LMFDB - Embedded newform 3174.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, 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("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3174 = 2 \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3174.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:	$25.3445176016$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.00000 q^{20} -2.00000 q^{21} -6.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +2.00000 q^{42} +12.0000 q^{43} +6.00000 q^{44} +2.00000 q^{45} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -2.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -4.00000 q^{61} -8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +6.00000 q^{66} +12.0000 q^{67} -4.00000 q^{70} -1.00000 q^{72} -10.0000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -2.00000 q^{78} +6.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -14.0000 q^{83} -2.00000 q^{84} -12.0000 q^{86} -6.00000 q^{87} -6.00000 q^{88} -2.00000 q^{90} -4.00000 q^{91} -8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	−2.00000	−0.436436
$22$	−6.00000	−1.27920
$23$	0	0
$23$	0	0
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	0	0
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	−2.00000	−0.316228
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	2.00000	0.308607
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	6.00000	0.904534
$45$	2.00000	0.298142
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	12.0000	1.61808
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−2.00000	−0.258199
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−8.00000	−1.01600
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	6.00000	0.738549
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	−4.00000	−0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	12.0000	1.36753
$78$	−2.00000	−0.226455
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−6.00000	−0.643268
$88$	−6.00000	−0.639602
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−2.00000	−0.210819
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−8.00000	−0.829561
$94$	8.00000	0.825137
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	3.00000	0.303046
$99$	6.00000	0.603023
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	2.00000	0.196116
$105$	−4.00000	−0.390360
$106$	2.00000	0.194257
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	−1.00000	−0.0962250
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	−12.0000	−1.14416
$111$	0	0
$112$	2.00000	0.188982
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	2.00000	0.182574
$121$	25.0000	2.27273
$122$	4.00000	0.362143
$123$	−10.0000	−0.901670
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	−2.00000	−0.178174
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	4.00000	0.350823
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−12.0000	−1.03664
$135$	−2.00000	−0.172133
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	4.00000	0.338062
$141$	8.00000	0.673722
$142$	0	0
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	12.0000	0.996546
$146$	10.0000	0.827606
$147$	3.00000	0.247436
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−1.00000	−0.0816497
$151$	−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
$153$	0	0
$154$	−12.0000	−0.966988
$155$	16.0000	1.28515
$156$	2.00000	0.160128
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	−6.00000	−0.477334
$159$	2.00000	0.158610
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	10.0000	0.780869
$165$	−12.0000	−0.934199
$166$	14.0000	1.08661
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	2.00000	0.154303
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	12.0000	0.914991
$173$	−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
$175$	−2.00000	−0.151186
$176$	6.00000	0.452267
$177$	12.0000	0.901975
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	2.00000	0.149071
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	4.00000	0.296500
$183$	4.00000	0.295689
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	−8.00000	−0.583460
$189$	−2.00000	−0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−6.00000	−0.430775
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−6.00000	−0.426401
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	1.00000	0.0707107
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	12.0000	0.842235
$204$	0	0
$205$	20.0000	1.39686
$206$	14.0000	0.975426
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	4.00000	0.276026
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	14.0000	0.957020
$215$	24.0000	1.63679
