LMFDB - Embedded newform 6003.2.a.c.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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{5} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{5} -3.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} +3.00000 q^{13} -1.00000 q^{16} +8.00000 q^{19} +3.00000 q^{20} -3.00000 q^{22} +1.00000 q^{23} +4.00000 q^{25} +3.00000 q^{26} -1.00000 q^{29} +7.00000 q^{31} +5.00000 q^{32} -5.00000 q^{37} +8.00000 q^{38} +9.00000 q^{40} -3.00000 q^{41} +2.00000 q^{43} +3.00000 q^{44} +1.00000 q^{46} +6.00000 q^{47} -7.00000 q^{49} +4.00000 q^{50} -3.00000 q^{52} +2.00000 q^{53} +9.00000 q^{55} -1.00000 q^{58} -1.00000 q^{59} +7.00000 q^{61} +7.00000 q^{62} +7.00000 q^{64} -9.00000 q^{65} -9.00000 q^{67} -3.00000 q^{71} -12.0000 q^{73} -5.00000 q^{74} -8.00000 q^{76} -14.0000 q^{79} +3.00000 q^{80} -3.00000 q^{82} -14.0000 q^{83} +2.00000 q^{86} +9.00000 q^{88} +10.0000 q^{89} -1.00000 q^{92} +6.00000 q^{94} -24.0000 q^{95} +14.0000 q^{97} -7.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−3.00000	−0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.00000	0.800000
$26$	3.00000	0.588348
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	9.00000	1.42302
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	0	0
$52$	−3.00000	−0.416025
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	0	0
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	7.00000	0.889001
$63$	0	0
$64$	7.00000	0.875000
$65$	−9.00000	−1.11631
$66$	0	0
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−3.00000	−0.331295
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	9.00000	0.959403
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	6.00000	0.618853
$95$	−24.0000	−2.46235
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	−9.00000	−0.882523
$105$	0	0
$106$	2.00000	0.194257
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	9.00000	0.858116
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	1.00000	0.0928477
$117$	0	0
$118$	−1.00000	−0.0920575
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	7.00000	0.633750
$123$	0	0
$124$	−7.00000	−0.628619
$125$	3.00000	0.268328
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−9.00000	−0.789352
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−9.00000	−0.777482
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	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
$142$	−3.00000	−0.251754
$143$	−9.00000	−0.752618
$144$	0	0
$145$	3.00000	0.249136
$146$	−12.0000	−0.993127
$147$	0	0
$148$	5.00000	0.410997
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−24.0000	−1.94666
$153$	0	0
$154$	0	0
$155$	−21.0000	−1.68676
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−14.0000	−1.11378
$159$	0	0
$160$	−15.0000	−1.18585
$161$	0	0
$162$	0	0
$163$	−17.0000	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$163$	−17.0000	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	−14.0000	−1.08661
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	10.0000	0.749532
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	15.0000	1.10282
$186$	0	0
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−24.0000	−1.74114
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	7.00000	0.500000
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	21.0000	1.48865	0.744325	−	0.667817i	$-0.232771\pi$
$199$	21.0000	1.48865	0.744325	+	0.667817i	$0.232771\pi$
$200$	−12.0000	−0.848528
$201$	0	0
$202$	−5.00000	−0.351799
$203$	0	0
$204$	0	0
$205$	9.00000	0.628587
$206$	1.00000	0.0696733
$207$	0	0
$208$	−3.00000	−0.208013
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−6.00000	−0.409197
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	−9.00000	−0.606780
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	28.0000	1.83434	0.917170	−	0.398495i	$-0.130467\pi$
$233$	28.0000	1.83434	0.917170	+	0.398495i	$0.130467\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	1.00000	0.0650945
$237$	0	0
$238$	0	0
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−7.00000	−0.448129
$245$	21.0000	1.34164
$246$	0	0
$247$	24.0000	1.52708
$248$	−21.0000	−1.33350
$249$	0	0
$250$	3.00000	0.189737
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	−5.00000	−0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	9.00000	0.558156
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	9.00000	0.549762
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	18.0000	1.07957
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	23.0000	1.36721	0.683604	−	0.729853i	$-0.260412\pi$
