LMFDB - Embedded newform 975.2.a.c.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
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)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -2.00000 q^{18} -2.00000 q^{19} -3.00000 q^{21} +2.00000 q^{22} +3.00000 q^{23} -2.00000 q^{26} +1.00000 q^{27} -6.00000 q^{28} -2.00000 q^{29} -6.00000 q^{31} +8.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +2.00000 q^{36} -11.0000 q^{37} +4.00000 q^{38} +1.00000 q^{39} -5.00000 q^{41} +6.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} -6.00000 q^{46} +10.0000 q^{47} -4.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} +2.00000 q^{52} -11.0000 q^{53} -2.00000 q^{54} -2.00000 q^{57} +4.00000 q^{58} +8.00000 q^{59} +13.0000 q^{61} +12.0000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +2.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +3.00000 q^{69} -5.00000 q^{71} -10.0000 q^{73} +22.0000 q^{74} -4.00000 q^{76} +3.00000 q^{77} -2.00000 q^{78} -3.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +12.0000 q^{83} -6.00000 q^{84} +8.00000 q^{86} -2.00000 q^{87} -15.0000 q^{89} -3.00000 q^{91} +6.00000 q^{92} -6.00000 q^{93} -20.0000 q^{94} +8.00000 q^{96} -17.0000 q^{97} -4.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−2.00000	−0.471405
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	2.00000	0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−6.00000	−1.13389
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	8.00000	1.41421
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	2.00000	0.333333
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	4.00000	0.648886
$39$	1.00000	0.160128
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	6.00000	0.925820
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−6.00000	−0.884652
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	2.00000	0.277350
$53$	−11.0000	−1.51097	−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000	−1.51097	−0.755483	+	0.655168i	$0.772598\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	4.00000	0.525226
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	12.0000	1.52400
$63$	−3.00000	−0.377964
$64$	−8.00000	−1.00000
$65$	0	0
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	3.00000	0.361158
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	22.0000	2.55745
$75$	0	0
$76$	−4.00000	−0.458831
$77$	3.00000	0.341882
$78$	−2.00000	−0.226455
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−6.00000	−0.654654
$85$	0	0
$86$	8.00000	0.862662
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	6.00000	0.625543
$93$	−6.00000	−0.622171
$94$	−20.0000	−2.06284
$95$	0	0
$96$	8.00000	0.816497
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−4.00000	−0.404061
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−2.00000	−0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	2.00000	0.192450
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	12.0000	1.13389
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	−4.00000	−0.371391
$117$	1.00000	0.0924500
$118$	−16.0000	−1.47292
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−26.0000	−2.35393
$123$	−5.00000	−0.450835
$124$	−12.0000	−1.07763
$125$	0	0
$126$	6.00000	0.534522
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	−2.00000	−0.174078
$133$	6.00000	0.520266
$134$	24.0000	2.07328
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−6.00000	−0.510754
$139$	−1.00000	−0.0848189	−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000	−0.0848189	−0.0424094	+	0.999100i	$0.513503\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	10.0000	0.839181
$143$	−1.00000	−0.0836242
$144$	−4.00000	−0.333333
$145$	0	0
$146$	20.0000	1.65521
$147$	2.00000	0.164957
$148$	−22.0000	−1.80839
