LMFDB - Embedded newform 354.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,2,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	354.a (trivial)

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:	yes
Analytic conductor:	$2.82670423155$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +1.00000 q^{21} +5.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -10.0000 q^{29} -8.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +9.00000 q^{37} -1.00000 q^{39} -5.00000 q^{41} -1.00000 q^{42} +3.00000 q^{43} -5.00000 q^{44} +6.00000 q^{46} -1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +10.0000 q^{58} -1.00000 q^{59} +6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +6.00000 q^{69} +1.00000 q^{71} -1.00000 q^{72} -9.00000 q^{74} +5.00000 q^{75} +5.00000 q^{77} +1.00000 q^{78} -3.00000 q^{79} +1.00000 q^{81} +5.00000 q^{82} +7.00000 q^{83} +1.00000 q^{84} -3.00000 q^{86} +10.0000 q^{87} +5.00000 q^{88} +16.0000 q^{89} -1.00000 q^{91} -6.00000 q^{92} +8.00000 q^{93} +1.00000 q^{96} -12.0000 q^{97} +6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	5.00000	1.06600
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	5.00000	0.870388
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	−1.00000	−0.154303
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	5.00000	0.707107
$51$	−1.00000	−0.140028
$52$	1.00000	0.138675
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	10.0000	1.31306
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	0.121268
$69$	6.00000	0.722315
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−9.00000	−1.04623
$75$	5.00000	0.577350
$76$	0	0
$77$	5.00000	0.569803
$78$	1.00000	0.113228
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.00000	0.552158
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−3.00000	−0.323498
$87$	10.0000	1.07211
$88$	5.00000	0.533002
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−6.00000	−0.625543
$93$	8.00000	0.829561
$94$	0	0
$95$	0	0
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	6.00000	0.606092
$99$	−5.00000	−0.502519
$100$	−5.00000	−0.500000
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	1.00000	0.0990148
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−10.0000	−0.928477
$117$	1.00000	0.0924500
$118$	1.00000	0.0920575
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	−6.00000	−0.543214
$123$	5.00000	0.450835
$124$	−8.00000	−0.718421
$125$	0	0
$126$	1.00000	0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−3.00000	−0.264135
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	5.00000	0.435194
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	−6.00000	−0.510754
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−1.00000	−0.0839181
$143$	−5.00000	−0.418121
$144$	1.00000	0.0833333
$145$	0	0
$146$	0	0
$147$	6.00000	0.494872
$148$	9.00000	0.739795
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	−5.00000	−0.408248
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	−5.00000	−0.402911
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	3.00000	0.238667
$159$	−4.00000	−0.317221
$160$	0	0
$161$	6.00000	0.472866
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−5.00000	−0.390434
$165$	0	0
$166$	−7.00000	−0.543305
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	−1.00000	−0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	3.00000	0.228748
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	−10.0000	−0.758098
$175$	5.00000	0.377964
$176$	−5.00000	−0.376889
$177$	1.00000	0.0751646
$178$	−16.0000	−1.19925
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	1.00000	0.0741249
$183$	−6.00000	−0.443533
$184$	6.00000	0.442326
$185$	0	0
$186$	−8.00000	−0.586588
$187$	−5.00000	−0.365636
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−6.00000	−0.428571
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	5.00000	0.355335
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	5.00000	0.353553
