LMFDB - Embedded newform 676.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,2,Mod(1,676)]

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, 0]))

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

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

chi := DirichletCharacter("676.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 676 = 2^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	676.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:	$5.39788717664$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	676.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -5.00000 q^{11} +6.00000 q^{15} +3.00000 q^{17} -3.00000 q^{19} +3.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} -1.00000 q^{29} -8.00000 q^{31} -15.0000 q^{33} +2.00000 q^{35} +3.00000 q^{37} +3.00000 q^{41} +1.00000 q^{43} +12.0000 q^{45} +4.00000 q^{47} -6.00000 q^{49} +9.00000 q^{51} -6.00000 q^{53} -10.0000 q^{55} -9.00000 q^{57} +5.00000 q^{59} -5.00000 q^{61} +6.00000 q^{63} +7.00000 q^{67} -3.00000 q^{69} -11.0000 q^{71} +14.0000 q^{73} -3.00000 q^{75} -5.00000 q^{77} -4.00000 q^{79} +9.00000 q^{81} +12.0000 q^{83} +6.00000 q^{85} -3.00000 q^{87} -9.00000 q^{89} -24.0000 q^{93} -6.00000 q^{95} -1.00000 q^{97} -30.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.00000	1.54919
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−15.0000	−2.61116
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	12.0000	1.78885
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−10.0000	−1.34840
$56$	0	0
$57$	−9.00000	−1.19208
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−11.0000	−1.30546	−0.652730	−	0.757591i	$-0.726376\pi$
$71$	−11.0000	−1.30546	−0.652730	+	0.757591i	$0.726376\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−24.0000	−2.48868
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	−30.0000	−3.01511
$100$	0	0
$101$	19.0000	1.89057	0.945285	−	0.326245i	$-0.105783\pi$
$101$	19.0000	1.89057	0.945285	+	0.326245i	$0.105783\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	9.00000	0.811503
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	18.0000	1.54919
$136$	0	0
$137$	15.0000	1.28154	0.640768	−	0.767734i	$-0.278616\pi$
$137$	15.0000	1.28154	0.640768	+	0.767734i	$0.278616\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	−18.0000	−1.42749
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−7.00000	−0.548282	−0.274141	−	0.961689i	$-0.588394\pi$
$163$	−7.00000	−0.548282	−0.274141	+	0.961689i	$0.588394\pi$
$164$	0	0
$165$	−30.0000	−2.33550
$166$	0	0
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−18.0000	−1.37649
$172$	0	0
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	15.0000	1.12747
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−15.0000	−1.10883
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	1.00000	0.0723575	0.0361787	−	0.999345i	$-0.488481\pi$
$191$	1.00000	0.0723575	0.0361787	+	0.999345i	$0.488481\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	0	0
$201$	21.0000	1.48123
$202$	0	0
$203$	−1.00000	−0.0701862
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	−33.0000	−2.26112
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	42.0000	2.83810
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	0	0
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	0	0
$248$	0	0
$249$	36.0000	2.28141
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	18.0000	1.12720
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−27.0000	−1.65237
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	−17.0000	−1.03268	−0.516338	−	0.856385i	$-0.672705\pi$
$271$	−17.0000	−1.03268	−0.516338	+	0.856385i	$0.672705\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	−48.0000	−2.87368
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	0	0
