LMFDB - Embedded newform 6003.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{4} -4.00000 q^{5} -4.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{4} -4.00000 q^{5} -4.00000 q^{7} -4.00000 q^{11} -5.00000 q^{13} +4.00000 q^{16} +5.00000 q^{17} +5.00000 q^{19} +8.00000 q^{20} -1.00000 q^{23} +11.0000 q^{25} +8.00000 q^{28} +1.00000 q^{29} -2.00000 q^{31} +16.0000 q^{35} +5.00000 q^{37} +2.00000 q^{41} +1.00000 q^{43} +8.00000 q^{44} -6.00000 q^{47} +9.00000 q^{49} +10.0000 q^{52} -2.00000 q^{53} +16.0000 q^{55} -9.00000 q^{59} -10.0000 q^{61} -8.00000 q^{64} +20.0000 q^{65} +8.00000 q^{67} -10.0000 q^{68} +3.00000 q^{71} +8.00000 q^{73} -10.0000 q^{76} +16.0000 q^{77} +13.0000 q^{79} -16.0000 q^{80} +6.00000 q^{83} -20.0000 q^{85} +9.00000 q^{89} +20.0000 q^{91} +2.00000 q^{92} -20.0000 q^{95} -6.00000 q^{97} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	8.00000	1.78885
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	0	0
$28$	8.00000	1.51186
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.0000	2.70449
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	8.00000	1.20605
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	10.0000	1.38675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	16.0000	2.15744
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	20.0000	2.48069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−10.0000	−1.21268
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	−10.0000	−1.14708
$77$	16.0000	1.82337
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	−16.0000	−1.78885
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−20.0000	−2.16930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	20.0000	2.09657
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	−20.0000	−2.05196
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	−22.0000	−2.20000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	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
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	−16.0000	−1.51186
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	−2.00000	−0.185695
$117$	0	0
$118$	0	0
$119$	−20.0000	−1.83340
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	−24.0000	−2.14663
$126$	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
$130$	0	0
$131$	−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
$133$	−20.0000	−1.73422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−32.0000	−2.70449
$141$	0	0
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	0	0
$175$	−44.0000	−3.32609
$176$	−16.0000	−1.20605
$177$	0	0
$178$	0	0
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	−20.0000	−1.46254
$188$	12.0000	0.875190
$189$	0	0
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	−18.0000	−1.28571
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	0	0
$208$	−20.0000	−1.38675
$209$	−20.0000	−1.38343
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	−32.0000	−2.15744
$221$	−25.0000	−1.68168
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	19.0000	1.25556	0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000	1.25556	0.627778	+	0.778393i	$0.283965\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	18.0000	1.17170
$237$	0	0
$238$	0	0
$239$	29.0000	1.87585	0.937927	−	0.346833i	$-0.112743\pi$
$239$	29.0000	1.87585	0.937927	+	0.346833i	$0.112743\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	0	0
$244$	20.0000	1.28037
$245$	−36.0000	−2.29996
$246$	0	0
$247$	−25.0000	−1.59071
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	−40.0000	−2.48069
$261$	0	0
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	−16.0000	−0.977356
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	20.0000	1.21268
$273$	0	0
$274$	0	0
$275$	−44.0000	−2.65330
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	−16.0000	−0.936329
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	20.0000	1.14708
$305$	40.0000	2.29039
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−32.0000	−1.82337
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	−26.0000	−1.46261
$317$	−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
$319$	−4.00000	−0.223957
$320$	32.0000	1.78885
$321$	0	0
$322$	0	0
$323$	25.0000	1.39104
$324$	0	0
$325$	−55.0000	−3.05085
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	−32.0000	−1.74835
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	0	0
$339$	0	0
$340$	40.0000	2.16930
$341$	8.00000	0.433224
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.0000	1.44944	0.724718	−	0.689046i	$-0.241970\pi$
$347$	27.0000	1.44944	0.724718	+	0.689046i	$0.241970\pi$
$348$	0	0
$349$	27.0000	1.44528	0.722638	−	0.691226i	$-0.242929\pi$
$349$	27.0000	1.44528	0.722638	+	0.691226i	$0.242929\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−18.0000	−0.953998
$357$	0	0
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	−40.0000	−2.09657
$365$	−32.0000	−1.67496
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	11.0000	0.565032	0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000	0.565032	0.282516	+	0.959263i	$0.408831\pi$
$380$	40.0000	2.05196
$381$	0	0
$382$	0	0
$383$	−10.0000	−0.510976	−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000	−0.510976	−0.255488	+	0.966812i	$0.582236\pi$
$384$	0	0
$385$	−64.0000	−3.26174
