LMFDB - Embedded newform 637.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, 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("637.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$637 = 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	637.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:	$5.08647060876$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	637.1

$q$ -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^{4} -3.00000 q^{8} -3.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{16} +7.00000 q^{17} -3.00000 q^{18} -7.00000 q^{19} -3.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} -1.00000 q^{26} -5.00000 q^{29} +5.00000 q^{32} +7.00000 q^{34} +3.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} +2.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +7.00000 q^{47} -5.00000 q^{50} +1.00000 q^{52} -3.00000 q^{53} -5.00000 q^{58} -7.00000 q^{59} -7.00000 q^{61} +7.00000 q^{64} -3.00000 q^{67} -7.00000 q^{68} -5.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +7.00000 q^{76} -6.00000 q^{79} +9.00000 q^{81} +2.00000 q^{86} +9.00000 q^{88} +6.00000 q^{92} +7.00000 q^{94} -14.0000 q^{97} +9.00000 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

)

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−3.00000	−0.707107
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	7.00000	1.20049
$35$	0	0
$36$	3.00000	0.500000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	0	0
$50$	−5.00000	−0.707107
$51$	0	0
$52$	1.00000	0.138675
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−5.00000	−0.656532
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	9.00000	1.06066
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	7.00000	0.802955
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	9.00000	0.959403
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	7.00000	0.721995
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	9.00000	0.904534
$100$	5.00000	0.500000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−3.00000	−0.291386
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	0	0
$116$	5.00000	0.464238
$117$	3.00000	0.277350
$118$	−7.00000	−0.644402
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−7.00000	−0.633750
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	0	0
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−21.0000	−1.80074
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	−5.00000	−0.419591
$143$	3.00000	0.250873
$144$	3.00000	0.250000
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	21.0000	1.70332
$153$	−21.0000	−1.69775
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	21.0000	1.60591
$172$	−2.00000	−0.152499
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	18.0000	1.32698
$185$	0	0
$186$	0	0
$187$	−21.0000	−1.53567
$188$	−7.00000	−0.510527
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	9.00000	0.639602
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	15.0000	1.06066
$201$	0	0
$202$	−14.0000	−0.985037
$203$	0	0
$204$	0	0
$205$	0	0
$206$	14.0000	0.975426
$207$	18.0000	1.25109
$208$	1.00000	0.0693375
$209$	21.0000	1.45260
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	4.00000	0.270914
$219$	0	0
$220$	0	0
$221$	−7.00000	−0.470871
$222$	0	0
$223$	21.0000	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$223$	21.0000	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	9.00000	0.598671
$227$	−28.0000	−1.85843	−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000	−1.85843	−0.929213	+	0.369546i	$0.879513\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	15.0000	0.984798
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	7.00000	0.455661
$237$	0	0
$238$	0	0
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	7.00000	0.445399
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.0000	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$251$	−14.0000	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	2.00000	0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	15.0000	0.928477
$262$	14.0000	0.864923
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	3.00000	0.183254
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	4.00000	0.241649
$275$	15.0000	0.904534
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	5.00000	0.296695
$285$	0	0
$286$	3.00000	0.177394
$287$	0	0
$288$	−15.0000	−0.883883
$289$	32.0000	1.88235
$290$	0	0
$291$	0	0