$285$	−18.0000	−1.06623
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−3.00000	−0.175863
$292$	0	0
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	0	0
$297$	−45.0000	−2.61116
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	57.0000	3.27456
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−24.0000	−1.36531
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	33.0000	1.84188
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	0	0
$326$	0	0
$327$	30.0000	1.65900
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	18.0000	0.986394
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	33.0000	1.79231
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	33.0000	1.77153	0.885766	−	0.464131i	$-0.153633\pi$
$347$	33.0000	1.77153	0.885766	+	0.464131i	$0.153633\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.00000	0.372572	0.186286	−	0.982496i	$-0.440355\pi$
$353$	7.00000	0.372572	0.186286	+	0.982496i	$0.440355\pi$
$354$	0	0
$355$	−22.0000	−1.16764
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	42.0000	2.20443
$364$	0	0
$365$	28.0000	1.46559
$366$	0	0
$367$	23.0000	1.20059	0.600295	−	0.799779i	$-0.295050\pi$
$367$	23.0000	1.20059	0.600295	+	0.799779i	$0.295050\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	−36.0000	−1.85903
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−15.0000	−0.768473
$382$	0	0
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	−10.0000	−0.509647
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	−9.00000	−0.450564
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	18.0000	0.894427
$406$	0	0
$407$	−15.0000	−0.743522
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	45.0000	2.21969
$412$	0	0
$413$	5.00000	0.246034
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−33.0000	−1.61602
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	24.0000	1.16692
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	−5.00000	−0.241967
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	−63.0000	−2.97980
$448$	0	0
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−15.0000	−0.706322
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	−48.0000	−2.22595
$466$	0	0
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−66.0000	−3.04112
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	29.0000	1.31412	0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000	1.31412	0.657058	+	0.753840i	$0.271801\pi$
$488$	0	0
$489$	−21.0000	−0.949653
$490$	0	0
$491$	31.0000	1.39901	0.699505	−	0.714628i	$-0.253404\pi$
$491$	31.0000	1.39901	0.699505	+	0.714628i	$0.253404\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−60.0000	−2.69680
$496$	0	0
$497$	−11.0000	−0.493417
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−27.0000	−1.20627
$502$	0	0
$503$	−23.0000	−1.02552	−0.512760	−	0.858532i	$-0.671377\pi$
$503$	−23.0000	−1.02552	−0.512760	+	0.858532i	$0.671377\pi$
$504$	0	0
$505$	38.0000	1.69098
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.00000	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$509$	−5.00000	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−27.0000	−1.19208
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−20.0000	−0.879599
$518$	0	0
$519$	33.0000	1.44854
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	15.0000	0.655904	0.327952	−	0.944694i	$-0.393642\pi$
$523$	15.0000	0.655904	0.327952	+	0.944694i	$0.393642\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	30.0000	1.30189
$532$	0	0
$533$	0	0
$534$	0	0
$535$	22.0000	0.951143
$536$	0	0
$537$	−63.0000	−2.71865
$538$	0	0
$539$	30.0000	1.29219
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	54.0000	2.31736
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	−30.0000	−1.28037
