LMFDB - Embedded newform 6013.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6013,2,Mod(1,6013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6013, 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("6013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6013 = 7 \cdot 859 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6013.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:	$48.0140467354$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{12} +6.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{18} +4.00000 q^{19} +1.00000 q^{21} -2.00000 q^{23} -5.00000 q^{25} +12.0000 q^{26} +5.00000 q^{27} -2.00000 q^{28} +8.00000 q^{29} -10.0000 q^{31} -8.00000 q^{32} -4.00000 q^{36} +8.00000 q^{38} -6.00000 q^{39} +2.00000 q^{41} +2.00000 q^{42} +1.00000 q^{43} -4.00000 q^{46} +12.0000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -10.0000 q^{50} +12.0000 q^{52} +3.00000 q^{53} +10.0000 q^{54} -4.00000 q^{57} +16.0000 q^{58} -8.00000 q^{59} +10.0000 q^{61} -20.0000 q^{62} +2.00000 q^{63} -8.00000 q^{64} -2.00000 q^{67} +2.00000 q^{69} +8.00000 q^{71} -9.00000 q^{73} +5.00000 q^{75} +8.00000 q^{76} -12.0000 q^{78} +15.0000 q^{79} +1.00000 q^{81} +4.00000 q^{82} +15.0000 q^{83} +2.00000 q^{84} +2.00000 q^{86} -8.00000 q^{87} -7.00000 q^{89} -6.00000 q^{91} -4.00000 q^{92} +10.0000 q^{93} +24.0000 q^{94} +8.00000 q^{96} +1.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	2.00000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−2.00000	−0.577350
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−4.00000	−0.942809
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	12.0000	2.35339
$27$	5.00000	0.962250
$28$	−2.00000	−0.377964
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−4.00000	−0.666667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	8.00000	1.29777
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	2.00000	0.308607
$43$	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
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	−10.0000	−1.41421
$51$	0	0
$52$	12.0000	1.66410
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	10.0000	1.36083
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	16.0000	2.10090
$59$	−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	$0.278858\pi$
$62$	−20.0000	−2.54000
$63$	2.00000	0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	8.00000	0.917663
$77$	0	0
$78$	−12.0000	−1.35873
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	2.00000	0.215666
$87$	−8.00000	−0.857690
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−4.00000	−0.417029
$93$	10.0000	1.03695
$94$	24.0000	2.47541
$95$	0	0
$96$	8.00000	0.816497
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	−10.0000	−1.00000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	10.0000	0.962250
$109$	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
$111$	0	0
$112$	4.00000	0.377964
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	16.0000	1.48556
$117$	−12.0000	−1.10940
$118$	−16.0000	−1.47292
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	20.0000	1.81071
$123$	−2.00000	−0.180334
$124$	−20.0000	−1.79605
$125$	0	0
$126$	4.00000	0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	4.00000	0.340503
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	16.0000	1.34269
$143$	0	0
$144$	8.00000	0.666667
$145$	0	0
$146$	−18.0000	−1.48969
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	10.0000	0.816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−12.0000	−0.960769
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	30.0000	2.38667
$159$	−3.00000	−0.237915
$160$	0	0
$161$	2.00000	0.157622
$162$	2.00000	0.157135
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	30.0000	2.32845
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−8.00000	−0.611775
$172$	2.00000	0.152499
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	−16.0000	−1.21296
$175$	5.00000	0.377964
$176$	0	0
$177$	8.00000	0.601317
$178$	−14.0000	−1.04934
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−12.0000	−0.889499
$183$	−10.0000	−0.739221
$184$	0	0
$185$	0	0
$186$	20.0000	1.46647
$187$	0	0
$188$	24.0000	1.75038
$189$	−5.00000	−0.363696
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	8.00000	0.577350
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	27.0000	1.91398	0.956990	−	0.290122i	$-0.0936959\pi$
$199$	27.0000	1.91398	0.956990	+	0.290122i	$0.0936959\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−30.0000	−2.11079
