LMFDB - Embedded newform 5415.2.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5415,2,Mod(1,5415)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5415.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5415 = 3 \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5415.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,-1,-1,-1,-2,-3,1,-1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.2389926945$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5415.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +5.00000 q^{32} +6.00000 q^{33} -6.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} +3.00000 q^{40} +2.00000 q^{42} -2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} +6.00000 q^{56} -4.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +6.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +8.00000 q^{69} +2.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} -1.00000 q^{75} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{84} +6.00000 q^{85} -2.00000 q^{86} +4.00000 q^{87} +18.0000 q^{88} -1.00000 q^{90} +8.00000 q^{92} -8.00000 q^{94} -5.00000 q^{96} +12.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	1.00000	0.235702
$19$	0	0
$19$	0	0
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	−6.00000	−1.27920
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	6.00000	1.04447
$34$	−6.00000	−1.02899
$35$	2.00000	0.338062
$36$	−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
$39$	0	0
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	2.00000	0.308607
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	−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
$51$	6.00000	0.840168
$52$	0	0
$53$	−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
$55$	6.00000	0.809040
$56$	6.00000	0.801784
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	6.00000	0.738549
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000	0.727607
$69$	8.00000	0.963087
$70$	2.00000	0.239046
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−3.00000	−0.353553
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	−4.00000	−0.464991
$75$	−1.00000	−0.115470
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−2.00000	−0.218218
$85$	6.00000	0.650791
$86$	−2.00000	−0.215666
$87$	4.00000	0.428845
$88$	18.0000	1.91881
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−3.00000	−0.303046
$99$	−6.00000	−0.603023
$100$	−1.00000	−0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	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	$0.371045\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	6.00000	0.572078
$111$	4.00000	0.379663
$112$	2.00000	0.188982
$113$	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
$115$	8.00000	0.746004
$116$	4.00000	0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	12.0000	1.10004
$120$	−3.00000	−0.273861
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−3.00000	−0.265165
$129$	2.00000	0.176090
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	8.00000	0.691095
$135$	1.00000	0.0860663
$136$	18.0000	1.54349
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	8.00000	0.681005
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	−16.0000	−1.34269
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	3.00000	0.247436
$148$	4.00000	0.328798
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−8.00000	−0.636446
$159$	2.00000	0.158610
$160$	−5.00000	−0.395285
$161$	16.0000	1.26098
$162$	1.00000	0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	−6.00000	−0.462910
$169$	−13.0000	−1.00000
$170$	6.00000	0.460179
$171$	0	0
$172$	2.00000	0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	4.00000	0.303239
$175$	−2.00000	−0.151186
$176$	6.00000	0.452267
$177$	12.0000	0.901975
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	1.00000	0.0745356
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	24.0000	1.76930
$185$	4.00000	0.294086
$186$	0	0
$187$	36.0000	2.63258
$188$	8.00000	0.583460
$189$	2.00000	0.145479
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−7.00000	−0.505181
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	3.00000	0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	−18.0000	−1.26648
$203$	8.00000	0.561490
$204$	−6.00000	−0.420084
$205$	0	0
$206$	8.00000	0.557386
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	−2.00000	−0.138013
$211$	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
