LMFDB - Embedded newform 731.2.s.a.386.1

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			386.1
Root			$-0.792772 - 3.22048i$ of defining polynomial
Character	$\chi$	$=$	731.386
Dual form			731.2.s.a.214.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +(-0.165842 + 0.248200i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(3.56441 - 2.07243i) q^{17} +(7.86853 + 5.25759i) q^{23} +(4.61940 + 1.91342i) q^{25} +(-2.59764 - 3.88764i) q^{31} +(2.29610 + 5.54328i) q^{36} +(11.3414 - 2.25594i) q^{41} +(2.50942 - 6.05828i) q^{43} +(-0.116472 - 0.585544i) q^{44} +(-7.48531 - 7.48531i) q^{47} +(6.46716 - 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 - 8.77779i) q^{53} +(-11.4652 - 4.74906i) q^{59} +(5.65685 + 5.65685i) q^{64} +11.7985i q^{67} +(-2.10997 + 7.97170i) q^{68} +(-2.35438 + 3.52358i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(-6.05828 + 2.50942i) q^{83} +(-18.5631 + 3.69244i) q^{92} +(3.08269 - 15.4977i) q^{97} +(0.497526 + 0.744600i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$e\left(\frac{13}{16}\right)$	$-1$

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	$-0.625000\pi$
$2$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$3$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$3$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$4$	−1.41421	+	1.41421i	−0.707107	+	0.707107i
$5$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$5$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$6$		0			0
$7$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$7$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$8$		0			0
$9$	1.14805	−	2.77164i	0.382683	−	0.923880i
$10$		0			0
$11$	−0.165842	+	0.248200i	−0.0500032	+	0.0748351i	−0.855629	−	0.517590i	$-0.826829\pi$
$11$	−0.165842	+	0.248200i	−0.0500032	+	0.0748351i	0.805626	+	0.592425i	$0.201829\pi$
$12$		0			0
$13$	−4.77872	−	4.77872i	−1.32538	−	1.32538i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	−4.77872	−	4.77872i	−1.32538	−	1.32538i	−0.416025	−	0.909353i	$-0.636577\pi$
$14$		0			0
$15$		0			0
$16$		−	4.00000i		−	1.00000i
$17$	3.56441	−	2.07243i	0.864496	−	0.502639i
$17$	3.56441	−	2.07243i	0.864496	−	0.502639i
$18$		0			0
$19$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$19$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.86853	+	5.25759i	1.64070	+	1.09628i	0.910902	+	0.412622i	$0.135387\pi$
$23$	7.86853	+	5.25759i	1.64070	+	1.09628i	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	$-0.437500\pi$
$29$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$30$		0			0
$31$	−2.59764	−	3.88764i	−0.466550	−	0.698241i	0.521349	−	0.853344i	$-0.325429\pi$
$31$	−2.59764	−	3.88764i	−0.466550	−	0.698241i	−0.987898	+	0.155103i	$0.950429\pi$
$32$		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	$-0.187500\pi$
$37$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.3414	−	2.25594i	1.77123	−	0.352319i	0.801781	−	0.597619i	$-0.203886\pi$
$41$	11.3414	−	2.25594i	1.77123	−	0.352319i	0.969447	+	0.245299i	$0.0788863\pi$
$42$		0			0
$43$	2.50942	−	6.05828i	0.382683	−	0.923880i
$43$	2.50942	−	6.05828i	0.382683	−	0.923880i
$44$	−0.116472	−	0.585544i	−0.0175588	−	0.0882740i
$45$		0			0
$46$		0			0
$47$	−7.48531	−	7.48531i	−1.09185	−	1.09185i	−0.995332	−	0.0965136i	$-0.969231\pi$
$47$	−7.48531	−	7.48531i	−1.09185	−	1.09185i	−0.0965136	−	0.995332i	$-0.530769\pi$
$48$		0			0
$49$	6.46716	−	2.67878i	0.923880	−	0.382683i
$50$		0			0
$51$		0			0
$52$	13.5163			1.87437
$53$	−3.63588	−	8.77779i	−0.499426	−	1.20572i	−0.949793	−	0.312878i	$-0.898707\pi$
$53$	−3.63588	−	8.77779i	−0.499426	−	1.20572i	0.450367	−	0.892844i	$-0.351293\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.4652	−	4.74906i	−1.49265	−	0.618274i	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−11.4652	−	4.74906i	−1.49265	−	0.618274i	−0.971891	+	0.235431i	$0.924350\pi$
$60$		0			0
$61$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$61$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$62$		0			0
$63$		0			0
$64$	5.65685	+	5.65685i	0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$			11.7985i			1.44142i	0.693236	+	0.720710i	$0.256184\pi$
$67$			11.7985i			1.44142i	−0.693236	+	0.720710i	$0.743816\pi$