$203$	−8.00000	−0.561490
$204$	0	0
$205$	0	0
$206$	32.0000	2.22955
$207$	4.00000	0.278019
$208$	−24.0000	−1.66410
$209$	0	0
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	6.00000	0.412082
$213$	−8.00000	−0.548151
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	10.0000	0.678844
$218$	20.0000	1.35457
$219$	9.00000	0.608164
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	8.00000	0.534522
$225$	10.0000	0.666667
$226$	36.0000	2.39468
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−8.00000	−0.529813
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−24.0000	−1.56893
$235$	0	0
$236$	−16.0000	−1.04151
$237$	−15.0000	−0.974355
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	−22.0000	−1.41421
$243$	−16.0000	−1.02640
$244$	20.0000	1.28037
$245$	0	0
$246$	−4.00000	−0.255031
$247$	24.0000	1.52708
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	−5.00000	−0.315597	−0.157799	−	0.987471i	$-0.550440\pi$
$251$	−5.00000	−0.315597	−0.157799	+	0.987471i	$0.550440\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	32.0000	2.00786
$255$	0	0
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	−16.0000	−0.990375
$262$	24.0000	1.48272
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	7.00000	0.428393
$268$	−4.00000	−0.244339
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	−10.0000	−0.604122
$275$	0	0
$276$	4.00000	0.240772
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	10.0000	0.599760
$279$	20.0000	1.19737
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	−24.0000	−1.42918
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	16.0000	0.942809
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−1.00000	−0.0586210
$292$	−18.0000	−1.05337
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	0	0
$297$	0	0
$298$	12.0000	0.695141
$299$	−12.0000	−0.693978
$300$	10.0000	0.577350
$301$	−1.00000	−0.0576390
$302$	32.0000	1.84139
$303$	15.0000	0.861727
$304$	−16.0000	−0.917663
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	26.0000	1.46726
$315$	0	0
$316$	30.0000	1.68763
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	6.00000	0.334887
$322$	4.00000	0.222911
$323$	0	0
$324$	2.00000	0.111111
$325$	−30.0000	−1.66410
$326$	−16.0000	−0.886158
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−12.0000	−0.661581
$330$	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
$333$	0	0
$334$	−16.0000	−0.875481
$335$	0	0
$336$	−4.00000	−0.218218
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	46.0000	2.50207
$339$	−18.0000	−0.977626
$340$	0	0
$341$	0	0
$342$	−16.0000	−0.865181
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−16.0000	−0.860165
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−16.0000	−0.857690
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	10.0000	0.534522
$351$	30.0000	1.60128
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	42.0000	2.21977
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	12.0000	0.630706
$363$	11.0000	0.577350
$364$	−12.0000	−0.628971
$365$	0	0
$366$	−20.0000	−1.04542
$367$	30.0000	1.56599	0.782994	−	0.622030i	$-0.213692\pi$
$367$	30.0000	1.56599	0.782994	+	0.622030i	$0.213692\pi$
$368$	8.00000	0.417029
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−3.00000	−0.155752
$372$	20.0000	1.03695
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.0000	2.47213
$378$	−10.0000	−0.514344
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	6.00000	0.306987
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	24.0000	1.22157
$387$	−2.00000	−0.101666
$388$	2.00000	0.101535
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12.0000	−0.605320
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	54.0000	2.70678
$399$	4.00000	0.200250
$400$	20.0000	1.00000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	4.00000	0.199502
$403$	−60.0000	−2.98881
$404$	−30.0000	−1.49256
$405$	0	0
$406$	−16.0000	−0.794067
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	32.0000	1.57653
$413$	8.00000	0.393654
$414$	8.00000	0.393179
$415$	0	0
$416$	−48.0000	−2.35339
$417$	−5.00000	−0.244851
$418$	0	0
$419$	1.00000	0.0488532	0.0244266	−	0.999702i	$-0.492224\pi$
$419$	1.00000	0.0488532	0.0244266	+	0.999702i	$0.492224\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	14.0000	0.681509
$423$	−24.0000	−1.16692
$424$	0	0
$425$	0	0
$426$	−16.0000	−0.775203
$427$	−10.0000	−0.483934
