LMFDB - Embedded newform 731.2.s.a.515.2

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			515.2
Root			$0.792772 + 3.22048i$ of defining polynomial
Character	$\chi$	$=$	731.515
Dual form			731.2.s.a.687.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +(5.50976 + 3.68150i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(-3.56441 + 2.07243i) q^{17} +(-0.868534 + 1.29985i) q^{23} +(-4.61940 - 1.91342i) q^{25} +(-8.40313 + 5.61479i) q^{31} +(-2.29610 - 5.54328i) q^{36} +(1.07359 + 5.39731i) q^{41} +(-2.50942 + 6.05828i) q^{43} +(-12.9984 + 2.58554i) q^{44} +(7.48531 + 7.48531i) q^{47} +(-6.46716 + 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 - 8.77779i) q^{53} +(-11.4652 - 4.74906i) q^{59} +(5.65685 + 5.65685i) q^{64} -11.7985i q^{67} +(2.10997 - 7.97170i) q^{68} +(14.3544 + 9.59129i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(6.05828 - 2.50942i) q^{83} +(-0.609977 - 3.06656i) q^{92} +(11.5348 + 2.29442i) q^{97} +(-16.5293 + 11.0445i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$e\left(\frac{5}{16}\right)$	$-1$

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		−0.555570	−	0.831470i	$-0.687500\pi$
$3$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$4$	−1.41421	+	1.41421i	−0.707107	+	0.707107i
$5$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$5$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$6$		0			0
$7$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$7$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$8$		0			0
$9$	−1.14805	+	2.77164i	−0.382683	+	0.923880i
$10$		0			0
$11$	5.50976	+	3.68150i	1.66125	+	1.11001i	0.855629	+	0.517590i	$0.173171\pi$
$11$	5.50976	+	3.68150i	1.66125	+	1.11001i	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
$23$	−0.868534	+	1.29985i	−0.181102	+	0.271038i	−0.910902	−	0.412622i	$-0.864613\pi$
$23$	−0.868534	+	1.29985i	−0.181102	+	0.271038i	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		−0.980785	−	0.195090i	$-0.937500\pi$
$29$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$30$		0			0
$31$	−8.40313	+	5.61479i	−1.50925	+	1.00845i	−0.521349	+	0.853344i	$0.674571\pi$
$31$	−8.40313	+	5.61479i	−1.50925	+	1.00845i	−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		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.07359	+	5.39731i	0.167667	+	0.842918i	0.969447	+	0.245299i	$0.0788863\pi$
$41$	1.07359	+	5.39731i	0.167667	+	0.842918i	−0.801781	+	0.597619i	$0.796114\pi$
$42$		0			0
$43$	−2.50942	+	6.05828i	−0.382683	+	0.923880i
$43$	−2.50942	+	6.05828i	−0.382683	+	0.923880i
$44$	−12.9984	+	2.58554i	−1.95958	+	0.389785i
$45$		0			0
$46$		0			0
$47$	7.48531	+	7.48531i	1.09185	+	1.09185i	0.995332	+	0.0965136i	$0.0307691\pi$
$47$	7.48531	+	7.48531i	1.09185	+	1.09185i	0.0965136	+	0.995332i	$0.469231\pi$
$48$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$61$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$62$		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	$-0.743816\pi$
$67$		−	11.7985i		−	1.44142i	0.693236	−	0.720710i	$-0.256184\pi$
$68$	2.10997	−	7.97170i	0.255872	−	0.966711i
$69$		0			0
$70$		0			0
$71$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$71$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$72$		0			0
$73$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$73$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	14.3544	+	9.59129i	1.61499	+	1.07910i	0.939942	+	0.341335i	$0.110879\pi$
$79$	14.3544	+	9.59129i	1.61499	+	1.07910i	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	−0.0245507	−	0.999699i	$-0.507816\pi$
$83$	6.05828	−	2.50942i	0.664983	−	0.275445i	0.689534	+	0.724254i	$0.257816\pi$
$84$		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
