LMFDB - Embedded newform 731.2.s.a.343.1

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			343.1
Root			$-3.22048 + 0.792772i$ of defining polynomial
Character	$\chi$	$=$	731.343
Dual form			731.2.s.a.601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +O(q^{10})$	$q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +(-0.955307 - 4.80265i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(2.07243 + 3.56441i) q^{17} +(5.47887 - 1.08981i) q^{23} +(-1.91342 + 4.61940i) q^{25} +(0.749168 - 3.76632i) q^{31} +(5.54328 - 2.29610i) q^{36} +(-2.38087 + 3.56322i) q^{41} +(6.05828 + 2.50942i) q^{43} +(-8.14298 - 5.44097i) q^{44} +(-6.16199 - 6.16199i) q^{47} +(-2.67878 - 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 - 4.22221i) q^{53} +(3.46524 - 8.36582i) q^{59} +(-5.65685 - 5.65685i) q^{64} +11.3488i q^{67} +(7.97170 + 2.10997i) q^{68} +(2.96615 + 14.9118i) q^{79} +(6.36396 + 6.36396i) q^{81} +(2.50942 + 6.05828i) q^{83} +(6.20706 - 9.28952i) q^{92} +(-16.2049 + 10.8277i) q^{97} +(2.86592 - 14.4080i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{13} + 56 q^{23} - 64 q^{59} + 96 q^{79}+O(q^{100}) $ 16 * q - 24 * q^13 + 56 * q^23 - 64 * q^59 + 96 * q^79

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$2$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$3$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$4$	1.41421	−	1.41421i	0.707107	−	0.707107i
$5$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$5$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$6$		0			0
$7$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$7$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$8$		0			0
$9$	2.77164	+	1.14805i	0.923880	+	0.382683i
$10$		0			0
$11$	−0.955307	−	4.80265i	−0.288036	−	1.44805i	−0.805626	−	0.592425i	$-0.798171\pi$
$11$	−0.955307	−	4.80265i	−0.288036	−	1.44805i	0.517590	−	0.855629i	$-0.326829\pi$
$12$		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
$17$	2.07243	+	3.56441i	0.502639	+	0.864496i
$17$	2.07243	+	3.56441i	0.502639	+	0.864496i
$18$		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
$23$	5.47887	−	1.08981i	1.14242	−	0.227242i	0.412622	−	0.910902i	$-0.364613\pi$
$23$	5.47887	−	1.08981i	1.14242	−	0.227242i	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		0.831470	−	0.555570i	$-0.187500\pi$
$29$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$30$		0			0
$31$	0.749168	−	3.76632i	0.134555	−	0.676451i	−0.853344	−	0.521349i	$-0.825429\pi$
$31$	0.749168	−	3.76632i	0.134555	−	0.676451i	0.987898	−	0.155103i	$-0.0495709\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.54328	−	2.29610i	0.923880	−	0.382683i
$37$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$37$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.38087	+	3.56322i	−0.371829	+	0.556481i	−0.969447	−	0.245299i	$-0.921114\pi$
$41$	−2.38087	+	3.56322i	−0.371829	+	0.556481i	0.597619	+	0.801781i	$0.296114\pi$
$42$		0			0
$43$	6.05828	+	2.50942i	0.923880	+	0.382683i
$43$	6.05828	+	2.50942i	0.923880	+	0.382683i
$44$	−8.14298	−	5.44097i	−1.22760	−	0.820257i
$45$		0			0
$46$		0			0
$47$	−6.16199	−	6.16199i	−0.898818	−	0.898818i	0.0965136	−	0.995332i	$-0.469231\pi$
$47$	−6.16199	−	6.16199i	−0.898818	−	0.898818i	−0.995332	+	0.0965136i	$0.969231\pi$
$48$		0			0
$49$	−2.67878	−	6.46716i	−0.382683	−	0.923880i
$50$		0			0
$51$		0			0
$52$	−13.5163			−1.87437
$53$	10.1933	−	4.22221i	1.40016	−	0.579965i	0.450367	−	0.892844i	$-0.351293\pi$
$53$	10.1933	−	4.22221i	1.40016	−	0.579965i	0.949793	+	0.312878i	$0.101293\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.46524	−	8.36582i	0.451135	−	1.08914i	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	3.46524	−	8.36582i	0.451135	−	1.08914i	0.971891	−	0.235431i	$-0.0756503\pi$
$60$		0			0