$201$	−4.00000	−0.282138
$202$	−17.0000	−1.19612
$203$	10.0000	0.701862
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	4.00000	0.274721
$213$	−1.00000	−0.0685189
$214$	6.00000	0.410152
$215$	0	0
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	10.0000	0.677285
$219$	0	0
$220$	0	0
$221$	1.00000	0.0672673
$222$	9.00000	0.604040
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	1.00000	0.0668153
$225$	−5.00000	−0.333333
$226$	−6.00000	−0.399114
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	−25.0000	−1.65205	−0.826023	−	0.563636i	$-0.809402\pi$
$229$	−25.0000	−1.65205	−0.826023	+	0.563636i	$0.809402\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	10.0000	0.656532
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	−1.00000	−0.0650945
$237$	3.00000	0.194871
$238$	1.00000	0.0648204
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	−14.0000	−0.899954
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−5.00000	−0.318788
$247$	0	0
$248$	8.00000	0.508001
$249$	−7.00000	−0.443607
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	−1.00000	−0.0629941
$253$	30.0000	1.88608
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	3.00000	0.186772
$259$	−9.00000	−0.559233
$260$	0	0
$261$	−10.0000	−0.618984
$262$	20.0000	1.23560
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	0	0
$267$	−16.0000	−0.979184
$268$	4.00000	0.244339
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	1.00000	0.0606339
$273$	1.00000	0.0605228
$274$	5.00000	0.302061
$275$	25.0000	1.50756
$276$	6.00000	0.361158
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−4.00000	−0.239904
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	0	0
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	1.00000	0.0593391
$285$	0	0
$286$	5.00000	0.295656
$287$	5.00000	0.295141
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−9.00000	−0.523114
$297$	5.00000	0.290129
$298$	−21.0000	−1.21650
$299$	−6.00000	−0.346989
$300$	5.00000	0.288675
$301$	−3.00000	−0.172917
$302$	10.0000	0.575435
$303$	−17.0000	−0.976624
$304$	0	0
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	5.00000	0.284901
$309$	−4.00000	−0.227552
$310$	0	0
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	1.00000	0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	−3.00000	−0.168763
$317$	32.0000	1.79730	0.898650	−	0.438667i	$-0.144549\pi$
$317$	32.0000	1.79730	0.898650	+	0.438667i	$0.144549\pi$
$318$	4.00000	0.224309
$319$	50.0000	2.79946
$320$	0	0
$321$	6.00000	0.334887
$322$	−6.00000	−0.334367
$323$	0	0
$324$	1.00000	0.0555556
$325$	−5.00000	−0.277350
$326$	−14.0000	−0.775388
$327$	10.0000	0.553001
$328$	5.00000	0.276079
$329$	0	0
$330$	0	0
$331$	34.0000	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$331$	34.0000	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$332$	7.00000	0.384175
$333$	9.00000	0.493197
$334$	0	0
$335$	0	0
$336$	1.00000	0.0545545
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	12.0000	0.652714
$339$	−6.00000	−0.325875
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	13.0000	0.701934
$344$	−3.00000	−0.161749
$345$	0	0
$346$	9.00000	0.483843
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	10.0000	0.536056
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	−5.00000	−0.267261
$351$	−1.00000	−0.0533761
$352$	5.00000	0.266501
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−1.00000	−0.0531494
$355$	0	0
$356$	16.0000	0.847998
$357$	1.00000	0.0529256
$358$	−5.00000	−0.264258
$359$	11.0000	0.580558	0.290279	−	0.956942i	$-0.406252\pi$
$359$	11.0000	0.580558	0.290279	+	0.956942i	$0.406252\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	22.0000	1.15629
$363$	−14.0000	−0.734809
$364$	−1.00000	−0.0524142
$365$	0	0
$366$	6.00000	0.313625
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	−6.00000	−0.312772
$369$	−5.00000	−0.260290
$370$	0	0
$371$	−4.00000	−0.207670
$372$	8.00000	0.414781
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	−1.00000	−0.0514344