$68$	−2.10997	+	7.97170i	−0.255872	+	0.966711i
$69$		0			0
$70$		0			0
$71$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$71$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$72$		0			0
$73$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$73$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.35438	+	3.52358i	−0.264889	+	0.396434i	−0.939942	−	0.341335i	$-0.889121\pi$
$79$	−2.35438	+	3.52358i	−0.264889	+	0.396434i	0.675053	+	0.737769i	$0.264121\pi$
$80$		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	$-0.742184\pi$
$83$	−6.05828	+	2.50942i	−0.664983	+	0.275445i	0.0245507	+	0.999699i	$0.492184\pi$
$84$		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
$92$	−18.5631	+	3.69244i	−1.93534	+	0.384963i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.08269	−	15.4977i	0.313000	−	1.57356i	−0.429093	−	0.903260i	$-0.641167\pi$
$97$	3.08269	−	15.4977i	0.313000	−	1.57356i	0.742093	−	0.670297i	$-0.233833\pi$
$98$		0			0
$99$	0.497526	+	0.744600i	0.0500032	+	0.0748351i
$100$	−9.23880	+	3.82683i	−0.923880	+	0.382683i
$101$		−	1.21270i		−	0.120668i	−0.998178	−	0.0603342i	$-0.980783\pi$
$101$		−	1.21270i		−	0.120668i	0.998178	−	0.0603342i	$-0.0192166\pi$
$102$		0			0
$103$	2.90887			0.286620			0.143310	−	0.989678i	$-0.454225\pi$
$103$	2.90887			0.286620			0.143310	+	0.989678i	$0.454225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.6772	+	3.11839i	1.51557	+	0.301466i	0.881640	−	0.471923i	$-0.156440\pi$
$107$	15.6772	+	3.11839i	1.51557	+	0.301466i	0.633932	+	0.773389i	$0.281440\pi$
$108$		0			0
$109$	−20.2630	+	4.03056i	−1.94084	+	0.386057i	−0.942133	+	0.335239i	$0.891183\pi$
$109$	−20.2630	+	4.03056i	−1.94084	+	0.386057i	−0.998708	+	0.0508181i	$0.983817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$113$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−18.7311	+	7.75867i	−1.73169	+	0.717290i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.17542	+	10.0804i	0.379583	+	0.916396i
$122$		0			0
$123$		0			0
$124$	9.17157	+	1.82434i	0.823631	+	0.163830i
$125$		0			0
$126$		0			0
$127$	20.6805	+	8.56613i	1.83509	+	0.760121i	0.962276	+	0.272075i	$0.0877098\pi$
$127$	20.6805	+	8.56613i	1.83509	+	0.760121i	0.872818	+	0.488046i	$0.162290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$131$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$132$		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
$139$	0.0999825	−	0.0668062i	0.00848041	−	0.00566643i	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	0.0999825	−	0.0668062i	0.00848041	−	0.00566643i	0.559803	+	0.828626i	$0.310877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.97859	−	0.393566i	0.165458	−	0.0329116i
$144$	−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	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$	−1.65191	−	12.2585i	−0.133549	−	0.991042i
$154$		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		−0.195090	−	0.980785i	$-0.562500\pi$
$163$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$164$	−12.8488	+	19.2295i	−1.00332	+	1.50157i
$165$		0			0
$166$		0			0
$167$	−6.28278	−	9.40284i	−0.486176	−	0.727613i	0.504566	−	0.863373i	$-0.331653\pi$
$167$	−6.28278	−	9.40284i	−0.486176	−	0.727613i	−0.990742	+	0.135760i	$0.956653\pi$
$168$		0			0
$169$			32.6723i			2.51326i
$170$		0			0
$171$		0			0
$172$	5.01885	+	12.1166i	0.382683	+	0.923880i
$173$	−21.0658	+	14.0757i	−1.60160	+	1.07016i	−0.651300	+	0.758820i	$0.725776\pi$
$173$	−21.0658	+	14.0757i	−1.60160	+	1.07016i	−0.950300	+	0.311335i	$0.899224\pi$
$174$		0			0
$175$		0			0
$176$	0.992800	+	0.663368i	0.0748351	+	0.0500032i
$177$		0			0
$178$		0			0
$179$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$179$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$180$		0			0
$181$	5.32439	−	7.96852i	0.395759	−	0.592295i	−0.579062	−	0.815283i	$-0.696581\pi$
$181$	5.32439	−	7.96852i	0.395759	−	0.592295i	0.974821	+	0.222988i	$0.0715812\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.0767508	+	1.22838i	−0.00561258	+	0.0898282i
$188$	21.1717			1.54410
$189$		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
$193$	−15.6371	−	10.4484i	−1.12558	−	0.752090i	−0.153831	−	0.988097i	$-0.549161\pi$