$61$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$61$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$62$		0			0
$63$		0			0
$64$	−5.65685	−	5.65685i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$			11.3488i			1.38647i	0.720710	+	0.693236i	$0.243816\pi$
$67$			11.3488i			1.38647i	−0.720710	+	0.693236i	$0.756184\pi$
$68$	7.97170	+	2.10997i	0.966711	+	0.255872i
$69$		0			0
$70$		0			0
$71$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$71$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$72$		0			0
$73$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$73$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.96615	+	14.9118i	0.333717	+	1.67771i	0.675053	+	0.737769i	$0.264121\pi$
$79$	2.96615	+	14.9118i	0.333717	+	1.67771i	−0.341335	+	0.939942i	$0.610879\pi$
$80$		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
$92$	6.20706	−	9.28952i	0.647130	−	0.968499i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.2049	+	10.8277i	−1.64535	+	1.09939i	−0.742093	+	0.670297i	$0.766167\pi$
$97$	−16.2049	+	10.8277i	−1.64535	+	1.09939i	−0.903260	+	0.429093i	$0.858833\pi$
$98$		0			0
$99$	2.86592	−	14.4080i	0.288036	−	1.44805i
$100$	3.82683	+	9.23880i	0.382683	+	0.923880i
$101$		−	20.0631i		−	1.99636i	−0.0603342	−	0.998178i	$-0.519217\pi$
$101$		−	20.0631i		−	1.99636i	0.0603342	−	0.998178i	$-0.480783\pi$
$102$		0			0
$103$	20.0883			1.97936			0.989678	−	0.143310i	$-0.0457745\pi$
$103$	20.0883			1.97936			0.989678	+	0.143310i	$0.0457745\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.4391	+	17.1198i	1.10585	+	1.65503i	0.633932	+	0.773389i	$0.281440\pi$
$107$	11.4391	+	17.1198i	1.10585	+	1.65503i	0.471923	+	0.881640i	$0.343560\pi$
$108$		0			0
$109$	−9.30560	+	13.9268i	−0.891315	+	1.33395i	0.0508181	+	0.998708i	$0.483817\pi$
$109$	−9.30560	+	13.9268i	−0.891315	+	1.33395i	−0.942133	+	0.335239i	$0.891183\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$113$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.75867	−	18.7311i	−0.717290	−	1.73169i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.9902	+	4.96649i	−1.09002	+	0.451499i
$122$		0			0
$123$		0			0
$124$	−4.26690	−	6.38587i	−0.383179	−	0.573468i
$125$		0			0
$126$		0			0
$127$	−1.00814	+	2.43387i	−0.0894583	+	0.215971i	−0.962276	−	0.272075i	$-0.912290\pi$
$127$	−1.00814	+	2.43387i	−0.0894583	+	0.215971i	0.872818	+	0.488046i	$0.162290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$131$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−16.2694	−	3.23618i	−1.37995	−	0.274489i	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−16.2694	−	3.23618i	−1.37995	−	0.274489i	−0.828626	+	0.559803i	$0.810877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−18.3854	+	27.5157i	−1.53746	+	2.30098i
$144$	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
$153$	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		−0.831470	−	0.555570i	$-0.812500\pi$
$163$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$164$	1.67210	+	8.40620i	0.130569	+	0.656414i
$165$		0			0
$166$		0			0
$167$	−4.76603	+	23.9604i	−0.368806	+	1.85411i	0.135760	+	0.990742i	$0.456653\pi$
$167$	−4.76603	+	23.9604i	−0.368806	+	1.85411i	−0.504566	+	0.863373i	$0.668347\pi$
$168$		0			0
$169$			32.6723i			2.51326i
$170$		0			0
$171$		0			0
$172$	12.1166	−	5.01885i	0.923880	−	0.382683i
$173$	12.6615	+	2.51852i	0.962634	+	0.191480i	0.651300	−	0.758820i	$-0.274224\pi$
$173$	12.6615	+	2.51852i	0.962634	+	0.191480i	0.311335	+	0.950300i	$0.399224\pi$
$174$		0			0
$175$		0			0
$176$	−19.2106	+	3.82123i	−1.44805	+	0.288036i
$177$		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
$181$	2.14636	+	10.7905i	0.159538	+	0.802050i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	2.14636	+	10.7905i	0.159538	+	0.802050i	−0.815283	+	0.579062i	$0.803419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	15.1388	−	13.3583i	1.10706	−	0.976854i
