LMFDB - Embedded newform 637.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [637,2,Mod(1,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,2,3,0,0,0,-3,-6,-6,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -3.00000 q^{9} -6.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} -4.00000 q^{16} -4.00000 q^{17} +6.00000 q^{18} -5.00000 q^{19} +6.00000 q^{20} +12.0000 q^{22} +3.00000 q^{23} +4.00000 q^{25} -2.00000 q^{26} -5.00000 q^{29} +3.00000 q^{31} +8.00000 q^{32} +8.00000 q^{34} -6.00000 q^{36} -4.00000 q^{37} +10.0000 q^{38} +6.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} -9.00000 q^{45} -6.00000 q^{46} -7.00000 q^{47} -8.00000 q^{50} +2.00000 q^{52} -9.00000 q^{53} -18.0000 q^{55} +10.0000 q^{58} -8.00000 q^{59} +10.0000 q^{61} -6.00000 q^{62} -8.00000 q^{64} +3.00000 q^{65} -6.00000 q^{67} -8.00000 q^{68} -8.00000 q^{71} +13.0000 q^{73} +8.00000 q^{74} -10.0000 q^{76} +3.00000 q^{79} -12.0000 q^{80} +9.00000 q^{81} -12.0000 q^{82} -15.0000 q^{83} -12.0000 q^{85} +2.00000 q^{86} -3.00000 q^{89} +18.0000 q^{90} +6.00000 q^{92} +14.0000 q^{94} -15.0000 q^{95} -7.00000 q^{97} +18.0000 q^{99} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−1.00000
$10$	−6.00000	−1.89737
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	6.00000	1.41421
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	6.00000	1.34164
$21$	0	0
$22$	12.0000	2.55841
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−2.00000	−0.392232
$27$	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
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	8.00000	1.37199
$35$	0	0
$36$	−6.00000	−1.00000
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	10.0000	1.62221
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−12.0000	−1.80907
$45$	−9.00000	−1.34164
$46$	−6.00000	−0.884652
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	0	0
$50$	−8.00000	−1.13137
$51$	0	0
$52$	2.00000	0.277350
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	3.00000	0.372104
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−8.00000	−0.970143
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−10.0000	−1.14708
$77$	0	0
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	−12.0000	−1.34164
$81$	9.00000	1.00000
$82$	−12.0000	−1.32518
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	2.00000	0.215666
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	18.0000	1.89737
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	14.0000	1.44399
$95$	−15.0000	−1.53897
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	18.0000	1.80907
$100$	8.00000	0.800000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	18.0000	1.74831
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	36.0000	3.43247
$111$	0	0
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	−10.0000	−0.928477
$117$	−3.00000	−0.277350
$118$	16.0000	1.47292
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−20.0000	−1.81071
$123$	0	0
$124$	6.00000	0.538816
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	−6.00000	−0.526235
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	0	0
$137$	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
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	0	0
$142$	16.0000	1.34269
$143$	−6.00000	−0.501745
$144$	12.0000	1.00000
$145$	−15.0000	−1.24568
$146$	−26.0000	−2.15178
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	0	0
$155$	9.00000	0.722897
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	24.0000	1.89737
$161$	0	0
$162$	−18.0000	−1.41421
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	30.0000	2.32845
$167$	−5.00000	−0.386912	−0.193456	−	0.981109i	$-0.561970\pi$
$167$	−5.00000	−0.386912	−0.193456	+	0.981109i	$0.561970\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	24.0000	1.84072
$171$	15.0000	1.14708
$172$	−2.00000	−0.152499
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	0	0
$176$	24.0000	1.80907
$177$	0	0
$178$	6.00000	0.449719
$179$	23.0000	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$179$	23.0000	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$180$	−18.0000	−1.34164
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	24.0000	1.75505
$188$	−14.0000	−1.02105
$189$	0	0
$190$	30.0000	2.17643