$193$	−15.6371	−	10.4484i	−1.12558	−	0.752090i	−0.971751	+	0.236007i	$0.924161\pi$
$194$		0			0
$195$		0			0
$196$	−5.35757	+	12.9343i	−0.382683	+	0.923880i
$197$	−5.33864	−	26.8391i	−0.380362	−	1.91221i	−0.408822	−	0.912614i	$-0.634060\pi$
$197$	−5.33864	−	26.8391i	−0.380362	−	1.91221i	0.0284595	−	0.999595i	$-0.490940\pi$
$198$		0			0
$199$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$199$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	23.6056	−	15.7728i	1.64070	−	1.09628i
$208$	−19.1149	+	19.1149i	−1.32538	+	1.32538i
$209$		0			0
$210$		0			0
$211$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$211$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$212$	17.5556	+	7.27176i	1.20572	+	0.499426i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−26.9369	−	7.12974i	−1.81197	−	0.479598i
$222$		0			0
$223$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$223$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$224$		0			0
$225$	10.6066	−	10.6066i	0.707107	−	0.707107i
$226$		0			0
$227$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$227$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$228$		0			0
$229$	8.65964	+	3.58694i	0.572245	+	0.237032i	0.649992	−	0.759941i	$-0.274772\pi$
$229$	8.65964	+	3.58694i	0.572245	+	0.237032i	−0.0777462	+	0.996973i	$0.524772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$233$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$234$		0			0
$235$		0			0
$236$	22.9305	−	9.49811i	1.49265	−	0.618274i
$237$		0			0
$238$		0			0
$239$	−10.5254			−0.680830			−0.340415	−	0.940275i	$-0.610568\pi$
$239$	−10.5254			−0.680830			−0.340415	+	0.940275i	$0.610568\pi$
$240$		0			0
$241$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$241$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$242$		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
$251$	12.2213	+	12.2213i	0.771400	+	0.771400i	0.978351	−	0.206951i	$-0.0663540\pi$
$251$	12.2213	+	12.2213i	0.771400	+	0.771400i	−0.206951	+	0.978351i	$0.566354\pi$
$252$		0			0
$253$	−2.60986	+	1.08104i	−0.164081	+	0.0679645i
$254$		0			0
$255$		0			0
$256$	−16.0000			−1.00000
$257$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$257$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−16.6856	−	16.6856i	−1.01924	−	1.01924i
$269$	10.5820	+	15.8370i	0.645195	+	0.965602i	0.999535	+	0.0304855i	$0.00970535\pi$
$269$	10.5820	+	15.8370i	0.645195	+	0.965602i	−0.354341	+	0.935116i	$0.615295\pi$
$270$		0			0
$271$			15.3188i			0.930548i	0.885167	+	0.465274i	$0.154044\pi$
$271$			15.3188i			0.930548i	−0.885167	+	0.465274i	$0.845956\pi$
$272$	−8.28973	−	14.2576i	−0.502639	−	0.864496i
$273$		0			0
$274$		0			0
$275$	−1.24100	+	0.829209i	−0.0748351	+	0.0500032i
$276$		0			0
$277$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$277$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$278$		0			0
$279$	−13.7574	+	2.73651i	−0.823631	+	0.163830i
$280$		0			0
$281$	−12.6171	+	30.4605i	−0.752676	+	1.81712i	−0.208792	+	0.977960i	$0.566953\pi$
$281$	−12.6171	+	30.4605i	−0.752676	+	1.81712i	−0.543884	+	0.839161i	$0.683047\pi$
$282$		0			0
$283$	−18.6923	+	27.9750i	−1.11114	+	1.66294i	−0.555409	+	0.831578i	$0.687438\pi$
$283$	−18.6923	+	27.9750i	−1.11114	+	1.66294i	−0.555732	+	0.831362i	$0.687562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.41004	−	14.7740i	0.494708	−	0.869059i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.11488	+	2.11488i	−0.123552	+	0.123552i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	−2.11488	+	2.11488i	−0.123552	+	0.123552i	0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.4770	−	62.7260i	−0.721563	−	3.62754i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	34.3208			1.95879			0.979395	−	0.201954i	$-0.0647291\pi$
$307$	34.3208			1.95879			0.979395	+	0.201954i	$0.0647291\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	19.6541	+	3.90944i	1.11448	+	0.221684i	0.717784	−	0.696265i	$-0.245156\pi$
$311$	19.6541	+	3.90944i	1.11448	+	0.221684i	0.396697	+	0.917950i	$0.370156\pi$
$312$		0			0
$313$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$313$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$314$		0			0