$188$	−17.4287			−1.27112
$189$		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
$193$	−27.2271	+	5.41580i	−1.95985	+	0.389838i	−0.971751	+	0.236007i	$0.924161\pi$
$193$	−27.2271	+	5.41580i	−1.95985	+	0.389838i	−0.988097	+	0.153831i	$0.950839\pi$
$194$		0			0
$195$		0			0
$196$	−12.9343	−	5.35757i	−0.923880	−	0.382683i
$197$	19.7681	+	13.2086i	1.40842	+	0.941074i	0.999595	+	0.0284595i	$0.00906017\pi$
$197$	19.7681	+	13.2086i	1.40842	+	0.941074i	0.408822	+	0.912614i	$0.365940\pi$
$198$		0			0
$199$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$199$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	16.4366	+	3.26944i	1.14242	+	0.227242i
$208$	−19.1149	+	19.1149i	−1.32538	+	1.32538i
$209$		0			0
$210$		0			0
$211$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$211$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$212$	8.44442	−	20.3866i	0.579965	−	1.40016i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.12974	−	26.9369i	0.479598	−	1.81197i
$222$		0			0
$223$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$223$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$	−10.6066	+	10.6066i	−0.707107	+	0.707107i
$226$		0			0
$227$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$227$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$228$		0			0
$229$	11.0127	−	26.5869i	0.727738	−	1.75691i	0.0777462	−	0.996973i	$-0.475228\pi$
$229$	11.0127	−	26.5869i	0.727738	−	1.75691i	0.649992	−	0.759941i	$-0.274772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$233$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$234$		0			0
$235$		0			0
$236$	−6.93047	−	16.7316i	−0.451135	−	1.08914i
$237$		0			0
$238$		0			0
$239$	10.5254			0.680830			0.340415	−	0.940275i	$-0.389432\pi$
$239$	10.5254			0.680830			0.340415	+	0.940275i	$0.389432\pi$
$240$		0			0
$241$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$241$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	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
$253$	−10.4680	−	25.2720i	−0.658117	−	1.58884i
$254$		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
$268$	16.0496	+	16.0496i	0.980384	+	0.980384i
$269$	1.05655	−	5.31162i	0.0644188	−	0.323855i	−0.935116	−	0.354341i	$-0.884705\pi$
$269$	1.05655	−	5.31162i	0.0644188	−	0.323855i	0.999535	+	0.0304855i	$0.00970535\pi$
$270$		0			0
$271$			29.1434i			1.77033i	0.465274	+	0.885167i	$0.345956\pi$
$271$			29.1434i			1.77033i	−0.465274	+	0.885167i	$0.654044\pi$
$272$	14.2576	−	8.28973i	0.864496	−	0.502639i
$273$		0			0
$274$		0			0
$275$	24.0133	+	4.77653i	1.44805	+	0.288036i
$276$		0			0
$277$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$277$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$278$		0			0
$279$	6.40035	−	9.57880i	0.383179	−	0.573468i
$280$		0			0
$281$	5.61715	+	2.32670i	0.335091	+	0.138799i	0.543884	−	0.839161i	$-0.316953\pi$
$281$	5.61715	+	2.32670i	0.335091	+	0.138799i	−0.208792	+	0.977960i	$0.566953\pi$
$282$		0			0
$283$	−4.64044	−	23.3291i	−0.275846	−	1.38677i	−0.831578	−	0.555409i	$-0.812562\pi$
$283$	−4.64044	−	23.3291i	−0.275846	−	1.38677i	0.555732	−	0.831362i	$-0.312438\pi$
$284$		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
$299$	−31.3899	−	20.9740i	−1.81532	−	1.21296i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.07703			0.403907			0.201954	−	0.979395i	$-0.435271\pi$
$307$	7.07703			0.403907			0.201954	+	0.979395i	$0.435271\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−19.2746	−	28.8465i	−1.09296	−	1.63573i	−0.696265	−	0.717784i	$-0.745156\pi$
$311$	−19.2746	−	28.8465i	−1.09296	−	1.63573i	−0.396697	−	0.917950i	$-0.629844\pi$
$312$		0			0