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	18.0000	0.764057
$556$	0	0
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−45.0000	−1.89990
$562$	0	0
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	0	0
$565$	22.0000	0.925547
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	7.00000	0.293455	0.146728	−	0.989177i	$-0.453126\pi$
$569$	7.00000	0.293455	0.146728	+	0.989177i	$0.453126\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	33.0000	1.37143
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	30.0000	1.24247
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.0000	−1.03186	−0.515930	−	0.856631i	$-0.672554\pi$
$587$	−25.0000	−1.03186	−0.515930	+	0.856631i	$0.672554\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	−27.0000	−1.11063
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	0	0
$597$	−57.0000	−2.33285
$598$	0	0
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	42.0000	1.71037
$604$	0	0
$605$	28.0000	1.13836
$606$	0	0
$607$	−27.0000	−1.09590	−0.547948	−	0.836512i	$-0.684591\pi$
$607$	−27.0000	−1.09590	−0.547948	+	0.836512i	$0.684591\pi$
$608$	0	0
$609$	−3.00000	−0.121566
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	7.00000	0.281809	0.140905	−	0.990023i	$-0.454999\pi$
$617$	7.00000	0.281809	0.140905	+	0.990023i	$0.454999\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	0	0
$623$	−9.00000	−0.360577
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	45.0000	1.79713
$628$	0	0
$629$	9.00000	0.358854
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	0	0
$633$	−39.0000	−1.55011
$634$	0	0
$635$	−10.0000	−0.396838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−66.0000	−2.61092
$640$	0	0
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	0	0
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	−29.0000	−1.14011	−0.570054	−	0.821607i	$-0.693078\pi$
$647$	−29.0000	−1.14011	−0.570054	+	0.821607i	$0.693078\pi$
$648$	0	0
$649$	−25.0000	−0.981336
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	84.0000	3.27715
$658$	0	0
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	−37.0000	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$661$	−37.0000	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−27.0000	−1.04388
$670$	0	0
$671$	25.0000	0.965114
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	63.0000	2.41417
$682$	0	0
$683$	−11.0000	−0.420903	−0.210452	−	0.977604i	$-0.567493\pi$
$683$	−11.0000	−0.420903	−0.210452	+	0.977604i	$0.567493\pi$
$684$	0	0
$685$	30.0000	1.14624
$686$	0	0
$687$	−18.0000	−0.686743
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	0	0
$693$	−30.0000	−1.13961
$694$	0	0
$695$	−22.0000	−0.834508
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	66.0000	2.49635
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−9.00000	−0.339441
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	19.0000	0.714569
$708$	0	0
$709$	−1.00000	−0.0375558	−0.0187779	−	0.999824i	$-0.505978\pi$
$709$	−1.00000	−0.0375558	−0.0187779	+	0.999824i	$0.505978\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	72.0000	2.68889
$718$	0	0
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−63.0000	−2.34300
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	−35.0000	−1.28924
$738$	0	0
$739$	15.0000	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$739$	15.0000	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	0	0
$745$	−42.0000	−1.53876
$746$	0	0
$747$	72.0000	2.63434
$748$	0	0
$749$	11.0000	0.401931
$750$	0	0
$751$	−9.00000	−0.328415	−0.164207	−	0.986426i	$-0.552507\pi$
$751$	−9.00000	−0.328415	−0.164207	+	0.986426i	$0.552507\pi$
$752$	0	0
$753$	27.0000	0.983935
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.00000	0.109037	0.0545184	−	0.998513i	$-0.482638\pi$
