LMFDB - Embedded newform 676.2.d.e.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,2,Mod(337,676)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("676.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.d (of order $2$, degree $1$, not 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:	$5.39788717664$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	676.337
Dual form			676.2.d.e.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35690 q^{3} +4.24698i q^{5} +2.13706i q^{7} -1.15883 q^{9} -3.35690i q^{11} +5.76271i q^{15} +0.0609989 q^{17} +6.18598i q^{19} +2.89977i q^{21} +5.09783 q^{23} -13.0368 q^{25} -5.64310 q^{27} -2.13706 q^{29} -2.85086i q^{31} -4.55496i q^{33} -9.07606 q^{35} -3.47219i q^{37} +5.39612i q^{41} +8.61356 q^{43} -4.92154i q^{45} +6.96077i q^{47} +2.43296 q^{49} +0.0827692 q^{51} +0.0217703 q^{53} +14.2567 q^{55} +8.39373i q^{57} +5.81163i q^{59} +10.6799 q^{61} -2.47650i q^{63} -7.40581i q^{67} +6.91723 q^{69} +0.0489173i q^{71} -4.51573i q^{73} -17.6896 q^{75} +7.17390 q^{77} +12.8170 q^{79} -4.18060 q^{81} +3.55496i q^{83} +0.259061i q^{85} -2.89977 q^{87} -2.34721i q^{89} -3.86831i q^{93} -26.2717 q^{95} -3.42327i q^{97} +3.89008i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{9} + 20 q^{17} - 6 q^{23} - 22 q^{25} - 42 q^{27} - 2 q^{29} - 24 q^{35} - 10 q^{43} - 24 q^{49} + 14 q^{51} - 6 q^{53} + 32 q^{55} + 16 q^{61} + 28 q^{69} - 14 q^{75} - 24 q^{77} + 18 q^{79}+ \cdots - 54 q^{95}+O(q^{100}) $ 6 * q + 10 * q^9 + 20 * q^17 - 6 * q^23 - 22 * q^25 - 42 * q^27 - 2 * q^29 - 24 * q^35 - 10 * q^43 - 24 * q^49 + 14 * q^51 - 6 * q^53 + 32 * q^55 + 16 * q^61 + 28 * q^69 - 14 * q^75 - 24 * q^77 + 18 * q^79 - 2 * q^81 + 28 * q^87 - 54 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$.

$n$	$339$	$509$
$\chi(n)$	$1$	$-1$

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
$2$	0	0
$3$	1.35690	0.783404	0.391702	−	0.920092i	$-0.371886\pi$
$3$	1.35690	0.783404	0.391702	+	0.920092i	$0.371886\pi$
$4$	0	0
$5$	4.24698i	1.89931i	0.313302	+	0.949654i	$0.398565\pi$
$5$	4.24698i	1.89931i	−0.313302	+	0.949654i	$0.601435\pi$
$6$	0	0
$7$	2.13706i	0.807734i	0.914818	+	0.403867i	$0.132334\pi$
$7$	2.13706i	0.807734i	−0.914818	+	0.403867i	$0.867666\pi$
$8$	0	0
$9$	−1.15883	−0.386278
$10$	0	0
$11$	− 3.35690i	− 1.01214i	−0.862492	−	0.506071i	$-0.831097\pi$
$11$	− 3.35690i	− 1.01214i	0.862492	−	0.506071i	$-0.168903\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.76271i	1.48793i
$16$	0	0
$17$	0.0609989	0.0147944	0.00739721	−	0.999973i	$-0.497645\pi$
$17$	0.0609989	0.0147944	0.00739721	+	0.999973i	$0.497645\pi$
$18$	0	0
$19$	6.18598i	1.41916i	0.704624	+	0.709581i	$0.251116\pi$
$19$	6.18598i	1.41916i	−0.704624	+	0.709581i	$0.748884\pi$
$20$	0	0
$21$	2.89977i	0.632782i
$22$	0	0
$23$	5.09783	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$23$	5.09783	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$24$	0	0
$25$	−13.0368	−2.60737
$26$	0	0
$27$	−5.64310	−1.08602
$28$	0	0
$29$	−2.13706	−0.396843	−0.198421	−	0.980117i	$-0.563581\pi$
$29$	−2.13706	−0.396843	−0.198421	+	0.980117i	$0.563581\pi$
$30$	0	0
$31$	− 2.85086i	− 0.512029i	−0.966673	−	0.256014i	$-0.917591\pi$
$31$	− 2.85086i	− 0.512029i	0.966673	−	0.256014i	$-0.0824094\pi$
$32$	0	0
$33$	− 4.55496i	− 0.792916i
$34$	0	0
$35$	−9.07606	−1.53413
$36$	0	0
$37$	− 3.47219i	− 0.570824i	−0.958405	−	0.285412i	$-0.907870\pi$
$37$	− 3.47219i	− 0.570824i	0.958405	−	0.285412i	$-0.0921305\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.39612i	0.842733i	0.906890	+	0.421367i	$0.138449\pi$
$41$	5.39612i	0.842733i	−0.906890	+	0.421367i	$0.861551\pi$
$42$	0	0
$43$	8.61356	1.31356	0.656778	−	0.754084i	$-0.271919\pi$
$43$	8.61356	1.31356	0.656778	+	0.754084i	$0.271919\pi$
$44$	0	0
$45$	− 4.92154i	− 0.733660i
$46$	0	0
$47$	6.96077i	1.01533i	0.861554	+	0.507666i	$0.169492\pi$
$47$	6.96077i	1.01533i	−0.861554	+	0.507666i	$0.830508\pi$
$48$	0	0
$49$	2.43296	0.347566
$50$	0	0
$51$	0.0827692	0.0115900
$52$	0	0
$53$	0.0217703	0.00299038	0.00149519	−	0.999999i	$-0.499524\pi$
$53$	0.0217703	0.00299038	0.00149519	+	0.999999i	$0.499524\pi$
$54$	0	0
$55$	14.2567	1.92237
$56$	0	0
$57$	8.39373i	1.11178i
$58$	0	0
$59$	5.81163i	0.756609i	0.925681	+	0.378305i	$0.123493\pi$
$59$	5.81163i	0.756609i	−0.925681	+	0.378305i	$0.876507\pi$
$60$	0	0
$61$	10.6799	1.36743	0.683713	−	0.729751i	$-0.260364\pi$
$61$	10.6799	1.36743	0.683713	+	0.729751i	$0.260364\pi$
$62$	0	0
$63$	− 2.47650i	− 0.312010i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.40581i	− 0.904764i	−0.891824	−	0.452382i	$-0.850574\pi$
$67$	− 7.40581i	− 0.904764i	0.891824	−	0.452382i	$-0.149426\pi$
$68$	0	0
$69$	6.91723	0.832737
$70$	0	0
$71$	0.0489173i	0.00580542i	0.999996	+	0.00290271i	$0.000923963\pi$
$71$	0.0489173i	0.00580542i	−0.999996	+	0.00290271i	$0.999076\pi$
$72$	0	0