$313$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$313$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$314$		0			0
$315$		0			0
$316$	25.2833	+	16.8937i	1.42229	+	0.950347i
$317$	2.72983	+	13.7238i	0.153323	+	0.770804i	0.978551	+	0.206005i	$0.0660464\pi$
$317$	2.72983	+	13.7238i	0.153323	+	0.770804i	−0.825228	+	0.564799i	$0.808954\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	31.2185	−	12.9311i	1.73169	−	0.717290i
$326$		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
$337$	6.55106	−	32.9344i	0.356859	−	1.79405i	−0.218163	−	0.975912i	$-0.570006\pi$
$337$	6.55106	−	32.9344i	0.356859	−	1.79405i	0.575022	−	0.818138i	$-0.304994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−18.8040			−1.01829
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$347$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−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
$359$	9.00692	−	3.73079i	0.475367	−	0.196903i	−0.132119	−	0.991234i	$-0.542178\pi$
$359$	9.00692	−	3.73079i	0.475367	−	0.196903i	0.607486	+	0.794330i	$0.292178\pi$
$360$		0			0
$361$	13.4350	−	13.4350i	0.707107	−	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.56297	+	5.05341i	0.394784	+	0.263786i	0.737079	−	0.675806i	$-0.236204\pi$
$367$	7.56297	+	5.05341i	0.394784	+	0.263786i	−0.342296	+	0.939592i	$0.611204\pi$
$368$	−4.35926	−	21.9155i	−0.227242	−	1.14242i
$369$	−10.6897	+	7.14260i	−0.556481	+	0.371829i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−15.5895	−	23.3313i	−0.800777	−	1.19845i	−0.976819	−	0.214065i	$-0.931330\pi$
$379$	−15.5895	−	23.3313i	−0.800777	−	1.19845i	0.176042	−	0.984383i	$-0.443670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$383$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.9104	+	13.9104i	0.707107	+	0.707107i
$388$	−7.60439	+	38.2298i	−0.386054	+	1.94083i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$	15.2391	+	17.2704i	0.770676	+	0.873400i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−16.3229	−	24.4289i	−0.820257	−	1.22760i
$397$	−37.7370	+	7.50636i	−1.89397	+	0.376734i	−0.997760	−	0.0669005i	$-0.978689\pi$
$397$	−37.7370	+	7.50636i	−1.89397	+	0.376734i	−0.896208	+	0.443634i	$0.853689\pi$
$398$		0			0
$399$		0			0
$400$	18.4776	+	7.65367i	0.923880	+	0.382683i
$401$	−31.0466	−	20.7447i	−1.55039	−	1.03594i	−0.976109	−	0.217281i	$-0.930281\pi$
$401$	−31.0466	−	20.7447i	−1.55039	−	1.03594i	−0.574283	−	0.818657i	$-0.694719\pi$
$402$		0			0
$403$	−21.5783	+	14.4181i	−1.07489	+	0.718218i
$404$	−28.3736	−	28.3736i	−1.41164	−	1.41164i
$405$		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
$412$	28.4091	−	28.4091i	1.39962	−	1.39962i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$419$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$	−10.0045	−	24.1531i	−0.486437	−	1.17436i
$424$		0			0
$425$	−20.4309	+	2.75319i	−0.991042	+	0.133549i
$426$		0			0
$427$		0			0
$428$	40.3882	+	8.03372i	1.95224	+	0.388325i
$429$		0			0
$430$		0			0
$431$	38.1848	−	7.59543i	1.83930	−	0.365859i	0.851848	−	0.523789i	$-0.175482\pi$
$431$	38.1848	−	7.59543i	1.83930	−	0.365859i	0.987450	+	0.157930i	$0.0504821\pi$
$432$		0			0
$433$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$433$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$434$		0			0
$435$		0			0
$436$	6.53539	+	32.8556i	0.312988	+	1.57350i
$437$		0			0
$438$		0			0
$439$	−0.819217	+	4.11848i	−0.0390991	+	0.196564i	−0.995398	−	0.0958262i	$-0.969451\pi$
$439$	−0.819217	+	4.11848i	−0.0390991	+	0.196564i	0.956299	+	0.292391i	$0.0944507\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$		0			0
$443$	−37.5579			−1.78443			−0.892215	−	0.451612i	$-0.850849\pi$
$443$	−37.5579			−1.78443			−0.892215	+	0.451612i	$0.850849\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$449$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$450$		0			0