$283$	23.0000	1.36721	0.683604	+	0.729853i	$0.260412\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−9.00000	−0.532181
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	3.00000	0.176166
$291$	0	0
$292$	12.0000	0.702247
$293$	−20.0000	−1.16841	−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000	−1.16841	−0.584206	+	0.811605i	$0.698594\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	15.0000	0.871857
$297$	0	0
$298$	3.00000	0.173785
$299$	3.00000	0.173494
$300$	0	0
$301$	0	0
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−8.00000	−0.458831
$305$	−21.0000	−1.20246
$306$	0	0
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	0	0
$309$	0	0
$310$	−21.0000	−1.19272
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	14.0000	0.787562
$317$	7.00000	0.393159	0.196580	−	0.980488i	$-0.437017\pi$
$317$	7.00000	0.393159	0.196580	+	0.980488i	$0.437017\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	−21.0000	−1.17394
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12.0000	0.665640
$326$	−17.0000	−0.941543
$327$	0	0
$328$	9.00000	0.496942
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	3.00000	0.164153
$335$	27.0000	1.47517
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	0	0
$341$	−21.0000	−1.13721
$342$	0	0
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−18.0000	−0.967686
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	0	0
$351$	0	0
$352$	−15.0000	−0.799503
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	0	0
$364$	0	0
$365$	36.0000	1.88433
$366$	0	0
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	15.0000	0.779813
$371$	0	0
$372$	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
$376$	−18.0000	−0.928279
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	−3.00000	−0.153493
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	0	0
$388$	−14.0000	−0.710742
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	0	0
$392$	21.0000	1.06066
$393$	0	0
$394$	−10.0000	−0.503793
$395$	42.0000	2.11325
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	21.0000	1.05263
$399$	0	0
$400$	−4.00000	−0.200000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	21.0000	1.04608
$404$	5.00000	0.248759
$405$	0	0
$406$	0	0
$407$	15.0000	0.743522
$408$	0	0
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	0	0
$415$	42.0000	2.06170
$416$	15.0000	0.735436
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	−3.00000	−0.146038
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	8.00000	0.386695
$429$	0	0
$430$	−6.00000	−0.289346
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	8.00000	0.382692
$438$	0	0
$439$	38.0000	1.81364	0.906821	−	0.421517i	$-0.138502\pi$
$439$	38.0000	1.81364	0.906821	+	0.421517i	$0.138502\pi$
$440$	−27.0000	−1.28717
$441$	0	0
$442$	0	0
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	−30.0000	−1.42214
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	−7.00000	−0.330350	−0.165175	−	0.986264i	$-0.552819\pi$
$449$	−7.00000	−0.330350	−0.165175	+	0.986264i	$0.552819\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−21.0000	−0.981266
$459$	0	0
$460$	3.00000	0.139876
$461$	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
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	28.0000	1.29707
$467$	25.0000	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$468$	0	0
$469$	0	0
$470$	−18.0000	−0.830278
$471$	0	0
$472$	3.00000	0.138086
$473$	−6.00000	−0.275880
$474$	0	0
$475$	32.0000	1.46826
$476$	0	0
$477$	0	0
$478$	−19.0000	−0.869040
$479$	23.0000	1.05090	0.525448	−	0.850825i	$-0.323898\pi$
$479$	23.0000	1.05090	0.525448	+	0.850825i	$0.323898\pi$
$480$	0	0
$481$	−15.0000	−0.683941
$482$	12.0000	0.546585
$483$	0	0
$484$	2.00000	0.0909091
$485$	−42.0000	−1.90712
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−21.0000	−0.950625
$489$	0	0
$490$	21.0000	0.948683
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	0	0
$494$	24.0000	1.07981
$495$	0	0
$496$	−7.00000	−0.314309
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	9.00000	0.401690
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	−3.00000	−0.133366
$507$	0	0
$508$	5.00000	0.221839
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	6.00000	0.264649
$515$	−3.00000	−0.132196
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	0	0
$520$	27.0000	1.18403
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	0	0
$533$	−9.00000	−0.389833
$534$	0	0
$535$	24.0000	1.03761
$536$	27.0000	1.16622
$537$	0	0
$538$	21.0000	0.905374
$539$	21.0000	0.904534
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−15.0000	−0.644305