$216$	1.00000	0.0680414
$217$	16.0000	1.08615
$218$	−16.0000	−1.08366
$219$	10.0000	0.675737
$220$	12.0000	0.809040
$221$	0	0
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	−2.00000	−0.133631
$225$	−1.00000	−0.0666667
$226$	−8.00000	−0.532152
$227$	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
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	−6.00000	−0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	2.00000	0.130744
$235$	−16.0000	−1.04372
$236$	−12.0000	−0.781133
$237$	−6.00000	−0.389742
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	−2.00000	−0.129099
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−25.0000	−1.60706
$243$	−1.00000	−0.0641500
$244$	−4.00000	−0.256074
$245$	−6.00000	−0.383326
$246$	10.0000	0.637577
$247$	0	0
$248$	−8.00000	−0.508001
$249$	14.0000	0.887214
$250$	12.0000	0.758947
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	−4.00000	−0.248069
$261$	6.00000	0.371391
$262$	8.00000	0.494242
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	6.00000	0.369274
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	2.00000	0.121716
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	−12.0000	−0.724947
$275$	−6.00000	−0.361814
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−12.0000	−0.719712
$279$	8.00000	0.478947
$280$	−4.00000	−0.239046
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	−8.00000	−0.476393
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	0	0
$285$	0	0
$286$	12.0000	0.709575
$287$	20.0000	1.18056
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−12.0000	−0.704664
$291$	−6.00000	−0.351726
$292$	−10.0000	−0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	−24.0000	−1.39733
$296$	0	0
$297$	−6.00000	−0.348155
$298$	−18.0000	−1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	24.0000	1.38334
$302$	12.0000	0.690522
$303$	6.00000	0.344691
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	12.0000	0.683763
$309$	14.0000	0.796432
$310$	−16.0000	−0.908739
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	−2.00000	−0.113228
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	16.0000	0.902932
$315$	4.00000	0.225374
$316$	6.00000	0.337526
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−2.00000	−0.112154
$319$	36.0000	2.01561
$320$	2.00000	0.111803
$321$	14.0000	0.781404
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	−16.0000	−0.884802
$328$	−10.0000	−0.552158
$329$	−16.0000	−0.882109
$330$	12.0000	0.660578
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−14.0000	−0.768350
$333$	0	0
$334$	−8.00000	−0.437741
$335$	24.0000	1.31126
$336$	−2.00000	−0.109109
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	9.00000	0.489535
$339$	−8.00000	−0.434500
$340$	0	0
$341$	48.0000	2.59935
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	−6.00000	−0.321634
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	2.00000	0.106904
$351$	2.00000	0.106752
$352$	−6.00000	−0.319801
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	0	0
$357$	0	0
$358$	16.0000	0.845626
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−2.00000	−0.105409
$361$	−19.0000	−1.00000
$362$	8.00000	0.420471
$363$	−25.0000	−1.31216
$364$	−4.00000	−0.209657
$365$	−20.0000	−1.04685
$366$	−4.00000	−0.209083
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−4.00000	−0.207670
$372$	−8.00000	−0.414781
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	8.00000	0.412568
$377$	−12.0000	−0.618031
$378$	2.00000	0.102869
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	24.0000	1.22315
$386$	−14.0000	−0.712581
$387$	12.0000	0.609994
$388$	6.00000	0.304604
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	3.00000	0.151523
$393$	8.00000	0.403547
$394$	6.00000	0.302276
$395$	12.0000	0.603786
$396$	6.00000	0.301511
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	2.00000	0.100251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	12.0000	0.598506
$403$	−16.0000	−0.797017
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	−12.0000	−0.595550
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−20.0000	−0.987730
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	−24.0000	−1.18096
$414$	0	0
$415$	−28.0000	−1.37447
$416$	2.00000	0.0980581
$417$	−12.0000	−0.587643
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	−4.00000	−0.195180
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−20.0000	−0.973585
$423$	−8.00000	−0.388973
$424$	2.00000	0.0971286
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−14.0000	−0.676716
$429$	12.0000	0.579365
$430$	−24.0000	−1.15738
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−16.0000	−0.768025
