LMFDB - Embedded newform 975.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{21} -4.00000 q^{22} +3.00000 q^{24} +1.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -10.0000 q^{29} +4.00000 q^{31} -5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} -1.00000 q^{39} +6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} -4.00000 q^{44} -1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +12.0000 q^{56} +10.0000 q^{58} +12.0000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} +8.00000 q^{67} +2.00000 q^{68} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +16.0000 q^{77} +1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{84} -12.0000 q^{86} -10.0000 q^{87} +12.0000 q^{88} -2.00000 q^{89} -4.00000 q^{91} +4.00000 q^{93} -5.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−4.00000	−0.617213
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	10.0000	1.31306
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	4.00000	0.503953
$64$	7.00000	0.875000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	0	0
$77$	16.0000	1.82337
$78$	1.00000	0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−10.0000	−1.07211
$88$	12.0000	1.27920
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−9.00000	−0.909137
$99$	4.00000	0.402015
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	2.00000	0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	6.00000	0.582772
$107$	−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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	−4.00000	−0.377964
$113$	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
$115$	0	0
$116$	10.0000	0.928477
$117$	−1.00000	−0.0924500
$118$	−12.0000	−1.10469
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	6.00000	0.541002
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−4.00000	−0.356348
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	3.00000	0.265165
$129$	12.0000	1.05654
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	9.00000	0.742307
$148$	−2.00000	−0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−16.0000	−1.28932
$155$	0	0
$156$	1.00000	0.0800641
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	12.0000	0.925820
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−12.0000	−0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−4.00000	−0.301511
$177$	12.0000	0.901975
$178$	2.00000	0.149906
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000	0.296500
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	−8.00000	−0.585018
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	7.00000	0.505181
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−4.00000	−0.284268
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	18.0000	1.26648
$203$	−40.0000	−2.80745
$204$	2.00000	0.140028
$205$	0	0
$206$	0	0
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	16.0000	1.08615
$218$	2.00000	0.135457
$219$	−2.00000	−0.135147
$220$	0	0
$221$	2.00000	0.134535
$222$	−2.00000	−0.134231
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	−30.0000	−1.96960
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−12.0000	−0.781133
$237$	8.00000	0.519656
$238$	8.00000	0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	0	0
$248$	12.0000	0.762001
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−4.00000	−0.251976
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	−12.0000	−0.747087
$259$	8.00000	0.497096
$260$	0	0
$261$	−10.0000	−0.618984
$262$	−4.00000	−0.247121
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	0	0
$267$	−2.00000	−0.122398
$268$	−8.00000	−0.488678
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	2.00000	0.121268
$273$	−4.00000	−0.242091
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	24.0000	1.41668
$288$	−5.00000	−0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	6.00000	0.348743
$297$	4.00000	0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	48.0000	2.76667
$302$	−4.00000	−0.230174
$303$	−18.0000	−1.03407
$304$	0	0
$305$	0	0
$306$	2.00000	0.114332
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−16.0000	−0.911685
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−3.00000	−0.169842
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	6.00000	0.336463
$319$	−40.0000	−2.23957
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	8.00000	0.443079
$327$	−2.00000	−0.110600
$328$	18.0000	0.993884
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	4.00000	0.219529