$315$		0			0
$316$	−1.65350	−	8.31270i	−0.0930166	−	0.467626i
$317$	18.3606	−	27.4786i	1.03123	−	1.54335i	0.206005	−	0.978551i	$-0.433954\pi$
$317$	18.3606	−	27.4786i	1.03123	−	1.54335i	0.825228	−	0.564799i	$-0.191046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	−12.9311	−	31.2185i	−0.717290	−	1.73169i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$331$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$332$	5.01885	−	12.1166i	0.275445	−	0.664983i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	19.0240	+	28.4714i	1.03630	+	1.55093i	0.818138	+	0.575022i	$0.195006\pi$
$337$	19.0240	+	28.4714i	1.03630	+	1.55093i	0.218163	+	0.975912i	$0.429994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.39571			0.0755819
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$347$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$348$		0			0
$349$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$349$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−21.7787	−	21.7787i	−1.15916	−	1.15916i	−0.984656	−	0.174509i	$-0.944166\pi$
$353$	−21.7787	−	21.7787i	−1.15916	−	1.15916i	−0.174509	−	0.984656i	$-0.555834\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.73079	−	9.00692i	−0.196903	−	0.475367i	0.794330	−	0.607486i	$-0.207822\pi$
$359$	−3.73079	−	9.00692i	−0.196903	−	0.475367i	−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
$367$	6.38915	+	32.1204i	0.333511	+	1.67667i	0.675806	+	0.737079i	$0.263796\pi$
$367$	6.38915	+	32.1204i	0.333511	+	1.67667i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	21.0303	−	31.4741i	1.09628	−	1.64070i
$369$	6.76783	−	34.0242i	0.352319	−	1.77123i
$370$		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
$379$	38.1805	+	7.59458i	1.96120	+	0.390107i	0.984383	+	0.176042i	$0.0563295\pi$
$379$	38.1805	+	7.59458i	1.96120	+	0.390107i	0.976819	+	0.214065i	$0.0686705\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$383$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.9104	−	13.9104i	−0.707107	−	0.707107i
$388$	17.5575	+	26.2767i	0.891349	+	1.33400i
$389$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$389$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$390$		0			0
$391$	38.9427	+	2.43319i	1.96942	+	0.123051i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.75663	−	0.349416i	−0.0882740	−	0.0175588i
$397$	28.7196	+	19.1898i	1.44139	+	0.963109i	0.997760	+	0.0669005i	$0.0213110\pi$
$397$	28.7196	+	19.1898i	1.44139	+	0.963109i	0.443634	+	0.896208i	$0.353689\pi$
$398$		0			0
$399$		0			0
$400$	7.65367	−	18.4776i	0.382683	−	0.923880i
$401$	−7.14894	−	35.9402i	−0.357001	−	1.79477i	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−7.14894	−	35.9402i	−0.357001	−	1.79477i	0.217281	−	0.976109i	$-0.430281\pi$
$402$		0			0
$403$	−6.16456	+	30.9913i	−0.307079	+	1.54379i
$404$	1.71502	+	1.71502i	0.0853254	+	0.0853254i
$405$		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
$412$	−4.11377	+	4.11377i	−0.202671	+	0.202671i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$419$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$420$		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
$423$	−29.3401	+	12.1531i	−1.42656	+	0.590902i
$424$		0			0
$425$	20.4309	−	2.75319i	0.991042	−	0.133549i
$426$		0			0
$427$		0			0
$428$	−26.5810	+	17.7608i	−1.28484	+	0.858502i
$429$		0			0
$430$		0			0
$431$	31.3742	+	20.9635i	1.51124	+	1.00978i	0.987450	+	0.157930i	$0.0504821\pi$
$431$	31.3742	+	20.9635i	1.51124	+	1.00978i	0.523789	+	0.851848i	$0.324518\pi$
$432$		0			0
$433$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$433$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$434$		0			0
$435$		0			0
$436$	22.9561	−	34.3562i	1.09940	−	1.64537i
$437$		0			0
$438$		0			0
$439$	−18.0289	−	26.9822i	−0.860473	−	1.28779i	−0.956299	−	0.292391i	$-0.905549\pi$
$439$	−18.0289	−	26.9822i	−0.860473	−	1.28779i	0.0958262	−	0.995398i	$-0.469451\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$	37.5579			1.78443			0.892215	−	0.451612i	$-0.149151\pi$
$443$	37.5579			1.78443			0.892215	+	0.451612i	$0.149151\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$449$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$450$		0			0