$73$	− 4.51573i	− 0.528526i	−0.964451	−	0.264263i	$-0.914871\pi$
$73$	− 4.51573i	− 0.528526i	0.964451	−	0.264263i	$-0.0851287\pi$
$74$	0	0
$75$	−17.6896	−2.04262
$76$	0	0
$77$	7.17390	0.817542
$78$	0	0
$79$	12.8170	1.44203	0.721013	−	0.692922i	$-0.243677\pi$
$79$	12.8170	1.44203	0.721013	+	0.692922i	$0.243677\pi$
$80$	0	0
$81$	−4.18060	−0.464512
$82$	0	0
$83$	3.55496i	0.390207i	0.980783	+	0.195104i	$0.0625043\pi$
$83$	3.55496i	0.390207i	−0.980783	+	0.195104i	$0.937496\pi$
$84$	0	0
$85$	0.259061i	0.0280991i
$86$	0	0
$87$	−2.89977	−0.310888
$88$	0	0
$89$	− 2.34721i	− 0.248803i	−0.992232	−	0.124402i	$-0.960299\pi$
$89$	− 2.34721i	− 0.248803i	0.992232	−	0.124402i	$-0.0397012\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 3.86831i	− 0.401125i
$94$	0	0
$95$	−26.2717	−2.69542
$96$	0	0
$97$	− 3.42327i	− 0.347581i	−0.984783	−	0.173790i	$-0.944399\pi$
$97$	− 3.42327i	− 0.347581i	0.984783	−	0.173790i	$-0.0556015\pi$
$98$	0	0
$99$	3.89008i	0.390968i
$100$	0	0
$101$	11.3787	1.13222	0.566110	−	0.824330i	$-0.308448\pi$
$101$	11.3787	1.13222	0.566110	+	0.824330i	$0.308448\pi$
$102$	0	0
$103$	6.07606	0.598692	0.299346	−	0.954145i	$-0.403231\pi$
$103$	6.07606	0.598692	0.299346	+	0.954145i	$0.403231\pi$
$104$	0	0
$105$	−12.3153	−1.20185
$106$	0	0
$107$	−16.0368	−1.55034	−0.775170	−	0.631753i	$-0.782336\pi$
$107$	−16.0368	−1.55034	−0.775170	+	0.631753i	$0.782336\pi$
$108$	0	0
$109$	− 14.9095i	− 1.42807i	−0.700111	−	0.714034i	$-0.746866\pi$
$109$	− 14.9095i	− 1.42807i	0.700111	−	0.714034i	$-0.253134\pi$
$110$	0	0
$111$	− 4.71140i	− 0.447186i
$112$	0	0
$113$	8.24459	0.775585	0.387793	−	0.921747i	$-0.373238\pi$
$113$	8.24459	0.775585	0.387793	+	0.921747i	$0.373238\pi$
$114$	0	0
$115$	21.6504i	2.01891i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.130359i	0.0119500i
$120$	0	0
$121$	−0.268750	−0.0244318
$122$	0	0
$123$	7.32198i	0.660201i
$124$	0	0
$125$	− 34.1323i	− 3.05288i
$126$	0	0
$127$	6.64310	0.589480	0.294740	−	0.955577i	$-0.404767\pi$
$127$	6.64310	0.589480	0.294740	+	0.955577i	$0.404767\pi$
$128$	0	0
$129$	11.6877	1.02905
$130$	0	0
$131$	7.86294	0.686988	0.343494	−	0.939155i	$-0.388390\pi$
$131$	7.86294	0.686988	0.343494	+	0.939155i	$0.388390\pi$
$132$	0	0
$133$	−13.2198	−1.14630
$134$	0	0
$135$	− 23.9661i	− 2.06268i
$136$	0	0
$137$	0.987918i	0.0844036i	0.999109	+	0.0422018i	$0.0134372\pi$
$137$	0.987918i	0.0844036i	−0.999109	+	0.0422018i	$0.986563\pi$
$138$	0	0
$139$	−12.0368	−1.02095	−0.510476	−	0.859892i	$-0.670531\pi$
$139$	−12.0368	−1.02095	−0.510476	+	0.859892i	$0.670531\pi$
$140$	0	0
$141$	9.44504i	0.795416i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 9.07606i	− 0.753726i
$146$	0	0
$147$	3.30127	0.272284
$148$	0	0
$149$	2.35929i	0.193280i	0.995319	+	0.0966402i	$0.0308096\pi$
$149$	2.35929i	0.193280i	−0.995319	+	0.0966402i	$0.969190\pi$
$150$	0	0
$151$	7.43967i	0.605431i	0.953081	+	0.302716i	$0.0978933\pi$
$151$	7.43967i	0.605431i	−0.953081	+	0.302716i	$0.902107\pi$
$152$	0	0
$153$	−0.0706876	−0.00571475
$154$	0	0
$155$	12.1075	0.972500
$156$	0	0
$157$	−10.9487	−0.873801	−0.436900	−	0.899510i	$-0.643924\pi$
$157$	−10.9487	−0.873801	−0.436900	+	0.899510i	$0.643924\pi$
$158$	0	0
$159$	0.0295400	0.00234267
$160$	0	0
$161$	10.8944i	0.858599i
$162$	0	0
$163$	− 7.02475i	− 0.550221i	−0.961413	−	0.275111i	$-0.911286\pi$
$163$	− 7.02475i	− 0.550221i	0.961413	−	0.275111i	$-0.0887145\pi$
$164$	0	0
$165$	19.3448	1.50599
$166$	0	0
$167$	15.4819i	1.19802i	0.800740	+	0.599012i	$0.204440\pi$
$167$	15.4819i	1.19802i	−0.800740	+	0.599012i	$0.795560\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 7.16852i	− 0.548191i
$172$	0	0
$173$	−13.7802	−1.04769	−0.523843	−	0.851815i	$-0.675502\pi$
$173$	−13.7802	−1.04769	−0.523843	+	0.851815i	$0.675502\pi$
$174$	0	0
$175$	− 27.8605i	− 2.10606i
$176$	0	0
$177$	7.88577i	0.592731i
$178$	0	0
$179$	−22.1226	−1.65352	−0.826760	−	0.562555i	$-0.809819\pi$
$179$	−22.1226	−1.65352	−0.826760	+	0.562555i	$0.809819\pi$
$180$	0	0
$181$	13.6407	1.01391	0.506953	−	0.861974i	$-0.330772\pi$
$181$	13.6407	1.01391	0.506953	+	0.861974i	$0.330772\pi$
$182$	0	0
$183$	14.4916	1.07125
$184$	0	0
$185$	14.7463	1.08417
$186$	0	0
$187$	− 0.204767i	− 0.0149740i
$188$	0	0
$189$	− 12.0597i	− 0.877212i
$190$	0	0
$191$	4.49157	0.324998	0.162499	−	0.986709i	$-0.448045\pi$
$191$	4.49157	0.324998	0.162499	+	0.986709i	$0.448045\pi$
$192$	0	0
$193$	− 25.7754i	− 1.85535i	−0.373385	−	0.927676i	$-0.621803\pi$
$193$	− 25.7754i	− 1.85535i	0.373385	−	0.927676i	$-0.378197\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.6233i	− 1.32685i	−0.748242	−	0.663426i	$-0.769102\pi$
$197$	− 18.6233i	− 1.32685i	0.748242	−	0.663426i	$-0.230898\pi$
$198$	0	0
$199$	11.9661	0.848258	0.424129	−	0.905602i	$-0.360580\pi$
$199$	11.9661	0.848258	0.424129	+	0.905602i	$0.360580\pi$
$200$	0	0
$201$	− 10.0489i	− 0.708796i
$202$	0	0
$203$	− 4.56704i	− 0.320543i