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	0	0
$336$	−4.00000	−0.218218
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−1.00000	−0.0543928
$339$	6.00000	0.325875
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	8.00000	0.431959
$344$	36.0000	1.94099
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	10.0000	0.536056
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−20.0000	−1.06600
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	2.00000	0.106000
$357$	−8.00000	−0.423405
$358$	−4.00000	−0.211407
$359$	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
$362$	10.0000	0.525588
$363$	5.00000	0.262432
$364$	4.00000	0.209657
$365$	0	0
$366$	2.00000	0.104542
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	−4.00000	−0.207390
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	10.0000	0.515026
$378$	−4.00000	−0.205738
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	18.0000	0.916176
$387$	12.0000	0.609994
$388$	10.0000	0.507673
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	27.0000	1.36371
$393$	4.00000	0.201773
$394$	18.0000	0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	−8.00000	−0.399004
$403$	−4.00000	−0.199254
$404$	18.0000	0.895533
$405$	0	0
$406$	40.0000	1.98517
$407$	8.00000	0.396545
$408$	−6.00000	−0.297044
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	12.0000	0.587643
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	2.00000	0.0955637
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−2.00000	−0.0951303
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	4.00000	0.189405
$447$	−6.00000	−0.283790
$448$	28.0000	1.32288
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	−6.00000	−0.282216
$453$	4.00000	0.187936
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	10.0000	0.467269
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	−16.0000	−0.744387
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	1.00000	0.0462250
$469$	32.0000	1.47762
$470$	0	0
$471$	18.0000	0.829396
$472$	36.0000	1.65703
$473$	48.0000	2.20704
$474$	−8.00000	−0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−6.00000	−0.271607
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−6.00000	−0.270501
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	4.00000	0.179244
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	12.0000	0.535586
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	12.0000	0.534522
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	−16.0000	−0.709885
$509$	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
$511$	−8.00000	−0.353899
$512$	11.0000	0.486136
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	−12.0000	−0.528271
$517$	0	0
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	10.0000	0.437688
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	24.0000	1.04645
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	−6.00000	−0.259889
$534$	2.00000	0.0865485
$535$	0	0
$536$	24.0000	1.03664
$537$	4.00000	0.172613
$538$	−22.0000	−0.948487
$539$	36.0000	1.55063
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	12.0000	0.515444
$543$	−10.0000	−0.429141
$544$	10.0000	0.428746
$545$	0	0
$546$	4.00000	0.171184
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	6.00000	0.256307
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000	1.36078
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−4.00000	−0.169334
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−8.00000	−0.337760
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	0	0
$566$	12.0000	0.504398
$567$	4.00000	0.167984
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	4.00000	0.167248
$573$	8.00000	0.334205
$574$	−24.0000	−1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	13.0000	0.540729
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−16.0000	−0.663792
$582$	10.0000	0.414513
$583$	−24.0000	−0.993978
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−9.00000	−0.371154
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	−2.00000	−0.0821995
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000	0.327418
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	−48.0000	−1.95633
$603$	8.00000	0.325785
$604$	−4.00000	−0.162758
$605$	0	0
$606$	18.0000	0.731200
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−40.0000	−1.62088
$610$	0	0
$611$	0	0
$612$	2.00000	0.0808452
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	48.0000	1.93398
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.00000	−0.320513