$451$	−1.32095	+	3.18906i	−0.0622012	+	0.150167i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$457$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.0377	+	24.2331i	0.467502	+	1.12865i	0.965250	+	0.261328i	$0.0841604\pi$
$461$	10.0377	+	24.2331i	0.467502	+	1.12865i	−0.497748	+	0.867322i	$0.665840\pi$
$462$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$467$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$468$	15.5173	−	37.4622i	0.717290	−	1.73169i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.08750	+	1.62756i	0.0500032	+	0.0748351i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−28.5030			−1.30506
$478$		0			0
$479$	−28.9678	+	19.3556i	−1.32357	+	0.884382i	−0.998127	−	0.0611793i	$-0.980514\pi$
$479$	−28.9678	+	19.3556i	−1.32357	+	0.884382i	−0.325444	+	0.945561i	$0.605514\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−20.1607	−	8.35084i	−0.916396	−	0.379583i
$485$		0			0
$486$		0			0
$487$	19.2710	−	28.8410i	0.873251	−	1.30691i	−0.0775113	−	0.996991i	$-0.524697\pi$
$487$	19.2710	−	28.8410i	0.873251	−	1.30691i	0.950762	−	0.309921i	$-0.100303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$491$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−15.5506	+	10.3906i	−0.698241	+	0.466550i
$497$		0			0
$498$		0			0
$499$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$499$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$503$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−41.3609	+	17.1323i	−1.83509	+	0.760121i
$509$		−	16.0932i		−	0.713316i	−0.934235	−	0.356658i	$-0.883916\pi$
$509$		−	16.0932i		−	0.713316i	0.934235	−	0.356658i	$-0.116084\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.09923	−	0.616476i	0.136304	−	0.0271126i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$521$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$522$		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
$527$	−17.3159	−	8.47372i	−0.754294	−	0.369121i
$528$		0			0
$529$	25.4699	+	61.4898i	1.10739	+	2.67347i
$530$		0			0
$531$	−26.3253	+	26.3253i	−1.14242	+	1.14242i
$532$		0			0
$533$	−64.9778	−	43.4168i	−2.81450	−	1.88059i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.407651	+	2.04940i	−0.0175588	+	0.0882740i
$540$		0			0
$541$	−5.12397	−	7.66857i	−0.220297	−	0.329698i	0.704816	−	0.709390i	$-0.251030\pi$
$541$	−5.12397	−	7.66857i	−0.220297	−	0.329698i	−0.925113	+	0.379693i	$0.876030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	35.8915	−	23.9819i	1.53461	−	1.02539i	0.553293	−	0.832987i	$-0.313371\pi$
$547$	35.8915	−	23.9819i	1.53461	−	1.02539i	0.981315	−	0.192406i	$-0.0616291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.0469184	+	0.235875i	−0.00198978	+	0.0100033i
$557$	19.6071	+	19.6071i	0.830780	+	0.830780i	0.987623	−	0.156844i	$-0.0501319\pi$
$557$	19.6071	+	19.6071i	0.830780	+	0.830780i	−0.156844	+	0.987623i	$0.550132\pi$
$558$		0			0
$559$	−40.9427	+	16.9590i	−1.73169	+	0.717290i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.6567	−	37.7986i	−0.659851	−	1.59302i	−0.798033	−	0.602614i	$-0.794126\pi$
$563$	−15.6567	−	37.7986i	−0.659851	−	1.59302i	0.138182	−	0.990407i	$-0.455874\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.4943	−	8.90324i	−0.901088	−	0.373243i	−0.116450	−	0.993197i	$-0.537151\pi$
$569$	−21.4943	−	8.90324i	−0.901088	−	0.373243i	−0.784639	+	0.619953i	$0.787151\pi$
$570$		0			0
$571$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$571$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$572$	−2.24156	+	3.35473i	−0.0937244	+	0.140268i
$573$		0			0
$574$		0			0
$575$	26.2879	+	39.3427i	1.09628	+	1.64070i
$576$	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
$583$	2.78163	+	0.553300i	0.115203	+	0.0229154i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$593$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$594$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$601$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$602$		0			0
$603$	32.7013	+	13.5453i	1.33170	+	0.551608i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$607$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			71.5404i			2.89422i