$204$	0	0
$205$	−22.9172	−1.60061
$206$	0	0
$207$	−5.90754	−0.410603
$208$	0	0
$209$	20.7657	1.43639
$210$	0	0
$211$	2.96077	0.203828	0.101914	−	0.994793i	$-0.467503\pi$
$211$	2.96077	0.203828	0.101914	+	0.994793i	$0.467503\pi$
$212$	0	0
$213$	0.0663757i	0.00454799i
$214$	0	0
$215$	36.5816i	2.49485i
$216$	0	0
$217$	6.09246	0.413583
$218$	0	0
$219$	− 6.12737i	− 0.414050i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 5.44265i	− 0.364467i	−0.983255	−	0.182233i	$-0.941667\pi$
$223$	− 5.44265i	− 0.364467i	0.983255	−	0.182233i	$-0.0583326\pi$
$224$	0	0
$225$	15.1075	1.00717
$226$	0	0
$227$	22.4480i	1.48993i	0.667105	+	0.744964i	$0.267533\pi$
$227$	22.4480i	1.48993i	−0.667105	+	0.744964i	$0.732467\pi$
$228$	0	0
$229$	16.6136i	1.09786i	0.835870	+	0.548928i	$0.184964\pi$
$229$	16.6136i	1.09786i	−0.835870	+	0.548928i	$0.815036\pi$
$230$	0	0
$231$	9.73423	0.640466
$232$	0	0
$233$	−9.73125	−0.637515	−0.318758	−	0.947836i	$-0.603266\pi$
$233$	−9.73125	−0.637515	−0.318758	+	0.947836i	$0.603266\pi$
$234$	0	0
$235$	−29.5623	−1.92843
$236$	0	0
$237$	17.3913	1.12969
$238$	0	0
$239$	− 2.82908i	− 0.182998i	−0.995805	−	0.0914991i	$-0.970834\pi$
$239$	− 2.82908i	− 0.182998i	0.995805	−	0.0914991i	$-0.0291659\pi$
$240$	0	0
$241$	− 18.3448i	− 1.18169i	−0.806784	−	0.590847i	$-0.798794\pi$
$241$	− 18.3448i	− 1.18169i	0.806784	−	0.590847i	$-0.201206\pi$
$242$	0	0
$243$	11.2567	0.722116
$244$	0	0
$245$	10.3327i	0.660134i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.82371i	0.305690i
$250$	0	0
$251$	2.53750	0.160166	0.0800828	−	0.996788i	$-0.474482\pi$
$251$	2.53750	0.160166	0.0800828	+	0.996788i	$0.474482\pi$
$252$	0	0
$253$	− 17.1129i	− 1.07588i
$254$	0	0
$255$	0.351519i	0.0220130i
$256$	0	0
$257$	23.4862	1.46503	0.732514	−	0.680752i	$-0.238347\pi$
$257$	23.4862	1.46503	0.732514	+	0.680752i	$0.238347\pi$
$258$	0	0
$259$	7.42029	0.461074
$260$	0	0
$261$	2.47650	0.153292
$262$	0	0
$263$	−15.8092	−0.974839	−0.487420	−	0.873168i	$-0.662062\pi$
$263$	−15.8092	−0.974839	−0.487420	+	0.873168i	$0.662062\pi$
$264$	0	0
$265$	0.0924579i	0.00567964i
$266$	0	0
$267$	− 3.18492i	− 0.194914i
$268$	0	0
$269$	−30.8582	−1.88145	−0.940727	−	0.339164i	$-0.889856\pi$
$269$	−30.8582	−1.88145	−0.940727	+	0.339164i	$0.889856\pi$
$270$	0	0
$271$	− 16.1933i	− 0.983671i	−0.870688	−	0.491836i	$-0.836326\pi$
$271$	− 16.1933i	− 0.983671i	0.870688	−	0.491836i	$-0.163674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.7633i	2.63903i
$276$	0	0
$277$	−13.1075	−0.787555	−0.393777	−	0.919206i	$-0.628832\pi$
$277$	−13.1075	−0.787555	−0.393777	+	0.919206i	$0.628832\pi$
$278$	0	0
$279$	3.30367i	0.197785i
$280$	0	0
$281$	− 14.2687i	− 0.851202i	−0.904911	−	0.425601i	$-0.860063\pi$
$281$	− 14.2687i	− 0.851202i	0.904911	−	0.425601i	$-0.139937\pi$
$282$	0	0
$283$	7.95646	0.472962	0.236481	−	0.971636i	$-0.424006\pi$
$283$	7.95646	0.472962	0.236481	+	0.971636i	$0.424006\pi$
$284$	0	0
$285$	−35.6480	−2.11161
$286$	0	0
$287$	−11.5319	−0.680704
$288$	0	0
$289$	−16.9963	−0.999781
$290$	0	0
$291$	− 4.64502i	− 0.272296i
$292$	0	0
$293$	− 19.6300i	− 1.14679i	−0.819278	−	0.573397i	$-0.805625\pi$
$293$	− 19.6300i	− 1.14679i	0.819278	−	0.573397i	$-0.194375\pi$
$294$	0	0
$295$	−24.6819	−1.43703
$296$	0	0
$297$	18.9433i	1.09920i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	18.4077i	1.06100i
$302$	0	0
$303$	15.4397	0.886986
$304$	0	0
$305$	45.3575i	2.59716i
$306$	0	0
$307$	15.6775i	0.894765i	0.894343	+	0.447382i	$0.147644\pi$
$307$	15.6775i	0.894765i	−0.894343	+	0.447382i	$0.852356\pi$
$308$	0	0
$309$	8.24459	0.469018
$310$	0	0
$311$	−16.4983	−0.935531	−0.467766	−	0.883853i	$-0.654941\pi$
$311$	−16.4983	−0.935531	−0.467766	+	0.883853i	$0.654941\pi$
$312$	0	0
$313$	−11.3937	−0.644012	−0.322006	−	0.946738i	$-0.604357\pi$
$313$	−11.3937	−0.644012	−0.322006	+	0.946738i	$0.604357\pi$
$314$	0	0
$315$	10.5176	0.592602
$316$	0	0
$317$	25.4470i	1.42924i	0.699511	+	0.714622i	$0.253401\pi$
$317$	25.4470i	1.42924i	−0.699511	+	0.714622i	$0.746599\pi$
$318$	0	0
$319$	7.17390i	0.401661i
$320$	0	0
$321$	−21.7603	−1.21454
$322$	0	0
$323$	0.377338i	0.0209957i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.2306i	− 1.11875i
$328$	0	0
$329$	−14.8756	−0.820119
$330$	0	0
$331$	8.95539i	0.492233i	0.969240	+	0.246117i	$0.0791546\pi$
$331$	8.95539i	0.492233i	−0.969240	+	0.246117i	$0.920845\pi$
$332$	0	0
$333$	4.02369i	0.220497i
$334$	0	0
$335$	31.4523	1.71842
$336$	0	0
$337$	−19.3110	−1.05194	−0.525968	−	0.850505i	$-0.676297\pi$
$337$	−19.3110	−1.05194	−0.525968	+	0.850505i	$0.676297\pi$
$338$	0	0
$339$	11.1870	0.607597
$340$	0	0
$341$	−9.57002	−0.518246
$342$	0	0
$343$	20.1588i	1.08847i
$344$	0	0
$345$	29.3773i	1.58162i
$346$	0	0
$347$	36.0411	1.93479	0.967395	−	0.253272i	$-0.0815068\pi$
$347$	36.0411	1.93479	0.967395	+	0.253272i	$0.0815068\pi$
$348$	0	0
