LMFDB - Embedded newform 1002.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1002,2,Mod(1,1002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1002, 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("1002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1002 = 2 \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1002.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:	$8.00101028253$
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	$\chi$	$=$	1002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{39} -10.0000 q^{41} +10.0000 q^{43} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +5.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} +12.0000 q^{53} +1.00000 q^{54} -6.00000 q^{58} +6.00000 q^{59} +6.00000 q^{61} +1.00000 q^{64} +10.0000 q^{67} +6.00000 q^{68} -4.00000 q^{69} -4.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} +5.00000 q^{75} -4.00000 q^{78} +1.00000 q^{81} +10.0000 q^{82} +14.0000 q^{83} -10.0000 q^{86} -6.00000 q^{87} -6.00000 q^{89} +4.00000 q^{92} -8.00000 q^{94} +1.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +O(q^{100})$

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
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	5.00000	0.707107
$51$	−6.00000	−0.840168
$52$	−4.00000	−0.554700
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	6.00000	0.727607
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	4.00000	0.464991
$75$	5.00000	0.577350
$76$	0	0
$77$	0	0
$78$	−4.00000	−0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	0	0
$86$	−10.0000	−1.07833
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	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
$100$	−5.00000	−0.500000
$101$	20.0000	1.99007	0.995037	−	0.0995037i	$-0.0317255\pi$
$101$	20.0000	1.99007	0.995037	+	0.0995037i	$0.0317255\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−6.00000	−0.543214
$123$	10.0000	0.901670
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	0	0
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	4.00000	0.340503
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	4.00000	0.335673
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	7.00000	0.577350
$148$	−4.00000	−0.328798
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−5.00000	−0.408248
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−14.0000	−1.08661
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	10.0000	0.762493
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	6.00000	0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	5.00000	0.353553
$201$	−10.0000	−0.705346
$202$	−20.0000	−1.40720
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	4.00000	0.278019
$208$	−4.00000	−0.277350
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	12.0000	0.824163
$213$	4.00000	0.274075
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	12.0000	0.812743
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−24.0000	−1.61441
$222$	−4.00000	−0.268462
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−10.0000	−0.665190
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000	0.707107
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	−18.0000	−1.11204
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	10.0000	0.610847
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	14.0000	0.845771
$275$	0	0
$276$	−4.00000	−0.240772
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	8.00000	0.476393
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−14.0000	−0.820695
$292$	10.0000	0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−16.0000	−0.925304
$300$	5.00000	0.288675
$301$	0	0
$302$	12.0000	0.690522
$303$	−20.0000	−1.14897
$304$	0	0
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−4.00000	−0.226455
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	12.0000	0.672927
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	−14.0000	−0.775388
$327$	12.0000	0.663602
$328$	10.0000	0.552158
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	14.0000	0.768350
$333$	−4.00000	−0.219199
$334$	−1.00000	−0.0547176
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−3.00000	−0.163178
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	18.0000	0.967686
$347$	22.0000	1.18102	0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000	1.18102	0.590511	+	0.807030i	$0.298926\pi$
$348$	−6.00000	−0.321634
$349$	−24.0000	−1.28469	−0.642345	−	0.766415i	$-0.722038\pi$
$349$	−24.0000	−1.28469	−0.642345	+	0.766415i	$0.722038\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−6.00000	−0.317999
$357$	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
$361$	−19.0000	−1.00000
$362$	−14.0000	−0.735824
$363$	11.0000	0.577350
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000	0.208514
$369$	−10.0000	−0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−8.00000	−0.409316
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	18.0000	0.916176
$387$	10.0000	0.508329
$388$	14.0000	0.710742
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	7.00000	0.353553
$393$	−18.0000	−0.907980
