LMFDB - Embedded newform 731.2.s.a.558.2

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			558.2
Root			$-2.83780 + 1.71665i$ of defining polynomial
Character	$\chi$	$=$	731.558
Dual form			731.2.s.a.300.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +O(q^{10})$	$q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +(2.75762 - 0.548525i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(3.98585 - 1.05499i) q^{17} +(-0.988290 - 4.96847i) q^{23} +(1.91342 - 4.61940i) q^{25} +(-0.443490 - 0.0882157i) q^{31} +(-5.54328 + 2.29610i) q^{36} +(-7.90673 - 5.28311i) q^{41} +(6.05828 + 2.50942i) q^{43} +(3.12414 - 4.67560i) q^{44} +(-0.935734 - 0.935734i) q^{47} +(2.67878 + 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} -16.3676i q^{67} +(4.14487 - 7.12882i) q^{68} +(14.0527 - 2.79526i) q^{79} +(6.36396 + 6.36396i) q^{81} +(2.50942 + 6.05828i) q^{83} +(-8.42413 - 5.62882i) q^{92} +(-2.67711 - 4.00657i) q^{97} +(-8.27287 - 1.64558i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$e\left(\frac{9}{16}\right)$	$-1$

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		0.195090	−	0.980785i	$-0.437500\pi$
$3$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$4$	1.41421	−	1.41421i	0.707107	−	0.707107i
$5$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$5$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$6$		0			0
$7$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$7$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$8$		0			0
$9$	−2.77164	−	1.14805i	−0.923880	−	0.382683i
$10$		0			0
$11$	2.75762	−	0.548525i	0.831455	−	0.165387i	0.239030	−	0.971012i	$-0.423171\pi$
$11$	2.75762	−	0.548525i	0.831455	−	0.165387i	0.592425	+	0.805626i	$0.298171\pi$
$12$		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
$23$	−0.988290	−	4.96847i	−0.206073	−	1.03600i	−0.935873	−	0.352337i	$-0.885387\pi$
$23$	−0.988290	−	4.96847i	−0.206073	−	1.03600i	0.729800	−	0.683660i	$-0.239613\pi$
$24$		0			0
$25$	1.91342	−	4.61940i	0.382683	−	0.923880i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$29$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$30$		0			0
$31$	−0.443490	−	0.0882157i	−0.0796532	−	0.0158440i	0.155103	−	0.987898i	$-0.450429\pi$
$31$	−0.443490	−	0.0882157i	−0.0796532	−	0.0158440i	−0.234756	+	0.972054i	$0.575429\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.54328	+	2.29610i	−0.923880	+	0.382683i
$37$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$37$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.90673	−	5.28311i	−1.23482	−	0.825083i	−0.245299	−	0.969447i	$-0.578886\pi$
$41$	−7.90673	−	5.28311i	−1.23482	−	0.825083i	−0.989525	+	0.144364i	$0.953886\pi$
$42$		0			0
$43$	6.05828	+	2.50942i	0.923880	+	0.382683i
$43$	6.05828	+	2.50942i	0.923880	+	0.382683i
$44$	3.12414	−	4.67560i	0.470981	−	0.704873i
$45$		0			0
$46$		0			0
$47$	−0.935734	−	0.935734i	−0.136491	−	0.136491i	0.635560	−	0.772051i	$-0.280769\pi$
$47$	−0.935734	−	0.935734i	−0.136491	−	0.136491i	−0.772051	+	0.635560i	$0.780769\pi$
$48$		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		0.555570	−	0.831470i	$-0.312500\pi$
$61$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$62$		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	$-0.506184\pi$
$67$		−	16.3676i		−	1.99962i	0.0194269	−	0.999811i	$-0.493816\pi$
$68$	4.14487	−	7.12882i	0.502639	−	0.864496i
$69$		0			0
$70$		0			0
$71$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$71$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$72$		0			0
$73$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$73$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	14.0527	−	2.79526i	1.58105	−	0.314491i	0.675053	−	0.737769i	$-0.264121\pi$
$79$	14.0527	−	2.79526i	1.58105	−	0.314491i	0.906000	+	0.423279i	$0.139121\pi$