$435$	−12.0000	−0.575356
$436$	16.0000	0.766261
$437$	0	0
$438$	−10.0000	−0.477818
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−12.0000	−0.572078
$441$	−3.00000	−0.142857
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	0	0
$446$	−12.0000	−0.568216
$447$	−18.0000	−0.851371
$448$	2.00000	0.0944911
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	1.00000	0.0471405
$451$	60.0000	2.82529
$452$	8.00000	0.376288
$453$	12.0000	0.563809
$454$	−10.0000	−0.469323
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−24.0000	−1.12145
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	12.0000	0.558291
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	6.00000	0.278543
$465$	−16.0000	−0.741982
$466$	−10.0000	−0.463241
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	−2.00000	−0.0924500
$469$	24.0000	1.10822
$470$	16.0000	0.738025
$471$	16.0000	0.737241
$472$	12.0000	0.552345
$473$	72.0000	3.31056
$474$	6.00000	0.275589
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	16.0000	0.731823
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−6.00000	−0.273293
$483$	0	0
$484$	25.0000	1.13636
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	4.00000	0.181071
$489$	12.0000	0.542659
$490$	6.00000	0.271052
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	−10.0000	−0.450835
$493$	0	0
$494$	0	0
$495$	12.0000	0.539360
$496$	8.00000	0.359211
$497$	0	0
$498$	−14.0000	−0.627355
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−12.0000	−0.536656
$501$	−8.00000	−0.357414
$502$	−6.00000	−0.267793
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−2.00000	−0.0890871
$505$	−12.0000	−0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	12.0000	0.532414
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−28.0000	−1.23383
$516$	−12.0000	−0.528271
$517$	−48.0000	−2.11104
$518$	0	0
$519$	18.0000	0.790112
$520$	4.00000	0.175412
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	−6.00000	−0.262613
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−8.00000	−0.349482
$525$	2.00000	0.0872872
$526$	4.00000	0.174408
$527$	0	0
$528$	−6.00000	−0.261116
$529$	0	0
$530$	4.00000	0.173749
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−20.0000	−0.866296
$534$	0	0
$535$	−28.0000	−1.21055
$536$	−12.0000	−0.518321
$537$	16.0000	0.690451
$538$	−18.0000	−0.776035
$539$	−18.0000	−0.775315
$540$	−2.00000	−0.0860663
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−28.0000	−1.20270
$543$	8.00000	0.343313
$544$	0	0
$545$	32.0000	1.37073
$546$	−4.00000	−0.171184
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	12.0000	0.512615
$549$	−4.00000	−0.170716
$550$	6.00000	0.255841
$551$	0	0
$552$	0	0
$553$	12.0000	0.510292
$554$	−26.0000	−1.10463
$555$	0	0
$556$	12.0000	0.508913
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−8.00000	−0.338667
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	0	0
$562$	16.0000	0.674919
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	8.00000	0.336861
$565$	16.0000	0.673125
$566$	24.0000	1.00880
$567$	2.00000	0.0839921
$568$	0	0
$569$	−16.0000	−0.670755	−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000	−0.670755	−0.335377	+	0.942084i	$0.608864\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	−20.0000	−0.834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	17.0000	0.707107
$579$	−14.0000	−0.581820
$580$	12.0000	0.498273
$581$	−28.0000	−1.16164
$582$	6.00000	0.248708
$583$	−12.0000	−0.496989
$584$	10.0000	0.413803
$585$	−4.00000	−0.165380
$586$	−6.00000	−0.247858
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	3.00000	0.123718
$589$	0	0
$590$	24.0000	0.988064
$591$	6.00000	0.246807
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	18.0000	0.737309
$597$	2.00000	0.0818546
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	−1.00000	−0.0408248
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−24.0000	−0.978167
$603$	12.0000	0.488678
$604$	−12.0000	−0.488273
$605$	50.0000	2.03279
$606$	−6.00000	−0.243733
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	8.00000	0.323911
$611$	16.0000	0.647291
$612$	0	0
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	20.0000	0.807134
$615$	−20.0000	−0.806478
$616$	−12.0000	−0.483494
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	−14.0000	−0.563163
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	−16.0000	−0.641542
$623$	0	0
$624$	2.00000	0.0800641
$625$	−19.0000	−0.760000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	−16.0000	−0.638470
$629$	0	0
$630$	−4.00000	−0.159364
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	−6.00000	−0.238667