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−20.0000	−0.962250
$433$	−37.0000	−1.77811	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−37.0000	−1.77811	−0.889053	+	0.457804i	$0.848636\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	20.0000	0.957826
$437$	−8.00000	−0.382692
$438$	18.0000	0.860073
$439$	−38.0000	−1.81364	−0.906821	−	0.421517i	$-0.861498\pi$
$439$	−38.0000	−1.81364	−0.906821	+	0.421517i	$0.861498\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−35.0000	−1.66290	−0.831450	−	0.555599i	$-0.812489\pi$
$443$	−35.0000	−1.66290	−0.831450	+	0.555599i	$0.812489\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	−6.00000	−0.283790
$448$	8.00000	0.377964
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	20.0000	0.942809
$451$	0	0
$452$	36.0000	1.69330
$453$	−16.0000	−0.751746
$454$	−16.0000	−0.750917
$455$	0	0
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	24.0000	1.12145
$459$	0	0
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	−32.0000	−1.48556
$465$	0	0
$466$	−36.0000	−1.66767
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	−24.0000	−1.10940
$469$	2.00000	0.0923514
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	0	0
$474$	−30.0000	−1.37795
$475$	−20.0000	−0.917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	0	0
$482$	−42.0000	−1.91305
$483$	−2.00000	−0.0910032
$484$	−22.0000	−1.00000
$485$	0	0
$486$	−32.0000	−1.45155
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−4.00000	−0.180334
$493$	0	0
$494$	48.0000	2.15962
$495$	0	0
$496$	40.0000	1.79605
$497$	−8.00000	−0.358849
$498$	−30.0000	−1.34433
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	−10.0000	−0.446322
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23.0000	−1.02147
$508$	32.0000	1.41977
$509$	44.0000	1.95027	0.975133	−	0.221621i	$-0.0711348\pi$
$509$	44.0000	1.95027	0.975133	+	0.221621i	$0.0711348\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	32.0000	1.41421
$513$	20.0000	0.883022
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	0	0
$518$	0	0
$519$	8.00000	0.351161
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−32.0000	−1.40060
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	24.0000	1.04844
$525$	−5.00000	−0.218218
$526$	8.00000	0.348817
$527$	0	0
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	16.0000	0.694341
$532$	−8.00000	−0.346844
$533$	12.0000	0.519778
$534$	14.0000	0.605839
$535$	0	0
$536$	0	0
$537$	−21.0000	−0.906217
$538$	−20.0000	−0.862261
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−56.0000	−2.40541
$543$	−6.00000	−0.257485
$544$	0	0
$545$	0	0
$546$	12.0000	0.513553
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−10.0000	−0.427179
$549$	−20.0000	−0.853579
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	−15.0000	−0.637865
$554$	−46.0000	−1.95435
$555$	0	0
$556$	10.0000	0.424094
$557$	37.0000	1.56774	0.783870	−	0.620925i	$-0.213243\pi$
$557$	37.0000	1.56774	0.783870	+	0.620925i	$0.213243\pi$
$558$	40.0000	1.69334
$559$	6.00000	0.253773
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	−24.0000	−1.01058
$565$	0	0
$566$	22.0000	0.924729
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	31.0000	1.29731	0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000	1.29731	0.648655	+	0.761083i	$0.275332\pi$
$572$	0	0
$573$	−3.00000	−0.125327
$574$	−4.00000	−0.166957
$575$	10.0000	0.417029
$576$	16.0000	0.666667
$577$	−19.0000	−0.790980	−0.395490	−	0.918470i	$-0.629425\pi$
$577$	−19.0000	−0.790980	−0.395490	+	0.918470i	$0.629425\pi$
$578$	−34.0000	−1.41421
$579$	−12.0000	−0.498703
$580$	0	0
$581$	−15.0000	−0.622305
$582$	−2.00000	−0.0829027
$583$	0	0
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	−31.0000	−1.27951	−0.639753	−	0.768580i	$-0.720964\pi$
$587$	−31.0000	−1.27951	−0.639753	+	0.768580i	$0.720964\pi$
$588$	−2.00000	−0.0824786
$589$	−40.0000	−1.64817
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	−27.0000	−1.10504
$598$	−24.0000	−0.981433
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	−2.00000	−0.0815139
$603$	4.00000	0.162893
$604$	32.0000	1.30206
$605$	0	0
$606$	30.0000	1.21867
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−32.0000	−1.29777
$609$	8.00000	0.324176
$610$	0	0
$611$	72.0000	2.91281
$612$	0	0
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	−56.0000	−2.25998
$615$	0	0
$616$	0	0
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	−32.0000	−1.28723
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−10.0000	−0.401286