$80$		0			0
$81$	6.36396	+	6.36396i	0.707107	+	0.707107i
$82$		0			0
$83$	2.50942	+	6.05828i	0.275445	+	0.664983i	0.999699	−	0.0245507i	$-0.00781552\pi$
$83$	2.50942	+	6.05828i	0.275445	+	0.664983i	−0.724254	+	0.689534i	$0.757816\pi$
$84$		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
$92$	−8.42413	−	5.62882i	−0.878276	−	0.586845i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.67711	−	4.00657i	−0.271819	−	0.406806i	0.670297	−	0.742093i	$-0.266167\pi$
$97$	−2.67711	−	4.00657i	−0.271819	−	0.406806i	−0.942116	+	0.335287i	$0.891167\pi$
$98$		0			0
$99$	−8.27287	−	1.64558i	−0.831455	−	0.165387i
$100$	−3.82683	−	9.23880i	−0.382683	−	0.923880i
$101$			15.0443i			1.49696i	0.663156	+	0.748481i	$0.269217\pi$
$101$			15.0443i			1.49696i	−0.663156	+	0.748481i	$0.730783\pi$
$102$		0			0
$103$	16.2614			1.60229			0.801143	−	0.598473i	$-0.204225\pi$
$103$	16.2614			1.60229			0.801143	+	0.598473i	$0.204225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.4579	+	10.9968i	−1.59105	+	1.06310i	−0.633932	+	0.773389i	$0.718560\pi$
$107$	−16.4579	+	10.9968i	−1.59105	+	1.06310i	−0.957114	+	0.289713i	$0.906440\pi$
$108$		0			0
$109$	16.8339	+	11.2480i	1.61239	+	1.07737i	0.942133	+	0.335239i	$0.108817\pi$
$109$	16.8339	+	11.2480i	1.61239	+	1.07737i	0.670259	+	0.742127i	$0.266183\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$113$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$114$		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
$121$	−2.85907	+	1.18427i	−0.259915	+	0.107661i
$122$		0			0
$123$		0			0
$124$	−0.751946	+	0.502434i	−0.0675267	+	0.0451199i
$125$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$131$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$132$		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
$139$	−4.25892	+	21.4110i	−0.361237	+	1.81606i	0.190086	+	0.981767i	$0.439123\pi$
$139$	−4.25892	+	21.4110i	−0.361237	+	1.81606i	−0.551323	+	0.834292i	$0.685877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.88071	+	3.92936i	0.491770	+	0.328590i
$144$	−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		0.555570	−	0.831470i	$-0.312500\pi$
$163$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$164$	−18.6533	+	3.71036i	−1.45657	+	0.289731i
$165$		0			0
$166$		0			0
$167$	−18.9699	−	3.77335i	−1.46794	−	0.291991i	−0.604564	−	0.796557i	$-0.706653\pi$
$167$	−18.9699	−	3.77335i	−1.46794	−	0.291991i	−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
$173$	−4.03800	+	20.3004i	−0.307003	+	1.54341i	0.451817	+	0.892111i	$0.350776\pi$
$173$	−4.03800	+	20.3004i	−0.307003	+	1.54341i	−0.758820	+	0.651300i	$0.774224\pi$
$174$		0			0
$175$		0			0
$176$	−2.19410	−	11.0305i	−0.165387	−	0.831455i
$177$		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
$181$	−26.3795	+	5.24721i	−1.96077	+	0.390022i	−0.974821	+	0.222988i	$0.928419\pi$
$181$	−26.3795	+	5.24721i	−1.96077	+	0.390022i	−0.985951	+	0.167034i	$0.946581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	10.4128	−	5.09560i	0.761458	−	0.372627i
$188$	−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
$193$	−2.28234	−	11.4741i	−0.164286	−	0.825923i	−0.971751	−	0.236007i	$-0.924161\pi$
$193$	−2.28234	−	11.4741i	−0.164286	−	0.825923i	0.807465	−	0.589915i	$-0.200839\pi$
$194$		0			0
$195$		0			0
$196$	12.9343	+	5.35757i	0.923880	+	0.382683i
$197$	−2.60600	+	3.90016i	−0.185670	+	0.277875i	−0.912614	−	0.408822i	$-0.865940\pi$
$197$	−2.60600	+	3.90016i	−0.185670	+	0.277875i	0.726944	+	0.686696i	$0.240940\pi$
$198$		0			0
$199$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$199$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.96487	+	14.9054i	−0.206073	+	1.03600i
$208$	7.11488	−	7.11488i	0.493328	−	0.493328i