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	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	$0.477302\pi$
$198$	−36.0000	−2.55841
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	−28.0000	−1.97007
$203$	0	0
$204$	0	0
$205$	18.0000	1.25717
$206$	−8.00000	−0.557386
$207$	−9.00000	−0.625543
$208$	−4.00000	−0.277350
$209$	30.0000	2.07514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−18.0000	−1.23625
$213$	0	0
$214$	8.00000	0.546869
$215$	−3.00000	−0.204598
$216$	0	0
$217$	0	0
$218$	4.00000	0.270914
$219$	0	0
$220$	−36.0000	−2.42712
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−15.0000	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$224$	0	0
$225$	−12.0000	−0.800000
$226$	6.00000	0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	6.00000	0.392232
$235$	−21.0000	−1.36989
$236$	−16.0000	−1.04151
$237$	0	0
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	−50.0000	−3.21412
$243$	0	0
$244$	20.0000	1.28037
$245$	0	0
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	0	0
$250$	6.00000	0.379473
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	8.00000	0.501965
$255$	0	0
$256$	16.0000	1.00000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	15.0000	0.928477
$262$	16.0000	0.988483
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	−27.0000	−1.65860
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	16.0000	0.970143
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−24.0000	−1.44725
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−36.0000	−2.15914
$279$	−9.00000	−0.538816
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	12.0000	0.709575
$287$	0	0
$288$	−24.0000	−1.41421
$289$	−1.00000	−0.0588235
$290$	30.0000	1.76166
$291$	0	0
$292$	26.0000	1.52153
$293$	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
$295$	−24.0000	−1.39733
$296$	0	0
$297$	0	0
$298$	36.0000	2.08542
$299$	3.00000	0.173494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	20.0000	1.14708
$305$	30.0000	1.71780
$306$	−24.0000	−1.37199
$307$	33.0000	1.88341	0.941705	−	0.336440i	$-0.109223\pi$
$307$	33.0000	1.88341	0.941705	+	0.336440i	$0.109223\pi$
$308$	0	0
$309$	0	0
$310$	−18.0000	−1.02233
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	6.00000	0.337526
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	−24.0000	−1.34164
$321$	0	0
$322$	0	0
$323$	20.0000	1.11283
$324$	18.0000	1.00000
$325$	4.00000	0.221880
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.0000	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$332$	−30.0000	−1.64646
$333$	12.0000	0.657596
$334$	10.0000	0.547176
$335$	−18.0000	−0.983445
$336$	0	0
$337$	17.0000	0.926049	0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000	0.926049	0.463025	+	0.886345i	$0.346764\pi$
$338$	−2.00000	−0.108786
$339$	0	0
$340$	−24.0000	−1.30158
$341$	−18.0000	−0.974755
$342$	−30.0000	−1.62221
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−16.0000	−0.860165
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	0	0
$352$	−48.0000	−2.55841
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−46.0000	−2.43118
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	28.0000	1.47165
$363$	0	0
$364$	0	0
$365$	39.0000	2.04135
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−12.0000	−0.625543
$369$	−18.0000	−0.937043
$370$	24.0000	1.24770
$371$	0	0
$372$	0	0
$373$	30.0000	1.55334	0.776671	−	0.629907i	$-0.216907\pi$
$373$	30.0000	1.55334	0.776671	+	0.629907i	$0.216907\pi$
$374$	−48.0000	−2.48202
$375$	0	0
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	−30.0000	−1.53897
$381$	0	0
$382$	16.0000	0.818631
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	−44.0000	−2.23954
$387$	3.00000	0.152499
$388$	−14.0000	−0.710742
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	−4.00000	−0.201517
$395$	9.00000	0.452839
$396$	36.0000	1.80907
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	−16.0000	−0.800000
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	28.0000	1.39305
$405$	27.0000	1.34164
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	13.0000	0.642809	0.321404	−	0.946942i	$-0.395845\pi$
$409$	13.0000	0.642809	0.321404	+	0.946942i	$0.395845\pi$
$410$	−36.0000	−1.77791
$411$	0	0
$412$	8.00000	0.394132
$413$	0	0
$414$	18.0000	0.884652
$415$	−45.0000	−2.20896