$149$	13.0000	1.06500	0.532501	−	0.846430i	$-0.321252\pi$
$149$	13.0000	1.06500	0.532501	+	0.846430i	$0.321252\pi$
$150$	0	0
$151$	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
$153$	1.00000	0.0808452
$154$	−6.00000	−0.483494
$155$	0	0
$156$	2.00000	0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	6.00000	0.477334
$159$	−11.0000	−0.872357
$160$	0	0
$161$	−9.00000	−0.709299
$162$	−2.00000	−0.157135
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−24.0000	−1.86276
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	4.00000	0.301511
$177$	8.00000	0.601317
$178$	30.0000	2.24860
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	6.00000	0.444750
$183$	13.0000	0.960988
$184$	0	0
$185$	0	0
$186$	12.0000	0.879883
$187$	−1.00000	−0.0731272
$188$	20.0000	1.45865
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−8.00000	−0.577350
$193$	13.0000	0.935760	0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000	0.935760	0.467880	+	0.883792i	$0.345018\pi$
$194$	34.0000	2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	2.00000	0.142134
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	6.00000	0.421117
$204$	2.00000	0.140028
$205$	0	0
$206$	−32.0000	−2.22955
$207$	3.00000	0.208514
$208$	−4.00000	−0.277350
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−22.0000	−1.51097
$213$	−5.00000	−0.342594
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	18.0000	1.22192
$218$	32.0000	2.16731
$219$	−10.0000	−0.675737
$220$	0	0
$221$	1.00000	0.0672673
$222$	22.0000	1.47654
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	28.0000	1.86253
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	−4.00000	−0.264906
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	27.0000	1.76883	0.884414	−	0.466702i	$-0.154558\pi$
$233$	27.0000	1.76883	0.884414	+	0.466702i	$0.154558\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	16.0000	1.04151
$237$	−3.00000	−0.194871
$238$	6.00000	0.388922
$239$	−13.0000	−0.840900	−0.420450	−	0.907316i	$-0.638128\pi$
$239$	−13.0000	−0.840900	−0.420450	+	0.907316i	$0.638128\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	20.0000	1.28565
$243$	1.00000	0.0641500
$244$	26.0000	1.66448
$245$	0	0
$246$	10.0000	0.637577
$247$	−2.00000	−0.127257
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−6.00000	−0.377964
$253$	−3.00000	−0.188608
$254$	20.0000	1.25491
$255$	0	0
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	33.0000	2.05052
$260$	0	0
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	−12.0000	−0.735767
$267$	−15.0000	−0.917985
$268$	−24.0000	−1.46603
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	−4.00000	−0.242536
$273$	−3.00000	−0.181568
$274$	36.0000	2.17484
$275$	0	0
$276$	6.00000	0.361158
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	2.00000	0.119952
$279$	−6.00000	−0.359211
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−20.0000	−1.19098
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	2.00000	0.118262
$287$	15.0000	0.885422
$288$	8.00000	0.471405
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−17.0000	−0.996558
$292$	−20.0000	−1.17041
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	−4.00000	−0.233285
$295$	0	0
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	−26.0000	−1.50614
$299$	3.00000	0.173494
$300$	0	0
$301$	12.0000	0.691669
$302$	−32.0000	−1.84139
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	−2.00000	−0.114332
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	6.00000	0.341882
$309$	16.0000	0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−20.0000	−1.12867
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	22.0000	1.23370
$319$	2.00000	0.111979
$320$	0	0
$321$	−9.00000	−0.502331
$322$	18.0000	1.00310
$323$	−2.00000	−0.111283
$324$	2.00000	0.111111
$325$	0	0
$326$	−26.0000	−1.44001