$612$	19.6723	+	15.0000i	0.795206	+	0.606339i
$613$	8.16043			0.329597			0.164798	−	0.986327i	$-0.447303\pi$
$613$	8.16043			0.329597			0.164798	+	0.986327i	$0.447303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.51744	+	1.89314i	0.383158	+	0.0762148i	0.382911	−	0.923785i	$-0.374922\pi$
$617$	9.51744	+	1.89314i	0.383158	+	0.0762148i	0.000246592		1.00000i	$0.499922\pi$
$618$		0			0
$619$	−46.2899	+	9.20763i	−1.86055	+	0.370086i	−0.992075	−	0.125649i	$-0.959899\pi$
$619$	−46.2899	+	9.20763i	−1.86055	+	0.370086i	−0.868474	+	0.495735i	$0.834899\pi$
$620$		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	$-0.625000\pi$
$631$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−43.7059	−	18.1036i	−1.73169	−	0.717290i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$641$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$642$		0			0
$643$	28.0519	+	41.9827i	1.10626	+	1.65563i	0.630978	+	0.775800i	$0.282654\pi$
$643$	28.0519	+	41.9827i	1.10626	+	1.65563i	0.475281	+	0.879834i	$0.342346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	3.08013	−	2.05808i	0.120906	−	0.0807866i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$653$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$654$		0			0
$655$		0			0
$656$	−9.02377	−	45.3656i	−0.352319	−	1.77123i
$657$		0			0
$658$		0			0
$659$	−2.60446	−	2.60446i	−0.101455	−	0.101455i	0.654557	−	0.756013i	$-0.272855\pi$
$659$	−2.60446	−	2.60446i	−0.101455	−	0.101455i	−0.756013	+	0.654557i	$0.772855\pi$
$660$		0			0
$661$	−6.05828	+	2.50942i	−0.235640	+	0.0976052i	−0.497379	−	0.867533i	$-0.665704\pi$
$661$	−6.05828	+	2.50942i	−0.235640	+	0.0976052i	0.261739	+	0.965139i	$0.415704\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	22.1828	+	4.41243i	0.858279	+	0.170722i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$673$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$674$		0			0
$675$		0			0
$676$	−46.2056	−	46.2056i	−1.77714	−	1.77714i
$677$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$677$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.1855	−	13.4875i	0.772378	−	0.516086i	−0.105819	−	0.994385i	$-0.533747\pi$
$683$	20.1855	−	13.4875i	0.772378	−	0.516086i	0.878197	+	0.478299i	$0.158747\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.2331	−	10.0377i	−0.923880	−	0.382683i
$689$	−24.5717	+	59.3214i	−0.936109	+	2.25997i
$690$		0			0
$691$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$691$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$692$	9.88545	−	49.6975i	0.375788	−	1.88922i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	35.7501	−	31.5454i	1.35413	−	1.19487i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−36.1149	+	36.1149i	−1.36404	+	1.36404i	−0.495342	+	0.868698i	$0.664957\pi$
$701$	−36.1149	+	36.1149i	−1.36404	+	1.36404i	−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$		0			0
$704$	−2.34217	+	0.465887i	−0.0882740	+	0.0175588i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.91327	+	34.7554i	−0.259633	+	1.30526i	0.602311	+	0.798262i	$0.294247\pi$
$709$	−6.91327	+	34.7554i	−0.259633	+	1.30526i	−0.861944	+	0.507003i	$0.830753\pi$
$710$		0			0
$711$	7.06315	+	10.5708i	0.264889	+	0.396434i
$712$		0			0
$713$		−	44.2473i		−	1.65708i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.66595	+	0.530291i	0.0994233	+	0.0197765i	0.244551	−	0.969636i	$-0.421359\pi$
$719$	2.66595	+	0.530291i	0.0994233	+	0.0197765i	−0.145128	+	0.989413i	$0.546359\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	3.73936	+	18.7990i	0.138972	+	0.698660i
$725$		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
$729$	−24.9447	+	10.3325i	−0.923880	+	0.382683i
$730$		0			0
$731$	−3.61077	−	26.7948i	−0.133549	−	0.991042i
$732$		0			0
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.92840	−	1.95669i	−0.107869	−	0.0720756i
$738$		0			0
$739$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$739$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$743$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			19.6723i			0.719772i
$748$	−1.62865	−	1.84574i	−0.0595495	−	0.0674868i
$749$		0			0
$750$		0			0