$416$	8.00000	0.392232
$417$	0	0
$418$	−60.0000	−2.93470
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	−12.0000	−0.584844	−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000	−0.584844	−0.292422	+	0.956289i	$0.594461\pi$
$422$	10.0000	0.486792
$423$	21.0000	1.02105
$424$	0	0
$425$	−16.0000	−0.776114
$426$	0	0
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	6.00000	0.289346
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	−15.0000	−0.717547
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	0	0
$442$	8.00000	0.380521
$443$	19.0000	0.902717	0.451359	−	0.892343i	$-0.350940\pi$
$443$	19.0000	0.902717	0.451359	+	0.892343i	$0.350940\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	30.0000	1.42054
$447$	0	0
$448$	0	0
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	24.0000	1.13137
$451$	−36.0000	−1.69517
$452$	−6.00000	−0.282216
$453$	0	0
$454$	40.0000	1.87729
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	28.0000	1.30835
$459$	0	0
$460$	18.0000	0.839254
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	−30.0000	−1.38972
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	42.0000	1.93732
$471$	0	0
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	−20.0000	−0.917663
$476$	0	0
$477$	27.0000	1.23625
$478$	8.00000	0.365911
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	−34.0000	−1.54866
$483$	0	0
$484$	50.0000	2.27273
$485$	−21.0000	−0.953561
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	10.0000	0.449921
$495$	54.0000	2.42712
$496$	−12.0000	−0.538816
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−6.00000	−0.268328
$501$	0	0
$502$	−52.0000	−2.32087
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	42.0000	1.86898
$506$	36.0000	1.60040
$507$	0	0
$508$	−8.00000	−0.354943
$509$	19.0000	0.842160	0.421080	−	0.907023i	$-0.361651\pi$
$509$	19.0000	0.842160	0.421080	+	0.907023i	$0.361651\pi$
$510$	0	0
$511$	0	0
$512$	−32.0000	−1.41421
$513$	0	0
$514$	−4.00000	−0.176432
$515$	12.0000	0.528783
$516$	0	0
$517$	42.0000	1.84716
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	−30.0000	−1.31306
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	30.0000	1.30806
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−14.0000	−0.608696
$530$	54.0000	2.34561
$531$	24.0000	1.04151
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	−32.0000	−1.37199
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−7.00000	−0.299298	−0.149649	−	0.988739i	$-0.547814\pi$
$547$	−7.00000	−0.299298	−0.149649	+	0.988739i	$0.547814\pi$
$548$	8.00000	0.341743
$549$	−30.0000	−1.28037
$550$	48.0000	2.04673
$551$	25.0000	1.06504
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	36.0000	1.52674
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	18.0000	0.762001
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	0	0
$562$	60.0000	2.53095
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	32.0000	1.34506
$567$	0	0
$568$	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
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	24.0000	1.00000
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	2.00000	0.0831890
$579$	0	0
$580$	−30.0000	−1.24568
$581$	0	0
$582$	0	0
$583$	54.0000	2.23645
$584$	0	0
$585$	−9.00000	−0.372104
$586$	−38.0000	−1.56977
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	48.0000	1.97613
$591$	0	0
$592$	16.0000	0.657596
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	0	0
$595$	0	0
$596$	−36.0000	−1.47462
$597$	0	0
$598$	−6.00000	−0.245358
$599$	11.0000	0.449448	0.224724	−	0.974422i	$-0.427852\pi$
$599$	11.0000	0.449448	0.224724	+	0.974422i	$0.427852\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	18.0000	0.733017
$604$	0	0
$605$	75.0000	3.04918
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	−60.0000	−2.42933
$611$	−7.00000	−0.283190
$612$	24.0000	0.970143
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	−66.0000	−2.66354
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	18.0000	0.722897
$621$	0	0
$622$	−12.0000	−0.481156
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	44.0000	1.75859
$627$	0	0
$628$	16.0000	0.638470
$629$	16.0000	0.637962
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	0	0
$633$	0	0
$634$	48.0000	1.90632
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	−60.0000	−2.37542