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	8.00000	0.341743
$549$	0	0
$550$	−12.0000	−0.511682
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	−17.0000	−0.722261
$555$	0	0
$556$	−18.0000	−0.763370
$557$	27.0000	1.14403	0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000	1.14403	0.572013	+	0.820244i	$0.306163\pi$
$558$	0	0
$559$	6.00000	0.253773
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	11.0000	0.463595	0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000	0.463595	0.231797	+	0.972764i	$0.425539\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	23.0000	0.966762
$567$	0	0
$568$	9.00000	0.377632
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	9.00000	0.376309
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−3.00000	−0.124568
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	36.0000	1.48969
$585$	0	0
$586$	−20.0000	−0.826192
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	3.00000	0.123508
$591$	0	0
$592$	5.00000	0.205499
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	0	0
$598$	3.00000	0.122679
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	0	0
$604$	8.00000	0.325515
$605$	6.00000	0.243935
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	−21.0000	−0.850265
$611$	18.0000	0.728202
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	29.0000	1.17034
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	21.0000	0.843380
$621$	0	0
$622$	−6.00000	−0.240578
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	0	0
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	42.0000	1.67067
$633$	0	0
$634$	7.00000	0.278006
$635$	15.0000	0.595257
$636$	0	0
$637$	−21.0000	−0.832050
$638$	3.00000	0.118771
$639$	0	0
$640$	9.00000	0.355756
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	12.0000	0.470679
$651$	0	0
$652$	17.0000	0.665771
$653$	−23.0000	−0.900060	−0.450030	−	0.893014i	$-0.648587\pi$
$653$	−23.0000	−0.900060	−0.450030	+	0.893014i	$0.648587\pi$
$654$	0	0
$655$	36.0000	1.40664
$656$	3.00000	0.117130
$657$	0	0
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	42.0000	1.62992
$665$	0	0
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−3.00000	−0.116073
$669$	0	0
$670$	27.0000	1.04310
$671$	−21.0000	−0.810696
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−29.0000	−1.11704
$675$	0	0
$676$	4.00000	0.153846
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−21.0000	−0.804132
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	6.00000	0.228582
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	12.0000	0.455514
$695$	−54.0000	−2.04834
$696$	0	0
$697$	0	0
$698$	25.0000	0.946264
$699$	0	0
$700$	0	0
$701$	11.0000	0.415464	0.207732	−	0.978186i	$-0.433392\pi$
$701$	11.0000	0.415464	0.207732	+	0.978186i	$0.433392\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	−21.0000	−0.791467
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	9.00000	0.337764
$711$	0	0
$712$	−30.0000	−1.12430
$713$	7.00000	0.262152
$714$	0	0
$715$	27.0000	1.00974
$716$	20.0000	0.747435
$717$	0	0
$718$	24.0000	0.895672
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	45.0000	1.67473
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−34.0000	−1.26099	−0.630495	−	0.776193i	$-0.717148\pi$
$727$	−34.0000	−1.26099	−0.630495	+	0.776193i	$0.717148\pi$
$728$	0	0
$729$	0	0
$730$	36.0000	1.33242
$731$	0	0
$732$	0	0
$733$	−25.0000	−0.923396	−0.461698	−	0.887037i	$-0.652760\pi$
$733$	−25.0000	−0.923396	−0.461698	+	0.887037i	$0.652760\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	5.00000	0.184302
$737$	27.0000	0.994558
$738$	0	0
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	−15.0000	−0.551411
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	6.00000	0.219676
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−3.00000	−0.109254
$755$	24.0000	0.873449
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	−14.0000	−0.508503
$759$	0	0
$760$	72.0000	2.61171
$761$	−50.0000	−1.81250	−0.906249	−	0.422744i	$-0.861067\pi$
$761$	−50.0000	−1.81250	−0.906249	+	0.422744i	$0.861067\pi$
$762$	0	0
$763$	0	0
$764$	3.00000	0.108536
$765$	0	0
$766$	−18.0000	−0.650366
$767$	−3.00000	−0.108324
$768$	0	0
$769$	27.0000	0.973645	0.486822	−	0.873501i	$-0.338156\pi$
$769$	27.0000	0.973645	0.486822	+	0.873501i	$0.338156\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	−42.0000	−1.50771
$777$	0	0
$778$	−20.0000	−0.717035
$779$	−24.0000	−0.859889
$780$	0	0
$781$	9.00000	0.322045
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	−6.00000	−0.214149
$786$	0	0