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−13.0000	−0.661683
$387$	3.00000	0.152499
$388$	−12.0000	−0.609208
$389$	−22.0000	−1.11544	−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000	−1.11544	−0.557722	+	0.830028i	$0.688325\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	6.00000	0.303046
$393$	20.0000	1.00887
$394$	0	0
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	−5.00000	−0.250000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	4.00000	0.199502
$403$	−8.00000	−0.398508
$404$	17.0000	0.845782
$405$	0	0
$406$	−10.0000	−0.496292
$407$	−45.0000	−2.23057
$408$	1.00000	0.0495074
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	4.00000	0.197066
$413$	1.00000	0.0492068
$414$	6.00000	0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−35.0000	−1.70986	−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000	−1.70986	−0.854931	+	0.518742i	$0.826401\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	3.00000	0.146038
$423$	0	0
$424$	−4.00000	−0.194257
$425$	−5.00000	−0.242536
$426$	1.00000	0.0484502
$427$	−6.00000	−0.290360
$428$	−6.00000	−0.290021
$429$	5.00000	0.241402
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	−1.00000	−0.0481125
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	0	0
$439$	−11.0000	−0.525001	−0.262501	−	0.964932i	$-0.584547\pi$
$439$	−11.0000	−0.525001	−0.262501	+	0.964932i	$0.584547\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−1.00000	−0.0475651
$443$	17.0000	0.807694	0.403847	−	0.914826i	$-0.367673\pi$
$443$	17.0000	0.807694	0.403847	+	0.914826i	$0.367673\pi$
$444$	−9.00000	−0.427121
$445$	0	0
$446$	9.00000	0.426162
$447$	−21.0000	−0.993266
$448$	−1.00000	−0.0472456
$449$	−31.0000	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$450$	5.00000	0.235702
$451$	25.0000	1.17720
$452$	6.00000	0.282216
$453$	10.0000	0.469841
$454$	−7.00000	−0.328526
$455$	0	0
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	25.0000	1.16817
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	5.00000	0.232621
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−20.0000	−0.926482
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	1.00000	0.0462250
$469$	−4.00000	−0.184703
$470$	0	0
$471$	6.00000	0.276465
$472$	1.00000	0.0460287
$473$	−15.0000	−0.689701
$474$	−3.00000	−0.137795
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	4.00000	0.183147
$478$	−8.00000	−0.365911
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	−15.0000	−0.683231
$483$	−6.00000	−0.273009
$484$	14.0000	0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	−7.00000	−0.317200	−0.158600	−	0.987343i	$-0.550698\pi$
$487$	−7.00000	−0.317200	−0.158600	+	0.987343i	$0.550698\pi$
$488$	−6.00000	−0.271607
$489$	−14.0000	−0.633102
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	5.00000	0.225417
$493$	−10.0000	−0.450377
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−1.00000	−0.0448561
$498$	7.00000	0.313678
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−30.0000	−1.33366
$507$	12.0000	0.532939
$508$	−8.00000	−0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	21.0000	0.926270
$515$	0	0
$516$	−3.00000	−0.132068
$517$	0	0
$518$	9.00000	0.395437
$519$	9.00000	0.395056
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	10.0000	0.437688
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−20.0000	−0.873704
$525$	−5.00000	−0.218218
$526$	−3.00000	−0.130806
$527$	−8.00000	−0.348485
$528$	5.00000	0.217597
$529$	13.0000	0.565217
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	0	0
$533$	−5.00000	−0.216574
$534$	16.0000	0.692388
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−5.00000	−0.215766
$538$	−3.00000	−0.129339
$539$	30.0000	1.29219
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	15.0000	0.644305
$543$	22.0000	0.944110
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−1.00000	−0.0427960
$547$	−30.0000	−1.28271	−0.641354	−	0.767245i	$-0.721627\pi$