$751$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$751$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$752$	−29.9413	+	29.9413i	−1.09185	+	1.09185i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$		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
$767$	32.0947	+	77.4836i	1.15887	+	2.79777i
$768$		0			0
$769$	−39.1416	+	39.1416i	−1.41148	+	1.41148i	−0.661812	+	0.749670i	$0.730212\pi$
$769$	−39.1416	+	39.1416i	−1.41148	+	1.41148i	−0.749670	+	0.661812i	$0.769788\pi$
$770$		0			0
$771$		0			0
$772$	36.8904	−	7.33796i	1.32771	−	0.264099i
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$		0			0
$775$	−4.56085	−	22.9289i	−0.163830	−	0.823631i
$776$		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
$787$	30.9493	+	6.15620i	1.10322	+	0.219445i	0.712923	−	0.701242i	$-0.247371\pi$
$787$	30.9493	+	6.15620i	1.10322	+	0.219445i	0.390301	+	0.920687i	$0.372371\pi$
$788$	45.5062	+	30.4063i	1.62109	+	1.08318i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	51.9513	−	21.5189i	1.84021	−	0.762240i	0.885545	−	0.464553i	$-0.153785\pi$
$797$	51.9513	−	21.5189i	1.84021	−	0.762240i	0.954664	−	0.297687i	$-0.0962151\pi$
$798$		0			0
$799$	−42.1935	−	11.1679i	−1.49270	−	0.395093i
$800$		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
$809$	9.25455	+	46.5258i	0.325373	+	1.63576i	0.703993	+	0.710207i	$0.251399\pi$
$809$	9.25455	+	46.5258i	0.325373	+	1.63576i	−0.378620	+	0.925552i	$0.623601\pi$
$810$		0			0
$811$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$811$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$812$		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
$821$	1.12766	+	0.224306i	0.0393556	+	0.00782832i	0.214729	−	0.976674i	$-0.431113\pi$
$821$	1.12766	+	0.224306i	0.0393556	+	0.00782832i	−0.175373	+	0.984502i	$0.556113\pi$
$822$		0			0
$823$	−25.7115	+	5.11433i	−0.896246	+	0.178274i	−0.621659	−	0.783288i	$-0.713541\pi$
$823$	−25.7115	+	5.11433i	−0.896246	+	0.178274i	−0.274587	+	0.961562i	$0.588541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−23.3431	+	34.9355i	−0.811721	+	1.21483i	0.161935	+	0.986801i	$0.448227\pi$
$827$	−23.3431	+	34.9355i	−0.811721	+	1.21483i	−0.973655	+	0.228024i	$0.926773\pi$
$828$	−11.0773	+	55.6894i	−0.384963	+	1.93534i
$829$		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
$832$		−	54.0650i		−	1.87437i
$833$	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	$-0.812500\pi$
$839$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$840$		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
$848$	−35.1112	+	14.5435i	−1.20572	+	0.499426i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	47.1112	−	31.4787i	1.61306	−	1.07781i	0.671475	−	0.741027i	$-0.265661\pi$
$853$	47.1112	−	31.4787i	1.61306	−	1.07781i	0.941582	−	0.336784i	$-0.109339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−35.0481	+	6.97151i	−1.19722	+	0.238142i	−0.753150	−	0.657849i	$-0.771466\pi$
$857$	−35.0481	+	6.97151i	−1.19722	+	0.238142i	−0.444072	+	0.895991i	$0.646466\pi$
$858$		0			0
$859$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$859$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$860$		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
$869$	−0.484098	−	1.16872i	−0.0164219	−	0.0396460i
$870$		0			0
$871$	56.3819	−	56.3819i	1.91043	−	1.91043i
$872$		0			0
$873$	−39.4151	−	26.3363i	−1.33400	−	0.891349i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.58557	−	28.0806i	−0.188611	−	0.948214i	−0.952887	−	0.303324i	$-0.901903\pi$
$877$	−5.58557	−	28.0806i	−0.188611	−	0.948214i	0.764276	−	0.644889i	$-0.223097\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.96791	−	14.9180i	−0.335827	−	0.502601i	0.624670	−	0.780889i	$-0.285233\pi$
$881$	−9.96791	−	14.9180i	−0.335827	−	0.502601i	−0.960498	+	0.278287i	$0.910233\pi$
$882$		0			0
$883$			36.7815i			1.23780i	0.785471	+	0.618899i	$0.212421\pi$
$883$			36.7815i			1.23780i	−0.785471	+	0.618899i	$0.787579\pi$
$884$	48.1775	−	28.0115i	1.62038	−	0.942130i
$885$		0			0
$886$		0			0
$887$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$887$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.63495	−	0.524123i	0.0882740	−	0.0175588i
$892$		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