$213$	16.0000	1.09630
$214$	12.0000	0.820303
$215$	2.00000	0.136399
$216$	3.00000	0.204124
$217$	0	0
$218$	6.00000	0.406371
$219$	−14.0000	−0.946032
$220$	−6.00000	−0.404520
$221$	0	0
$222$	4.00000	0.268462
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−10.0000	−0.668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	8.00000	0.527504
$231$	−12.0000	−0.789542
$232$	12.0000	0.787839
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−1.00000	−0.0645497
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	2.00000	0.125988
$253$	48.0000	3.01773
$254$	−4.00000	−0.250982
$255$	−6.00000	−0.375735
$256$	−17.0000	−1.06250
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	2.00000	0.124515
$259$	8.00000	0.497096
$260$	0	0
$261$	−4.00000	−0.247594
$262$	−2.00000	−0.123560
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−18.0000	−1.10782
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	1.00000	0.0608581
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−10.0000	−0.604122
$275$	−6.00000	−0.361814
$276$	−8.00000	−0.481543
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	−6.00000	−0.358569
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	8.00000	0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	19.0000	1.11765
$290$	4.00000	0.234888
$291$	−12.0000	−0.703452
$292$	−14.0000	−0.819288
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	3.00000	0.174964
$295$	12.0000	0.698667
$296$	12.0000	0.697486
$297$	6.00000	0.348155
$298$	6.00000	0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	16.0000	0.920697
$303$	18.0000	1.03407
$304$	0	0
$305$	−2.00000	−0.114520
$306$	−6.00000	−0.342997
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−12.0000	−0.683763
$309$	−8.00000	−0.455104
$310$	0	0
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−2.00000	−0.112867
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	2.00000	0.112154
$319$	24.0000	1.34374
$320$	−7.00000	−0.391312
$321$	−12.0000	−0.669775
$322$	16.0000	0.891645
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−22.0000	−1.21847
$327$	−6.00000	−0.331801
$328$	0	0
$329$	16.0000	0.882109
$330$	−6.00000	−0.330289
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	−16.0000	−0.875481
$335$	−8.00000	−0.437087
$336$	−2.00000	−0.109109
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	−13.0000	−0.707107
$339$	−6.00000	−0.325875
$340$	−6.00000	−0.325396
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	6.00000	0.323498
$345$	−8.00000	−0.430706
$346$	−6.00000	−0.322562
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	−4.00000	−0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	−30.0000	−1.59901
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	12.0000	0.637793
$355$	16.0000	0.849192
$356$	0	0
$357$	−12.0000	−0.635107
$358$	20.0000	1.05703
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	3.00000	0.158114
$361$	0	0
$362$	−10.0000	−0.525588
$363$	−25.0000	−1.31216
$364$	0	0
$365$	−14.0000	−0.732793
$366$	−2.00000	−0.104542
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	4.00000	0.207950
$371$	4.00000	0.207670
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	36.0000	1.86152
$375$	1.00000	0.0516398
$376$	24.0000	1.23771
$377$	0	0
$378$	2.00000	0.102869
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	10.0000	0.511645
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	3.00000	0.153093
$385$	−12.0000	−0.611577
$386$	−24.0000	−1.22157
$387$	−2.00000	−0.101666
$388$	−12.0000	−0.609208
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	9.00000	0.454569
$393$	2.00000	0.100887
$394$	6.00000	0.302276
$395$	8.00000	0.402524
$396$	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	$0.451890\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	18.0000	0.895533
$405$	−1.00000	−0.0496904
$406$	8.00000	0.397033
$407$	24.0000	1.18964
$408$	−18.0000	−0.891133
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	−8.00000	−0.394132
$413$	24.0000	1.18096
$414$	−8.00000	−0.393179
$415$	0	0
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	2.00000	0.0975900
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	20.0000	0.973585
$423$	−8.00000	−0.388973
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	16.0000	0.775203
$427$	−4.00000	−0.193574
$428$	−12.0000	−0.580042
$429$	0	0
$430$	2.00000	0.0964486
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−6.00000	−0.287348
$437$	0	0
$438$	−14.0000	−0.668946