$547$	−30.0000	−1.28271	−0.641354	+	0.767245i	$0.721627\pi$
$548$	−5.00000	−0.213589
$549$	6.00000	0.256074
$550$	−25.0000	−1.06600
$551$	0	0
$552$	−6.00000	−0.255377
$553$	3.00000	0.127573
$554$	26.0000	1.10463
$555$	0	0
$556$	4.00000	0.169638
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	8.00000	0.338667
$559$	3.00000	0.126886
$560$	0	0
$561$	5.00000	0.211100
$562$	7.00000	0.295277
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	−31.0000	−1.30303
$567$	−1.00000	−0.0419961
$568$	−1.00000	−0.0419591
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−5.00000	−0.209061
$573$	12.0000	0.501307
$574$	−5.00000	−0.208696
$575$	30.0000	1.25109
$576$	1.00000	0.0416667
$577$	−23.0000	−0.957503	−0.478751	−	0.877951i	$-0.658910\pi$
$577$	−23.0000	−0.957503	−0.478751	+	0.877951i	$0.658910\pi$
$578$	16.0000	0.665512
$579$	−13.0000	−0.540262
$580$	0	0
$581$	−7.00000	−0.290409
$582$	−12.0000	−0.497416
$583$	−20.0000	−0.828315
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	0	0
$591$	0	0
$592$	9.00000	0.369898
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	21.0000	0.860194
$597$	4.00000	0.163709
$598$	6.00000	0.245358
$599$	−33.0000	−1.34834	−0.674172	−	0.738575i	$-0.735499\pi$
$599$	−33.0000	−1.34834	−0.674172	+	0.738575i	$0.735499\pi$
$600$	−5.00000	−0.204124
$601$	−48.0000	−1.95796	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000	−1.95796	−0.978980	+	0.203954i	$0.934621\pi$
$602$	3.00000	0.122271
$603$	4.00000	0.162893
$604$	−10.0000	−0.406894
$605$	0	0
$606$	17.0000	0.690578
$607$	37.0000	1.50178	0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000	1.50178	0.750892	+	0.660425i	$0.229624\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	0	0
$611$	0	0
$612$	1.00000	0.0404226
$613$	37.0000	1.49442	0.747208	−	0.664590i	$-0.231394\pi$
$613$	37.0000	1.49442	0.747208	+	0.664590i	$0.231394\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−5.00000	−0.201456
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	4.00000	0.160904
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	−7.00000	−0.280674
$623$	−16.0000	−0.641026
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−6.00000	−0.239426
$629$	9.00000	0.358854
$630$	0	0
$631$	−27.0000	−1.07485	−0.537427	−	0.843311i	$-0.680603\pi$
$631$	−27.0000	−1.07485	−0.537427	+	0.843311i	$0.680603\pi$
$632$	3.00000	0.119334
$633$	3.00000	0.119239
$634$	−32.0000	−1.27088
$635$	0	0
$636$	−4.00000	−0.158610
$637$	−6.00000	−0.237729
$638$	−50.0000	−1.97952
$639$	1.00000	0.0395594
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	−6.00000	−0.236801
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−1.00000	−0.0392837
$649$	5.00000	0.196267
$650$	5.00000	0.196116
$651$	−8.00000	−0.313545
$652$	14.0000	0.548282
$653$	−48.0000	−1.87839	−0.939193	−	0.343391i	$-0.888424\pi$
$653$	−48.0000	−1.87839	−0.939193	+	0.343391i	$0.888424\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−5.00000	−0.195217
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	−34.0000	−1.32145
$663$	−1.00000	−0.0388368
$664$	−7.00000	−0.271653
$665$	0	0
$666$	−9.00000	−0.348743
$667$	60.0000	2.32321
$668$	0	0
$669$	9.00000	0.347960
$670$	0	0
$671$	−30.0000	−1.15814
$672$	−1.00000	−0.0385758
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−2.00000	−0.0770371
$675$	5.00000	0.192450
$676$	−12.0000	−0.461538
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	6.00000	0.230429
$679$	12.0000	0.460518
$680$	0	0
$681$	−7.00000	−0.268241
$682$	−40.0000	−1.53168
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	0	0
$686$	−13.0000	−0.496342
$687$	25.0000	0.953809
$688$	3.00000	0.114374
$689$	4.00000	0.152388
$690$	0	0
$691$	25.0000	0.951045	0.475522	−	0.879704i	$-0.342259\pi$
$691$	25.0000	0.951045	0.475522	+	0.879704i	$0.342259\pi$
$692$	−9.00000	−0.342129
$693$	5.00000	0.189934
$694$	36.0000	1.36654
$695$	0	0
$696$	−10.0000	−0.379049
$697$	−5.00000	−0.189389
$698$	7.00000	0.264954