$92$	−0.609977	−	3.06656i	−0.0635945	−	0.319711i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.5348	+	2.29442i	1.17119	+	0.232963i	0.742093	−	0.670297i	$-0.233833\pi$
$97$	11.5348	+	2.29442i	1.17119	+	0.232963i	0.429093	+	0.903260i	$0.358833\pi$
$98$		0			0
$99$	−16.5293	+	11.0445i	−1.66125	+	1.11001i
$100$	9.23880	−	3.82683i	0.923880	−	0.382683i
$101$			1.21270i			0.120668i	0.998178	+	0.0603342i	$0.0192166\pi$
$101$			1.21270i			0.120668i	−0.998178	+	0.0603342i	$0.980783\pi$
$102$		0			0
$103$	−2.90887			−0.286620			−0.143310	−	0.989678i	$-0.545775\pi$
$103$	−2.90887			−0.286620			−0.143310	+	0.989678i	$0.545775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.56231	+	12.8816i	−0.247708	+	1.24531i	0.633932	+	0.773389i	$0.281440\pi$
$107$	−2.56231	+	12.8816i	−0.247708	+	1.24531i	−0.881640	+	0.471923i	$0.843560\pi$
$108$		0			0
$109$	0.590659	+	2.96944i	0.0565749	+	0.284421i	0.998708	−	0.0508181i	$-0.0161829\pi$
$109$	0.590659	+	2.96944i	0.0565749	+	0.284421i	−0.942133	+	0.335239i	$0.891183\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$113$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$114$		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
$121$	12.5945	+	30.4057i	1.14495	+	2.76415i
$122$		0			0
$123$		0			0
$124$	3.94331	−	19.8243i	0.354119	−	1.78028i
$125$		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		−0.980785	−	0.195090i	$-0.937500\pi$
$131$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$132$		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
$139$	−13.1000	−	19.6055i	−1.11113	−	1.66292i	−0.559803	−	0.828626i	$-0.689123\pi$
$139$	−13.1000	−	19.6055i	−1.11113	−	1.66292i	−0.551323	−	0.834292i	$-0.685877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.73672	−	43.9225i	−0.730601	−	3.67298i
$144$	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		0.980785	−	0.195090i	$-0.0625000\pi$
$163$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$164$	−9.15123	−	6.11466i	−0.714591	−	0.477475i
$165$		0			0
$166$		0			0
$167$	19.3236	−	12.9116i	1.49531	−	0.999133i	0.504566	−	0.863373i	$-0.331653\pi$
$167$	19.3236	−	12.9116i	1.49531	−	0.999133i	0.990742	−	0.135760i	$-0.0433475\pi$
$168$		0			0
$169$			32.6723i			2.51326i
$170$		0			0
$171$		0			0
$172$	−5.01885	−	12.1166i	−0.382683	−	0.923880i
$173$	3.93274	+	5.88576i	0.299001	+	0.447486i	0.950300	−	0.311335i	$-0.100776\pi$
$173$	3.93274	+	5.88576i	0.299001	+	0.447486i	−0.651300	+	0.758820i	$0.725776\pi$
$174$		0			0
$175$		0			0
$176$	14.7260	−	22.0390i	1.11001	−	1.66125i
$177$		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
$181$	20.9054	+	13.9685i	1.55388	+	1.03827i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	20.9054	+	13.9685i	1.55388	+	1.03827i	0.579062	+	0.815283i	$0.303419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−27.2687	−	1.70378i	−1.99409	−	0.124593i
$188$	−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
$193$	−11.3629	+	17.0058i	−0.817921	+	1.22410i	0.153831	+	0.988097i	$0.450839\pi$
$193$	−11.3629	+	17.0058i	−0.817921	+	1.22410i	−0.971751	+	0.236007i	$0.924161\pi$
$194$		0			0
$195$		0			0
$196$	5.35757	−	12.9343i	0.382683	−	0.923880i
$197$	−6.13753	+	1.22083i	−0.437281	+	0.0869806i	−0.408822	−	0.912614i	$-0.634060\pi$
$197$	−6.13753	+	1.22083i	−0.437281	+	0.0869806i	−0.0284595	+	0.999595i	$0.509060\pi$
$198$		0			0
$199$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$199$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.60560	−	3.89956i	−0.181102	−	0.271038i
$208$	−19.1149	+	19.1149i	−1.32538	+	1.32538i
$209$		0			0
$210$		0			0
$211$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$211$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$212$	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		0.555570	−	0.831470i	$-0.312500\pi$