$349$	− 23.2416i	− 1.24409i	−0.782980	−	0.622047i	$-0.786301\pi$
$349$	− 23.2416i	− 1.24409i	0.782980	−	0.622047i	$-0.213699\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.2349i	1.28989i	0.764228	+	0.644947i	$0.223120\pi$
$353$	24.2349i	1.28989i	−0.764228	+	0.644947i	$0.776880\pi$
$354$	0	0
$355$	−0.207751	−0.0110263
$356$	0	0
$357$	0.176883i	0.00936164i
$358$	0	0
$359$	− 17.7332i	− 0.935921i	−0.883749	−	0.467960i	$-0.844989\pi$
$359$	− 17.7332i	− 0.935921i	0.883749	−	0.467960i	$-0.155011\pi$
$360$	0	0
$361$	−19.2664	−1.01402
$362$	0	0
$363$	−0.364666	−0.0191400
$364$	0	0
$365$	19.1782	1.00383
$366$	0	0
$367$	30.2083	1.57686	0.788431	−	0.615123i	$-0.210894\pi$
$367$	30.2083	1.57686	0.788431	+	0.615123i	$0.210894\pi$
$368$	0	0
$369$	− 6.25321i	− 0.325529i
$370$	0	0
$371$	0.0465244i	0.00241543i
$372$	0	0
$373$	−0.681268	−0.0352747	−0.0176374	−	0.999844i	$-0.505614\pi$
$373$	−0.681268	−0.0352747	−0.0176374	+	0.999844i	$0.505614\pi$
$374$	0	0
$375$	− 46.3139i	− 2.39164i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	36.4959i	1.87467i	0.348433	+	0.937334i	$0.386714\pi$
$379$	36.4959i	1.87467i	−0.348433	+	0.937334i	$0.613286\pi$
$380$	0	0
$381$	9.01400	0.461801
$382$	0	0
$383$	− 5.20775i	− 0.266104i	−0.991109	−	0.133052i	$-0.957522\pi$
$383$	− 5.20775i	− 0.266104i	0.991109	−	0.133052i	$-0.0424777\pi$
$384$	0	0
$385$	30.4674i	1.55276i
$386$	0	0
$387$	−9.98169	−0.507398
$388$	0	0
$389$	17.5254	0.888574	0.444287	−	0.895885i	$-0.353457\pi$
$389$	17.5254	0.888574	0.444287	+	0.895885i	$0.353457\pi$
$390$	0	0
$391$	0.310962	0.0157260
$392$	0	0
$393$	10.6692	0.538189
$394$	0	0
$395$	54.4336i	2.73885i
$396$	0	0
$397$	29.8049i	1.49587i	0.663774	+	0.747933i	$0.268954\pi$
$397$	29.8049i	1.49587i	−0.663774	+	0.747933i	$0.731046\pi$
$398$	0	0
$399$	−17.9379	−0.898020
$400$	0	0
$401$	− 16.3341i	− 0.815684i	−0.913053	−	0.407842i	$-0.866281\pi$
$401$	− 16.3341i	− 0.815684i	0.913053	−	0.407842i	$-0.133719\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 17.7549i	− 0.882250i
$406$	0	0
$407$	−11.6558	−0.577755
$408$	0	0
$409$	− 9.49827i	− 0.469659i	−0.972037	−	0.234830i	$-0.924547\pi$
$409$	− 9.49827i	− 0.469659i	0.972037	−	0.234830i	$-0.0754532\pi$
$410$	0	0
$411$	1.34050i	0.0661221i
$412$	0	0
$413$	−12.4198	−0.611139
$414$	0	0
$415$	−15.0978	−0.741124
$416$	0	0
$417$	−16.3327	−0.799817
$418$	0	0
$419$	12.4450	0.607980	0.303990	−	0.952675i	$-0.401681\pi$
$419$	12.4450	0.607980	0.303990	+	0.952675i	$0.401681\pi$
$420$	0	0
$421$	3.65578i	0.178172i	0.996024	+	0.0890858i	$0.0283945\pi$
$421$	3.65578i	0.178172i	−0.996024	+	0.0890858i	$0.971605\pi$
$422$	0	0
$423$	− 8.06638i	− 0.392201i
$424$	0	0
$425$	−0.795233	−0.0385745
$426$	0	0
$427$	22.8237i	1.10452i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 35.6219i	− 1.71585i	−0.513777	−	0.857924i	$-0.671754\pi$
$431$	− 35.6219i	− 1.71585i	0.513777	−	0.857924i	$-0.328246\pi$
$432$	0	0
$433$	34.6926	1.66722	0.833610	−	0.552353i	$-0.186270\pi$
$433$	34.6926	1.66722	0.833610	+	0.552353i	$0.186270\pi$
$434$	0	0
$435$	− 12.3153i	− 0.590472i
$436$	0	0
$437$	31.5351i	1.50853i
$438$	0	0
$439$	−9.48858	−0.452865	−0.226433	−	0.974027i	$-0.572706\pi$
$439$	−9.48858	−0.452865	−0.226433	+	0.974027i	$0.572706\pi$
$440$	0	0
$441$	−2.81940	−0.134257
$442$	0	0
$443$	14.1849	0.673946	0.336973	−	0.941514i	$-0.390597\pi$
$443$	14.1849	0.673946	0.336973	+	0.941514i	$0.390597\pi$
$444$	0	0
$445$	9.96854	0.472554
$446$	0	0
$447$	3.20131i	0.151417i
$448$	0	0
$449$	6.99894i	0.330300i	0.986268	+	0.165150i	$0.0528109\pi$
$449$	6.99894i	0.330300i	−0.986268	+	0.165150i	$0.947189\pi$
$450$	0	0
$451$	18.1142	0.852966
$452$	0	0
$453$	10.0949i	0.474297i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.21313i	0.150304i	0.997172	+	0.0751519i	$0.0239442\pi$
$457$	3.21313i	0.150304i	−0.997172	+	0.0751519i	$0.976056\pi$
$458$	0	0
$459$	−0.344223	−0.0160670
$460$	0	0
$461$	15.5278i	0.723202i	0.932333	+	0.361601i	$0.117770\pi$
$461$	15.5278i	0.723202i	−0.932333	+	0.361601i	$0.882230\pi$
$462$	0	0
$463$	− 32.0834i	− 1.49104i	−0.666483	−	0.745520i	$-0.732201\pi$
$463$	− 32.0834i	− 1.49104i	0.666483	−	0.745520i	$-0.267799\pi$
$464$	0	0
$465$	16.4286	0.761860
$466$	0	0
$467$	−7.24459	−0.335239	−0.167620	−	0.985852i	$-0.553608\pi$
$467$	−7.24459	−0.335239	−0.167620	+	0.985852i	$0.553608\pi$
$468$	0	0
$469$	15.8267	0.730809
$470$	0	0
$471$	−14.8562	−0.684539
$472$	0	0
$473$	− 28.9148i	− 1.32951i
$474$	0	0
$475$	− 80.6456i	− 3.70027i
$476$	0	0
$477$	−0.0252281	−0.00115512
$478$	0	0
$479$	− 5.78209i	− 0.264190i	−0.991237	−	0.132095i	$-0.957830\pi$
$479$	− 5.78209i	− 0.264190i	0.991237	−	0.132095i	$-0.0421704\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.7826i	0.672630i
$484$	0	0
$485$	14.5386	0.660162
$486$	0	0
$487$	4.14782i	0.187956i	0.995574	+	0.0939778i	$0.0299583\pi$
$487$	4.14782i	0.187956i	−0.995574	+	0.0939778i	$0.970042\pi$