$699$	−20.0000	−0.756469
$700$	5.00000	0.188982
$701$	−47.0000	−1.77517	−0.887583	−	0.460648i	$-0.847617\pi$
$701$	−47.0000	−1.77517	−0.887583	+	0.460648i	$0.847617\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	−5.00000	−0.188445
$705$	0	0
$706$	24.0000	0.903252
$707$	−17.0000	−0.639351
$708$	1.00000	0.0375823
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	−16.0000	−0.599625
$713$	48.0000	1.79761
$714$	−1.00000	−0.0374241
$715$	0	0
$716$	5.00000	0.186859
$717$	−8.00000	−0.298765
$718$	−11.0000	−0.410516
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	19.0000	0.707107
$723$	−15.0000	−0.557856
$724$	−22.0000	−0.817624
$725$	50.0000	1.85695
$726$	14.0000	0.519589
$727$	11.0000	0.407967	0.203984	−	0.978974i	$-0.434611\pi$
$727$	11.0000	0.407967	0.203984	+	0.978974i	$0.434611\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.00000	0.110959
$732$	−6.00000	−0.221766
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	6.00000	0.221163
$737$	−20.0000	−0.736709
$738$	5.00000	0.184053
$739$	37.0000	1.36107	0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000	1.36107	0.680534	+	0.732717i	$0.261748\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	−24.0000	−0.878702
$747$	7.00000	0.256117
$748$	−5.00000	−0.182818
$749$	6.00000	0.219235
$750$	0	0
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	10.0000	0.364179
$755$	0	0
$756$	1.00000	0.0363696
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	−14.0000	−0.508503
$759$	−30.0000	−1.08893
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	−8.00000	−0.289809
$763$	10.0000	0.362024
$764$	−12.0000	−0.434145
$765$	0	0
$766$	21.0000	0.758761
$767$	−1.00000	−0.0361079
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	21.0000	0.756297
$772$	13.0000	0.467880
$773$	−2.00000	−0.0719350	−0.0359675	−	0.999353i	$-0.511451\pi$
$773$	−2.00000	−0.0719350	−0.0359675	+	0.999353i	$0.511451\pi$
$774$	−3.00000	−0.107833
$775$	40.0000	1.43684
$776$	12.0000	0.430775
$777$	9.00000	0.322873
$778$	22.0000	0.788738
$779$	0	0
$780$	0	0
$781$	−5.00000	−0.178914
$782$	6.00000	0.214560
$783$	10.0000	0.357371
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−20.0000	−0.713376
$787$	−46.0000	−1.63972	−0.819861	−	0.572562i	$-0.805950\pi$
$787$	−46.0000	−1.63972	−0.819861	+	0.572562i	$0.805950\pi$
$788$	0	0
$789$	−3.00000	−0.106803
$790$	0	0
$791$	−6.00000	−0.213335
$792$	5.00000	0.177667
$793$	6.00000	0.213066
$794$	22.0000	0.780751
$795$	0	0
$796$	−4.00000	−0.141776
$797$	21.0000	0.743858	0.371929	−	0.928261i	$-0.378696\pi$
$797$	21.0000	0.743858	0.371929	+	0.928261i	$0.378696\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	16.0000	0.565332
$802$	−16.0000	−0.564980
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	−3.00000	−0.105605
$808$	−17.0000	−0.598058
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	10.0000	0.350931
$813$	15.0000	0.526073
$814$	45.0000	1.57725
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	0	0
$818$	−26.0000	−0.909069
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−21.0000	−0.732905	−0.366453	−	0.930437i	$-0.619428\pi$
$821$	−21.0000	−0.732905	−0.366453	+	0.930437i	$0.619428\pi$
$822$	−5.00000	−0.174395
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−4.00000	−0.139347
$825$	−25.0000	−0.870388
$826$	−1.00000	−0.0347945
$827$	26.0000	0.904109	0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000	0.904109	0.452054	+	0.891990i	$0.350691\pi$
$828$	−6.00000	−0.208514
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	1.00000	0.0346688
$833$	−6.00000	−0.207888
$834$	4.00000	0.138509
$835$	0	0
$836$	0	0
$837$	8.00000	0.276520
$838$	35.0000	1.20905
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−1.00000	−0.0344623
$843$	7.00000	0.241093
$844$	−3.00000	−0.103264
$845$	0	0
$846$	0	0
$847$	−14.0000	−0.481046
$848$	4.00000	0.137361
$849$	−31.0000	−1.06392
$850$	5.00000	0.171499
$851$	−54.0000	−1.85110
$852$	−1.00000	−0.0342594