$624$	1.00000	0.0400320
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−4.00000	−0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	24.0000	0.954669
$633$	−20.0000	−0.794929
$634$	26.0000	1.03259
$635$	0	0
$636$	6.00000	0.237915
$637$	−9.00000	−0.356593
$638$	40.0000	1.58362
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	12.0000	0.473602
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	3.00000	0.117851
$649$	48.0000	1.88416
$650$	0	0
$651$	16.0000	0.627089
$652$	8.00000	0.313304
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	16.0000	0.621858
$663$	2.00000	0.0776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−8.00000	−0.309529
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−8.00000	−0.308837
$672$	−20.0000	−0.771517
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−6.00000	−0.230429
$679$	−40.0000	−1.53506
$680$	0	0
$681$	20.0000	0.766402
$682$	−16.0000	−0.612672
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	−10.0000	−0.381524
$688$	−12.0000	−0.457496
$689$	6.00000	0.228582
$690$	0	0
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	6.00000	0.228086
$693$	16.0000	0.607790
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−30.0000	−1.13715
$697$	−12.0000	−0.454532
$698$	26.0000	0.984115
$699$	14.0000	0.529529
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	1.00000	0.0377426
$703$	0	0
$704$	28.0000	1.05529
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	−72.0000	−2.70784
$708$	−12.0000	−0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	−6.00000	−0.224860
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−24.0000	−0.896296
$718$	−24.0000	−0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	19.0000	0.707107
$723$	10.0000	0.371904
$724$	10.0000	0.371647
$725$	0	0
$726$	−5.00000	−0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−12.0000	−0.444750
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	2.00000	0.0739221
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	−6.00000	−0.220863
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−26.0000	−0.951928
$747$	−4.00000	−0.146352
$748$	8.00000	0.292509
$749$	−48.0000	−1.75388
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	−10.0000	−0.364179
$755$	0	0
$756$	−4.00000	−0.145479
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−16.0000	−0.579619
$763$	−8.00000	−0.289619
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−12.0000	−0.433295
$768$	−17.0000	−0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−26.0000	−0.936367
$772$	18.0000	0.647834
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−30.0000	−1.07694
$777$	8.00000	0.286998
$778$	−22.0000	−0.788738
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−10.0000	−0.357371
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−4.00000	−0.142675
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	18.0000	0.641223
$789$	−24.0000	−0.854423
$790$	0	0
$791$	24.0000	0.853342
$792$	12.0000	0.426401
$793$	2.00000	0.0710221
$794$	38.0000	1.34857
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	−22.0000	−0.776847
$803$	−8.00000	−0.282314
$804$	−8.00000	−0.282138
$805$	0	0
$806$	4.00000	0.140894
$807$	22.0000	0.774437
$808$	−54.0000	−1.89971
$809$	−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
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	40.0000	1.40372
$813$	−12.0000	−0.420858
$814$	−8.00000	−0.280400
$815$	0	0
$816$	2.00000	0.0700140
$817$	0	0
$818$	−34.0000	−1.18878
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	6.00000	0.209274
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	−48.0000	−1.67013
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−7.00000	−0.242681
$833$	−18.0000	−0.623663
$834$	−12.0000	−0.415526
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	−4.00000	−0.138178
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	10.0000	0.344623
$843$	−10.0000	−0.344418
$844$	20.0000	0.688428
$845$	0	0
$846$	0	0
$847$	20.0000	0.687208
$848$	6.00000	0.206041
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	−36.0000	−1.23045
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	4.00000	0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	34.0000	1.15537
$867$	−13.0000	−0.441503
$868$	−16.0000	−0.543075
$869$	32.0000	1.08553
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−6.00000	−0.203186
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	−32.0000	−1.07995
$879$	6.00000	0.202375