$327$	−16.0000	−0.884802
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	24.0000	1.31717
$333$	−11.0000	−0.602796
$334$	−24.0000	−1.31322
$335$	0	0
$336$	12.0000	0.654654
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	−2.00000	−0.108786
$339$	−14.0000	−0.760376
$340$	0	0
$341$	6.00000	0.324918
$342$	4.00000	0.216295
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	−12.0000	−0.645124
$347$	19.0000	1.01997	0.509987	−	0.860182i	$-0.329650\pi$
$347$	19.0000	1.01997	0.509987	+	0.860182i	$0.329650\pi$
$348$	−4.00000	−0.214423
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−8.00000	−0.426401
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−16.0000	−0.850390
$355$	0	0
$356$	−30.0000	−1.59000
$357$	−3.00000	−0.158777
$358$	−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
$361$	−15.0000	−0.789474
$362$	14.0000	0.735824
$363$	−10.0000	−0.524864
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−26.0000	−1.35904
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	−12.0000	−0.625543
$369$	−5.00000	−0.260290
$370$	0	0
$371$	33.0000	1.71327
$372$	−12.0000	−0.622171
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	6.00000	0.308607
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	−10.0000	−0.512316
$382$	16.0000	0.818631
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	0	0
$386$	−26.0000	−1.32337
$387$	−4.00000	−0.203331
$388$	−34.0000	−1.72609
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	0	0
$396$	−2.00000	−0.100504
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	−8.00000	−0.401004
$399$	6.00000	0.300376
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	24.0000	1.19701
$403$	−6.00000	−0.298881
$404$	0	0
$405$	0	0
$406$	−12.0000	−0.595550
$407$	11.0000	0.545250
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	32.0000	1.57653
$413$	−24.0000	−1.18096
$414$	−6.00000	−0.294884
$415$	0	0
$416$	8.00000	0.392232
$417$	−1.00000	−0.0489702
$418$	−4.00000	−0.195646
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	8.00000	0.389434
$423$	10.0000	0.486217
$424$	0	0
$425$	0	0
$426$	10.0000	0.484502
$427$	−39.0000	−1.88734
$428$	−18.0000	−0.870063
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−4.00000	−0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−36.0000	−1.72806
$435$	0	0
$436$	−32.0000	−1.53252
$437$	−6.00000	−0.287019
$438$	20.0000	0.955637
$439$	−17.0000	−0.811366	−0.405683	−	0.914014i	$-0.632966\pi$
$439$	−17.0000	−0.811366	−0.405683	+	0.914014i	$0.632966\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−2.00000	−0.0951303
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	−22.0000	−1.04407
$445$	0	0
$446$	16.0000	0.757622
$447$	13.0000	0.614879
$448$	24.0000	1.13389
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	−28.0000	−1.31701
$453$	16.0000	0.751746
$454$	44.0000	2.06502
$455$	0	0
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	−36.0000	−1.68217
$459$	1.00000	0.0466760
$460$	0	0
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	−6.00000	−0.279145
$463$	−27.0000	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$463$	−27.0000	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	−54.0000	−2.50150
$467$	23.0000	1.06431	0.532157	−	0.846646i	$-0.321382\pi$
$467$	23.0000	1.06431	0.532157	+	0.846646i	$0.321382\pi$
$468$	2.00000	0.0924500
$469$	36.0000	1.66233
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	4.00000	0.183920
$474$	6.00000	0.275589
$475$	0	0
$476$	−6.00000	−0.275010
$477$	−11.0000	−0.503655
$478$	26.0000	1.18921
$479$	9.00000	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$479$	9.00000	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	4.00000	0.182195
$483$	−9.00000	−0.409514
$484$	−20.0000	−0.909091
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	0	0