$209$		0			0
$210$		0			0
$211$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$211$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$212$	−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		−0.195090	−	0.980785i	$-0.562500\pi$
$227$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$228$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$233$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$234$		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		0.195090	−	0.980785i	$-0.437500\pi$
$241$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$242$		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
$253$	−5.45066	−	13.1591i	−0.342680	−	0.827303i
$254$		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
$269$	−31.3480	−	6.23550i	−1.91132	−	0.380185i	−0.911784	−	0.410671i	$-0.865295\pi$
$269$	−31.3480	−	6.23550i	−1.91132	−	0.380185i	−0.999535	+	0.0304855i	$0.990295\pi$
$270$		0			0
$271$			31.4395i			1.90981i	0.296909	+	0.954906i	$0.404044\pi$
$271$			31.4395i			1.90981i	−0.296909	+	0.954906i	$0.595956\pi$
$272$	−4.21995	−	15.9434i	−0.255872	−	0.966711i
$273$		0			0
$274$		0			0
$275$	2.74263	−	13.7881i	0.165387	−	0.831455i
$276$		0			0
$277$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$277$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$278$		0			0
$279$	1.12792	+	0.753651i	0.0675267	+	0.0451199i
$280$		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
$283$	30.4844	−	6.06372i	1.81211	−	0.360451i	0.831362	−	0.555732i	$-0.187562\pi$
$283$	30.4844	−	6.06372i	1.81211	−	0.360451i	0.980747	+	0.195281i	$0.0625619\pi$
$284$		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
$299$	7.07962	−	10.5954i	0.409425	−	0.612748i
$300$		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	$-0.185271\pi$
$307$	29.2727			1.67068			0.835340	+	0.549734i	$0.185271\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.9199	−	10.6373i	0.902733	−	0.603187i	−0.0152162	−	0.999884i	$-0.504844\pi$
$311$	15.9199	−	10.6373i	0.902733	−	0.603187i	0.917950	+	0.396697i	$0.129844\pi$
$312$		0			0
$313$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$313$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$314$		0			0
$315$		0			0
$316$	15.9204	−	23.8266i	0.895594	−	1.34035i
$317$	−24.9691	+	4.96667i	−1.40241	+	0.278956i	−0.837608	−	0.546272i	$-0.816046\pi$
$317$	−24.9691	+	4.96667i	−1.40241	+	0.278956i	−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
$337$	−35.9996	−	7.16077i	−1.96103	−	0.390072i	−0.985113	−	0.171909i	$-0.945006\pi$
$337$	−35.9996	−	7.16077i	−1.96103	−	0.390072i	−0.975912	−	0.218163i	$-0.929994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.27137			−0.0688484
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$347$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$348$		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	$-0.207822\pi$
$359$	33.8316	−	14.0135i	1.78556	−	0.739605i	0.991234	−	0.132119i	$-0.0421781\pi$
$360$		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
$367$	−12.5818	+	18.8300i	−0.656765	+	0.982919i	0.342296	+	0.939592i	$0.388796\pi$
$367$	−12.5818	+	18.8300i	−0.656765	+	0.982919i	−0.999061	+	0.0433266i	$0.986204\pi$
$368$	−19.8739	+	3.95316i	−1.03600	+	0.206073i
$369$	15.8493	+	23.7202i	0.825083	+	1.23482i
$370$		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
$379$	−11.8069	+	7.88911i	−0.606479	+	0.405236i	−0.820544	−	0.571583i	$-0.806330\pi$
$379$	−11.8069	+	7.88911i	−0.606479	+	0.405236i	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
$388$	−9.45215	−	1.88015i	−0.479860	−	0.0954501i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$	−9.18085	−	18.7609i	−0.464295	−	0.948781i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−14.0268	+	9.37241i	−0.704873	+	0.470981i