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−18.0000	−0.858116
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−6.00000	−0.283790
$448$	−14.0000	−0.661438
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	28.0000	1.31411
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	2.00000	0.0934539
$459$	6.00000	0.280056
$460$	−8.00000	−0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−12.0000	−0.558291
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	22.0000	1.01913
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	8.00000	0.369012
$471$	2.00000	0.0921551
$472$	36.0000	1.65703
$473$	12.0000	0.551761
$474$	8.00000	0.367452
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−2.00000	−0.0915737
$478$	6.00000	0.274434
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	5.00000	0.228218
$481$	0	0
$482$	−18.0000	−0.819878
$483$	−16.0000	−0.728025
$484$	−25.0000	−1.13636
$485$	−12.0000	−0.544892
$486$	−1.00000	−0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	22.0000	0.994874
$490$	3.00000	0.135526
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	1.00000	0.0447214
$501$	16.0000	0.714827
$502$	−6.00000	−0.267793
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	6.00000	0.267261
$505$	18.0000	0.800989
$506$	48.0000	2.13386
$507$	13.0000	0.577350
$508$	4.00000	0.177471
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	−6.00000	−0.265684
$511$	−28.0000	−1.23865
$512$	−11.0000	−0.486136
$513$	0	0
$514$	22.0000	0.970378
$515$	−8.00000	−0.352522
$516$	−2.00000	−0.0880451
$517$	48.0000	2.11104
$518$	8.00000	0.351500
$519$	6.00000	0.263371
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	−4.00000	−0.175075
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	2.00000	0.0873704
$525$	2.00000	0.0872872
$526$	−12.0000	−0.523225
$527$	0	0
$528$	−6.00000	−0.261116
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−24.0000	−1.03664
$537$	−20.0000	−0.863064
$538$	−12.0000	−0.517357
$539$	18.0000	0.775315
$540$	−1.00000	−0.0430331
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−20.0000	−0.859074
$543$	10.0000	0.429141
$544$	−30.0000	−1.28624
$545$	−6.00000	−0.257012
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	10.0000	0.427179
$549$	2.00000	0.0853579
$550$	−6.00000	−0.255841
$551$	0	0
$552$	−24.0000	−1.02151
$553$	16.0000	0.680389
$554$	−2.00000	−0.0849719
$555$	−4.00000	−0.169791
$556$	16.0000	0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	−36.0000	−1.51992
$562$	4.00000	0.168730
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−8.00000	−0.336861
$565$	−6.00000	−0.252422
$566$	−14.0000	−0.588464
$567$	−2.00000	−0.0839921
$568$	48.0000	2.01404
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	−10.0000	−0.417756
$574$	0	0
$575$	−8.00000	−0.333623
$576$	7.00000	0.291667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	19.0000	0.790296
$579$	24.0000	0.997406
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−12.0000	−0.497416
$583$	12.0000	0.496989
$584$	−42.0000	−1.73797
$585$	0	0
$586$	−30.0000	−1.23929
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	−3.00000	−0.123718
$589$	0	0
$590$	12.0000	0.494032
$591$	−6.00000	−0.246807
$592$	4.00000	0.164399
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	6.00000	0.246183
$595$	−12.0000	−0.491952
$596$	−6.00000	−0.245770
$597$	20.0000	0.818546
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	3.00000	0.122474
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	8.00000	0.325785
$604$	−16.0000	−0.651031
$605$	−25.0000	−1.01639
$606$	18.0000	0.731200
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	6.00000	0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−36.0000	−1.45048
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	−8.00000	−0.321807
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	34.0000	1.36328
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	2.00000	0.0798087
$629$	24.0000	0.956943
$630$	2.00000	0.0796819
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	24.0000	0.954669
$633$	−20.0000	−0.794929
$634$	6.00000	0.238290
$635$	4.00000	0.158735
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	24.0000	0.950169
$639$	−16.0000	−0.632950
$640$	3.00000	0.118585
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	−12.0000	−0.473602
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	−16.0000	−0.630488
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	−3.00000	−0.117851