$451$	19.3873	+	8.03050i	0.912915	+	0.378142i
$452$		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	$0.625000\pi$
$468$	−37.4622	−	15.5173i	−1.73169	−	0.717290i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.26437	−	31.4931i	0.288036	−	1.44805i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	33.0995			1.51552
$478$		0			0
$479$	42.5397	+	8.46167i	1.94369	+	0.386624i	0.998127	+	0.0611793i	$0.0194862\pi$
$479$	42.5397	+	8.46167i	1.94369	+	0.386624i	0.945561	+	0.325444i	$0.105514\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−9.93298	+	23.9803i	−0.451499	+	1.09002i
$485$		0			0
$486$		0			0
$487$	1.02019	+	5.12883i	0.0462292	+	0.232410i	0.996991	−	0.0775113i	$-0.0246974\pi$
$487$	1.02019	+	5.12883i	0.0462292	+	0.232410i	−0.950762	+	0.309921i	$0.899697\pi$
$488$		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
$496$	−15.0653	−	2.99667i	−0.676451	−	0.134555i
$497$		0			0
$498$		0			0
$499$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$499$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$503$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.01629	+	4.86775i	0.0894583	+	0.215971i
$509$			42.1546i			1.86847i	0.356658	+	0.934235i	$0.383916\pi$
$509$			42.1546i			1.86847i	−0.356658	+	0.934235i	$0.616084\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−23.7073	+	35.4805i	−1.04265	+	1.56043i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$521$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	14.9773	−	5.13511i	0.652422	−	0.223689i
$528$		0			0
$529$	7.58106	−	3.14018i	0.329611	−	0.136529i
$530$		0			0
$531$	19.2088	−	19.2088i	0.833589	−	0.833589i
$532$		0			0
$533$	28.4051	−	5.65013i	1.23036	−	0.244734i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−28.5004	+	19.0434i	−1.22760	+	0.820257i
$540$		0			0
$541$	7.56216	−	38.0176i	0.325123	−	1.63450i	−0.379693	−	0.925113i	$-0.623970\pi$
$541$	7.56216	−	38.0176i	0.325123	−	1.63450i	0.704816	−	0.709390i	$-0.251030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	42.4330	+	8.44044i	1.81430	+	0.360887i	0.981315	−	0.192406i	$-0.0616291\pi$
$547$	42.4330	+	8.44044i	1.81430	+	0.360887i	0.832987	+	0.553293i	$0.186629\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−27.5850	+	18.4317i	−1.16986	+	0.781678i
$557$	−27.0104	−	27.0104i	−1.14447	−	1.14447i	−0.987623	−	0.156844i	$-0.949868\pi$
$557$	−27.0104	−	27.0104i	−1.14447	−	1.14447i	−0.156844	−	0.987623i	$-0.550132\pi$
$558$		0			0
$559$	−16.9590	−	40.9427i	−0.717290	−	1.73169i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.2141	−	9.20140i	0.936214	−	0.387793i	0.138182	−	0.990407i	$-0.455874\pi$
$563$	22.2141	−	9.20140i	0.936214	−	0.387793i	0.798033	+	0.602614i	$0.205874\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.90324	+	21.4943i	−0.373243	+	0.901088i	0.619953	+	0.784639i	$0.287151\pi$
$569$	−8.90324	+	21.4943i	−0.373243	+	0.901088i	−0.993197	+	0.116450i	$0.962849\pi$
$570$		0			0
$571$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$571$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$572$	12.9122	+	64.9139i	0.539885	+	2.71419i
$573$		0			0
$574$		0			0
$575$	−5.44907	+	27.3943i	−0.227242	+	1.14242i
$576$	−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
$583$	−30.0155	−	44.9214i	−1.24312	−	1.86046i
$584$		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	$-0.362902\pi$
$599$	32.4577	−	32.4577i	1.32618	−	1.32618i	0.908671	−	0.417514i	$-0.137098\pi$
$600$		0			0
$601$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$601$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$602$		0			0
$603$	−13.0290	+	31.4547i	−0.530580	+	1.28093i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$607$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			58.8928i			2.38255i