$488$	0	0
$489$	− 9.53186i	− 0.431046i
$490$	0	0
$491$	−7.19375	−0.324649	−0.162325	−	0.986737i	$-0.551899\pi$
$491$	−7.19375	−0.324649	−0.162325	+	0.986737i	$0.551899\pi$
$492$	0	0
$493$	−0.130359	−0.00587105
$494$	0	0
$495$	−16.5211	−0.742569
$496$	0	0
$497$	−0.104539	−0.00468924
$498$	0	0
$499$	− 41.5066i	− 1.85809i	−0.369964	−	0.929046i	$-0.620630\pi$
$499$	− 41.5066i	− 1.85809i	0.369964	−	0.929046i	$-0.379370\pi$
$500$	0	0
$501$	21.0073i	0.938537i
$502$	0	0
$503$	−25.8605	−1.15306	−0.576532	−	0.817074i	$-0.695595\pi$
$503$	−25.8605	−1.15306	−0.576532	+	0.817074i	$0.695595\pi$
$504$	0	0
$505$	48.3250i	2.15043i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.3924i	1.56874i	0.620293	+	0.784370i	$0.287014\pi$
$509$	35.3924i	1.56874i	−0.620293	+	0.784370i	$0.712986\pi$
$510$	0	0
$511$	9.65040	0.426909
$512$	0	0
$513$	− 34.9081i	− 1.54123i
$514$	0	0
$515$	25.8049i	1.13710i
$516$	0	0
$517$	23.3666	1.02766
$518$	0	0
$519$	−18.6983	−0.820762
$520$	0	0
$521$	−34.2620	−1.50105	−0.750524	−	0.660844i	$-0.770199\pi$
$521$	−34.2620	−1.50105	−0.750524	+	0.660844i	$0.770199\pi$
$522$	0	0
$523$	−14.6028	−0.638536	−0.319268	−	0.947664i	$-0.603437\pi$
$523$	−14.6028	−0.638536	−0.319268	+	0.947664i	$0.603437\pi$
$524$	0	0
$525$	− 37.8039i	− 1.64990i
$526$	0	0
$527$	− 0.173899i	− 0.00757516i
$528$	0	0
$529$	2.98792	0.129909
$530$	0	0
$531$	− 6.73471i	− 0.292261i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 68.1081i	− 2.94457i
$536$	0	0
$537$	−30.0180	−1.29537
$538$	0	0
$539$	− 8.16719i	− 0.351786i
$540$	0	0
$541$	− 18.8866i	− 0.811999i	−0.913873	−	0.406000i	$-0.866923\pi$
$541$	− 18.8866i	− 0.811999i	0.913873	−	0.406000i	$-0.133077\pi$
$542$	0	0
$543$	18.5090	0.794298
$544$	0	0
$545$	63.3202	2.71234
$546$	0	0
$547$	−34.6752	−1.48260	−0.741301	−	0.671172i	$-0.765791\pi$
$547$	−34.6752	−1.48260	−0.741301	+	0.671172i	$0.765791\pi$
$548$	0	0
$549$	−12.3763	−0.528206
$550$	0	0
$551$	− 13.2198i	− 0.563184i
$552$	0	0
$553$	27.3907i	1.16477i
$554$	0	0
$555$	20.0092	0.849344
$556$	0	0
$557$	4.25667i	0.180361i	0.995925	+	0.0901804i	$0.0287444\pi$
$557$	4.25667i	0.180361i	−0.995925	+	0.0901804i	$0.971256\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 0.277848i	− 0.0117307i
$562$	0	0
$563$	−21.1371	−0.890821	−0.445411	−	0.895326i	$-0.646942\pi$
$563$	−21.1371	−0.890821	−0.445411	+	0.895326i	$0.646942\pi$
$564$	0	0
$565$	35.0146i	1.47307i
$566$	0	0
$567$	− 8.93422i	− 0.375202i
$568$	0	0
$569$	−35.4795	−1.48738	−0.743689	−	0.668526i	$-0.766926\pi$
$569$	−35.4795	−1.48738	−0.743689	+	0.668526i	$0.766926\pi$
$570$	0	0
$571$	−3.94033	−0.164898	−0.0824488	−	0.996595i	$-0.526274\pi$
$571$	−3.94033	−0.164898	−0.0824488	+	0.996595i	$0.526274\pi$
$572$	0	0
$573$	6.09459	0.254605
$574$	0	0
$575$	−66.4596	−2.77156
$576$	0	0
$577$	− 16.6256i	− 0.692135i	−0.938210	−	0.346067i	$-0.887517\pi$
$577$	− 16.6256i	− 0.692135i	0.938210	−	0.346067i	$-0.112483\pi$
$578$	0	0
$579$	− 34.9745i	− 1.45349i
$580$	0	0
$581$	−7.59717	−0.315184
$582$	0	0
$583$	− 0.0730805i	− 0.00302669i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.00298i	− 0.330318i	−0.986267	−	0.165159i	$-0.947186\pi$
$587$	− 8.00298i	− 0.330318i	0.986267	−	0.165159i	$-0.0528138\pi$
$588$	0	0
$589$	17.6353	0.726651
$590$	0	0
$591$	− 25.2698i	− 1.03946i
$592$	0	0
$593$	− 15.8345i	− 0.650243i	−0.945672	−	0.325122i	$-0.894595\pi$
$593$	− 15.8345i	− 0.650243i	0.945672	−	0.325122i	$-0.105405\pi$
$594$	0	0
$595$	−0.553630	−0.0226966
$596$	0	0
$597$	16.2368	0.664529
$598$	0	0
$599$	25.9226	1.05917	0.529585	−	0.848257i	$-0.322348\pi$
$599$	25.9226	1.05917	0.529585	+	0.848257i	$0.322348\pi$
$600$	0	0
$601$	20.0616	0.818329	0.409165	−	0.912461i	$-0.365820\pi$
$601$	20.0616	0.818329	0.409165	+	0.912461i	$0.365820\pi$
$602$	0	0
$603$	8.58211i	0.349490i
$604$	0	0
$605$	− 1.14138i	− 0.0464035i
$606$	0	0
$607$	41.7614	1.69504	0.847521	−	0.530762i	$-0.178094\pi$
$607$	41.7614	1.69504	0.847521	+	0.530762i	$0.178094\pi$
$608$	0	0
$609$	− 6.19700i	− 0.251115i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 12.4179i	− 0.501554i	−0.968045	−	0.250777i	$-0.919314\pi$
$613$	− 12.4179i	− 0.501554i	0.968045	−	0.250777i	$-0.0806861\pi$
$614$	0	0
$615$	−31.0963	−1.25392
$616$	0	0
$617$	− 17.2198i	− 0.693244i	−0.938005	−	0.346622i	$-0.887329\pi$
$617$	− 17.2198i	− 0.693244i	0.938005	−	0.346622i	$-0.112671\pi$
$618$	0	0
$619$	− 42.6136i	− 1.71278i	−0.516326	−	0.856392i	$-0.672701\pi$
$619$	− 42.6136i	− 1.71278i	0.516326	−	0.856392i	$-0.327299\pi$
$620$	0	0
$621$	−28.7676	−1.15440
$622$	0	0
$623$	5.01613	0.200967
$624$	0	0
$625$	79.7749	3.19100
$626$	0	0
$627$	28.1769	1.12528
$628$	0	0
$629$	− 0.211800i	− 0.00844501i
$630$	0	0
$631$	26.8605i	1.06930i	0.845073	+	0.534651i	$0.179557\pi$
$631$	26.8605i	1.06930i	−0.845073	+	0.534651i	$0.820443\pi$
$632$	0	0
$633$	4.01746	0.159680
$634$	0	0