$227$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$228$		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		−0.980785	−	0.195090i	$-0.937500\pi$
$233$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$234$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$241$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$242$		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
$253$	−9.57082	+	3.96436i	−0.601712	+	0.249237i
$254$		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
$269$	22.2052	−	14.8370i	1.35388	−	0.904631i	0.354341	−	0.935116i	$-0.384705\pi$
$269$	22.2052	−	14.8370i	1.35388	−	0.904631i	0.999535	+	0.0304855i	$0.00970535\pi$
$270$		0			0
$271$		−	15.3188i		−	0.930548i	−0.885167	−	0.465274i	$-0.845956\pi$
$271$		−	15.3188i		−	0.930548i	0.885167	−	0.465274i	$-0.154044\pi$
$272$	8.28973	+	14.2576i	0.502639	+	0.864496i
$273$		0			0
$274$		0			0
$275$	−18.4075	−	27.5488i	−1.11001	−	1.66125i
$276$		0			0
$277$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$277$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$278$		0			0
$279$	−5.91496	−	29.7365i	−0.354119	−	1.78028i
$280$		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
$283$	−0.00543901	−	0.00363423i	−0.000323316	−	0.000216033i	0.555409	−	0.831578i	$-0.312562\pi$
$283$	−0.00543901	−	0.00363423i	−0.000323316	−	0.000216033i	−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
$299$	10.3621	−	2.06115i	0.599256	−	0.119199i
$300$		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	$-0.935271\pi$
$307$	−34.3208			−1.95879			−0.979395	+	0.201954i	$0.935271\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.66244	+	28.4670i	−0.321087	+	1.61422i	0.396697	+	0.917950i	$0.370156\pi$
$311$	−5.66244	+	28.4670i	−0.321087	+	1.61422i	−0.717784	+	0.696265i	$0.754844\pi$
$312$		0			0
$313$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$313$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$314$		0			0
$315$		0			0
$316$	−33.8643	+	6.73603i	−1.90502	+	0.378931i
$317$	11.0250	+	7.36664i	0.619223	+	0.413752i	0.825228	−	0.564799i	$-0.191046\pi$
$317$	11.0250	+	7.36664i	0.619223	+	0.413752i	−0.206005	+	0.978551i	$0.566046\pi$
$318$		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
$337$	−11.0141	+	7.35937i	−0.599975	+	0.400890i	−0.818138	−	0.575022i	$-0.804994\pi$
$337$	−11.0141	+	7.35937i	−0.599975	+	0.400890i	0.218163	+	0.975912i	$0.429994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−66.9701			−3.62663
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$347$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$348$		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	0.991234	−	0.132119i	$-0.0421781\pi$
$359$	3.73079	+	9.00692i	0.196903	+	0.475367i	−0.794330	+	0.607486i	$0.792178\pi$
$360$		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
$367$	−19.5040	+	3.87959i	−1.01810	+	0.202513i	−0.675806	−	0.737079i	$-0.736204\pi$
$367$	−19.5040	+	3.87959i	−1.01810	+	0.202513i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	5.19941	+	3.47413i	0.271038	+	0.181102i
$369$	−16.1919	−	3.22077i	−0.842918	−	0.167667i
$370$		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
$379$	−0.147240	+	0.740225i	−0.00756320	+	0.0380228i	−0.984383	−	0.176042i	$-0.943670\pi$
$379$	−0.147240	+	0.740225i	−0.00756320	+	0.0380228i	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
$388$	−19.5575	+	13.0679i	−0.992883	+	0.663423i
$389$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$389$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$390$		0			0
$391$	0.401953	−	6.43319i	0.0203276	−	0.325340i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	7.75663	−	38.9952i	0.389785	−	1.95958i