$489$	13.0000	0.587880
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−10.0000	−0.450835
$493$	−2.00000	−0.0900755
$494$	4.00000	0.179969
$495$	0	0
$496$	24.0000	1.07763
$497$	15.0000	0.672842
$498$	−24.0000	−1.07547
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	0	0
$506$	6.00000	0.266733
$507$	1.00000	0.0444116
$508$	−20.0000	−0.887357
$509$	−7.00000	−0.310270	−0.155135	−	0.987893i	$-0.549581\pi$
$509$	−7.00000	−0.310270	−0.155135	+	0.987893i	$0.549581\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	−32.0000	−1.41421
$513$	−2.00000	−0.0883022
$514$	−36.0000	−1.58789
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−10.0000	−0.439799
$518$	−66.0000	−2.89987
$519$	6.00000	0.263371
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	4.00000	0.175075
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	−6.00000	−0.261364
$528$	4.00000	0.174078
$529$	−14.0000	−0.608696
$530$	0	0
$531$	8.00000	0.347170
$532$	12.0000	0.520266
$533$	−5.00000	−0.216574
$534$	30.0000	1.29823
$535$	0	0
$536$	0	0
$537$	2.00000	0.0863064
$538$	8.00000	0.344904
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−44.0000	−1.88996
$543$	−7.00000	−0.300399
$544$	8.00000	0.342997
$545$	0	0
$546$	6.00000	0.256776
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	−36.0000	−1.53784
$549$	13.0000	0.554826
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	9.00000	0.382719
$554$	−36.0000	−1.52949
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	12.0000	0.508001
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	−60.0000	−2.53095
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	20.0000	0.842152
$565$	0	0
$566$	−24.0000	−1.00880
$567$	−3.00000	−0.125988
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	−2.00000	−0.0836242
$573$	−8.00000	−0.334205
$574$	−30.0000	−1.25218
$575$	0	0
$576$	−8.00000	−0.333333
$577$	19.0000	0.790980	0.395490	−	0.918470i	$-0.370575\pi$
$577$	19.0000	0.790980	0.395490	+	0.918470i	$0.370575\pi$
$578$	32.0000	1.33102
$579$	13.0000	0.540262
$580$	0	0
$581$	−36.0000	−1.49353
$582$	34.0000	1.40935
$583$	11.0000	0.455573
$584$	0	0
$585$	0	0
$586$	48.0000	1.98286
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	4.00000	0.164957
$589$	12.0000	0.494451
$590$	0	0
$591$	0	0
$592$	44.0000	1.80839
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	26.0000	1.06500
$597$	4.00000	0.163709
$598$	−6.00000	−0.245358
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	−24.0000	−0.978167
$603$	−12.0000	−0.488678
$604$	32.0000	1.30206
$605$	0	0
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	−16.0000	−0.648886
$609$	6.00000	0.243132
$610$	0	0
$611$	10.0000	0.404557
$612$	2.00000	0.0808452
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	−32.0000	−1.28723
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	−48.0000	−1.92462
$623$	45.0000	1.80289
$624$	−4.00000	−0.160128
$625$	0	0
$626$	−20.0000	−0.799361
$627$	2.00000	0.0798723
$628$	20.0000	0.798087
$629$	−11.0000	−0.438599
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	56.0000	2.22404
$635$	0	0
$636$	−22.0000	−0.872357
$637$	2.00000	0.0792429
$638$	−4.00000	−0.158362
$639$	−5.00000	−0.197797
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	18.0000	0.710403
$643$	−15.0000	−0.591542	−0.295771	−	0.955259i	$-0.595577\pi$
$643$	−15.0000	−0.591542	−0.295771	+	0.955259i	$0.595577\pi$
$644$	−18.0000	−0.709299
$645$	0	0
$646$	4.00000	0.157378
$647$	−47.0000	−1.84776	−0.923880	−	0.382682i	$-0.875001\pi$
$647$	−47.0000	−1.84776	−0.923880	+	0.382682i	$0.875001\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	18.0000	0.705476
$652$	26.0000	1.01824
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	32.0000	1.25130
$655$	0	0
$656$	20.0000	0.780869
$657$	−10.0000	−0.390137
$658$	60.0000	2.33904