$635$	28.2131i	1.11960i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 0.0566871i	− 0.00224251i
$640$	0	0
$641$	−33.3045	−1.31545	−0.657725	−	0.753258i	$-0.728481\pi$
$641$	−33.3045	−1.31545	−0.657725	+	0.753258i	$0.728481\pi$
$642$	0	0
$643$	30.6746i	1.20969i	0.796344	+	0.604843i	$0.206764\pi$
$643$	30.6746i	1.20969i	−0.796344	+	0.604843i	$0.793236\pi$
$644$	0	0
$645$	49.6375i	1.95447i
$646$	0	0
$647$	3.25608	0.128010	0.0640048	−	0.997950i	$-0.479613\pi$
$647$	3.25608	0.128010	0.0640048	+	0.997950i	$0.479613\pi$
$648$	0	0
$649$	19.5090	0.765796
$650$	0	0
$651$	8.26683	0.324003
$652$	0	0
$653$	14.8183	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$653$	14.8183	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$654$	0	0
$655$	33.3937i	1.30480i
$656$	0	0
$657$	5.23298i	0.204158i
$658$	0	0
$659$	30.5392	1.18964	0.594818	−	0.803860i	$-0.297224\pi$
$659$	30.5392	1.18964	0.594818	+	0.803860i	$0.297224\pi$
$660$	0	0
$661$	19.3676i	0.753314i	0.926353	+	0.376657i	$0.122927\pi$
$661$	19.3676i	0.753314i	−0.926353	+	0.376657i	$0.877073\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 56.1444i	− 2.17718i
$666$	0	0
$667$	−10.8944	−0.421833
$668$	0	0
$669$	− 7.38511i	− 0.285525i
$670$	0	0
$671$	− 35.8514i	− 1.38403i
$672$	0	0
$673$	−41.5459	−1.60148	−0.800738	−	0.599015i	$-0.795559\pi$
$673$	−41.5459	−1.60148	−0.800738	+	0.599015i	$0.795559\pi$
$674$	0	0
$675$	73.5682	2.83164
$676$	0	0
$677$	−40.6064	−1.56063	−0.780315	−	0.625387i	$-0.784941\pi$
$677$	−40.6064	−1.56063	−0.780315	+	0.625387i	$0.784941\pi$
$678$	0	0
$679$	7.31575	0.280753
$680$	0	0
$681$	30.4596i	1.16722i
$682$	0	0
$683$	2.40581i	0.0920559i	0.998940	+	0.0460279i	$0.0146563\pi$
$683$	2.40581i	0.0920559i	−0.998940	+	0.0460279i	$0.985344\pi$
$684$	0	0
$685$	−4.19567	−0.160308
$686$	0	0
$687$	22.5429i	0.860064i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.81269i	0.259167i	0.991569	+	0.129583i	$0.0413640\pi$
$691$	6.81269i	0.259167i	−0.991569	+	0.129583i	$0.958636\pi$
$692$	0	0
$693$	−8.31336	−0.315798
$694$	0	0
$695$	− 51.1202i	− 1.93910i
$696$	0	0
$697$	0.329158i	0.0124677i
$698$	0	0
$699$	−13.2043	−0.499432
$700$	0	0
$701$	12.7802	0.482700	0.241350	−	0.970438i	$-0.422410\pi$
$701$	12.7802	0.482700	0.241350	+	0.970438i	$0.422410\pi$
$702$	0	0
$703$	21.4789	0.810092
$704$	0	0
$705$	−40.1129	−1.51074
$706$	0	0
$707$	24.3169i	0.914532i
$708$	0	0
$709$	4.94139i	0.185578i	0.995686	+	0.0927890i	$0.0295782\pi$
$709$	4.94139i	0.185578i	−0.995686	+	0.0927890i	$0.970422\pi$
$710$	0	0
$711$	−14.8528	−0.557023
$712$	0	0
$713$	− 14.5332i	− 0.544272i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.83877i	− 0.143362i
$718$	0	0
$719$	37.6155	1.40282	0.701410	−	0.712758i	$-0.252554\pi$
$719$	37.6155	1.40282	0.701410	+	0.712758i	$0.252554\pi$
$720$	0	0
$721$	12.9849i	0.483584i
$722$	0	0
$723$	− 24.8920i	− 0.925744i
$724$	0	0
$725$	27.8605	1.03471
$726$	0	0
$727$	17.1879	0.637464	0.318732	−	0.947845i	$-0.396743\pi$
$727$	17.1879	0.637464	0.318732	+	0.947845i	$0.396743\pi$
$728$	0	0
$729$	27.8159	1.03022
$730$	0	0
$731$	0.525418	0.0194333
$732$	0	0
$733$	− 19.2597i	− 0.711371i	−0.934606	−	0.355686i	$-0.884247\pi$
$733$	− 19.2597i	− 0.711371i	0.934606	−	0.355686i	$-0.115753\pi$
$734$	0	0
$735$	14.0204i	0.517152i
$736$	0	0
$737$	−24.8605	−0.915750
$738$	0	0
$739$	− 18.0785i	− 0.665027i	−0.943099	−	0.332513i	$-0.892103\pi$
$739$	− 18.0785i	− 0.665027i	0.943099	−	0.332513i	$-0.107897\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.8219i	1.20412i	0.798451	+	0.602059i	$0.205653\pi$
$743$	32.8219i	1.20412i	−0.798451	+	0.602059i	$0.794347\pi$
$744$	0	0
$745$	−10.0199	−0.367099
$746$	0	0
$747$	− 4.11960i	− 0.150728i
$748$	0	0
$749$	− 34.2717i	− 1.25226i
$750$	0	0
$751$	33.1702	1.21040	0.605198	−	0.796075i	$-0.293094\pi$
$751$	33.1702	1.21040	0.605198	+	0.796075i	$0.293094\pi$
$752$	0	0
$753$	3.44312	0.125474
$754$	0	0
$755$	−31.5961	−1.14990
$756$	0	0
$757$	−0.736627	−0.0267732	−0.0133866	−	0.999910i	$-0.504261\pi$
$757$	−0.736627	−0.0267732	−0.0133866	+	0.999910i	$0.504261\pi$
$758$	0	0
$759$	− 23.2204i	− 0.842848i
$760$	0	0
$761$	− 36.3817i	− 1.31883i	−0.751777	−	0.659417i	$-0.770803\pi$
$761$	− 36.3817i	− 1.31883i	0.751777	−	0.659417i	$-0.229197\pi$
$762$	0	0
$763$	31.8625	1.15350
$764$	0	0
$765$	− 0.300209i	− 0.0108541i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 11.4523i	− 0.412982i	−0.978449	−	0.206491i	$-0.933796\pi$
$769$	− 11.4523i	− 0.412982i	0.978449	−	0.206491i	$-0.0662044\pi$
$770$	0	0
$771$	31.8683	1.14771
$772$	0	0
$773$	7.57732i	0.272537i	0.990672	+	0.136269i	$0.0435110\pi$
$773$	7.57732i	0.272537i	−0.990672	+	0.136269i	$0.956489\pi$
$774$	0	0
$775$	37.1661i	1.33505i
$776$	0	0
$777$	10.0686	0.361207
$778$	0	0
$779$	−33.3803	−1.19597
$780$	0	0
$781$	0.164210	0.00587591
$782$	0	0
$783$	12.0597	0.430977
$784$	0	0
$785$	− 46.4989i	− 1.65962i
$786$	0	0