$649$	72.0000	2.82625
$650$	0	0
$651$	0	0
$652$	22.0000	0.861586
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−6.00000	−0.234619
$655$	2.00000	0.0781465
$656$	0	0
$657$	14.0000	0.546192
$658$	16.0000	0.623745
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	6.00000	0.233550
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−4.00000	−0.154997
$667$	32.0000	1.23904
$668$	16.0000	0.619059
$669$	8.00000	0.309298
$670$	−8.00000	−0.309067
$671$	−12.0000	−0.463255
$672$	10.0000	0.385758
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	12.0000	0.462223
$675$	−1.00000	−0.0384900
$676$	13.0000	0.500000
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−6.00000	−0.230429
$679$	−24.0000	−0.921035
$680$	−18.0000	−0.690268
$681$	−28.0000	−1.07296
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	20.0000	0.763604
$687$	−2.00000	−0.0763048
$688$	2.00000	0.0762493
$689$	0	0
$690$	−8.00000	−0.304555
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	6.00000	0.228086
$693$	12.0000	0.455842
$694$	8.00000	0.303676
$695$	16.0000	0.606915
$696$	−12.0000	−0.454859
$697$	0	0
$698$	−2.00000	−0.0757011
$699$	−22.0000	−0.832116
$700$	2.00000	0.0755929
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	0	0
$704$	−42.0000	−1.58293
$705$	−8.00000	−0.301297
$706$	−34.0000	−1.27961
$707$	36.0000	1.35392
$708$	−12.0000	−0.450988
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	0	0
$713$	0	0
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−20.0000	−0.747435
$717$	−6.00000	−0.224074
$718$	10.0000	0.373197
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	1.00000	0.0372678
$721$	−16.0000	−0.595871
$722$	0	0
$723$	18.0000	0.669427
$724$	10.0000	0.371647
$725$	−4.00000	−0.148556
$726$	−25.0000	−0.927837
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−14.0000	−0.518163
$731$	12.0000	0.443836
$732$	2.00000	0.0739221
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−14.0000	−0.516749
$735$	−3.00000	−0.110657
$736$	−40.0000	−1.47442
$737$	−48.0000	−1.76810
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	4.00000	0.146845
$743$	−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
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	−36.0000	−1.31629
$749$	−24.0000	−0.876941
$750$	1.00000	0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	8.00000	0.291730
$753$	6.00000	0.218652
$754$	0	0
$755$	−16.0000	−0.582300
$756$	−2.00000	−0.0727393
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−20.0000	−0.726433
$759$	−48.0000	−1.74229
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	4.00000	0.144905
$763$	−12.0000	−0.434429
$764$	−10.0000	−0.361787
$765$	6.00000	0.216930
$766$	−8.00000	−0.289052
$767$	0	0
$768$	17.0000	0.613435
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	−12.0000	−0.432450
$771$	−22.0000	−0.792311
$772$	24.0000	0.863779
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−36.0000	−1.29232
$777$	−8.00000	−0.286998
$778$	−14.0000	−0.501924
$779$	0	0
$780$	0	0
$781$	96.0000	3.43515
$782$	48.0000	1.71648
$783$	4.00000	0.142948
$784$	3.00000	0.107143
$785$	2.00000	0.0713831
$786$	2.00000	0.0713376
$787$	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
$789$	12.0000	0.427211
$790$	8.00000	0.284627
$791$	−12.0000	−0.426671
$792$	18.0000	0.639602
$793$	0	0
$794$	6.00000	0.212932
$795$	−2.00000	−0.0709327
$796$	20.0000	0.708881
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	5.00000	0.176777
$801$	0	0
$802$	−4.00000	−0.141245
$803$	−84.0000	−2.96430
$804$	8.00000	0.282138
$805$	−16.0000	−0.563926
$806$	0	0
$807$	12.0000	0.422420
$808$	54.0000	1.89971
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	−1.00000	−0.0351364
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−8.00000	−0.280745
$813$	20.0000	0.701431
$814$	24.0000	0.841200
$815$	22.0000	0.770626
$816$	−6.00000	−0.210042
$817$	0	0
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	10.0000	0.348790
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−24.0000	−0.836080
$825$	6.00000	0.208893
$826$	24.0000	0.835067
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	8.00000	0.278019
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	18.0000	0.623663
$834$	16.0000	0.554035
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	−30.0000	−1.03633
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	6.00000	0.207020
$841$	−13.0000	−0.448276