$639$	24.0000	0.949425
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	−40.0000	−1.57378
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	−8.00000	−0.313786
$651$	0	0
$652$	−8.00000	−0.313304
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−24.0000	−0.937043
$657$	−39.0000	−1.52153
$658$	0	0
$659$	17.0000	0.662226	0.331113	−	0.943591i	$-0.392576\pi$
$659$	17.0000	0.662226	0.331113	+	0.943591i	$0.392576\pi$
$660$	0	0
$661$	33.0000	1.28355	0.641776	−	0.766892i	$-0.278198\pi$
$661$	33.0000	1.28355	0.641776	+	0.766892i	$0.278198\pi$
$662$	−44.0000	−1.71011
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−24.0000	−0.929981
$667$	−15.0000	−0.580802
$668$	−10.0000	−0.386912
$669$	0	0
$670$	36.0000	1.39080
$671$	−60.0000	−2.31627
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	2.00000	0.0769231
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	36.0000	1.37851
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	30.0000	1.14708
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−11.0000	−0.418460	−0.209230	−	0.977866i	$-0.567096\pi$
$691$	−11.0000	−0.418460	−0.209230	+	0.977866i	$0.567096\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	64.0000	2.42941
$695$	54.0000	2.04834
$696$	0	0
$697$	−24.0000	−0.909065
$698$	22.0000	0.832712
$699$	0	0
$700$	0	0
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	48.0000	1.80907
$705$	0	0
$706$	−20.0000	−0.752710
$707$	0	0
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	48.0000	1.80141
$711$	−9.00000	−0.337526
$712$	0	0
$713$	9.00000	0.337053
$714$	0	0
$715$	−18.0000	−0.673162
$716$	46.0000	1.71910
$717$	0	0
$718$	−40.0000	−1.49279
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	36.0000	1.34164
$721$	0	0
$722$	−12.0000	−0.446594
$723$	0	0
$724$	−28.0000	−1.04061
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−78.0000	−2.88691
$731$	4.00000	0.147945
$732$	0	0
$733$	−51.0000	−1.88373	−0.941864	−	0.335994i	$-0.890928\pi$
$733$	−51.0000	−1.88373	−0.941864	+	0.335994i	$0.890928\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	24.0000	0.884652
$737$	36.0000	1.32608
$738$	36.0000	1.32518
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−54.0000	−1.97841
$746$	−60.0000	−2.19676
$747$	45.0000	1.64646
$748$	48.0000	1.75505
$749$	0	0
$750$	0	0
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	28.0000	1.02105
$753$	0	0
$754$	10.0000	0.364179
$755$	0	0
$756$	0	0
$757$	−15.0000	−0.545184	−0.272592	−	0.962130i	$-0.587881\pi$
$757$	−15.0000	−0.545184	−0.272592	+	0.962130i	$0.587881\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	0	0
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	36.0000	1.30158
$766$	−72.0000	−2.60147
$767$	−8.00000	−0.288863
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	44.0000	1.58359
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	−6.00000	−0.215666
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	−60.0000	−2.15110
$779$	−30.0000	−1.07486
$780$	0	0
$781$	48.0000	1.71758
$782$	24.0000	0.858238
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	−37.0000	−1.31891	−0.659454	−	0.751745i	$-0.729212\pi$
$787$	−37.0000	−1.31891	−0.659454	+	0.751745i	$0.729212\pi$
$788$	4.00000	0.142494
$789$	0	0
$790$	−18.0000	−0.640411
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	32.0000	1.13137
$801$	9.00000	0.317999
$802$	64.0000	2.25992
$803$	−78.0000	−2.75256
$804$	0	0
$805$	0	0
$806$	−6.00000	−0.211341
$807$	0	0
$808$	0	0
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	−54.0000	−1.89737
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	0	0
$814$	−48.0000	−1.68240
$815$	−12.0000	−0.420342
$816$	0	0
$817$	5.00000	0.174928
$818$	−26.0000	−0.909069
$819$	0	0
$820$	36.0000	1.25717
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	−18.0000	−0.625543
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	90.0000	3.12395
$831$	0	0
$832$	−8.00000	−0.277350
$833$	0	0
$834$	0	0
$835$	−15.0000	−0.519096
$836$	60.0000	2.07514
$837$	0	0
$838$	−20.0000	−0.690889
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	24.0000	0.827095
$843$	0	0
$844$	−10.0000	−0.344214
$845$	3.00000	0.103203
$846$	−42.0000	−1.44399
$847$	0	0
$848$	36.0000	1.23625
$849$	0	0
$850$	32.0000	1.09759