$622$	42.0000	1.68405
$623$	7.00000	0.280449
$624$	24.0000	0.960769
$625$	25.0000	1.00000
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	26.0000	1.03751
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−7.00000	−0.278225
$634$	34.0000	1.35031
$635$	0	0
$636$	−6.00000	−0.237915
$637$	6.00000	0.237729
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	12.0000	0.473602
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	41.0000	1.61188	0.805938	−	0.592000i	$-0.201661\pi$
$647$	41.0000	1.61188	0.805938	+	0.592000i	$0.201661\pi$
$648$	0	0
$649$	0	0
$650$	−60.0000	−2.35339
$651$	−10.0000	−0.391931
$652$	−16.0000	−0.626608
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	−8.00000	−0.312348
$657$	18.0000	0.702247
$658$	−24.0000	−0.935617
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−33.0000	−1.28355	−0.641776	−	0.766892i	$-0.721802\pi$
$661$	−33.0000	−1.28355	−0.641776	+	0.766892i	$0.721802\pi$
$662$	44.0000	1.71011
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	−16.0000	−0.619059
$669$	12.0000	0.463947
$670$	0	0
$671$	0	0
$672$	−8.00000	−0.308607
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	44.0000	1.69482
$675$	−25.0000	−0.962250
$676$	46.0000	1.76923
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−36.0000	−1.38257
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	−16.0000	−0.611775
$685$	0	0
$686$	−2.00000	−0.0763604
$687$	−12.0000	−0.457829
$688$	−4.00000	−0.152499
$689$	18.0000	0.685745
$690$	0	0
$691$	−27.0000	−1.02713	−0.513564	−	0.858051i	$-0.671675\pi$
$691$	−27.0000	−1.02713	−0.513564	+	0.858051i	$0.671675\pi$
$692$	−16.0000	−0.608229
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−68.0000	−2.57384
$699$	18.0000	0.680823
$700$	10.0000	0.377964
$701$	29.0000	1.09531	0.547657	−	0.836703i	$-0.315520\pi$
$701$	29.0000	1.09531	0.547657	+	0.836703i	$0.315520\pi$
$702$	60.0000	2.26455
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−48.0000	−1.80650
$707$	15.0000	0.564133
$708$	16.0000	0.601317
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	0	0
$716$	42.0000	1.56961
$717$	12.0000	0.448148
$718$	36.0000	1.34351
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	−6.00000	−0.223297
$723$	21.0000	0.780998
$724$	12.0000	0.445976
$725$	−40.0000	−1.48556
$726$	22.0000	0.816497
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	−20.0000	−0.739221
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	60.0000	2.21464
$735$	0	0
$736$	16.0000	0.589768
$737$	0	0
$738$	−8.00000	−0.294484
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	−6.00000	−0.220267
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	0	0
$746$	−36.0000	−1.31805
$747$	−30.0000	−1.09764
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	−48.0000	−1.75038
$753$	5.00000	0.182210
$754$	96.0000	3.49611
$755$	0	0
$756$	−10.0000	−0.363696
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−56.0000	−2.03401
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−32.0000	−1.15924
$763$	−10.0000	−0.362024
$764$	6.00000	0.217072
$765$	0	0
$766$	48.0000	1.73431
$767$	−48.0000	−1.73318
$768$	−16.0000	−0.577350
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	24.0000	0.863779
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−4.00000	−0.143777
$775$	50.0000	1.79605
$776$	0	0
$777$	0	0
$778$	38.0000	1.36237
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	40.0000	1.42948
$784$	−4.00000	−0.142857
$785$	0	0
$786$	−24.0000	−0.856052
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−24.0000	−0.854965
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	60.0000	2.13066
$794$	12.0000	0.425864
$795$	0	0
$796$	54.0000	1.91398
$797$	−28.0000	−0.991811	−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000	−0.991811	−0.495905	+	0.868377i	$0.665164\pi$
$798$	8.00000	0.283197
$799$	0	0
$800$	40.0000	1.41421
$801$	14.0000	0.494666
$802$	−44.0000	−1.55369
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−120.000	−4.22682
$807$	10.0000	0.352017
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	−16.0000	−0.561490
$813$	28.0000	0.982003
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	20.0000	0.699284
$819$	12.0000	0.419314
$820$	0	0
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	10.0000	0.348790
$823$	38.0000	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$823$	38.0000	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$824$	0	0
$825$	0	0
$826$	16.0000	0.556711