$397$	11.0409	−	16.5238i	0.554126	−	0.829307i	−0.443634	−	0.896208i	$-0.646311\pi$
$397$	11.0409	−	16.5238i	0.554126	−	0.829307i	0.997760	+	0.0669005i	$0.0213110\pi$
$398$		0			0
$399$		0			0
$400$	−7.65367	+	18.4776i	−0.382683	+	0.923880i
$401$	−15.8511	+	3.15297i	−0.791564	+	0.157452i	−0.574283	−	0.818657i	$-0.694719\pi$
$401$	−15.8511	+	3.15297i	−0.791564	+	0.157452i	−0.217281	+	0.976109i	$0.569719\pi$
$402$		0			0
$403$	66.9877	+	13.3247i	3.33690	+	0.663750i
$404$	−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		−0.831470	−	0.555570i	$-0.812500\pi$
$419$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$420$		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
$428$	−14.5937	−	21.8410i	−0.705413	−	1.05573i
$429$		0			0
$430$		0			0
$431$	9.62585	−	14.4061i	0.463661	−	0.693917i	−0.523789	−	0.851848i	$-0.675482\pi$
$431$	9.62585	−	14.4061i	0.463661	−	0.693917i	0.987450	+	0.157930i	$0.0504821\pi$
$432$		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
$436$	−5.03475	−	3.36411i	−0.241121	−	0.161112i
$437$		0			0
$438$		0			0
$439$	−22.0445	+	14.7296i	−1.05213	+	0.703008i	−0.956299	−	0.292391i	$-0.905549\pi$
$439$	−22.0445	+	14.7296i	−1.05213	+	0.703008i	−0.0958262	+	0.995398i	$0.530549\pi$
$440$		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		−0.195090	−	0.980785i	$-0.562500\pi$
$449$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$450$		0			0
$451$	−13.9550	+	33.6903i	−0.657114	+	1.58641i
$452$		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	$-0.915840\pi$
$461$	−10.0377	−	24.2331i	−0.467502	−	1.12865i	0.497748	−	0.867322i	$-0.334160\pi$
$462$		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
$473$	−36.1299	+	24.1412i	−1.66125	+	1.11001i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	28.5030			1.30506
$478$		0			0
$479$	−14.7224	−	22.0336i	−0.672683	−	1.00674i	−0.998127	−	0.0611793i	$-0.980514\pi$
$479$	−14.7224	−	22.0336i	−0.672683	−	1.00674i	0.325444	−	0.945561i	$-0.394486\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−60.8114	−	25.1889i	−2.76415	−	1.14495i
$485$		0			0
$486$		0			0
$487$	22.6920	+	15.1623i	1.02827	+	0.687070i	0.950762	−	0.309921i	$-0.100303\pi$
$487$	22.6920	+	15.1623i	1.02827	+	0.687070i	0.0775113	+	0.996991i	$0.475303\pi$
$488$		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
$496$	22.4592	+	33.6125i	1.00845	+	1.50925i
$497$		0			0
$498$		0			0
$499$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$499$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$503$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$504$		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	$0.116084\pi$
$509$			16.0932i			0.713316i	−0.934235	+	0.356658i	$0.883916\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	13.6851	+	68.7995i	0.601869	+	3.02580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$521$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$522$		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
$527$	18.3159	−	37.4284i	0.797854	−	1.63040i
$528$		0			0
$529$	7.86645	+	18.9913i	0.342020	+	0.825709i
$530$		0			0
$531$	26.3253	−	26.3253i	1.14242	−	1.14242i
$532$		0			0
$533$	20.6618	−	30.9226i	0.894963	−	1.33941i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−45.4944	−	9.04940i	−1.95958	−	0.389785i
$540$		0			0
$541$	37.9112	−	25.3314i	1.62993	−	1.08908i	0.704816	−	0.709390i	$-0.251030\pi$
$541$	37.9112	−	25.3314i	1.62993	−	1.08908i	0.925113	−	0.379693i	$-0.123970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	10.0106	+	14.9819i	0.428022	+	0.640580i	0.981315	−	0.192406i	$-0.0616291\pi$
$547$	10.0106	+	14.9819i	0.428022	+	0.640580i	−0.553293	+	0.832987i	$0.686629\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	46.2526	+	9.20020i	1.96155	+	0.390176i