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	1.00000	0.0388368
$664$	0	0
$665$	0	0
$666$	22.0000	0.852483
$667$	−6.00000	−0.232321
$668$	24.0000	0.928588
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−13.0000	−0.501859
$672$	−24.0000	−0.925820
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	2.00000	0.0769231
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	28.0000	1.07533
$679$	51.0000	1.95720
$680$	0	0
$681$	−22.0000	−0.843042
$682$	−12.0000	−0.459504
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−30.0000	−1.14541
$687$	18.0000	0.686743
$688$	16.0000	0.609994
$689$	−11.0000	−0.419067
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	12.0000	0.456172
$693$	3.00000	0.113961
$694$	−38.0000	−1.44246
$695$	0	0
$696$	0	0
$697$	−5.00000	−0.189389
$698$	16.0000	0.605609
$699$	27.0000	1.02123
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	−2.00000	−0.0754851
$703$	22.0000	0.829746
$704$	8.00000	0.301511
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	16.0000	0.601317
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	−18.0000	−0.674105
$714$	6.00000	0.224544
$715$	0	0
$716$	4.00000	0.149487
$717$	−13.0000	−0.485494
$718$	−48.0000	−1.79134
$719$	−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
$721$	−48.0000	−1.78761
$722$	30.0000	1.11648
$723$	−2.00000	−0.0743808
$724$	−14.0000	−0.520306
$725$	0	0
$726$	20.0000	0.742270
$727$	−38.0000	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	26.0000	0.960988
$733$	49.0000	1.80986	0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000	1.80986	0.904928	+	0.425564i	$0.139924\pi$
$734$	72.0000	2.65757
$735$	0	0
$736$	24.0000	0.884652
$737$	12.0000	0.442026
$738$	10.0000	0.368105
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	−66.0000	−2.42294
$743$	34.0000	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$743$	34.0000	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$744$	0	0
$745$	0	0
$746$	8.00000	0.292901
$747$	12.0000	0.439057
$748$	−2.00000	−0.0731272
$749$	27.0000	0.986559
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	−40.0000	−1.45865
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	−6.00000	−0.218218
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	28.0000	1.01701
$759$	−3.00000	−0.108893
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	20.0000	0.724524
$763$	48.0000	1.73772
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−60.0000	−2.16789
$767$	8.00000	0.288863
$768$	16.0000	0.577350
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	26.0000	0.935760
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	0	0
$777$	33.0000	1.18387
$778$	0	0
$779$	10.0000	0.358287
$780$	0	0
$781$	5.00000	0.178914
$782$	−6.00000	−0.214560
$783$	−2.00000	−0.0714742
$784$	−8.00000	−0.285714
$785$	0	0
$786$	12.0000	0.428026
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	0	0
$789$	8.00000	0.284808
$790$	0	0
$791$	42.0000	1.49335
$792$	0	0
$793$	13.0000	0.461644
$794$	−58.0000	−2.05834
$795$	0	0
$796$	8.00000	0.283552
$797$	47.0000	1.66483	0.832413	−	0.554156i	$-0.186959\pi$
$797$	47.0000	1.66483	0.832413	+	0.554156i	$0.186959\pi$
$798$	−12.0000	−0.424795
$799$	10.0000	0.353775
$800$	0	0
$801$	−15.0000	−0.529999
$802$	−60.0000	−2.11867
$803$	10.0000	0.352892
$804$	−24.0000	−0.846415
$805$	0	0
$806$	12.0000	0.422682
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	12.0000	0.421117
$813$	22.0000	0.771574
$814$	−22.0000	−0.771100
$815$	0	0
$816$	−4.00000	−0.140028
$817$	8.00000	0.279885
$818$	−4.00000	−0.139857
$819$	−3.00000	−0.104828
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	36.0000	1.25564
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	0	0
$826$	48.0000	1.67013
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	6.00000	0.208514