$397$	−7.70930	−	38.7573i	−0.386919	−	1.94517i	−0.320018	−	0.947411i	$-0.603689\pi$
$397$	−7.70930	−	38.7573i	−0.386919	−	1.94517i	−0.0669005	−	0.997760i	$-0.521311\pi$
$398$		0			0
$399$		0			0
$400$	−18.4776	−	7.65367i	−0.923880	−	0.382683i
$401$	−22.2448	+	33.2918i	−1.11085	+	1.66251i	−0.536572	+	0.843854i	$0.680281\pi$
$401$	−22.2448	+	33.2918i	−1.11085	+	1.66251i	−0.574283	+	0.818657i	$0.694719\pi$
$402$		0			0
$403$	−0.631934	−	0.945756i	−0.0314789	−	0.0471114i
$404$	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		0.980785	−	0.195090i	$-0.0625000\pi$
$419$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$420$		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
$428$	−7.72314	+	38.8268i	−0.373312	+	1.87677i
$429$		0			0
$430$		0			0
$431$	0.305758	+	1.53715i	0.0147278	+	0.0740419i	0.987450	−	0.157930i	$-0.0504821\pi$
$431$	0.305758	+	1.53715i	0.0147278	+	0.0740419i	−0.972722	+	0.231972i	$0.925482\pi$
$432$		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
$436$	39.7138	−	7.89957i	1.90195	−	0.378321i
$437$		0			0
$438$		0			0
$439$	19.4539	+	3.86962i	0.928484	+	0.184687i	0.636093	−	0.771612i	$-0.280549\pi$
$439$	19.4539	+	3.86962i	0.928484	+	0.184687i	0.292391	+	0.956299i	$0.405549\pi$
$440$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$449$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$450$		0			0
$451$	−24.7017	−	10.2318i	−1.16316	−	0.481796i
$452$		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	0.261328	−	0.965250i	$-0.415840\pi$
$461$	24.2331	−	10.0377i	1.12865	−	0.467502i	0.867322	+	0.497748i	$0.165840\pi$
$462$		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
$473$	18.0829	+	3.59692i	0.831455	+	0.165387i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.25020			0.148816
$478$		0			0
$479$	8.25782	−	41.5149i	0.377310	−	1.89686i	−0.0611793	−	0.998127i	$-0.519486\pi$
$479$	8.25782	−	41.5149i	0.377310	−	1.89686i	0.438489	−	0.898737i	$-0.355514\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.36853	+	5.71814i	−0.107661	+	0.259915i
$485$		0			0
$486$		0			0
$487$	21.1874	−	4.21443i	0.960092	−	0.190974i	0.309921	−	0.950762i	$-0.399697\pi$
$487$	21.1874	−	4.21443i	0.960092	−	0.190974i	0.650171	+	0.759788i	$0.274697\pi$
$488$		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
$496$	−0.352863	+	1.77396i	−0.0158440	+	0.0796532i
$497$		0			0
$498$		0			0
$499$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$499$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$503$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$504$		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	$0.133916\pi$
$509$			18.4282i			0.816817i	−0.912799	+	0.408408i	$0.866084\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−3.09368	−	2.06713i	−0.136060	−	0.0909122i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$521$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$522$		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
$527$	−1.86075	+	0.116262i	−0.0810556	+	0.00506445i
$528$		0			0
$529$	−2.45974	+	1.01886i	−0.106945	+	0.0442981i
$530$		0			0
$531$	32.1975	−	32.1975i	1.39725	−	1.39725i
$532$		0			0
$533$	−4.66669	−	23.4610i	−0.202137	−	1.01621i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.9345	+	16.3646i	0.470981	+	0.704873i
$540$		0			0
$541$	−37.8536	−	7.52955i	−1.62745	−	0.323720i	−0.704816	−	0.709390i	$-0.748970\pi$
$541$	−37.8536	−	7.52955i	−1.62745	−	0.323720i	−0.922637	+	0.385670i	$0.873970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.0249681	+	0.125523i	−0.00106756	+	0.00536697i	−0.981315	−	0.192406i	$-0.938371\pi$
$547$	−0.0249681	+	0.125523i	−0.00106756	+	0.00536697i	0.980248	+	0.197773i	$0.0633709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	24.2567	+	36.3028i	1.02872	+	1.53958i