$292$	−14.0000	−0.819288
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	−24.0000	−1.39497
$297$	0	0
$298$	−6.00000	−0.347571
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	−3.00000	−0.172631
$303$	0	0
$304$	7.00000	0.401478
$305$	0	0
$306$	−21.0000	−1.20049
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	6.00000	0.337526
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	15.0000	0.839839
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−49.0000	−2.72643
$324$	−9.00000	−0.500000
$325$	5.00000	0.277350
$326$	−13.0000	−0.720003
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−24.0000	−1.31519
$334$	−7.00000	−0.383023
$335$	0	0
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	0	0
$342$	21.0000	1.13555
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	7.00000	0.376322
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−15.0000	−0.799503
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−10.0000	−0.528516
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−7.00000	−0.367912
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−21.0000	−1.08588
$375$	0	0
$376$	−21.0000	−1.08299
$377$	5.00000	0.257513
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	−20.0000	−1.02329
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	−6.00000	−0.304997
$388$	14.0000	0.710742
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	−42.0000	−2.12403
$392$	0	0
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	−9.00000	−0.452267
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−14.0000	−0.701757
$399$	0	0
$400$	5.00000	0.250000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	0	0
$412$	−14.0000	−0.689730
$413$	0	0
$414$	18.0000	0.884652
$415$	0	0
$416$	−5.00000	−0.245145
$417$	0	0
$418$	21.0000	1.02714
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	−26.0000	−1.26566
$423$	−21.0000	−1.02105
$424$	9.00000	0.437079
$425$	−35.0000	−1.69775
$426$	0	0
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	21.0000	1.00920	0.504598	−	0.863355i	$-0.331641\pi$
$433$	21.0000	1.00920	0.504598	+	0.863355i	$0.331641\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	42.0000	2.00913
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	0	0
$442$	−7.00000	−0.332956
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	0	0
$446$	21.0000	0.994379
$447$	0	0
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	15.0000	0.707107
$451$	0	0
$452$	−9.00000	−0.423324
$453$	0	0
$454$	−28.0000	−1.31411
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−27.0000	−1.25075
$467$	14.0000	0.647843	0.323921	−	0.946084i	$-0.394999\pi$
$467$	14.0000	0.647843	0.323921	+	0.946084i	$0.394999\pi$
$468$	−3.00000	−0.138675
$469$	0	0
$470$	0	0
$471$	0	0
$472$	21.0000	0.966603
$473$	−6.00000	−0.275880
$474$	0	0
$475$	35.0000	1.60591
$476$	0	0
$477$	9.00000	0.412082
$478$	−19.0000	−0.869040
$479$	7.00000	0.319838	0.159919	−	0.987130i	$-0.448877\pi$
$479$	7.00000	0.319838	0.159919	+	0.987130i	$0.448877\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	28.0000	1.27537
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	25.0000	1.13286	0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000	1.13286	0.566429	+	0.824110i	$0.308325\pi$
$488$	21.0000	0.950625
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−35.0000	−1.57632
$494$	7.00000	0.314945
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	0	0
$502$	−14.0000	−0.624851
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	0	0
$506$	18.0000	0.800198
$507$	0	0
$508$	−2.00000	−0.0887357
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−14.0000	−0.617514
$515$	0	0
$516$	0	0
$517$	−21.0000	−0.923579
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	15.0000	0.656532
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	21.0000	0.911322
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	−21.0000	−0.905374
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	−7.00000	−0.300676
$543$	0	0
$544$	35.0000	1.50061
$545$	0	0
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−4.00000	−0.170872
$549$	21.0000	0.896258
$550$	15.0000	0.639602
$551$	35.0000	1.49105
$552$	0	0
$553$	0	0
$554$	−17.0000	−0.722261
$555$	0	0
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	15.0000	0.629386
$569$	1.00000	0.0419222	0.0209611	−	0.999780i	$-0.493327\pi$
$569$	1.00000	0.0419222	0.0209611	+	0.999780i	$0.493327\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−3.00000	−0.125436