$787$	30.5066i	1.08744i	0.839265	+	0.543722i	$0.182985\pi$
$787$	30.5066i	1.08744i	−0.839265	+	0.543722i	$0.817015\pi$
$788$	0	0
$789$	−21.4515	−0.763693
$790$	0	0
$791$	17.6192i	0.626467i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.125456i	0.00444946i
$796$	0	0
$797$	−36.2435	−1.28381	−0.641906	−	0.766784i	$-0.721856\pi$
$797$	−36.2435	−1.28381	−0.641906	+	0.766784i	$0.721856\pi$
$798$	0	0
$799$	0.424600i	0.0150213i
$800$	0	0
$801$	2.72002i	0.0961073i
$802$	0	0
$803$	−15.1588	−0.534944
$804$	0	0
$805$	−46.2683	−1.63074
$806$	0	0
$807$	−41.8713	−1.47394
$808$	0	0
$809$	22.3236	0.784857	0.392429	−	0.919782i	$-0.371635\pi$
$809$	22.3236	0.784857	0.392429	+	0.919782i	$0.371635\pi$
$810$	0	0
$811$	− 39.4922i	− 1.38676i	−0.720573	−	0.693379i	$-0.756121\pi$
$811$	− 39.4922i	− 1.38676i	0.720573	−	0.693379i	$-0.243879\pi$
$812$	0	0
$813$	− 21.9726i	− 0.770612i
$814$	0	0
$815$	29.8340	1.04504
$816$	0	0
$817$	53.2833i	1.86415i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.3830i	0.397269i	0.980074	+	0.198634i	$0.0636506\pi$
$821$	11.3830i	0.397269i	−0.980074	+	0.198634i	$0.936349\pi$
$822$	0	0
$823$	−25.6848	−0.895317	−0.447659	−	0.894205i	$-0.647742\pi$
$823$	−25.6848	−0.895317	−0.447659	+	0.894205i	$0.647742\pi$
$824$	0	0
$825$	59.3822i	2.06742i
$826$	0	0
$827$	36.7251i	1.27706i	0.769598	+	0.638529i	$0.220457\pi$
$827$	36.7251i	1.27706i	−0.769598	+	0.638529i	$0.779543\pi$
$828$	0	0
$829$	11.5808	0.402217	0.201109	−	0.979569i	$-0.435546\pi$
$829$	11.5808	0.402217	0.201109	+	0.979569i	$0.435546\pi$
$830$	0	0
$831$	−17.7855	−0.616974
$832$	0	0
$833$	0.148408	0.00514203
$834$	0	0
$835$	−65.7512	−2.27541
$836$	0	0
$837$	16.0877i	0.556071i
$838$	0	0
$839$	− 13.4577i	− 0.464612i	−0.972643	−	0.232306i	$-0.925373\pi$
$839$	− 13.4577i	− 0.464612i	0.972643	−	0.232306i	$-0.0746271\pi$
$840$	0	0
$841$	−24.4330	−0.842516
$842$	0	0
$843$	− 19.3612i	− 0.666835i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.574335i	− 0.0197344i
$848$	0	0
$849$	10.7961	0.370521
$850$	0	0
$851$	− 17.7006i	− 0.606770i
$852$	0	0
$853$	5.34050i	0.182855i	0.995812	+	0.0914277i	$0.0291430\pi$
$853$	5.34050i	0.182855i	−0.995812	+	0.0914277i	$0.970857\pi$
$854$	0	0
$855$	30.4446	1.04118
$856$	0	0
$857$	−49.3381	−1.68536	−0.842679	−	0.538417i	$-0.819023\pi$
$857$	−49.3381	−1.68536	−0.842679	+	0.538417i	$0.819023\pi$
$858$	0	0
$859$	−4.92500	−0.168039	−0.0840194	−	0.996464i	$-0.526776\pi$
$859$	−4.92500	−0.168039	−0.0840194	+	0.996464i	$0.526776\pi$
$860$	0	0
$861$	−15.6475	−0.533266
$862$	0	0
$863$	− 40.0315i	− 1.36269i	−0.731964	−	0.681343i	$-0.761396\pi$
$863$	− 40.0315i	− 1.36269i	0.731964	−	0.681343i	$-0.238604\pi$
$864$	0	0
$865$	− 58.5241i	− 1.98988i
$866$	0	0
$867$	−23.0622	−0.783233
$868$	0	0
$869$	− 43.0253i	− 1.45953i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	3.96700i	0.134263i
$874$	0	0
$875$	72.9428	2.46592
$876$	0	0
$877$	− 28.9379i	− 0.977165i	−0.872518	−	0.488582i	$-0.837514\pi$
$877$	− 28.9379i	− 0.977165i	0.872518	−	0.488582i	$-0.162486\pi$
$878$	0	0
$879$	− 26.6358i	− 0.898404i
$880$	0	0
$881$	−21.6692	−0.730053	−0.365027	−	0.930997i	$-0.618940\pi$
$881$	−21.6692	−0.730053	−0.365027	+	0.930997i	$0.618940\pi$
$882$	0	0
$883$	−11.7614	−0.395802	−0.197901	−	0.980222i	$-0.563412\pi$
$883$	−11.7614	−0.395802	−0.197901	+	0.980222i	$0.563412\pi$
$884$	0	0
$885$	−33.4907	−1.12578
$886$	0	0
$887$	26.8485	0.901483	0.450742	−	0.892654i	$-0.351160\pi$
$887$	26.8485	0.901483	0.450742	+	0.892654i	$0.351160\pi$
$888$	0	0
$889$	14.1967i	0.476143i
$890$	0	0
$891$	14.0339i	0.470152i
$892$	0	0
$893$	−43.0592	−1.44092
$894$	0	0
$895$	− 93.9542i	− 3.14054i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.09246i	0.203195i
$900$	0	0
$901$	0.00132796	4.42409e−5 0
$902$	0	0
$903$	24.9774i	0.831195i
$904$	0	0
$905$	57.9318i	1.92572i
$906$	0	0
$907$	−16.4306	−0.545568	−0.272784	−	0.962075i	$-0.587944\pi$
$907$	−16.4306	−0.545568	−0.272784	+	0.962075i	$0.587944\pi$
$908$	0	0
$909$	−13.1860	−0.437351
$910$	0	0
$911$	26.0519	0.863138	0.431569	−	0.902080i	$-0.357960\pi$
$911$	26.0519	0.863138	0.431569	+	0.902080i	$0.357960\pi$
$912$	0	0
$913$	11.9336	0.394945
$914$	0	0
$915$	61.5454i	2.03463i
$916$	0	0
$917$	16.8036i	0.554904i
$918$	0	0
$919$	4.36957	0.144139	0.0720694	−	0.997400i	$-0.477040\pi$
$919$	4.36957	0.144139	0.0720694	+	0.997400i	$0.477040\pi$
$920$	0	0
$921$	21.2728i	0.700963i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	45.2664i	1.48835i
$926$	0	0
$927$	−7.04115	−0.231262
$928$	0	0
$929$	− 32.2868i	− 1.05930i	−0.848218	−	0.529648i	$-0.822324\pi$
$929$	− 32.2868i	− 1.05930i	0.848218	−	0.529648i	$-0.177676\pi$
$930$	0	0
$931$	15.0502i	0.493252i
$932$	0	0
$933$	−22.3864	−0.732899
$934$	0	0
$935$	0.869641	0.0284403
$936$	0	0
$937$	−48.5730	−1.58681	−0.793405	−	0.608693i	$-0.791694\pi$
$937$	−48.5730	−1.58681	−0.793405	+	0.608693i	$0.791694\pi$