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	10.0000	0.498755
$403$	0	0
$404$	20.0000	0.995037
$405$	0	0
$406$	0	0
$407$	0	0
$408$	6.00000	0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	8.00000	0.394132
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	−12.0000	−0.582772
$425$	−30.0000	−1.45521
$426$	−4.00000	−0.193801
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−1.00000	−0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	−12.0000	−0.574696
$437$	0	0
$438$	10.0000	0.477818
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	24.0000	1.14156
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	16.0000	0.757622
$447$	−4.00000	−0.189194
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	5.00000	0.235702
$451$	0	0
$452$	10.0000	0.470360
$453$	12.0000	0.563809
$454$	−10.0000	−0.469323
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	22.0000	1.02799
$459$	−6.00000	−0.280056
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	−6.00000	−0.276172
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000	0.549442
$478$	−8.00000	−0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	10.0000	0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−6.00000	−0.271607
$489$	−14.0000	−0.633102
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	10.0000	0.450835
$493$	36.0000	1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	14.0000	0.627355
$499$	−18.0000	−0.805791	−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	24.0000	1.07117
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	−8.00000	−0.354943
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	−6.00000	−0.262613
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	18.0000	0.786334
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	40.0000	1.73259
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−10.0000	−0.431934
$537$	−20.0000	−0.863064
$538$	24.0000	1.03471
$539$	0	0
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−20.0000	−0.859074
$543$	−14.0000	−0.600798
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	−14.0000	−0.598050
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	4.00000	0.170251
$553$	0	0
$554$	−12.0000	−0.509831
$555$	0	0
$556$	10.0000	0.424094
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−16.0000	−0.672530
$567$	0	0
$568$	4.00000	0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−20.0000	−0.834058
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−19.0000	−0.790296
$579$	18.0000	0.748054
$580$	0	0
$581$	0	0
$582$	14.0000	0.580319
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	18.0000	0.743573
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	0	0
$591$	24.0000	0.987228
$592$	−4.00000	−0.164399
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	24.0000	0.982255
$598$	16.0000	0.654289
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−5.00000	−0.204124
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	−12.0000	−0.488273
$605$	0	0
$606$	20.0000	0.812444
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.0000	−1.29458
$612$	6.00000	0.242536
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	8.00000	0.321807
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−8.00000	−0.320771
$623$	0	0
$624$	4.00000	0.160128
$625$	25.0000	1.00000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	6.00000	0.239426
$629$	−24.0000	−0.956943
$630$	0	0
$631$	48.0000	1.91085	0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000	1.91085	0.955425	+	0.295234i	$0.0953977\pi$
$632$	0	0
$633$	0	0
$634$	−30.0000	−1.19145
$635$	0	0
$636$	−12.0000	−0.475831
$637$	28.0000	1.10940
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−12.0000	−0.473602
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−20.0000	−0.784465
$651$	0	0
$652$	14.0000	0.548282
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−12.0000	−0.469237
$655$	0	0
$656$	−10.0000	−0.390434
$657$	10.0000	0.390137
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	−36.0000	−1.40024	−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000	−1.40024	−0.700119	+	0.714026i	$0.746870\pi$
$662$	−10.0000	−0.388661
$663$	24.0000	0.932083
$664$	−14.0000	−0.543305
$665$	0	0
$666$	4.00000	0.154997
$667$	24.0000	0.929284
$668$	1.00000	0.0386912
$669$	16.0000	0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−2.00000	−0.0770371
$675$	5.00000	0.192450
$676$	3.00000	0.115385
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	0	0
$681$	−10.0000	−0.383201
$682$	0	0
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.0000	0.839352
$688$	10.0000	0.381246
$689$	−48.0000	−1.82865
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−22.0000	−0.835109
$695$	0	0
$696$	6.00000	0.227429
$697$	−60.0000	−2.27266
$698$	24.0000	0.908413
$699$	10.0000	0.378235
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	0	0
$708$	−6.00000	−0.225494
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	−8.00000	−0.298765
$718$	−24.0000	−0.895672
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	0	0
$722$	19.0000	0.707107
$723$	10.0000	0.371904