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	0	0
$855$	45.0000	1.53897
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	−6.00000	−0.204598
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	24.0000	0.816024
$866$	24.0000	0.815553
$867$	0	0
$868$	0	0
$869$	−18.0000	−0.610608
$870$	0	0
$871$	−6.00000	−0.203302
$872$	0	0
$873$	21.0000	0.710742
$874$	30.0000	1.01477
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	−44.0000	−1.48493
$879$	0	0
$880$	72.0000	2.42712
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	−38.0000	−1.27663
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	0	0
$890$	18.0000	0.603361
$891$	−54.0000	−1.80907
$892$	−30.0000	−1.00447
$893$	35.0000	1.17123
$894$	0	0
$895$	69.0000	2.30642
$896$	0	0
$897$	0	0
$898$	−72.0000	−2.40267
$899$	−15.0000	−0.500278
$900$	−24.0000	−0.800000
$901$	36.0000	1.19933
$902$	72.0000	2.39734
$903$	0	0
$904$	0	0
$905$	−42.0000	−1.39613
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	−40.0000	−1.32745
$909$	−42.0000	−1.39305
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	90.0000	2.97857
$914$	0	0
$915$	0	0
$916$	−28.0000	−0.925146
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	−44.0000	−1.44906
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−16.0000	−0.526077
$926$	28.0000	0.920137
$927$	−12.0000	−0.394132
$928$	−40.0000	−1.31306
$929$	−5.00000	−0.164045	−0.0820223	−	0.996630i	$-0.526138\pi$
$929$	−5.00000	−0.164045	−0.0820223	+	0.996630i	$0.526138\pi$
$930$	0	0
$931$	0	0
$932$	30.0000	0.982683
$933$	0	0
$934$	−44.0000	−1.43972
$935$	72.0000	2.35465
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	0	0
$939$	0	0
$940$	−42.0000	−1.36989
$941$	−55.0000	−1.79295	−0.896474	−	0.443096i	$-0.853880\pi$
$941$	−55.0000	−1.79295	−0.896474	+	0.443096i	$0.853880\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	32.0000	1.04151
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	13.0000	0.421998
$950$	40.0000	1.29777
$951$	0	0
$952$	0	0
$953$	−39.0000	−1.26333	−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000	−1.26333	−0.631667	+	0.775240i	$0.717629\pi$
$954$	−54.0000	−1.74831
$955$	−24.0000	−0.776622
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−22.0000	−0.710788
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	8.00000	0.257930
$963$	12.0000	0.386695
$964$	34.0000	1.09507
$965$	66.0000	2.12462
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	42.0000	1.34854
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	0	0
$974$	52.0000	1.66619
$975$	0	0
$976$	−40.0000	−1.28037
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	6.00000	0.191565
$982$	24.0000	0.765871
$983$	−17.0000	−0.542216	−0.271108	−	0.962549i	$-0.587390\pi$
$983$	−17.0000	−0.542216	−0.271108	+	0.962549i	$0.587390\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	−40.0000	−1.27386
$987$	0	0
$988$	−10.0000	−0.318142
$989$	−3.00000	−0.0953945
$990$	−108.000	−3.43247
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	24.0000	0.762001
$993$	0	0
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.a.1.1		1
3.2	odd	2		5733.2.a.l.1.1		1
7.2	even	3		637.2.e.d.508.1		2
7.3	odd	6		637.2.e.e.79.1		2
7.4	even	3		637.2.e.d.79.1		2
7.5	odd	6		637.2.e.e.508.1		2
7.6	odd	2		91.2.a.a.1.1	✓	1
13.12	even	2		8281.2.a.l.1.1		1
21.20	even	2		819.2.a.f.1.1		1
28.27	even	2		1456.2.a.g.1.1		1
35.34	odd	2		2275.2.a.h.1.1		1
56.13	odd	2		5824.2.a.s.1.1		1
56.27	even	2		5824.2.a.t.1.1		1
91.34	even	4		1183.2.c.b.337.1		2
91.83	even	4		1183.2.c.b.337.2		2
91.90	odd	2		1183.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.a.1.1	✓	1	7.6	odd	2
637.2.a.a.1.1		1	1.1	even	1	trivial
637.2.e.d.79.1		2	7.4	even	3
637.2.e.d.508.1		2	7.2	even	3
637.2.e.e.79.1		2	7.3	odd	6
637.2.e.e.508.1		2	7.5	odd	6
819.2.a.f.1.1		1	21.20	even	2
1183.2.a.b.1.1		1	91.90	odd	2
1183.2.c.b.337.1		2	91.34	even	4
1183.2.c.b.337.2		2	91.83	even	4
1456.2.a.g.1.1		1	28.27	even	2
2275.2.a.h.1.1		1	35.34	odd	2
5733.2.a.l.1.1		1	3.2	odd	2
5824.2.a.s.1.1		1	56.13	odd	2
5824.2.a.t.1.1		1	56.27	even	2
8281.2.a.l.1.1		1	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

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L-functions

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Representations

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