$938$	0	0
$939$	−15.4601	−0.504522
$940$	0	0
$941$	4.82849i	0.157404i	0.996898	+	0.0787022i	$0.0250776\pi$
$941$	4.82849i	0.157404i	−0.996898	+	0.0787022i	$0.974922\pi$
$942$	0	0
$943$	27.5086i	0.895802i
$944$	0	0
$945$	51.2172	1.66609
$946$	0	0
$947$	35.1065i	1.14081i	0.821365	+	0.570403i	$0.193213\pi$
$947$	35.1065i	1.14081i	−0.821365	+	0.570403i	$0.806787\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	34.5289i	1.11968i
$952$	0	0
$953$	28.1866	0.913053	0.456526	−	0.889710i	$-0.349093\pi$
$953$	28.1866	0.913053	0.456526	+	0.889710i	$0.349093\pi$
$954$	0	0
$955$	19.0756i	0.617272i
$956$	0	0
$957$	9.73423i	0.314663i
$958$	0	0
$959$	−2.11124	−0.0681756
$960$	0	0
$961$	22.8726	0.737827
$962$	0	0
$963$	18.5840	0.598862
$964$	0	0
$965$	109.468	3.52388
$966$	0	0
$967$	44.6674i	1.43641i	0.695834	+	0.718203i	$0.255035\pi$
$967$	44.6674i	1.43641i	−0.695834	+	0.718203i	$0.744965\pi$
$968$	0	0
$969$	0.512009i	0.0164481i
$970$	0	0
$971$	−14.9108	−0.478510	−0.239255	−	0.970957i	$-0.576903\pi$
$971$	−14.9108	−0.478510	−0.239255	+	0.970957i	$0.576903\pi$
$972$	0	0
$973$	− 25.7235i	− 0.824657i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.2127i	− 1.12655i	−0.826269	−	0.563276i	$-0.809541\pi$
$977$	− 35.2127i	− 1.12655i	0.826269	−	0.563276i	$-0.190459\pi$
$978$	0	0
$979$	−7.87933	−0.251824
$980$	0	0
$981$	17.2776i	0.551631i
$982$	0	0
$983$	30.5646i	0.974861i	0.873162	+	0.487430i	$0.162066\pi$
$983$	30.5646i	0.974861i	−0.873162	+	0.487430i	$0.837934\pi$
$984$	0	0
$985$	79.0926	2.52010
$986$	0	0
$987$	−20.1847	−0.642485
$988$	0	0
$989$	43.9105	1.39627
$990$	0	0
$991$	62.5739	1.98772	0.993862	−	0.110626i	$-0.0352856\pi$
$991$	62.5739	1.98772	0.993862	+	0.110626i	$0.0352856\pi$
$992$	0	0
$993$	12.1515i	0.385617i
$994$	0	0
$995$	50.8200i	1.61110i
$996$	0	0
$997$	−1.88444	−0.0596809	−0.0298405	−	0.999555i	$-0.509500\pi$
$997$	−1.88444	−0.0596809	−0.0298405	+	0.999555i	$0.509500\pi$
$998$	0	0
$999$	19.5939i	0.619924i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.2.d.e.337.4		6
3.2	odd	2		6084.2.b.p.4393.1		6
4.3	odd	2		2704.2.f.n.337.4		6
13.2	odd	12		676.2.e.g.529.2		6
13.3	even	3		676.2.h.e.485.4		12
13.4	even	6		676.2.h.e.361.3		12
13.5	odd	4		676.2.a.h.1.2	yes	3
13.6	odd	12		676.2.e.g.653.2		6
13.7	odd	12		676.2.e.f.653.2		6
13.8	odd	4		676.2.a.g.1.2	✓	3
13.9	even	3		676.2.h.e.361.4		12
13.10	even	6		676.2.h.e.485.3		12
13.11	odd	12		676.2.e.f.529.2		6
13.12	even	2	inner	676.2.d.e.337.3		6
39.5	even	4		6084.2.a.x.1.1		3
39.8	even	4		6084.2.a.bc.1.3		3
39.38	odd	2		6084.2.b.p.4393.6		6
52.31	even	4		2704.2.a.y.1.2		3
52.47	even	4		2704.2.a.x.1.2		3
52.51	odd	2		2704.2.f.n.337.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
676.2.a.g.1.2	✓	3	13.8	odd	4
676.2.a.h.1.2	yes	3	13.5	odd	4
676.2.d.e.337.3		6	13.12	even	2	inner
676.2.d.e.337.4		6	1.1	even	1	trivial
676.2.e.f.529.2		6	13.11	odd	12
676.2.e.f.653.2		6	13.7	odd	12
676.2.e.g.529.2		6	13.2	odd	12
676.2.e.g.653.2		6	13.6	odd	12
676.2.h.e.361.3		12	13.4	even	6
676.2.h.e.361.4		12	13.9	even	3
676.2.h.e.485.3		12	13.10	even	6
676.2.h.e.485.4		12	13.3	even	3
2704.2.a.x.1.2		3	52.47	even	4
2704.2.a.y.1.2		3	52.31	even	4
2704.2.f.n.337.3		6	52.51	odd	2
2704.2.f.n.337.4		6	4.3	odd	2
6084.2.a.x.1.1		3	39.5	even	4
6084.2.a.bc.1.3		3	39.8	even	4
6084.2.b.p.4393.1		6	3.2	odd	2
6084.2.b.p.4393.6		6	39.38	odd	2

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:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.35690 q^{3} +4.24698i q^{5} +2.13706i q^{7} -1.15883 q^{9} -3.35690i q^{11} +5.76271i q^{15} +0.0609989 q^{17} +6.18598i q^{19} +2.89977i q^{21} +5.09783 q^{23} -13.0368 q^{25} -5.64310 q^{27} -2.13706 q^{29} -2.85086i q^{31} -4.55496i q^{33} -9.07606 q^{35} -3.47219i q^{37} +5.39612i q^{41} +8.61356 q^{43} -4.92154i q^{45} +6.96077i q^{47} +2.43296 q^{49} +0.0827692 q^{51} +0.0217703 q^{53} +14.2567 q^{55} +8.39373i q^{57} +5.81163i q^{59} +10.6799 q^{61} -2.47650i q^{63} -7.40581i q^{67} +6.91723 q^{69} +0.0489173i q^{71} -4.51573i q^{73} -17.6896 q^{75} +7.17390 q^{77} +12.8170 q^{79} -4.18060 q^{81} +3.55496i q^{83} +0.259061i q^{85} -2.89977 q^{87} -2.34721i q^{89} -3.86831i q^{93} -26.2717 q^{95} -3.42327i q^{97} +3.89008i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{9} + 20 q^{17} - 6 q^{23} - 22 q^{25} - 42 q^{27} - 2 q^{29} - 24 q^{35} - 10 q^{43} - 24 q^{49} + 14 q^{51} - 6 q^{53} + 32 q^{55} + 16 q^{61} + 28 q^{69} - 14 q^{75} - 24 q^{77} + 18 q^{79}+ \cdots - 54 q^{95}+O(q^{100}) \) 6 * q + 10 * q^9 + 20 * q^17 - 6 * q^23 - 22 * q^25 - 42 * q^27 - 2 * q^29 - 24 * q^35 - 10 * q^43 - 24 * q^49 + 14 * q^51 - 6 * q^53 + 32 * q^55 + 16 * q^61 + 28 * q^69 - 14 * q^75 - 24 * q^77 + 18 * q^79 - 2 * q^81 + 28 * q^87 - 54 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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