$573$	0	0
$574$	0	0
$575$	30.0000	1.25109
$576$	−21.0000	−0.875000
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	32.0000	1.33102
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.00000	0.372742
$584$	−42.0000	−1.73797
$585$	0	0
$586$	14.0000	0.578335
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	6.00000	0.245358
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	9.00000	0.366508
$604$	3.00000	0.122068
$605$	0	0
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−35.0000	−1.41944
$609$	0	0
$610$	0	0
$611$	−7.00000	−0.283190
$612$	21.0000	0.848875
$613$	32.0000	1.29247	0.646234	−	0.763139i	$-0.276343\pi$
$613$	32.0000	1.29247	0.646234	+	0.763139i	$0.276343\pi$
$614$	21.0000	0.847491
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−7.00000	−0.279330
$629$	56.0000	2.23287
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	18.0000	0.716002
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	0	0
$638$	15.0000	0.593856
$639$	15.0000	0.593391
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	0	0
$646$	−49.0000	−1.92788
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	−27.0000	−1.06066
$649$	21.0000	0.824322
$650$	5.00000	0.196116
$651$	0	0
$652$	13.0000	0.509119
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−42.0000	−1.63858
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−24.0000	−0.929981
$667$	30.0000	1.16160
$668$	7.00000	0.270838
$669$	0	0
$670$	0	0
$671$	21.0000	0.810696
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	23.0000	0.885927
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−35.0000	−1.34516	−0.672580	−	0.740025i	$-0.734814\pi$
$677$	−35.0000	−1.34516	−0.672580	+	0.740025i	$0.734814\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	−21.0000	−0.802955
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	3.00000	0.114291
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−7.00000	−0.266100
$693$	0	0
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	0	0
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−56.0000	−2.11208
$704$	−21.0000	−0.791467
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	18.0000	0.675053
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	10.0000	0.373718
$717$	0	0
$718$	8.00000	0.298557
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	0	0
$722$	30.0000	1.11648
$723$	0	0
$724$	7.00000	0.260153
$725$	25.0000	0.928477
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	14.0000	0.517809
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	−30.0000	−1.10581
$737$	9.00000	0.331519
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	0	0
$746$	15.0000	0.549189
$747$	0	0
$748$	21.0000	0.767836
$749$	0	0
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−7.00000	−0.255264
$753$	0	0
$754$	5.00000	0.182089
$755$	0	0
$756$	0	0
$757$	9.00000	0.327111	0.163555	−	0.986534i	$-0.447704\pi$
$757$	9.00000	0.327111	0.163555	+	0.986534i	$0.447704\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	20.0000	0.723575
$765$	0	0
$766$	0	0
$767$	7.00000	0.252755
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−6.00000	−0.215666
$775$	0	0
$776$	42.0000	1.50771
$777$	0	0
$778$	−3.00000	−0.107555
$779$	0	0
$780$	0	0
$781$	15.0000	0.536742
$782$	−42.0000	−1.50192
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−27.0000	−0.959403
$793$	7.00000	0.248577
$794$	14.0000	0.496841
$795$	0	0
$796$	14.0000	0.496217
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	49.0000	1.73350
$800$	−25.0000	−0.883883
$801$	0	0
$802$	22.0000	0.776847
$803$	−42.0000	−1.48215
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	42.0000	1.47755
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	−14.0000	−0.489798
$818$	−28.0000	−0.978997
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−42.0000	−1.46314
$825$	0	0
$826$	0	0
$827$	−5.00000	−0.173867	−0.0869335	−	0.996214i	$-0.527707\pi$
$827$	−5.00000	−0.173867	−0.0869335	+	0.996214i	$0.527707\pi$
$828$	−18.0000	−0.625543
$829$	35.0000	1.21560	0.607800	−	0.794090i	$-0.292052\pi$
$829$	35.0000	1.21560	0.607800	+	0.794090i	$0.292052\pi$
$830$	0	0
$831$	0	0
$832$	−7.00000	−0.242681
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−21.0000	−0.726300
$837$	0	0
$838$	14.0000	0.483622
$839$	7.00000	0.241667	0.120833	−	0.992673i	$-0.461443\pi$
$839$	7.00000	0.241667	0.120833	+	0.992673i	$0.461443\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	30.0000	1.03387