$724$	14.0000	0.520306
$725$	−30.0000	−1.11417
$726$	−11.0000	−0.408248
$727$	−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
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.0000	2.21918
$732$	−6.00000	−0.221766
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	10.0000	0.368105
$739$	42.0000	1.54499	0.772497	−	0.635018i	$-0.219007\pi$
$739$	42.0000	1.54499	0.772497	+	0.635018i	$0.219007\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	−16.0000	−0.585802
$747$	14.0000	0.512233
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−52.0000	−1.89751	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000	−1.89751	−0.948753	+	0.316017i	$0.897654\pi$
$752$	8.00000	0.291730
$753$	24.0000	0.874609
$754$	24.0000	0.874028
$755$	0	0
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	34.0000	1.23494
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	−18.0000	−0.647834
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	24.0000	0.860442
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−24.0000	−0.858238
$783$	−6.00000	−0.214423
$784$	−7.00000	−0.250000
$785$	0	0
$786$	18.0000	0.642039
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	−24.0000	−0.854965
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−24.0000	−0.852265
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−24.0000	−0.850657
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	−10.0000	−0.352673
$805$	0	0
$806$	0	0
$807$	24.0000	0.844840
$808$	−20.0000	−0.703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	46.0000	1.61528	0.807639	−	0.589677i	$-0.200745\pi$
$811$	46.0000	1.61528	0.807639	+	0.589677i	$0.200745\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	0	0
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	−14.0000	−0.488306
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	4.00000	0.139010
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	−4.00000	−0.138675
$833$	−42.0000	−1.45521
$834$	10.0000	0.346272
$835$	0	0
$836$	0	0
$837$	0	0
$838$	20.0000	0.690889
$839$	32.0000	1.10476	0.552381	−	0.833592i	$-0.313719\pi$
$839$	32.0000	1.10476	0.552381	+	0.833592i	$0.313719\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−14.0000	−0.482472
$843$	−22.0000	−0.757720
$844$	0	0
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	12.0000	0.412082
$849$	−16.0000	−0.549119
$850$	30.0000	1.02899
$851$	−16.0000	−0.548473
$852$	4.00000	0.137038
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−19.0000	−0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−40.0000	−1.35535
$872$	12.0000	0.406371
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−30.0000	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$877$	−30.0000	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$878$	−40.0000	−1.34993
$879$	18.0000	0.607125
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	7.00000	0.235702
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	6.00000	0.201574
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−4.00000	−0.134231
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	4.00000	0.133780
$895$	0	0
$896$	0	0
$897$	16.0000	0.534224
$898$	−10.0000	−0.333704
$899$	0	0
$900$	−5.00000	−0.166667
$901$	72.0000	2.39867
$902$	0	0
$903$	0	0
$904$	−10.0000	−0.332595
$905$	0	0
$906$	−12.0000	−0.398673
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	10.0000	0.331862
$909$	20.0000	0.663358
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	0	0
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−22.0000	−0.726900
$917$	0	0
$918$	6.00000	0.198030
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	−10.0000	−0.329332
$923$	16.0000	0.526646
$924$	0	0
$925$	20.0000	0.657596
$926$	4.00000	0.131448
$927$	8.00000	0.262754
$928$	−6.00000	−0.196960
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	0	0
$932$	−10.0000	−0.327561
$933$	−8.00000	−0.261908
$934$	−28.0000	−0.916188
$935$	0	0
$936$	4.00000	0.130744
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	6.00000	0.195491
$943$	−40.0000	−1.30258
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−16.0000	−0.515861
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	0	0
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	16.0000	0.512673
$975$	−20.0000	−0.640513
$976$	6.00000	0.192055
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	14.0000	0.447671
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	−8.00000	−0.255290
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	0	0
$989$	40.0000	1.27193
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	0	0
$996$	−14.0000	−0.443607
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	18.0000	0.569780
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1002.2.a.a.1.1	✓	1
3.2	odd	2		3006.2.a.e.1.1		1
4.3	odd	2		8016.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.a.1.1	✓	1	1.1	even	1	trivial
3006.2.a.e.1.1		1	3.2	odd	2
8016.2.a.h.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1002 = 2 \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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