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	42.0000	1.49429
$791$	0	0
$792$	0	0
$793$	21.0000	0.745732
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−21.0000	−0.744325
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	0	0
$800$	20.0000	0.707107
$801$	0	0
$802$	27.0000	0.953403
$803$	36.0000	1.27041
$804$	0	0
$805$	0	0
$806$	21.0000	0.739693
$807$	0	0
$808$	15.0000	0.527698
$809$	31.0000	1.08990	0.544951	−	0.838468i	$-0.316548\pi$
$809$	31.0000	1.08990	0.544951	+	0.838468i	$0.316548\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	15.0000	0.525750
$815$	51.0000	1.78645
$816$	0	0
$817$	16.0000	0.559769
$818$	−38.0000	−1.32864
$819$	0	0
$820$	−9.00000	−0.314294
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−9.00000	−0.313720	−0.156860	−	0.987621i	$-0.550137\pi$
$823$	−9.00000	−0.313720	−0.156860	+	0.987621i	$0.550137\pi$
$824$	−3.00000	−0.104510
$825$	0	0
$826$	0	0
$827$	35.0000	1.21707	0.608535	−	0.793527i	$-0.291758\pi$
$827$	35.0000	1.21707	0.608535	+	0.793527i	$0.291758\pi$
$828$	0	0
$829$	24.0000	0.833554	0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000	0.833554	0.416777	+	0.909009i	$0.363160\pi$
$830$	42.0000	1.45784
$831$	0	0
$832$	21.0000	0.728044
$833$	0	0
$834$	0	0
$835$	−9.00000	−0.311458
$836$	24.0000	0.830057
$837$	0	0
$838$	−6.00000	−0.207267
$839$	−33.0000	−1.13929	−0.569643	−	0.821892i	$-0.692919\pi$
$839$	−33.0000	−1.13929	−0.569643	+	0.821892i	$0.692919\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	9.00000	0.310160
$843$	0	0
$844$	3.00000	0.103264
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	40.0000	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$853$	40.0000	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$854$	0	0
$855$	0	0
$856$	24.0000	0.820303
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	−14.0000	−0.475739
$867$	0	0
$868$	0	0
$869$	42.0000	1.42475
$870$	0	0
$871$	−27.0000	−0.914860
$872$	−6.00000	−0.203186
$873$	0	0
$874$	8.00000	0.270604
$875$	0	0
$876$	0	0
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	38.0000	1.28244
$879$	0	0
$880$	−9.00000	−0.303390
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	0	0
$886$	−30.0000	−1.00787
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	0	0
$890$	−30.0000	−1.00560
$891$	0	0
$892$	−8.00000	−0.267860
$893$	48.0000	1.60626
$894$	0	0
$895$	60.0000	2.00558
$896$	0	0
$897$	0	0
$898$	−7.00000	−0.233593
$899$	−7.00000	−0.233463
$900$	0	0
$901$	0	0
$902$	9.00000	0.299667
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−30.0000	−0.997234
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	6.00000	0.199117
$909$	0	0
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	−16.0000	−0.529233
$915$	0	0
$916$	21.0000	0.693860
$917$	0	0
$918$	0	0
$919$	−55.0000	−1.81428	−0.907141	−	0.420826i	$-0.861740\pi$
$919$	−55.0000	−1.81428	−0.907141	+	0.420826i	$0.861740\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	3.00000	0.0987997
$923$	−9.00000	−0.296239
$924$	0	0
$925$	−20.0000	−0.657596
$926$	4.00000	0.131448
$927$	0	0
$928$	−5.00000	−0.164133
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	0	0
$931$	−56.0000	−1.83533
$932$	−28.0000	−0.917170
$933$	0	0
$934$	25.0000	0.818025
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	18.0000	0.587095
$941$	−11.0000	−0.358590	−0.179295	−	0.983795i	$-0.557382\pi$
$941$	−11.0000	−0.358590	−0.179295	+	0.983795i	$0.557382\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	1.00000	0.0325472
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	32.0000	1.03822
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	9.00000	0.291233
$956$	19.0000	0.614504
$957$	0	0
$958$	23.0000	0.743096
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	−15.0000	−0.483619
$963$	0	0
$964$	−12.0000	−0.386494
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	−42.0000	−1.34854
$971$	−1.00000	−0.0320915	−0.0160458	−	0.999871i	$-0.505108\pi$
$971$	−1.00000	−0.0320915	−0.0160458	+	0.999871i	$0.505108\pi$
$972$	0	0
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−7.00000	−0.224065
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	−21.0000	−0.670820
$981$	0	0
$982$	−30.0000	−0.957338
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	0	0
$988$	−24.0000	−0.763542
$989$	2.00000	0.0635963
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	35.0000	1.11125
$993$	0	0
$994$	0	0
$995$	−63.0000	−1.99723
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.a.1.1	✓	1	3.2	odd	2
6003.2.a.c.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

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