$386$	0	0
$387$	0	0
$388$	12.0000	0.609208
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−52.0000	−2.61640
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	44.0000	2.20000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	24.0000	1.19404
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−36.0000	−1.78009	−0.890043	−	0.455877i	$-0.849326\pi$
$409$	−36.0000	−1.78009	−0.890043	+	0.455877i	$0.849326\pi$
$410$	0	0
$411$	0	0
$412$	−32.0000	−1.57653
$413$	36.0000	1.77144
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	55.0000	2.66789
$426$	0	0
$427$	40.0000	1.93574
$428$	36.0000	1.74013
$429$	0	0
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	0	0
$436$	−28.0000	−1.34096
$437$	−5.00000	−0.239182
$438$	0	0
$439$	−27.0000	−1.28864	−0.644320	−	0.764756i	$-0.722859\pi$
$439$	−27.0000	−1.28864	−0.644320	+	0.764756i	$0.722859\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	0	0
$448$	32.0000	1.51186
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	20.0000	0.940721
$453$	0	0
$454$	0	0
$455$	−80.0000	−3.75046
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	0	0
$459$	0	0
$460$	−8.00000	−0.373002
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	−39.0000	−1.81248	−0.906242	−	0.422760i	$-0.861061\pi$
$463$	−39.0000	−1.81248	−0.906242	+	0.422760i	$0.861061\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	55.0000	2.52357
$476$	40.0000	1.83340
$477$	0	0
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	0	0
$483$	0	0
$484$	−10.0000	−0.454545
$485$	24.0000	1.08978
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	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
$493$	5.00000	0.225189
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−12.0000	−0.538274
$498$	0	0
$499$	15.0000	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$499$	15.0000	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$500$	48.0000	2.14663
$501$	0	0
$502$	0	0
$503$	−37.0000	−1.64975	−0.824874	−	0.565316i	$-0.808754\pi$
$503$	−37.0000	−1.64975	−0.824874	+	0.565316i	$0.808754\pi$
$504$	0	0
$505$	48.0000	2.13597
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−43.0000	−1.90594	−0.952971	−	0.303062i	$-0.901991\pi$
$509$	−43.0000	−1.90594	−0.952971	+	0.303062i	$0.901991\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−64.0000	−2.82018
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	0	0
$527$	−10.0000	−0.435607
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	40.0000	1.73422
$533$	−10.0000	−0.433148
$534$	0	0
$535$	72.0000	3.11283
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−36.0000	−1.55063
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−56.0000	−2.39878
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	0	0
$551$	5.00000	0.213007
$552$	0	0
$553$	−52.0000	−2.21126
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	64.0000	2.70449
$561$	0	0
$562$	0	0
$563$	−1.00000	−0.0421450	−0.0210725	−	0.999778i	$-0.506708\pi$
$563$	−1.00000	−0.0421450	−0.0210725	+	0.999778i	$0.506708\pi$
$564$	0	0
$565$	40.0000	1.68281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−40.0000	−1.67248
$573$	0	0
$574$	0	0
$575$	−11.0000	−0.458732
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	0	0
$580$	8.00000	0.332182
$581$	−24.0000	−0.995688
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	0	0
$592$	20.0000	0.821995
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	80.0000	3.27968
$596$	40.0000	1.63846
$597$	0	0
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	0	0
$604$	−38.0000	−1.54620
$605$	−20.0000	−0.813116
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0000	1.21367
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	41.0000	1.64793	0.823965	−	0.566641i	$-0.191757\pi$
$619$	41.0000	1.64793	0.823965	+	0.566641i	$0.191757\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	0	0
$623$	−36.0000	−1.44231
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	0	0
$628$	46.0000	1.83560
$629$	25.0000	0.996815
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−45.0000	−1.78296
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.0000	1.54041	0.770204	−	0.637798i	$-0.220155\pi$
$641$	39.0000	1.54041	0.770204	+	0.637798i	$0.220155\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	0	0
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−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
$655$	32.0000	1.25034
$656$	8.00000	0.312348
$657$	0	0
$658$	0	0
$659$	−29.0000	−1.12968	−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000	−1.12968	−0.564840	+	0.825201i	$0.691062\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	80.0000	3.10227
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−42.0000	−1.62503
$669$	0	0
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	0	0
$676$	−24.0000	−0.923077
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	24.0000	0.921035
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	10.0000	0.380970
$690$	0	0
$691$	52.0000	1.97817	0.989087	−	0.147335i	$-0.0470696\pi$
$691$	52.0000	1.97817	0.989087	+	0.147335i	$0.0470696\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	0	0
$700$	88.0000	3.32609
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	25.0000	0.942893
$704$	32.0000	1.20605
$705$	0	0
$706$	0	0
$707$	48.0000	1.80523