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	6.00000	0.205076
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	−5.00000	−0.170697
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−5.00000	−0.170400
$862$	−26.0000	−0.885564
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	31.0000	1.05342
$867$	16.0000	0.543388
$868$	8.00000	0.271538
$869$	15.0000	0.508840
$870$	0	0
$871$	4.00000	0.135535
$872$	10.0000	0.338643
$873$	−12.0000	−0.406138
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	11.0000	0.371232
$879$	12.0000	0.404750
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	6.00000	0.202031
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	1.00000	0.0336336
$885$	0	0
$886$	−17.0000	−0.571126
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	9.00000	0.302020
$889$	8.00000	0.268311
$890$	0	0
$891$	−5.00000	−0.167506
$892$	−9.00000	−0.301342
$893$	0	0
$894$	21.0000	0.702345
$895$	0	0
$896$	1.00000	0.0334077
$897$	6.00000	0.200334
$898$	31.0000	1.03448
$899$	80.0000	2.66815
$900$	−5.00000	−0.166667
$901$	4.00000	0.133259
$902$	−25.0000	−0.832409
$903$	3.00000	0.0998337
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−10.0000	−0.332228
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	7.00000	0.232303
$909$	17.0000	0.563854
$910$	0	0
$911$	1.00000	0.0331315	0.0165657	−	0.999863i	$-0.494727\pi$
$911$	1.00000	0.0331315	0.0165657	+	0.999863i	$0.494727\pi$
$912$	0	0
$913$	−35.0000	−1.15833
$914$	4.00000	0.132308
$915$	0	0
$916$	−25.0000	−0.826023
$917$	20.0000	0.660458
$918$	1.00000	0.0330049
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	20.0000	0.658665
$923$	1.00000	0.0329154
$924$	−5.00000	−0.164488
$925$	−45.0000	−1.47959
$926$	−20.0000	−0.657241
$927$	4.00000	0.131377
$928$	10.0000	0.328266
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	0	0
$932$	20.0000	0.655122
$933$	−7.00000	−0.229170
$934$	−13.0000	−0.425373
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	4.00000	0.130605
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−6.00000	−0.195491
$943$	30.0000	0.976934
$944$	−1.00000	−0.0325472
$945$	0	0
$946$	15.0000	0.487692
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	3.00000	0.0974355
$949$	0	0
$950$	0	0
$951$	−32.0000	−1.03767
$952$	1.00000	0.0324102
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	−4.00000	−0.129505
$955$	0	0
$956$	8.00000	0.258738
$957$	−50.0000	−1.61627
$958$	28.0000	0.904639
$959$	5.00000	0.161458
$960$	0	0
$961$	33.0000	1.06452
$962$	−9.00000	−0.290172
$963$	−6.00000	−0.193347
$964$	15.0000	0.483117
$965$	0	0
$966$	6.00000	0.193047
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	−14.0000	−0.449977
$969$	0	0
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	−1.00000	−0.0320750
$973$	−4.00000	−0.128234
$974$	7.00000	0.224294
$975$	5.00000	0.160128
$976$	6.00000	0.192055
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	14.0000	0.447671
$979$	−80.0000	−2.55681
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−30.0000	−0.957338
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	−5.00000	−0.159394
$985$	0	0
$986$	10.0000	0.318465
$987$	0	0
$988$	0	0
$989$	−18.0000	−0.572367
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	8.00000	0.254000
$993$	−34.0000	−1.07896
$994$	1.00000	0.0317181
$995$	0	0
$996$	−7.00000	−0.221803
$997$	−20.0000	−0.633406	−0.316703	−	0.948525i	$-0.602576\pi$
$997$	−20.0000	−0.633406	−0.316703	+	0.948525i	$0.602576\pi$
$998$	−24.0000	−0.759707
$999$	−9.00000	−0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.a.a.1.1	✓	1
3.2	odd	2		1062.2.a.k.1.1		1
4.3	odd	2		2832.2.a.e.1.1		1
5.4	even	2		8850.2.a.be.1.1		1
12.11	even	2		8496.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.a.a.1.1	✓	1	1.1	even	1	trivial
1062.2.a.k.1.1		1	3.2	odd	2
2832.2.a.e.1.1		1	4.3	odd	2
8496.2.a.j.1.1		1	12.11	even	2
8850.2.a.be.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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