$557$	−32.9636	−	32.9636i	−1.39671	−	1.39671i	−0.809260	−	0.587450i	$-0.800132\pi$
$557$	−32.9636	−	32.9636i	−1.39671	−	1.39671i	−0.587450	−	0.809260i	$-0.699868\pi$
$558$		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	$0.462849\pi$
$569$	−15.9388	+	38.4796i	−0.668189	+	1.61315i	−0.784639	+	0.619953i	$0.787151\pi$
$570$		0			0
$571$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$571$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$572$	13.8735	−	2.75962i	0.580082	−	0.115385i
$573$		0			0
$574$		0			0
$575$	−24.8423	−	4.94145i	−1.03600	−	0.206073i
$576$	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
$583$	−2.53277	+	1.69234i	−0.104897	+	0.0700897i
$584$		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		−0.195090	−	0.980785i	$-0.562500\pi$
$601$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$602$		0			0
$603$	−18.7908	+	45.3651i	−0.765222	+	1.84741i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$607$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$608$		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	$-0.302697\pi$
$613$	28.7652			1.16182			0.580909	+	0.813969i	$0.302697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	40.5148	−	27.0712i	1.63107	−	1.08984i	0.707281	−	0.706932i	$-0.249922\pi$
$617$	40.5148	−	27.0712i	1.63107	−	1.08984i	0.923785	−	0.382911i	$-0.125078\pi$
$618$		0			0
$619$	−2.90893	−	1.94369i	−0.116920	−	0.0781233i	0.495735	−	0.868474i	$-0.334899\pi$
$619$	−2.90893	−	1.94369i	−0.116920	−	0.0781233i	−0.612655	+	0.790350i	$0.709899\pi$
$620$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$641$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$642$		0			0
$643$	23.2538	+	4.62547i	0.917040	+	0.182411i	0.630978	−	0.775800i	$-0.282654\pi$
$643$	23.2538	+	4.62547i	0.917040	+	0.182411i	0.286062	+	0.958211i	$0.407654\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−8.32554	+	41.8553i	−0.326806	+	1.64296i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$653$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$654$		0			0
$655$		0			0
$656$	−21.1324	+	31.6269i	−0.825083	+	1.23482i
$657$		0			0
$658$		0			0
$659$	−23.7632	−	23.7632i	−0.925684	−	0.925684i	0.0717399	−	0.997423i	$-0.477145\pi$
$659$	−23.7632	−	23.7632i	−0.925684	−	0.925684i	−0.997423	+	0.0717399i	$0.977145\pi$
$660$		0			0
$661$	2.50942	+	6.05828i	0.0976052	+	0.235640i	0.965139	−	0.261739i	$-0.0842959\pi$
$661$	2.50942	+	6.05828i	0.0976052	+	0.235640i	−0.867533	+	0.497379i	$0.834296\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−32.1638	+	21.4912i	−1.24446	+	0.831519i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$673$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$674$		0			0
$675$		0			0
$676$	−9.43608	−	9.43608i	−0.362926	−	0.362926i
$677$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$677$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.53059	+	32.8315i	−0.249886	+	1.25626i	0.628311	+	0.777962i	$0.283747\pi$
$683$	−6.53059	+	32.8315i	−0.249886	+	1.25626i	−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		0.980785	−	0.195090i	$-0.0625000\pi$
$691$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$692$	22.9985	+	34.4197i	0.874272	+	1.30844i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−37.0887	−	12.7162i	−1.40483	−	0.481660i
$698$		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
$704$	−18.7024	−	12.4965i	−0.704873	−	0.470981i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.6404	+	39.8702i	1.00050	+	1.49736i	0.861944	+	0.507003i	$0.169247\pi$
$709$	26.6404	+	39.8702i	1.00050	+	1.49736i	0.138558	+	0.990354i	$0.455753\pi$
$710$		0			0
$711$	−42.1581	−	8.38577i	−1.58105	−	0.314491i