$633$	−20.0000	−0.794929
$634$	−22.0000	−0.873732
$635$	24.0000	0.952411
$636$	2.00000	0.0793052
$637$	6.00000	0.237729
$638$	−36.0000	−1.42525
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	−14.0000	−0.552536
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	0	0
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	−1.00000	−0.0392837
$649$	−72.0000	−2.82625
$650$	−2.00000	−0.0784465
$651$	−16.0000	−0.627089
$652$	−12.0000	−0.469956
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	16.0000	0.625650
$655$	−16.0000	−0.625172
$656$	10.0000	0.390434
$657$	−10.0000	−0.390137
$658$	16.0000	0.623745
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	−12.0000	−0.467099
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	14.0000	0.543305
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	−12.0000	−0.463947
$670$	−24.0000	−0.927201
$671$	−24.0000	−0.926510
$672$	2.00000	0.0771517
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−10.0000	−0.385186
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	8.00000	0.307238
$679$	12.0000	0.460518
$680$	0	0
$681$	−10.0000	−0.383201
$682$	−48.0000	−1.83801
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	20.0000	0.763604
$687$	−24.0000	−0.915657
$688$	12.0000	0.457496
$689$	4.00000	0.152388
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−18.0000	−0.684257
$693$	12.0000	0.455842
$694$	8.00000	0.303676
$695$	24.0000	0.910372
$696$	6.00000	0.227429
$697$	0	0
$698$	−26.0000	−0.984115
$699$	−10.0000	−0.378235
$700$	−2.00000	−0.0755929
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	−2.00000	−0.0754851
$703$	0	0
$704$	6.00000	0.226134
$705$	16.0000	0.602595
$706$	2.00000	0.0752710
$707$	−12.0000	−0.451306
$708$	12.0000	0.450988
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−24.0000	−0.897549
$716$	−16.0000	−0.597948
$717$	16.0000	0.597531
$718$	24.0000	0.895672
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	2.00000	0.0745356
$721$	−28.0000	−1.04277
$722$	19.0000	0.707107
$723$	−6.00000	−0.223142
$724$	−8.00000	−0.297318
$725$	−6.00000	−0.222834
$726$	25.0000	0.927837
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	20.0000	0.740233
$731$	0	0
$732$	4.00000	0.147844
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	−14.0000	−0.516749
$735$	6.00000	0.221313
$736$	0	0
$737$	72.0000	2.65215
$738$	−10.0000	−0.368105
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	8.00000	0.293294
$745$	36.0000	1.31894
$746$	−32.0000	−1.17160
$747$	−14.0000	−0.512233
$748$	0	0
$749$	−28.0000	−1.02310
$750$	−12.0000	−0.438178
$751$	38.0000	1.38664	0.693320	−	0.720630i	$-0.256147\pi$
$751$	38.0000	1.38664	0.693320	+	0.720630i	$0.256147\pi$
$752$	−8.00000	−0.291730
$753$	−6.00000	−0.218652
$754$	12.0000	0.437014
$755$	−24.0000	−0.873449
$756$	−2.00000	−0.0727393
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	12.0000	0.434714
$763$	32.0000	1.15848
$764$	0	0
$765$	0	0
$766$	4.00000	0.144526
$767$	24.0000	0.866590
$768$	−1.00000	−0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−24.0000	−0.864900
$771$	6.00000	0.216085
$772$	14.0000	0.503871
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	−12.0000	−0.431331
$775$	−8.00000	−0.287368
$776$	−6.00000	−0.215387
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	4.00000	0.143223
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	−3.00000	−0.107143
$785$	−32.0000	−1.14213
$786$	−8.00000	−0.285351
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−6.00000	−0.213741
$789$	4.00000	0.142404
$790$	−12.0000	−0.426941
$791$	16.0000	0.568895
$792$	−6.00000	−0.213201
$793$	8.00000	0.284088
$794$	−6.00000	−0.212932
$795$	4.00000	0.141865
$796$	−2.00000	−0.0708881
$797$	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
$800$	1.00000	0.0353553
$801$	0	0
$802$	−32.0000	−1.12996
$803$	−60.0000	−2.11735
$804$	−12.0000	−0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	−2.00000	−0.0702728
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	12.0000	0.421117
$813$	−28.0000	−0.982003
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	0	0
$818$	10.0000	0.349642
$819$	−4.00000	−0.139771
$820$	20.0000	0.698430
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	12.0000	0.418548
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	14.0000	0.487713
$825$	6.00000	0.208893
$826$	24.0000	0.835067
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	42.0000	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$830$	28.0000	0.971894
$831$	−26.0000	−0.901930
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	12.0000	0.415526
$835$	16.0000	0.553703
$836$	0	0
$837$	−8.00000	−0.276520