$843$	0	0
$844$	26.0000	0.894957
$845$	0	0
$846$	−21.0000	−0.721995
$847$	0	0
$848$	3.00000	0.103020
$849$	0	0
$850$	−35.0000	−1.20049
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	0	0
$856$	−24.0000	−0.820303
$857$	21.0000	0.717346	0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000	0.717346	0.358673	+	0.933463i	$0.383229\pi$
$858$	0	0
$859$	−56.0000	−1.91070	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−56.0000	−1.91070	−0.955348	+	0.295484i	$0.904519\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	0	0
$865$	0	0
$866$	21.0000	0.713609
$867$	0	0
$868$	0	0
$869$	18.0000	0.610608
$870$	0	0
$871$	3.00000	0.101651
$872$	−12.0000	−0.406371
$873$	42.0000	1.42148
$874$	42.0000	1.42067
$875$	0	0
$876$	0	0
$877$	−48.0000	−1.62084	−0.810422	−	0.585846i	$-0.800762\pi$
$877$	−48.0000	−1.62084	−0.810422	+	0.585846i	$0.800762\pi$
$878$	14.0000	0.472477
$879$	0	0
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−40.0000	−1.34611	−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000	−1.34611	−0.673054	+	0.739594i	$0.735018\pi$
$884$	7.00000	0.235435
$885$	0	0
$886$	−20.0000	−0.671913
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−27.0000	−0.904534
$892$	−21.0000	−0.703132
$893$	−49.0000	−1.63972
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−12.0000	−0.400445
$899$	0	0
$900$	−15.0000	−0.500000
$901$	−21.0000	−0.699611
$902$	0	0
$903$	0	0
$904$	−27.0000	−0.898007
$905$	0	0
$906$	0	0
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	28.0000	0.929213
$909$	42.0000	1.39305
$910$	0	0
$911$	−54.0000	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	0	0
$922$	−28.0000	−0.922131
$923$	5.00000	0.164577
$924$	0	0
$925$	−40.0000	−1.31519
$926$	16.0000	0.525793
$927$	−42.0000	−1.37946
$928$	−25.0000	−0.820665
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	0	0
$932$	27.0000	0.884414
$933$	0	0
$934$	14.0000	0.458094
$935$	0	0
$936$	−9.00000	−0.294174
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	0	0
$944$	7.00000	0.227831
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	35.0000	1.13555
$951$	0	0
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	9.00000	0.291386
$955$	0	0
$956$	19.0000	0.614504
$957$	0	0
$958$	7.00000	0.226160
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	−24.0000	−0.773389
$964$	−28.0000	−0.901819
$965$	0	0
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	0	0
$974$	25.0000	0.801052
$975$	0	0
$976$	7.00000	0.224065
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	30.0000	0.957338
$983$	−7.00000	−0.223265	−0.111633	−	0.993750i	$-0.535608\pi$
$983$	−7.00000	−0.223265	−0.111633	+	0.993750i	$0.535608\pi$
$984$	0	0
$985$	0	0
$986$	−35.0000	−1.11463
$987$	0	0
$988$	−7.00000	−0.222700
$989$	−12.0000	−0.381578
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.00000	−0.221692	−0.110846	−	0.993838i	$-0.535356\pi$
$997$	−7.00000	−0.221692	−0.110846	+	0.993838i	$0.535356\pi$
$998$	8.00000	0.253236
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.c.1.1		1
3.2	odd	2		5733.2.a.c.1.1		1
7.2	even	3		91.2.e.a.53.1	✓	2
7.3	odd	6		637.2.e.a.79.1		2
7.4	even	3		91.2.e.a.79.1	yes	2
7.5	odd	6		637.2.e.a.508.1		2
7.6	odd	2		637.2.a.d.1.1		1
13.12	even	2		8281.2.a.f.1.1		1
21.2	odd	6		819.2.j.b.235.1		2
21.11	odd	6		819.2.j.b.352.1		2
21.20	even	2		5733.2.a.d.1.1		1
28.11	odd	6		1456.2.r.g.625.1		2
28.23	odd	6		1456.2.r.g.417.1		2
91.25	even	6		1183.2.e.b.170.1		2
91.51	even	6		1183.2.e.b.508.1		2
91.90	odd	2		8281.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.a.53.1	✓	2	7.2	even	3
91.2.e.a.79.1	yes	2	7.4	even	3
637.2.a.c.1.1		1	1.1	even	1	trivial
637.2.a.d.1.1		1	7.6	odd	2
637.2.e.a.79.1		2	7.3	odd	6
637.2.e.a.508.1		2	7.5	odd	6
819.2.j.b.235.1		2	21.2	odd	6
819.2.j.b.352.1		2	21.11	odd	6
1183.2.e.b.170.1		2	91.25	even	6
1183.2.e.b.508.1		2	91.51	even	6
1456.2.r.g.417.1		2	28.23	odd	6
1456.2.r.g.625.1		2	28.11	odd	6
5733.2.a.c.1.1		1	3.2	odd	2
5733.2.a.d.1.1		1	21.20	even	2
8281.2.a.e.1.1		1	91.90	odd	2
8281.2.a.f.1.1		1	13.12	even	2

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Classical	Maass
Hilbert	Bianchi

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

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	637.2.a.c.1.1

Level	$637$
Weight	$2$
Character	637.1
Self dual	yes
Analytic conductor	$5.086$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$