$712$		0			0
$713$			2.29065i			0.0857855i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.9540	−	9.99195i	0.557690	−	0.372637i	−0.244551	−	0.969636i	$-0.578641\pi$
$719$	14.9540	−	9.99195i	0.557690	−	0.372637i	0.802242	+	0.597000i	$0.203641\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−29.8856	+	44.7269i	−1.11069	+	1.66226i
$725$		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
$737$	−8.97805	−	45.1357i	−0.330711	−	1.66260i
$738$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$743$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	19.6723i		−	0.719772i
$748$	7.51964	−	21.9322i	0.274945	−	0.801919i
$749$		0			0
$750$		0			0
$751$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$751$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$752$	−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	−0.998068	−	0.0621249i	$-0.980212\pi$
$769$	−25.9545	+	25.9545i	−0.935943	+	0.935943i	0.0621249	+	0.998068i	$0.480212\pi$
$770$		0			0
$771$		0			0
$772$	−19.4545	−	12.9991i	−0.700184	−	0.467848i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$		0			0
$775$	−1.25609	+	1.87986i	−0.0451199	+	0.0675267i
$776$		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
$787$	9.47882	−	6.33354i	0.337883	−	0.225766i	−0.375040	−	0.927009i	$-0.622371\pi$
$787$	9.47882	−	6.33354i	0.337883	−	0.225766i	0.712923	+	0.701242i	$0.247371\pi$
$788$	1.83021	+	9.20110i	0.0651986	+	0.327776i
$789$		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
$809$	−26.2004	+	39.2117i	−0.921158	+	1.37861i	0.00439415	+	0.999990i	$0.498601\pi$
$809$	−26.2004	+	39.2117i	−0.921158	+	1.37861i	−0.925552	+	0.378620i	$0.876399\pi$
$810$		0			0
$811$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$811$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$812$		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
$821$	43.6466	−	29.1638i	1.52328	−	1.01782i	0.538776	−	0.842449i	$-0.318887\pi$
$821$	43.6466	−	29.1638i	1.52328	−	1.01782i	0.984502	−	0.175373i	$-0.0561132\pi$
$822$		0			0
$823$	47.5468	+	31.7698i	1.65738	+	1.10742i	0.874089	+	0.485765i	$0.161459\pi$
$823$	47.5468	+	31.7698i	1.65738	+	1.10742i	0.783288	+	0.621659i	$0.213541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−51.3592	+	10.2160i	−1.78593	+	0.355244i	−0.973655	−	0.228024i	$-0.926773\pi$
$827$	−51.3592	+	10.2160i	−1.78593	+	0.355244i	−0.812279	+	0.583269i	$0.801773\pi$
$828$	16.8865	+	25.2724i	0.586845	+	0.878276i
$829$		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		−0.195090	−	0.980785i	$-0.562500\pi$
$839$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$840$		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
$853$	−1.67082	+	8.39979i	−0.0572078	+	0.287603i	−0.998790	−	0.0491804i	$-0.984339\pi$
$853$	−1.67082	+	8.39979i	−0.0572078	+	0.287603i	0.941582	+	0.336784i	$0.109339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−42.2080	−	28.2025i	−1.44180	−	0.963379i	−0.997727	−	0.0673876i	$-0.978534\pi$
$857$	−42.2080	−	28.2025i	−1.44180	−	0.963379i	−0.444072	−	0.895991i	$-0.646466\pi$
$858$		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
$869$	37.2188	−	15.4165i	1.26256	−	0.522970i
$870$		0			0
$871$	29.1134	−	29.1134i	0.986470	−	0.986470i
$872$		0			0
$873$	2.82022	+	14.1782i	0.0954501	+	0.479860i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	32.7000	−	48.9390i	1.10420	−	1.65255i	0.459310	−	0.888276i	$-0.348097\pi$
$877$	32.7000	−	48.9390i	1.10420	−	1.65255i	0.644889	−	0.764276i	$-0.276903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37.4964	−	7.45849i	−1.26328	−	0.251283i	−0.482396	−	0.875953i	$-0.660233\pi$
$881$	−37.4964	−	7.45849i	−1.26328	−	0.251283i	−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		0.195090	−	0.980785i	$-0.437500\pi$