$901$	−31.1511	−	23.7525i	−1.03780	−	0.791311i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−43.7591	−	29.2389i	−1.45300	−	0.970861i	−0.996712	−	0.0810272i	$-0.974180\pi$
$907$	−43.7591	−	29.2389i	−1.45300	−	0.970861i	−0.456284	−	0.889834i	$-0.650820\pi$
$908$		0			0
$909$	−3.36117	−	1.39224i	−0.111483	−	0.0461778i
$910$		0			0
$911$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$911$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$912$		0			0
$913$	0.381879	−	1.91983i	0.0126383	−	0.0635372i
$914$		0			0
$915$		0			0
$916$	−17.3193	+	7.17388i	−0.572245	+	0.237032i
$917$		0			0
$918$		0			0
$919$	−32.7872			−1.08155			−0.540775	−	0.841167i	$-0.681869\pi$
$919$	−32.7872			−1.08155			−0.540775	+	0.841167i	$0.681869\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	3.33953	−	8.06234i	0.109685	−	0.264802i
$928$		0			0
$929$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$929$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$930$		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	$-0.625000\pi$
$937$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−39.1427	−	26.1543i	−1.27602	−	0.852606i	−0.281744	−	0.959490i	$-0.590913\pi$
$941$	−39.1427	−	26.1543i	−1.27602	−	0.852606i	−0.994272	+	0.106883i	$0.965913\pi$
$942$		0			0
$943$	101.101	+	41.8774i	3.29230	+	1.36372i
$944$	−18.9962	+	45.8609i	−0.618274	+	1.49265i
$945$		0			0
$946$		0			0
$947$	6.09797	−	30.6565i	0.198157	−	0.996204i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	6.09797	−	30.6565i	0.198157	−	0.996204i	0.943966	−	0.330044i	$-0.107063\pi$
$948$		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
$956$	14.8851	−	14.8851i	0.481419	−	0.481419i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.49716	−	8.44289i	0.112812	−	0.272351i
$962$		0			0
$963$	26.6412	−	39.8714i	0.858502	−	1.28484i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	25.6920	−	10.6420i	0.826201	−	0.342223i	0.0708031	−	0.997490i	$-0.477444\pi$
$967$	25.6920	−	10.6420i	0.826201	−	0.342223i	0.755397	+	0.655267i	$0.227444\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.46955	+	13.2047i	0.175526	+	0.423757i	0.987019	−	0.160605i	$-0.0513446\pi$
$971$	5.46955	+	13.2047i	0.175526	+	0.423757i	−0.811493	+	0.584363i	$0.801345\pi$
$972$		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	$-0.691955\pi$
$977$	−48.4663	−	20.0754i	−1.55057	−	0.642268i	−0.983420	+	0.181344i	$0.941955\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−12.0917	+	60.7889i	−0.386057	+	1.94084i
$982$		0			0
$983$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$983$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	51.5974	−	34.4763i	1.64070	−	1.09628i
$990$		0			0
$991$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$991$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$997$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$998$		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.386.1	yes	16
17.10	odd	16	inner	731.2.s.a.214.1	✓	16
43.42	odd	2	CM	731.2.s.a.386.1	yes	16
731.214	even	16	inner	731.2.s.a.214.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.214.1	✓	16	17.10	odd	16	inner
731.2.s.a.214.1	✓	16	731.214	even	16	inner
731.2.s.a.386.1	yes	16	1.1	even	1	trivial
731.2.s.a.386.1	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)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.41421i) q^{4} +(1.14805 - 2.77164i) q^{9} +(-0.165842 + 0.248200i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(3.56441 - 2.07243i) q^{17} +(7.86853 + 5.25759i) q^{23} +(4.61940 + 1.91342i) q^{25} +(-2.59764 - 3.88764i) q^{31} +(2.29610 + 5.54328i) q^{36} +(11.3414 - 2.25594i) q^{41} +(2.50942 - 6.05828i) q^{43} +(-0.116472 - 0.585544i) q^{44} +(-7.48531 - 7.48531i) q^{47} +(6.46716 - 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 - 8.77779i) q^{53} +(-11.4652 - 4.74906i) q^{59} +(5.65685 + 5.65685i) q^{64} +11.7985i q^{67} +(-2.10997 + 7.97170i) q^{68} +(-2.35438 + 3.52358i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(-6.05828 + 2.50942i) q^{83} +(-18.5631 + 3.69244i) q^{92} +(3.08269 - 15.4977i) q^{97} +(0.497526 + 0.744600i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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