$557$	−19.6071	−	19.6071i	−0.830780	−	0.830780i	0.156844	−	0.987623i	$-0.449868\pi$
$557$	−19.6071	−	19.6071i	−0.830780	−	0.830780i	−0.987623	+	0.156844i	$0.949868\pi$
$558$		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	0.784639	−	0.619953i	$-0.212849\pi$
$569$	21.4943	+	8.90324i	0.901088	+	0.373243i	0.116450	+	0.993197i	$0.462849\pi$
$570$		0			0
$571$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$571$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$572$	74.4713	+	49.7601i	3.11380	+	2.08058i
$573$		0			0
$574$		0			0
$575$	6.49926	−	4.34267i	0.271038	−	0.181102i
$576$	−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
$583$	12.2826	−	61.7490i	0.508695	−	2.55738i
$584$		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		0.555570	−	0.831470i	$-0.312500\pi$
$601$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$602$		0			0
$603$	32.7013	+	13.5453i	1.33170	+	0.551608i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$607$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$608$		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	$-0.552697\pi$
$613$	−8.16043			−0.329597			−0.164798	+	0.986327i	$0.552697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.50519	−	47.7858i	0.382665	−	1.92379i	−0.000246592	−	1.00000i	$-0.500078\pi$
$617$	9.50519	−	47.7858i	0.382665	−	1.92379i	0.382911	−	0.923785i	$-0.374922\pi$
$618$		0			0
$619$	3.07516	+	15.4599i	0.123601	+	0.621385i	0.992075	+	0.125649i	$0.0401014\pi$
$619$	3.07516	+	15.4599i	0.123601	+	0.621385i	−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		−0.980785	−	0.195090i	$-0.937500\pi$
$641$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$642$		0			0
$643$	3.94809	−	2.63803i	0.155697	−	0.104034i	−0.475281	−	0.879834i	$-0.657654\pi$
$643$	3.94809	−	2.63803i	0.155697	−	0.104034i	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
$649$	−45.6870	−	68.3754i	−1.79337	−	2.68397i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$653$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$654$		0			0
$655$		0			0
$656$	21.5892	−	4.29437i	0.842918	−	0.167667i
$657$		0			0
$658$		0			0
$659$	2.60446	+	2.60446i	0.101455	+	0.101455i	0.756013	−	0.654557i	$-0.227145\pi$
$659$	2.60446	+	2.60446i	0.101455	+	0.101455i	−0.654557	+	0.756013i	$0.727145\pi$
$660$		0			0
$661$	6.05828	−	2.50942i	0.235640	−	0.0976052i	−0.261739	−	0.965139i	$-0.584296\pi$
$661$	6.05828	−	2.50942i	0.235640	−	0.0976052i	0.497379	+	0.867533i	$0.334296\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−9.06793	+	45.5876i	−0.350849	+	1.76384i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$673$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$674$		0			0
$675$		0			0
$676$	−46.2056	−	46.2056i	−1.77714	−	1.77714i
$677$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$677$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.7165	+	38.4875i	0.984016	+	1.47268i	0.878197	+	0.478299i	$0.158747\pi$
$683$	25.7165	+	38.4875i	0.984016	+	1.47268i	0.105819	+	0.994385i	$0.466253\pi$
$684$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$691$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$692$	−13.8855	−	2.76199i	−0.527846	−	0.104995i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−15.0123	−	17.0133i	−0.568631	−	0.644424i
$698$		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
$704$	10.3422	+	51.9936i	0.389785	+	1.95958i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−38.9888	−	7.75535i	−1.46426	−	0.291258i	−0.602311	−	0.798262i	$-0.705753\pi$
$709$	−38.9888	−	7.75535i	−1.46426	−	0.291258i	−0.861944	+	0.507003i	$0.830753\pi$
$710$		0			0
$711$	−43.0632	+	28.7739i	−1.61499	+	1.07910i
$712$		0			0
$713$		−	15.7995i		−	0.591695i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.4489	−	52.5303i	0.389679	−	1.95905i	0.145128	−	0.989413i	$-0.453641\pi$