$757$	3.00000	0.109037	0.0545184	+	0.998513i	$0.482638\pi$
$758$	0	0
$759$	15.0000	0.544466
$760$	0	0
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	36.0000	1.30158
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.0000	−0.757279	−0.378640	−	0.925544i	$-0.623608\pi$
$769$	−21.0000	−0.757279	−0.378640	+	0.925544i	$0.623608\pi$
$770$	0	0
$771$	9.00000	0.324127
$772$	0	0
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	55.0000	1.96805
$782$	0	0
$783$	−9.00000	−0.321634
$784$	0	0
$785$	−44.0000	−1.57043
$786$	0	0
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	0	0
$789$	45.0000	1.60204
$790$	0	0
$791$	11.0000	0.391115
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−36.0000	−1.27679
$796$	0	0
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	−54.0000	−1.90800
$802$	0	0
$803$	−70.0000	−2.47025
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	45.0000	1.58408
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	−51.0000	−1.78865
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	−3.00000	−0.104957
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.0000	1.64031	0.820156	−	0.572140i	$-0.193887\pi$
$821$	47.0000	1.64031	0.820156	+	0.572140i	$0.193887\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	0	0
$825$	15.0000	0.522233
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	0	0
$831$	−27.0000	−0.936620
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	−72.0000	−2.48868
$838$	0	0
$839$	−43.0000	−1.48452	−0.742262	−	0.670109i	$-0.766247\pi$
$839$	−43.0000	−1.48452	−0.742262	+	0.670109i	$0.766247\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	57.0000	1.95623
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	−36.0000	−1.23117
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	0	0
$879$	57.0000	1.92256
$880$	0	0
$881$	51.0000	1.71823	0.859117	−	0.511780i	$-0.171014\pi$
$881$	51.0000	1.71823	0.859117	+	0.511780i	$0.171014\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	30.0000	1.00844
$886$	0	0
$887$	−53.0000	−1.77957	−0.889783	−	0.456384i	$-0.849144\pi$
$887$	−53.0000	−1.77957	−0.889783	+	0.456384i	$0.849144\pi$
$888$	0	0
$889$	−5.00000	−0.167695
$890$	0	0
$891$	−45.0000	−1.50756
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−42.0000	−1.40391
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	−18.0000	−0.599667
$902$	0	0
$903$	3.00000	0.0998337
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	3.00000	0.0996134	0.0498067	−	0.998759i	$-0.484139\pi$
$907$	3.00000	0.0996134	0.0498067	+	0.998759i	$0.484139\pi$
$908$	0	0
$909$	114.000	3.78114
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	−30.0000	−0.991769
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	33.0000	1.08857	0.544285	−	0.838901i	$-0.316801\pi$
$919$	33.0000	1.08857	0.544285	+	0.838901i	$0.316801\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−3.00000	−0.0986394
$926$	0	0
$927$	−48.0000	−1.57653
$928$	0	0
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	−72.0000	−2.35717
$934$	0	0
$935$	−30.0000	−0.981105
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	42.0000	1.37062
$940$	0	0
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	0	0
$945$	18.0000	0.585540
$946$	0	0
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	90.0000	2.91845
$952$	0	0
$953$	−1.00000	−0.0323932	−0.0161966	−	0.999869i	$-0.505156\pi$
$953$	−1.00000	−0.0323932	−0.0161966	+	0.999869i	$0.505156\pi$
$954$	0	0
$955$	2.00000	0.0647185
$956$	0	0
$957$	15.0000	0.484881
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	66.0000	2.12682
$964$	0	0
$965$	22.0000	0.708205
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	−27.0000	−0.867365
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	−11.0000	−0.352644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.0000	1.24772	0.623860	−	0.781536i	$-0.285563\pi$