$827$	21.0000	0.730242	0.365121	−	0.930960i	$-0.381028\pi$
$827$	21.0000	0.730242	0.365121	+	0.930960i	$0.381028\pi$
$828$	8.00000	0.278019
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	0	0
$831$	23.0000	0.797861
$832$	−48.0000	−1.66410
$833$	0	0
$834$	−10.0000	−0.346272
$835$	0	0
$836$	0	0
$837$	−50.0000	−1.72825
$838$	2.00000	0.0690889
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	68.0000	2.34343
$843$	−9.00000	−0.309976
$844$	14.0000	0.481900
$845$	0	0
$846$	−48.0000	−1.65027
$847$	11.0000	0.377964
$848$	−12.0000	−0.412082
$849$	−11.0000	−0.377519
$850$	0	0
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−29.0000	−0.992941	−0.496471	−	0.868054i	$-0.665371\pi$
$853$	−29.0000	−0.992941	−0.496471	+	0.868054i	$0.665371\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	0	0
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	0	0
$859$	1.00000	0.0341196
$859$	1.00000	0.0341196
$860$	0	0
$861$	2.00000	0.0681598
$862$	48.0000	1.63489
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	−40.0000	−1.36083
$865$	0	0
$866$	−74.0000	−2.51462
$867$	17.0000	0.577350
$868$	20.0000	0.678844
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	−16.0000	−0.541208
$875$	0	0
$876$	18.0000	0.608164
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	−76.0000	−2.56488
$879$	−6.00000	−0.202375
$880$	0	0
$881$	48.0000	1.61716	0.808581	−	0.588386i	$-0.200236\pi$
$881$	48.0000	1.61716	0.808581	+	0.588386i	$0.200236\pi$
$882$	−4.00000	−0.134687
$883$	53.0000	1.78359	0.891796	−	0.452438i	$-0.149446\pi$
$883$	53.0000	1.78359	0.891796	+	0.452438i	$0.149446\pi$
$884$	0	0
$885$	0	0
$886$	−70.0000	−2.35170
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	−24.0000	−0.803579
$893$	48.0000	1.60626
$894$	−12.0000	−0.401340
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	30.0000	1.00111
$899$	−80.0000	−2.66815
$900$	20.0000	0.666667
$901$	0	0
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	0	0
$906$	−32.0000	−1.06313
$907$	40.0000	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$908$	−16.0000	−0.530979
$909$	30.0000	0.995037
$910$	0	0
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	16.0000	0.529813
$913$	0	0
$914$	66.0000	2.18309
$915$	0	0
$916$	24.0000	0.792982
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	6.00000	0.197599
$923$	48.0000	1.57994
$924$	0	0
$925$	0	0
$926$	−38.0000	−1.24876
$927$	−32.0000	−1.05102
$928$	−64.0000	−2.10090
$929$	−19.0000	−0.623370	−0.311685	−	0.950186i	$-0.600893\pi$
$929$	−19.0000	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−36.0000	−1.17922
$933$	−21.0000	−0.687509
$934$	−64.0000	−2.09414
$935$	0	0
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	4.00000	0.130605
$939$	1.00000	0.0326338
$940$	0	0
$941$	−13.0000	−0.423788	−0.211894	−	0.977293i	$-0.567963\pi$
$941$	−13.0000	−0.423788	−0.211894	+	0.977293i	$0.567963\pi$
$942$	−26.0000	−0.847126
$943$	−4.00000	−0.130258
$944$	32.0000	1.04151
$945$	0	0
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	−30.0000	−0.974355
$949$	−54.0000	−1.75291
$950$	−40.0000	−1.29777
$951$	−17.0000	−0.551263
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	28.0000	0.904639
$959$	5.00000	0.161458
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	12.0000	0.386695
$964$	−42.0000	−1.35273
$965$	0	0
$966$	−4.00000	−0.128698
$967$	−62.0000	−1.99379	−0.996893	−	0.0787703i	$-0.974901\pi$
$967$	−62.0000	−1.99379	−0.996893	+	0.0787703i	$0.974901\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	−32.0000	−1.02640
$973$	−5.00000	−0.160293
$974$	−26.0000	−0.833094
$975$	30.0000	0.960769
$976$	−40.0000	−1.28037
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	16.0000	0.511624
$979$	0	0
$980$	0	0
$981$	−20.0000	−0.638551
$982$	−66.0000	−2.10614
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.0000	0.381964
$988$	48.0000	1.52708
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	43.0000	1.36594	0.682970	−	0.730446i	$-0.260688\pi$
$991$	43.0000	1.36594	0.682970	+	0.730446i	$0.260688\pi$
$992$	80.0000	2.54000
$993$	−22.0000	−0.698149
$994$	−16.0000	−0.507489
$995$	0	0
$996$	−30.0000	−0.950586
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6013.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6013.2.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6013 = 7 \cdot 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6013.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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