$719$	10.4489	−	52.5303i	0.389679	−	1.95905i	0.244551	−	0.969636i	$-0.421359\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−49.3191	+	9.81018i	−1.83293	+	0.364593i
$725$		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
$737$	43.4363	−	65.0071i	1.60000	−	2.39457i
$738$		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		−0.980785	−	0.195090i	$-0.937500\pi$
$743$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			19.6723i			0.719772i
$748$	40.9733	−	36.1543i	1.49813	−	1.32193i
$749$		0			0
$750$		0			0
$751$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$751$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$752$	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	$-0.269788\pi$
$769$	39.1416	−	39.1416i	1.41148	−	1.41148i	0.749670	−	0.661812i	$-0.230212\pi$
$770$		0			0
$771$		0			0
$772$	−7.98025	−	40.1194i	−0.287215	−	1.44393i
$773$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$		0			0
$775$	49.5608	−	9.85827i	1.78028	−	0.354119i
$776$		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
$787$	9.05068	−	45.5008i	0.322622	−	1.62193i	−0.390301	−	0.920687i	$-0.627629\pi$
$787$	9.05068	−	45.5008i	0.322622	−	1.62193i	0.712923	−	0.701242i	$-0.247371\pi$
$788$	6.95327	−	10.4063i	0.247700	−	0.370709i
$789$		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
$809$	−30.7927	+	6.12504i	−1.08261	+	0.215345i	−0.703993	−	0.710207i	$-0.748601\pi$
$809$	−30.7927	+	6.12504i	−1.08261	+	0.215345i	−0.378620	+	0.925552i	$0.623601\pi$
$810$		0			0
$811$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$811$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$812$		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
$821$	−11.1776	+	56.1938i	−0.390102	+	1.96118i	−0.175373	+	0.984502i	$0.556113\pi$
$821$	−11.1776	+	56.1938i	−0.390102	+	1.96118i	−0.214729	+	0.976674i	$0.568887\pi$
$822$		0			0
$823$	−9.95680	−	50.0562i	−0.347072	−	1.74485i	−0.621659	−	0.783288i	$-0.713541\pi$
$823$	−9.95680	−	50.0562i	−0.347072	−	1.74485i	0.274587	−	0.961562i	$-0.411459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.6569	−	21.8206i	−1.13559	−	0.758777i	−0.161935	−	0.986801i	$-0.551773\pi$
$827$	−32.6569	−	21.8206i	−1.13559	−	0.758777i	−0.973655	+	0.228024i	$0.926773\pi$
$828$	9.19968	+	1.82993i	0.319711	+	0.0635945i
$829$		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		0.555570	−	0.831470i	$-0.312500\pi$
$839$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$840$		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
$853$	7.88878	+	11.8064i	0.270107	+	0.404243i	0.941582	−	0.336784i	$-0.109339\pi$
$853$	7.88878	+	11.8064i	0.270107	+	0.404243i	−0.671475	+	0.741027i	$0.734339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.04813	+	45.4880i	0.309078	+	1.55384i	0.753150	+	0.657849i	$0.228534\pi$
$857$	9.04813	+	45.4880i	0.309078	+	1.55384i	−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
$869$	43.7788	+	105.691i	1.48509	+	3.58534i
$870$		0			0
$871$	−56.3819	+	56.3819i	−1.91043	+	1.91043i
$872$		0			0
$873$	−19.6019	+	29.3363i	−0.663423	+	0.992883i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	50.8524	−	10.1152i	1.71716	−	0.341565i	0.764276	−	0.644889i	$-0.223097\pi$
$877$	50.8524	−	10.1152i	1.71716	−	0.341565i	0.952887	+	0.303324i	$0.0980966\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47.0504	−	31.4381i	1.58517	−	1.05918i	0.624670	−	0.780889i	$-0.285233\pi$
$881$	47.0504	−	31.4381i	1.58517	−	1.05918i	0.960498	−	0.278287i	$-0.0897667\pi$
$882$		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		−0.555570	−	0.831470i	$-0.687500\pi$
$887$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−11.6349	−	58.4928i	−0.389785	−	1.95958i