$887$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	21.0402	+	14.0586i	0.704873	+	0.470981i
$892$		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
$907$	−11.6729	−	58.6837i	−0.387593	−	1.94856i	−0.306566	−	0.951849i	$-0.599180\pi$
$907$	−11.6729	−	58.6837i	−0.387593	−	1.94856i	−0.0810272	−	0.996712i	$-0.525820\pi$
$908$		0			0
$909$	17.2716	−	41.6973i	0.572863	−	1.38301i
$910$		0			0
$911$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$911$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$912$		0			0
$913$	10.2432	+	15.3300i	0.338999	+	0.507348i
$914$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$929$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$930$		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
$941$	−3.57639	−	17.9797i	−0.116587	−	0.586122i	−0.994272	−	0.106883i	$-0.965913\pi$
$941$	−3.57639	−	17.9797i	−0.116587	−	0.586122i	0.877685	−	0.479238i	$-0.159087\pi$
$942$		0			0
$943$	−18.4348	+	44.5056i	−0.600320	+	1.44930i
$944$	56.0907	+	23.2335i	1.82560	+	0.756187i
$945$		0			0
$946$		0			0
$947$	−4.77148	−	7.14102i	−0.155052	−	0.232052i	0.745808	−	0.666160i	$-0.232063\pi$
$947$	−4.77148	−	7.14102i	−0.155052	−	0.232052i	−0.900861	+	0.434109i	$0.857063\pi$
$948$		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
$961$	−28.4514	−	11.7849i	−0.917786	−	0.380159i
$962$		0			0
$963$	58.2402	−	11.5847i	1.87677	−	0.373312i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−21.2886	−	51.3952i	−0.684594	−	1.65276i	−0.755397	−	0.655267i	$-0.772556\pi$
$967$	−21.2886	−	51.3952i	−0.684594	−	1.65276i	0.0708031	−	0.997490i	$-0.477444\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	56.0432	−	23.2138i	1.79851	−	0.744968i	0.811493	−	0.584363i	$-0.198655\pi$
$971$	56.0432	−	23.2138i	1.79851	−	0.744968i	0.987019	−	0.160605i	$-0.0513446\pi$
$972$		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	$-0.558045\pi$
$977$	20.0754	−	48.4663i	0.642268	−	1.55057i	0.823612	−	0.567153i	$-0.191955\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−33.7441	−	50.5016i	−1.07737	−	1.61239i
$982$		0			0
$983$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$983$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.48065	−	32.5804i	0.206073	−	1.03600i
$990$		0			0
$991$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$991$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$997$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$998$		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.558.2	yes	16
17.11	odd	16	inner	731.2.s.a.300.2	✓	16
43.42	odd	2	CM	731.2.s.a.558.2	yes	16
731.300	even	16	inner	731.2.s.a.300.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.300.2	✓	16	17.11	odd	16	inner
731.2.s.a.300.2	✓	16	731.300	even	16	inner
731.2.s.a.558.2	yes	16	1.1	even	1	trivial
731.2.s.a.558.2	yes	16	43.42	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.41421i) q^{4} +(-2.77164 - 1.14805i) q^{9} +(2.75762 - 0.548525i) q^{11} +(1.77872 + 1.77872i) q^{13} -4.00000i q^{16} +(3.98585 - 1.05499i) q^{17} +(-0.988290 - 4.96847i) q^{23} +(1.91342 - 4.61940i) q^{25} +(-0.443490 - 0.0882157i) q^{31} +(-5.54328 + 2.29610i) q^{36} +(-7.90673 - 5.28311i) q^{41} +(6.05828 + 2.50942i) q^{43} +(3.12414 - 4.67560i) q^{44} +(-0.935734 - 0.935734i) q^{47} +(2.67878 + 6.46716i) q^{49} +5.03098 q^{52} +(-1.00093 + 0.414599i) q^{53} +(-5.80838 + 14.0227i) q^{59} +(-5.65685 - 5.65685i) q^{64} -16.3676i q^{67} +(4.14487 - 7.12882i) q^{68} +(14.0527 - 2.79526i) q^{79} +(6.36396 + 6.36396i) q^{81} +(2.50942 + 6.05828i) q^{83} +(-8.42413 - 5.62882i) q^{92} +(-2.67711 - 4.00657i) q^{97} +(-8.27287 - 1.64558i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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