$612$	19.6723	+	15.0000i	0.795206	+	0.606339i
$613$	−48.8406			−1.97265			−0.986327	−	0.164798i	$-0.947303\pi$
$613$	−48.8406			−1.97265			−0.986327	+	0.164798i	$0.947303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.3282	+	22.9402i	0.617089	+	0.923539i	1.00000		0.000246592i	$-7.84928e-5\pi$
$617$	15.3282	+	22.9402i	0.617089	+	0.923539i	−0.382911	+	0.923785i	$0.625078\pi$
$618$		0			0
$619$	24.7335	−	37.0163i	0.994123	−	1.48781i	0.125649	−	0.992075i	$-0.459899\pi$
$619$	24.7335	−	37.0163i	0.994123	−	1.48781i	0.868474	−	0.495735i	$-0.165101\pi$
$620$		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
$637$	−18.1036	+	43.7059i	−0.717290	+	1.73169i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$641$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$642$		0			0
$643$	−6.31034	+	31.7242i	−0.248856	+	1.25108i	0.630978	+	0.775800i	$0.282654\pi$
$643$	−6.31034	+	31.7242i	−0.248856	+	1.25108i	−0.879834	+	0.475281i	$0.842346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−43.4885	−	8.65040i	−1.70707	−	0.339558i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$653$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$654$		0			0
$655$		0			0
$656$	14.2529	+	9.52346i	0.556481	+	0.371829i
$657$		0			0
$658$		0			0
$659$	−36.2107	−	36.2107i	−1.41057	−	1.41057i	−0.756013	−	0.654557i	$-0.772855\pi$
$659$	−36.2107	−	36.2107i	−1.41057	−	1.41057i	−0.654557	−	0.756013i	$-0.727145\pi$
$660$		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
$668$	27.1450	+	40.6254i	1.05027	+	1.57184i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$673$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$674$		0			0
$675$		0			0
$676$	46.2056	+	46.2056i	1.77714	+	1.77714i
$677$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$677$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	48.9386	+	9.73449i	1.87258	+	0.372480i	0.994385	−	0.105819i	$-0.0337466\pi$
$683$	48.9386	+	9.73449i	1.87258	+	0.372480i	0.878197	+	0.478299i	$0.158747\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	10.0377	−	24.2331i	0.382683	−	0.923880i
$689$	−68.8878	−	28.5342i	−2.62441	−	1.08707i
$690$		0			0
$691$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$691$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$692$	21.4678	−	14.3443i	0.816082	−	0.545289i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17.6350	−	1.10185i	−0.667972	−	0.0417357i
$698$		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
$704$	−21.7639	+	32.5719i	−0.820257	+	1.22760i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−44.2064	+	29.5378i	−1.66021	+	1.10931i	−0.798262	+	0.602311i	$0.794247\pi$
$709$	−44.2064	+	29.5378i	−1.66021	+	1.10931i	−0.861944	+	0.507003i	$0.830753\pi$
$710$		0			0
$711$	−8.89844	+	44.7355i	−0.333717	+	1.67771i
$712$		0			0
$713$		−	21.4516i		−	0.803370i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.9729	−	29.8915i	−0.744862	−	1.11476i	−0.989413	−	0.145128i	$-0.953641\pi$
$719$	−19.9729	−	29.8915i	−0.744862	−	1.11476i	0.244551	−	0.969636i	$-0.421359\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	18.2955	+	12.2246i	0.679946	+	0.454325i
$725$		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
$731$	3.61077	+	26.7948i	0.133549	+	0.991042i
$732$		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
$737$	54.5042	−	10.8416i	2.00769	−	0.399354i
$738$		0			0
$739$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$739$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$743$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			19.6723i			0.719772i
$748$	2.51805	−	40.3010i	0.0920691	−	1.47355i
$749$		0			0
$750$		0			0
$751$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$751$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$752$	−24.6479	+	24.6479i	−0.898818	+	0.898818i