$892$		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
$907$	−16.2758	+	24.3584i	−0.540427	+	0.808807i	−0.996712	−	0.0810272i	$-0.974180\pi$
$907$	−16.2758	+	24.3584i	−0.540427	+	0.808807i	0.456284	+	0.889834i	$0.349180\pi$
$908$		0			0
$909$	−3.36117	−	1.39224i	−0.111483	−	0.0461778i
$910$		0			0
$911$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$911$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$912$		0			0
$913$	42.6181	+	8.47727i	1.41045	+	0.280557i
$914$		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		−0.831470	−	0.555570i	$-0.812500\pi$
$929$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$930$		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
$941$	−21.8573	+	32.7118i	−0.712528	+	1.06637i	0.281744	+	0.959490i	$0.409087\pi$
$941$	−21.8573	+	32.7118i	−0.712528	+	1.06637i	−0.994272	+	0.106883i	$0.965913\pi$
$942$		0			0
$943$	−7.94815	−	3.29223i	−0.258828	−	0.107210i
$944$	−18.9962	+	45.8609i	−0.618274	+	1.49265i
$945$		0			0
$946$		0			0
$947$	−52.0000	−	10.3435i	−1.68977	−	0.336117i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	−52.0000	−	10.3435i	−1.68977	−	0.336117i	−0.943966	+	0.330044i	$0.892937\pi$
$948$		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
$961$	27.2235	−	65.7234i	0.878178	−	2.12011i
$962$		0			0
$963$	−32.7615	−	21.8905i	−1.05573	−	0.705413i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−25.6920	+	10.6420i	−0.826201	+	0.342223i	−0.755397	−	0.655267i	$-0.772556\pi$
$967$	−25.6920	+	10.6420i	−0.826201	+	0.342223i	−0.0708031	+	0.997490i	$0.522556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.46955	−	13.2047i	−0.175526	−	0.423757i	0.811493	−	0.584363i	$-0.198655\pi$
$971$	−5.46955	−	13.2047i	−0.175526	−	0.423757i	−0.987019	+	0.160605i	$0.948655\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	48.4663	+	20.0754i	1.55057	+	0.642268i	0.983420	−	0.181344i	$-0.0580446\pi$
$977$	48.4663	+	20.0754i	1.55057	+	0.642268i	0.567153	+	0.823612i	$0.308045\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−8.90833	−	1.77198i	−0.284421	−	0.0565749i
$982$		0			0
$983$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$983$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5.69536	−	8.52370i	−0.181102	−	0.271038i
$990$		0			0
$991$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$991$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$997$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$998$		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.515.2	✓	16
17.7	odd	16	inner	731.2.s.a.687.2	yes	16
43.42	odd	2	CM	731.2.s.a.515.2	✓	16
731.687	even	16	inner	731.2.s.a.687.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.515.2	✓	16	1.1	even	1	trivial
731.2.s.a.515.2	✓	16	43.42	odd	2	CM
731.2.s.a.687.2	yes	16	17.7	odd	16	inner
731.2.s.a.687.2	yes	16	731.687	even	16	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.41421i) q^{4} +(-1.14805 + 2.77164i) q^{9} +(5.50976 + 3.68150i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(-3.56441 + 2.07243i) q^{17} +(-0.868534 + 1.29985i) q^{23} +(-4.61940 - 1.91342i) q^{25} +(-8.40313 + 5.61479i) q^{31} +(-2.29610 - 5.54328i) q^{36} +(1.07359 + 5.39731i) q^{41} +(-2.50942 + 6.05828i) q^{43} +(-12.9984 + 2.58554i) q^{44} +(7.48531 + 7.48531i) q^{47} +(-6.46716 + 2.67878i) q^{49} +13.5163 q^{52} +(-3.63588 - 8.77779i) q^{53} +(-11.4652 - 4.74906i) q^{59} +(5.65685 + 5.65685i) q^{64} -11.7985i q^{67} +(2.10997 - 7.97170i) q^{68} +(14.3544 + 9.59129i) q^{79} +(-6.36396 - 6.36396i) q^{81} +(6.05828 - 2.50942i) q^{83} +(-0.609977 - 3.06656i) q^{92} +(11.5348 + 2.29442i) q^{97} +(-16.5293 + 11.0445i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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