4.61940i) q^{25} +(2.17064 - 10.9126i) q^{31} +(5.54328 - 2.29610i) q^{36} +(4.76537 - 7.13187i) q^{41} +(-6.05828 - 2.50942i) q^{43} +(9.99074 + 6.67560i) q^{44} +(0.935734 + 0.935734i) q^{47} +(-2.67878 - 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} +16.3676i q^{67} +(-4.14487 + 7.12882i) q^{68} +(-2.05270 - 10.3196i) q^{79} +(6.36396 + 6.36396i) q^{81} +(-2.50942 - 6.05828i) q^{83} +(9.05001 - 13.5443i) q^{92} +(15.8804 - 10.6110i) q^{97} +(-3.51624 + 17.6773i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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