$842$	−6.00000	−0.206774
$843$	−4.00000	−0.137767
$844$	−20.0000	−0.688428
$845$	13.0000	0.447214
$846$	−8.00000	−0.275046
$847$	−50.0000	−1.71802
$848$	2.00000	0.0686803
$849$	14.0000	0.480479
$850$	−6.00000	−0.205798
$851$	32.0000	1.09695
$852$	−16.0000	−0.548151
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−36.0000	−1.23045
$857$	−46.0000	−1.57133	−0.785665	−	0.618652i	$-0.787679\pi$
$857$	−46.0000	−1.57133	−0.785665	+	0.618652i	$0.787679\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−36.0000	−1.22616
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−5.00000	−0.170103
$865$	6.00000	0.204006
$866$	16.0000	0.543702
$867$	−19.0000	−0.645274
$868$	0	0
$869$	48.0000	1.62829
$870$	−4.00000	−0.135613
$871$	0	0
$872$	−18.0000	−0.609557
$873$	12.0000	0.406138
$874$	0	0
$875$	2.00000	0.0676123
$876$	14.0000	0.473016
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	−24.0000	−0.809961
$879$	30.0000	1.01187
$880$	−6.00000	−0.202260
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−3.00000	−0.101015
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	−12.0000	−0.403148
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	−12.0000	−0.402694
$889$	8.00000	0.268311
$890$	0	0
$891$	−6.00000	−0.201008
$892$	8.00000	0.267860
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−20.0000	−0.668526
$896$	6.00000	0.200446
$897$	0	0
$898$	8.00000	0.266963
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	12.0000	0.399778
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−18.0000	−0.598671
$905$	10.0000	0.332411
$906$	−16.0000	−0.531564
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−28.0000	−0.929213
$909$	−18.0000	−0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	2.00000	0.0661180
$916$	−2.00000	−0.0660819
$917$	4.00000	0.132092
$918$	6.00000	0.198030
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−24.0000	−0.791257
$921$	−8.00000	−0.263609
$922$	30.0000	0.987997
$923$	0	0
$924$	12.0000	0.394771
$925$	−4.00000	−0.131519
$926$	−14.0000	−0.460069
$927$	8.00000	0.262754
$928$	−20.0000	−0.656532
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	0	0
$932$	−22.0000	−0.720634
$933$	−34.0000	−1.11311
$934$	12.0000	0.392652
$935$	−36.0000	−1.17733
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	−16.0000	−0.522419
$939$	−10.0000	−0.326338
$940$	−8.00000	−0.260931
$941$	−44.0000	−1.43436	−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000	−1.43436	−0.717180	+	0.696888i	$0.754567\pi$
$942$	2.00000	0.0651635
$943$	0	0
$944$	12.0000	0.390567
$945$	−2.00000	−0.0650600
$946$	12.0000	0.390154
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	−36.0000	−1.16677
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	−2.00000	−0.0647524
$955$	−10.0000	−0.323592
$956$	−6.00000	−0.194054
$957$	−24.0000	−0.775810
$958$	18.0000	0.581554
$959$	20.0000	0.645834
$960$	7.00000	0.225924
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.0000	0.386695
$964$	18.0000	0.579741
$965$	24.0000	0.772587
$966$	−16.0000	−0.514792
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	−75.0000	−2.41059
$969$	0	0
$970$	−12.0000	−0.385297
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	1.00000	0.0320750
$973$	32.0000	1.02587
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	22.0000	0.703482
$979$	0	0
$980$	−3.00000	−0.0958315
$981$	6.00000	0.191565
$982$	−26.0000	−0.829693
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	24.0000	0.764316
$987$	−16.0000	−0.509286
$988$	0	0
$989$	16.0000	0.508770
$990$	6.00000	0.190693
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	32.0000	1.01498
$995$	20.0000	0.634043
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−24.0000	−0.759707
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5415.2.a.h.1.1		1
19.18	odd	2		285.2.a.a.1.1	✓	1
57.56	even	2		855.2.a.c.1.1		1
76.75	even	2		4560.2.a.h.1.1		1
95.18	even	4		1425.2.c.c.799.2		2
95.37	even	4		1425.2.c.c.799.1		2
95.94	odd	2		1425.2.a.g.1.1		1
285.284	even	2		4275.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.a.1.1	✓	1	19.18	odd	2
855.2.a.c.1.1		1	57.56	even	2
1425.2.a.g.1.1		1	95.94	odd	2
1425.2.c.c.799.1		2	95.37	even	4
1425.2.c.c.799.2		2	95.18	even	4
4275.2.a.h.1.1		1	285.284	even	2
4560.2.a.h.1.1		1	76.75	even	2
5415.2.a.h.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5415.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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