$880$	0	0
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	−9.00000	−0.303046
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	6.00000	0.201347
$889$	64.0000	2.14649
$890$	0	0
$891$	4.00000	0.134005
$892$	4.00000	0.133930
$893$	0	0
$894$	6.00000	0.200670
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	−22.0000	−0.734150
$899$	−40.0000	−1.33407
$900$	0	0
$901$	12.0000	0.399778
$902$	−24.0000	−0.799113
$903$	48.0000	1.59734
$904$	18.0000	0.598671
$905$	0	0
$906$	−4.00000	−0.132891
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	−20.0000	−0.663723
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	2.00000	0.0661541
$915$	0	0
$916$	10.0000	0.330409
$917$	16.0000	0.528367
$918$	2.00000	0.0660098
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	38.0000	1.25146
$923$	0	0
$924$	−16.0000	−0.526361
$925$	0	0
$926$	4.00000	0.131448
$927$	0	0
$928$	50.0000	1.64133
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	−32.0000	−1.04484
$939$	6.00000	0.195803
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−18.0000	−0.586472
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	−8.00000	−0.259828
$949$	2.00000	0.0649227
$950$	0	0
$951$	−26.0000	−0.843108
$952$	−24.0000	−0.777844
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	−40.0000	−1.29302
$958$	24.0000	0.775405
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	15.0000	0.482118
$969$	0	0
$970$	0	0
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	−1.00000	−0.0320750
$973$	48.0000	1.53881
$974$	12.0000	0.384505
$975$	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	$0.265500\pi$
$978$	8.00000	0.255812
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	12.0000	0.382935
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	−20.0000	−0.636930
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	−20.0000	−0.635001
$993$	−16.0000	−0.507745
$994$	0	0
$995$	0	0
$996$	4.00000	0.126745
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	24.0000	0.759707
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.f.1.1		1
3.2	odd	2		2925.2.a.p.1.1		1
5.2	odd	4		975.2.c.f.274.1		2
5.3	odd	4		975.2.c.f.274.2		2
5.4	even	2		39.2.a.a.1.1	✓	1
15.2	even	4		2925.2.c.e.2224.2		2
15.8	even	4		2925.2.c.e.2224.1		2
15.14	odd	2		117.2.a.a.1.1		1
20.19	odd	2		624.2.a.i.1.1		1
35.34	odd	2		1911.2.a.f.1.1		1
40.19	odd	2		2496.2.a.e.1.1		1
40.29	even	2		2496.2.a.q.1.1		1
45.4	even	6		1053.2.e.b.703.1		2
45.14	odd	6		1053.2.e.d.703.1		2
45.29	odd	6		1053.2.e.d.352.1		2
45.34	even	6		1053.2.e.b.352.1		2
55.54	odd	2		4719.2.a.c.1.1		1
60.59	even	2		1872.2.a.h.1.1		1
65.4	even	6		507.2.e.b.484.1		2
65.9	even	6		507.2.e.a.484.1		2
65.19	odd	12		507.2.j.e.361.1		4
65.24	odd	12		507.2.j.e.316.1		4
65.29	even	6		507.2.e.a.22.1		2
65.34	odd	4		507.2.b.a.337.2		2
65.44	odd	4		507.2.b.a.337.1		2
65.49	even	6		507.2.e.b.22.1		2
65.54	odd	12		507.2.j.e.316.2		4
65.59	odd	12		507.2.j.e.361.2		4
65.64	even	2		507.2.a.a.1.1		1
105.104	even	2		5733.2.a.e.1.1		1
120.29	odd	2		7488.2.a.bl.1.1		1
120.59	even	2		7488.2.a.by.1.1		1
195.44	even	4		1521.2.b.b.1351.2		2
195.164	even	4		1521.2.b.b.1351.1		2
195.194	odd	2		1521.2.a.e.1.1		1
260.259	odd	2		8112.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.a.1.1	✓	1	5.4	even	2
117.2.a.a.1.1		1	15.14	odd	2
507.2.a.a.1.1		1	65.64	even	2
507.2.b.a.337.1		2	65.44	odd	4
507.2.b.a.337.2		2	65.34	odd	4
507.2.e.a.22.1		2	65.29	even	6
507.2.e.a.484.1		2	65.9	even	6
507.2.e.b.22.1		2	65.49	even	6
507.2.e.b.484.1		2	65.4	even	6
507.2.j.e.316.1		4	65.24	odd	12
507.2.j.e.316.2		4	65.54	odd	12
507.2.j.e.361.1		4	65.19	odd	12
507.2.j.e.361.2		4	65.59	odd	12
624.2.a.i.1.1		1	20.19	odd	2
975.2.a.f.1.1		1	1.1	even	1	trivial
975.2.c.f.274.1		2	5.2	odd	4
975.2.c.f.274.2		2	5.3	odd	4
1053.2.e.b.352.1		2	45.34	even	6
1053.2.e.b.703.1		2	45.4	even	6
1053.2.e.d.352.1		2	45.29	odd	6
1053.2.e.d.703.1		2	45.14	odd	6
1521.2.a.e.1.1		1	195.194	odd	2
1521.2.b.b.1351.1		2	195.164	even	4
1521.2.b.b.1351.2		2	195.44	even	4
1872.2.a.h.1.1		1	60.59	even	2
1911.2.a.f.1.1		1	35.34	odd	2
2496.2.a.e.1.1		1	40.19	odd	2
2496.2.a.q.1.1		1	40.29	even	2
2925.2.a.p.1.1		1	3.2	odd	2
2925.2.c.e.2224.1		2	15.8	even	4
2925.2.c.e.2224.2		2	15.2	even	4
4719.2.a.c.1.1		1	55.54	odd	2
5733.2.a.e.1.1		1	105.104	even	2
7488.2.a.bl.1.1		1	120.29	odd	2
7488.2.a.by.1.1		1	120.59	even	2
8112.2.a.s.1.1		1	260.259	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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