$838$	−26.0000	−0.898155
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	4.00000	0.137849
$843$	16.0000	0.551069
$844$	20.0000	0.688428
$845$	−18.0000	−0.619219
$846$	8.00000	0.275046
$847$	50.0000	1.71802
$848$	−2.00000	−0.0686803
$849$	24.0000	0.823678
$850$	0	0
$851$	0	0
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	14.0000	0.478510
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	−12.0000	−0.409673
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	24.0000	0.818393
$861$	−20.0000	−0.681598
$862$	−12.0000	−0.408722
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	1.00000	0.0340207
$865$	−36.0000	−1.22404
$866$	−2.00000	−0.0679628
$867$	17.0000	0.577350
$868$	16.0000	0.543075
$869$	36.0000	1.22122
$870$	12.0000	0.406838
$871$	−24.0000	−0.813209
$872$	−16.0000	−0.541828
$873$	6.00000	0.203069
$874$	0	0
$875$	−24.0000	−0.811348
$876$	10.0000	0.337869
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	16.0000	0.539974
$879$	−6.00000	−0.202375
$880$	12.0000	0.404520
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	3.00000	0.101015
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	−36.0000	−1.20944
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	6.00000	0.201008
$892$	12.0000	0.401790
$893$	0	0
$894$	18.0000	0.602010
$895$	−32.0000	−1.06964
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	48.0000	1.60089
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	−60.0000	−1.99778
$903$	−24.0000	−0.798670
$904$	−8.00000	−0.266076
$905$	−16.0000	−0.531858
$906$	−12.0000	−0.398673
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	10.0000	0.331862
$909$	−6.00000	−0.199007
$910$	8.00000	0.265197
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	−84.0000	−2.77999
$914$	26.0000	0.860004
$915$	8.00000	0.264472
$916$	24.0000	0.792982
$917$	−16.0000	−0.528367
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	30.0000	0.987997
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	40.0000	1.31448
$927$	−14.0000	−0.459820
$928$	−6.00000	−0.196960
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	10.0000	0.327561
$933$	−16.0000	−0.523816
$934$	18.0000	0.588978
$935$	0	0
$936$	2.00000	0.0653720
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	−24.0000	−0.783628
$939$	−26.0000	−0.848478
$940$	−16.0000	−0.521862
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	−16.0000	−0.521308
$943$	0	0
$944$	−12.0000	−0.390567
$945$	−4.00000	−0.130120
$946$	−72.0000	−2.34092
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	−6.00000	−0.194871
$949$	20.0000	0.649227
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	2.00000	0.0647524
$955$	0	0
$956$	−16.0000	−0.517477
$957$	−36.0000	−1.16371
$958$	24.0000	0.775405
$959$	24.0000	0.775000
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	0	0
$963$	−14.0000	−0.451144
$964$	6.00000	0.193247
$965$	28.0000	0.901352
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	−12.0000	−0.385297
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	−1.00000	−0.0320750
$973$	24.0000	0.769405
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	−4.00000	−0.128037
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	−6.00000	−0.191663
$981$	16.0000	0.510841
$982$	32.0000	1.02116
$983$	−52.0000	−1.65854	−0.829271	−	0.558846i	$-0.811244\pi$
$983$	−52.0000	−1.65854	−0.829271	+	0.558846i	$0.811244\pi$
$984$	10.0000	0.318788
$985$	−12.0000	−0.382352
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	0	0
$990$	−12.0000	−0.381385
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−8.00000	−0.254000
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−4.00000	−0.126809
$996$	14.0000	0.443607
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3174.2.a.b.1.1		1
3.2	odd	2		9522.2.a.i.1.1		1
23.22	odd	2		138.2.a.a.1.1	✓	1
69.68	even	2		414.2.a.d.1.1		1
92.91	even	2		1104.2.a.e.1.1		1
115.22	even	4		3450.2.d.j.2899.1		2
115.68	even	4		3450.2.d.j.2899.2		2
115.114	odd	2		3450.2.a.y.1.1		1
161.160	even	2		6762.2.a.q.1.1		1
184.45	odd	2		4416.2.a.z.1.1		1
184.91	even	2		4416.2.a.m.1.1		1
276.275	odd	2		3312.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.a.a.1.1	✓	1	23.22	odd	2
414.2.a.d.1.1		1	69.68	even	2
1104.2.a.e.1.1		1	92.91	even	2
3174.2.a.b.1.1		1	1.1	even	1	trivial
3312.2.a.n.1.1		1	276.275	odd	2
3450.2.a.y.1.1		1	115.114	odd	2
3450.2.d.j.2899.1		2	115.22	even	4
3450.2.d.j.2899.2		2	115.68	even	4
4416.2.a.m.1.1		1	184.91	even	2
4416.2.a.z.1.1		1	184.45	odd	2
6762.2.a.q.1.1		1	161.160	even	2
9522.2.a.i.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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