$753$		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
$767$	−56.5373	+	23.4185i	−2.04144	+	0.845594i
$768$		0			0
$769$	−2.43637	+	2.43637i	−0.0878579	+	0.0878579i	−0.749670	−	0.661812i	$-0.769788\pi$
$769$	−2.43637	+	2.43637i	−0.0878579	+	0.0878579i	0.661812	+	0.749670i	$0.269788\pi$
$770$		0			0
$771$		0			0
$772$	−30.8458	+	46.1640i	−1.11016	+	1.66148i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$		0			0
$775$	15.9647	+	10.6673i	0.573468	+	0.383179i
$776$		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
$787$	−5.82852	−	8.72299i	−0.207764	−	0.310941i	0.712923	−	0.701242i	$-0.247371\pi$
$787$	−5.82852	−	8.72299i	−0.207764	−	0.310941i	−0.920687	+	0.390301i	$0.872371\pi$
$788$	46.6361	−	9.27649i	1.66134	−	0.330461i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.95129	−	4.71083i	−0.0691182	−	0.166866i	0.885545	−	0.464553i	$-0.153785\pi$
$797$	−1.95129	−	4.71083i	−0.0691182	−	0.166866i	−0.954664	+	0.297687i	$0.903785\pi$
$798$		0			0
$799$	9.19354	−	34.7342i	0.325244	−	1.22881i
$800$		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
$809$	−9.43130	−	6.30180i	−0.331587	−	0.221559i	0.378620	−	0.925552i	$-0.376399\pi$
$809$	−9.43130	−	6.30180i	−0.331587	−	0.221559i	−0.710207	+	0.703993i	$0.751399\pi$
$810$		0			0
$811$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$811$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.9597	−	34.3617i	−0.801300	−	1.19923i	−0.976674	−	0.214729i	$-0.931113\pi$
$821$	−22.9597	−	34.3617i	−0.801300	−	1.19923i	0.175373	−	0.984502i	$-0.443887\pi$
$822$		0			0
$823$	−9.75114	+	14.5936i	−0.339903	+	0.508701i	−0.961562	−	0.274587i	$-0.911459\pi$
$823$	−9.75114	+	14.5936i	−0.339903	+	0.508701i	0.621659	+	0.783288i	$0.286459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.378050	+	1.90059i	0.0131461	+	0.0660898i	0.986801	−	0.161935i	$-0.0517734\pi$
$827$	0.378050	+	1.90059i	0.0131461	+	0.0660898i	−0.973655	+	0.228024i	$0.926773\pi$
$828$	27.8686	−	18.6212i	0.968499	−	0.647130i
$829$		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		0.980785	−	0.195090i	$-0.0625000\pi$
$839$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$840$		0			0
$841$	11.0978	−	26.7925i	0.382683	−	0.923880i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−16.8888	−	40.7733i	−0.579965	−	1.40016i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	49.1426	+	9.77506i	1.68261	+	0.334692i	0.941582	−	0.336784i	$-0.109339\pi$
$853$	49.1426	+	9.77506i	1.68261	+	0.334692i	0.741027	+	0.671475i	$0.234339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−32.2582	+	48.2779i	−1.10192	+	1.64914i	−0.444072	+	0.895991i	$0.646466\pi$
$857$	−32.2582	+	48.2779i	−1.10192	+	1.64914i	−0.657849	+	0.753150i	$0.728534\pi$
$858$		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
$869$	68.7827	−	28.4907i	2.33329	−	0.966482i
$870$		0			0
$871$	54.2326	−	54.2326i	1.83760	−	1.83760i
$872$		0			0
$873$	−57.3448	+	11.4066i	−1.94083	+	0.386054i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−13.6507	−	9.12111i	−0.460952	−	0.307998i	0.303324	−	0.952887i	$-0.401903\pi$
$877$	−13.6507	−	9.12111i	−0.460952	−	0.307998i	−0.764276	+	0.644889i	$0.776903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−10.2812	+	51.6872i	−0.346383	+	1.74139i	0.278287	+	0.960498i	$0.410233\pi$
$881$	−10.2812	+	51.6872i	−0.346383	+	1.74139i	−0.624670	+	0.780889i	$0.714767\pi$
$882$		0			0
$883$		−	36.7815i		−	1.23780i	−0.785471	−	0.618899i	$-0.787579\pi$
$883$		−	36.7815i		−	1.23780i	0.785471	−	0.618899i	$-0.212421\pi$
$884$	−28.0115	−	48.1775i	−0.942130	−	1.62038i
$885$		0			0
$886$		0			0
$887$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$887$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	24.4844	−	36.6434i	0.820257	−	1.22760i