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.00000	0.0749006
$714$	0	0
$715$	−80.0000	−2.99183
$716$	14.0000	0.523205
$717$	0	0
$718$	0	0
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	0	0
$721$	−64.0000	−2.38348
$722$	0	0
$723$	0	0
$724$	−12.0000	−0.445976
$725$	11.0000	0.408530
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.00000	0.184932
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.0000	−1.17874
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	40.0000	1.47043
$741$	0	0
$742$	0	0
$743$	−13.0000	−0.476924	−0.238462	−	0.971152i	$-0.576643\pi$
$743$	−13.0000	−0.476924	−0.238462	+	0.971152i	$0.576643\pi$
$744$	0	0
$745$	80.0000	2.93097
$746$	0	0
$747$	0	0
$748$	40.0000	1.46254
$749$	72.0000	2.63082
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	−24.0000	−0.875190
$753$	0	0
$754$	0	0
$755$	−76.0000	−2.76592
$756$	0	0
$757$	7.00000	0.254419	0.127210	−	0.991876i	$-0.459398\pi$
$757$	7.00000	0.254419	0.127210	+	0.991876i	$0.459398\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	0	0
$763$	−56.0000	−2.02734
$764$	42.0000	1.51951
$765$	0	0
$766$	0	0
$767$	45.0000	1.62486
$768$	0	0
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	0	0
$771$	0	0
$772$	−28.0000	−1.00774
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	−22.0000	−0.790263
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.0000	0.358287
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	36.0000	1.28571
$785$	92.0000	3.28362
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	0	0
$791$	40.0000	1.42224
$792$	0	0
$793$	50.0000	1.77555
$794$	0	0
$795$	0	0
$796$	4.00000	0.141776
$797$	−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
$799$	−30.0000	−1.06132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32.0000	−1.12926
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	8.00000	0.280745
$813$	0	0
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	5.00000	0.174928
$818$	0	0
$819$	0	0
$820$	16.0000	0.558744
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	0	0
$832$	40.0000	1.38675
$833$	45.0000	1.55916
$834$	0	0
$835$	−84.0000	−2.90694
$836$	40.0000	1.38343
$837$	0	0
$838$	0	0
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	−36.0000	−1.23917
$845$	−48.0000	−1.65125
$846$	0	0
$847$	−20.0000	−0.687208
$848$	−8.00000	−0.274721
$849$	0	0
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	0	0
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	0	0
$868$	−16.0000	−0.543075
$869$	−52.0000	−1.76398
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	96.0000	3.24539
$876$	0	0
$877$	−33.0000	−1.11433	−0.557165	−	0.830402i	$-0.688111\pi$
$877$	−33.0000	−1.11433	−0.557165	+	0.830402i	$0.688111\pi$
$878$	0	0
$879$	0	0
$880$	64.0000	2.15744
$881$	−43.0000	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$881$	−43.0000	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$882$	0	0
$883$	51.0000	1.71629	0.858143	−	0.513410i	$-0.171618\pi$
$883$	51.0000	1.71629	0.858143	+	0.513410i	$0.171618\pi$
$884$	50.0000	1.68168
$885$	0	0
$886$	0	0
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	−34.0000	−1.13840
$893$	−30.0000	−1.00391
$894$	0	0
$895$	28.0000	0.935937
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.00000	−0.0667037
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	−36.0000	−1.19470
$909$	0	0
$910$	0	0
$911$	−9.00000	−0.298183	−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000	−0.298183	−0.149092	+	0.988823i	$0.547635\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	−38.0000	−1.25556
$917$	32.0000	1.05673
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.0000	−0.493731
$924$	0	0
$925$	55.0000	1.80839
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.00000	0.229663	0.114831	−	0.993385i	$-0.463367\pi$
$929$	7.00000	0.229663	0.114831	+	0.993385i	$0.463367\pi$
$930$	0	0
$931$	45.0000	1.47482
$932$	−18.0000	−0.589610
$933$	0	0
$934$	0	0
$935$	80.0000	2.61628
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	−48.0000	−1.56559
$941$	−36.0000	−1.17357	−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000	−1.17357	−0.586783	+	0.809744i	$0.699606\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−36.0000	−1.17170
$945$	0	0
$946$	0	0
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	84.0000	2.71818
$956$	−58.0000	−1.87585
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	52.0000	1.67481
$965$	−56.0000	−1.80270
$966$	0	0
$967$	−30.0000	−0.964735	−0.482367	−	0.875969i	$-0.660223\pi$
$967$	−30.0000	−0.964735	−0.482367	+	0.875969i	$0.660223\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−40.0000	−1.28037
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	72.0000	2.29996
$981$	0	0
$982$	0	0
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	0	0
$988$	50.0000	1.59071
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	60.0000	1.90022	0.950110	−	0.311916i	$-0.100971\pi$
$997$	60.0000	1.90022	0.950110	+	0.311916i	$0.100971\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.a.1.1		1
3.2	odd	2		2001.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.b.1.1	✓	1	3.2	odd	2
6003.2.a.a.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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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