$977$	39.0000	1.24772	0.623860	+	0.781536i	$0.285563\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	60.0000	1.91565
$982$	0	0
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	27.0000	0.857683	0.428842	−	0.903380i	$-0.358922\pi$
$991$	27.0000	0.857683	0.428842	+	0.903380i	$0.358922\pi$
$992$	0	0
$993$	51.0000	1.61844
$994$	0	0
$995$	−38.0000	−1.20468
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	0	0
$999$	27.0000	0.854242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	676.2.a.e.1.1		1
3.2	odd	2		6084.2.a.f.1.1		1
4.3	odd	2		2704.2.a.b.1.1		1
13.2	odd	12		676.2.h.b.485.1		4
13.3	even	3		52.2.e.a.9.1	✓	2
13.4	even	6		676.2.e.a.653.1		2
13.5	odd	4		676.2.d.d.337.1		2
13.6	odd	12		676.2.h.b.361.1		4
13.7	odd	12		676.2.h.b.361.2		4
13.8	odd	4		676.2.d.d.337.2		2
13.9	even	3		52.2.e.a.29.1	yes	2
13.10	even	6		676.2.e.a.529.1		2
13.11	odd	12		676.2.h.b.485.2		4
13.12	even	2		676.2.a.d.1.1		1
39.5	even	4		6084.2.b.l.4393.2		2
39.8	even	4		6084.2.b.l.4393.1		2
39.29	odd	6		468.2.l.a.217.1		2
39.35	odd	6		468.2.l.a.289.1		2
39.38	odd	2		6084.2.a.k.1.1		1
52.3	odd	6		208.2.i.d.113.1		2
52.31	even	4		2704.2.f.a.337.1		2
52.35	odd	6		208.2.i.d.81.1		2
52.47	even	4		2704.2.f.a.337.2		2
52.51	odd	2		2704.2.a.a.1.1		1
65.3	odd	12		1300.2.bb.f.1049.1		4
65.9	even	6		1300.2.i.f.601.1		2
65.22	odd	12		1300.2.bb.f.549.1		4
65.29	even	6		1300.2.i.f.1101.1		2
65.42	odd	12		1300.2.bb.f.1049.2		4
65.48	odd	12		1300.2.bb.f.549.2		4
91.3	odd	6		2548.2.l.a.373.1		2
91.9	even	3		2548.2.l.h.1537.1		2
91.16	even	3		2548.2.i.a.165.1		2
91.48	odd	6		2548.2.k.d.393.1		2
91.55	odd	6		2548.2.k.d.1569.1		2
91.61	odd	6		2548.2.l.a.1537.1		2
91.68	odd	6		2548.2.i.h.165.1		2
91.74	even	3		2548.2.i.a.1745.1		2
91.81	even	3		2548.2.l.h.373.1		2
91.87	odd	6		2548.2.i.h.1745.1		2
104.3	odd	6		832.2.i.a.321.1		2
104.29	even	6		832.2.i.j.321.1		2
104.35	odd	6		832.2.i.a.705.1		2
104.61	even	6		832.2.i.j.705.1		2
156.35	even	6		1872.2.t.f.289.1		2
156.107	even	6		1872.2.t.f.1153.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.e.a.9.1	✓	2	13.3	even	3
52.2.e.a.29.1	yes	2	13.9	even	3
208.2.i.d.81.1		2	52.35	odd	6
208.2.i.d.113.1		2	52.3	odd	6
468.2.l.a.217.1		2	39.29	odd	6
468.2.l.a.289.1		2	39.35	odd	6
676.2.a.d.1.1		1	13.12	even	2
676.2.a.e.1.1		1	1.1	even	1	trivial
676.2.d.d.337.1		2	13.5	odd	4
676.2.d.d.337.2		2	13.8	odd	4
676.2.e.a.529.1		2	13.10	even	6
676.2.e.a.653.1		2	13.4	even	6
676.2.h.b.361.1		4	13.6	odd	12
676.2.h.b.361.2		4	13.7	odd	12
676.2.h.b.485.1		4	13.2	odd	12
676.2.h.b.485.2		4	13.11	odd	12
832.2.i.a.321.1		2	104.3	odd	6
832.2.i.a.705.1		2	104.35	odd	6
832.2.i.j.321.1		2	104.29	even	6
832.2.i.j.705.1		2	104.61	even	6
1300.2.i.f.601.1		2	65.9	even	6
1300.2.i.f.1101.1		2	65.29	even	6
1300.2.bb.f.549.1		4	65.22	odd	12
1300.2.bb.f.549.2		4	65.48	odd	12
1300.2.bb.f.1049.1		4	65.3	odd	12
1300.2.bb.f.1049.2		4	65.42	odd	12
1872.2.t.f.289.1		2	156.35	even	6
1872.2.t.f.1153.1		2	156.107	even	6
2548.2.i.a.165.1		2	91.16	even	3
2548.2.i.a.1745.1		2	91.74	even	3
2548.2.i.h.165.1		2	91.68	odd	6
2548.2.i.h.1745.1		2	91.87	odd	6
2548.2.k.d.393.1		2	91.48	odd	6
2548.2.k.d.1569.1		2	91.55	odd	6
2548.2.l.a.373.1		2	91.3	odd	6
2548.2.l.a.1537.1		2	91.61	odd	6
2548.2.l.h.373.1		2	91.81	even	3
2548.2.l.h.1537.1		2	91.9	even	3
2704.2.a.a.1.1		1	52.51	odd	2
2704.2.a.b.1.1		1	4.3	odd	2
2704.2.f.a.337.1		2	52.31	even	4
2704.2.f.a.337.2		2	52.47	even	4
6084.2.a.f.1.1		1	3.2	odd	2
6084.2.a.k.1.1		1	39.38	odd	2
6084.2.b.l.4393.1		2	39.8	even	4
6084.2.b.l.4393.2		2	39.5	even	4

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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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