$892$		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
$901$	36.1747	+	27.5829i	1.20515	+	0.918920i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	56.8160	−	11.3014i	1.88655	−	0.375257i	0.889834	−	0.456284i	$-0.150820\pi$
$907$	56.8160	−	11.3014i	1.88655	−	0.375257i	0.996712	+	0.0810272i	$0.0258201\pi$
$908$		0			0
$909$	23.0335	−	55.6078i	0.763973	−	1.84439i
$910$		0			0
$911$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$911$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$912$		0			0
$913$	26.6986	−	17.8394i	0.883593	−	0.590398i
$914$		0			0
$915$		0			0
$916$	−22.0253	−	53.1739i	−0.727738	−	1.75691i
$917$		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
$927$	55.6774	+	23.0623i	1.82869	+	0.757467i
$928$		0			0
$929$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$929$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$937$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−59.9330	+	11.9214i	−1.95376	+	0.388627i	−0.959490	+	0.281744i	$0.909087\pi$
$941$	−59.9330	+	11.9214i	−1.95376	+	0.388627i	−0.994272	+	0.106883i	$0.965913\pi$
$942$		0			0
$943$	−9.16120	+	22.1171i	−0.298330	+	0.720232i
$944$	−33.4633	−	13.8609i	−1.08914	−	0.451135i
$945$		0			0
$946$		0			0
$947$	−12.7945	+	8.54900i	−0.415765	+	0.277805i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$	−12.7945	+	8.54900i	−0.415765	+	0.277805i	0.330044	+	0.943966i	$0.392937\pi$
$948$		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
$961$	15.0163	+	6.21997i	0.484398	+	0.200644i
$962$		0			0
$963$	12.0506	+	60.5824i	0.388325	+	1.95224i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.6420	+	25.6920i	0.342223	+	0.826201i	0.997490	+	0.0708031i	$0.0225562\pi$
$967$	10.6420	+	25.6920i	0.342223	+	0.826201i	−0.655267	+	0.755397i	$0.727444\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.2047	+	5.46955i	−0.423757	+	0.175526i	−0.584363	−	0.811493i	$-0.698655\pi$
$971$	−13.2047	+	5.46955i	−0.423757	+	0.175526i	0.160605	+	0.987019i	$0.448655\pi$
$972$		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
$981$	−41.7805	+	27.9168i	−1.33395	+	0.891315i
$982$		0			0
$983$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$983$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	35.9273	+	7.14639i	1.14242	+	0.227242i
$990$		0			0
$991$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$991$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$997$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.a.343.1	✓	16
17.6	odd	16	inner	731.2.s.a.601.1	yes	16
43.42	odd	2	CM	731.2.s.a.343.1	✓	16
731.601	even	16	inner	731.2.s.a.601.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.a.343.1	✓	16	1.1	even	1	trivial
731.2.s.a.343.1	✓	16	43.42	odd	2	CM
731.2.s.a.601.1	yes	16	17.6	odd	16	inner
731.2.s.a.601.1	yes	16	731.601	even	16	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.41421i) q^{4} +(2.77164 + 1.14805i) q^{9} +(-0.955307 - 4.80265i) q^{11} +(-4.77872 - 4.77872i) q^{13} -4.00000i q^{16} +(2.07243 + 3.56441i) q^{17} +(5.47887 - 1.08981i) q^{23} +(-1.91342 + 4.61940i) q^{25} +(0.749168 - 3.76632i) q^{31} +(5.54328 - 2.29610i) q^{36} +(-2.38087 + 3.56322i) q^{41} +(6.05828 + 2.50942i) q^{43} +(-8.14298 - 5.44097i) q^{44} +(-6.16199 - 6.16199i) q^{47} +(-2.67878 - 6.46716i) q^{49} -13.5163 q^{52} +(10.1933 - 4.22221i) q^{53} +(3.46524 - 8.36582i) q^{59} +(-5.65685 - 5.65685i) q^{64} +11.3488i q^{67} +(7.97170 + 2.10997i) q^{68} +(2.96615 + 14.9118i) q^{79} +(6.36396 + 6.36396i) q^{81} +(2.50942 + 6.05828i) q^{83} +(6.20706 - 9.28952i) q^{92} +(-16.2049 + 10.8277i) q^{97} +(2.86592 - 14.4080i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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