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	−8.00000	−0.277350
$833$	2.00000	0.0692959
$834$	2.00000	0.0692543
$835$	0	0
$836$	4.00000	0.138343
$837$	−6.00000	−0.207390
$838$	52.0000	1.79631
$839$	−5.00000	−0.172619	−0.0863096	−	0.996268i	$-0.527507\pi$
$839$	−5.00000	−0.172619	−0.0863096	+	0.996268i	$0.527507\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−8.00000	−0.275698
$843$	30.0000	1.03325
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−20.0000	−0.687614
$847$	30.0000	1.03081
$848$	44.0000	1.51097
$849$	12.0000	0.411839
$850$	0	0
$851$	−33.0000	−1.13123
$852$	−10.0000	−0.342594
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	78.0000	2.66911
$855$	0	0
$856$	0	0
$857$	−29.0000	−0.990621	−0.495311	−	0.868716i	$-0.664946\pi$
$857$	−29.0000	−0.990621	−0.495311	+	0.868716i	$0.664946\pi$
$858$	2.00000	0.0682789
$859$	−29.0000	−0.989467	−0.494734	−	0.869045i	$-0.664734\pi$
$859$	−29.0000	−0.989467	−0.494734	+	0.869045i	$0.664734\pi$
$860$	0	0
$861$	15.0000	0.511199
$862$	48.0000	1.63489
$863$	−34.0000	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	8.00000	0.271851
$867$	−16.0000	−0.543388
$868$	36.0000	1.22192
$869$	3.00000	0.101768
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−17.0000	−0.575363
$874$	12.0000	0.405906
$875$	0	0
$876$	−20.0000	−0.675737
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	34.0000	1.14744
$879$	−24.0000	−0.809500
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−4.00000	−0.134687
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	18.0000	0.604722
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−16.0000	−0.535720
$893$	−20.0000	−0.669274
$894$	−26.0000	−0.869570
$895$	0	0
$896$	0	0
$897$	3.00000	0.100167
$898$	26.0000	0.867631
$899$	12.0000	0.400222
$900$	0	0
$901$	−11.0000	−0.366463
$902$	−10.0000	−0.332964
$903$	12.0000	0.399335
$904$	0	0
$905$	0	0
$906$	−32.0000	−1.06313
$907$	−6.00000	−0.199227	−0.0996134	−	0.995026i	$-0.531761\pi$
$907$	−6.00000	−0.199227	−0.0996134	+	0.995026i	$0.531761\pi$
$908$	−44.0000	−1.46019
$909$	0	0
$910$	0	0
$911$	44.0000	1.45779	0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000	1.45779	0.728893	+	0.684628i	$0.240035\pi$
$912$	8.00000	0.264906
$913$	−12.0000	−0.397142
$914$	−22.0000	−0.727695
$915$	0	0
$916$	36.0000	1.18947
$917$	18.0000	0.594412
$918$	−2.00000	−0.0660098
$919$	37.0000	1.22052	0.610259	−	0.792202i	$-0.291065\pi$
$919$	37.0000	1.22052	0.610259	+	0.792202i	$0.291065\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	−30.0000	−0.987997
$923$	−5.00000	−0.164577
$924$	6.00000	0.197386
$925$	0	0
$926$	54.0000	1.77455
$927$	16.0000	0.525509
$928$	−16.0000	−0.525226
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	54.0000	1.76883
$933$	24.0000	0.785725
$934$	−46.0000	−1.50517
$935$	0	0
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	−72.0000	−2.35088
$939$	10.0000	0.326338
$940$	0	0
$941$	37.0000	1.20617	0.603083	−	0.797679i	$-0.293939\pi$
$941$	37.0000	1.20617	0.603083	+	0.797679i	$0.293939\pi$
$942$	−20.0000	−0.651635
$943$	−15.0000	−0.488467
$944$	−32.0000	−1.04151
$945$	0	0
$946$	−8.00000	−0.260102
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−6.00000	−0.194871
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	1.00000	0.0323932	0.0161966	−	0.999869i	$-0.494844\pi$
$953$	1.00000	0.0323932	0.0161966	+	0.999869i	$0.494844\pi$
$954$	22.0000	0.712276
$955$	0	0
$956$	−26.0000	−0.840900
$957$	2.00000	0.0646508
$958$	−18.0000	−0.581554
$959$	54.0000	1.74375
$960$	0	0
$961$	5.00000	0.161290
$962$	22.0000	0.709308
$963$	−9.00000	−0.290021
$964$	−4.00000	−0.128831
$965$	0	0
$966$	18.0000	0.579141
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	−2.00000	−0.0642493
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	2.00000	0.0641500
$973$	3.00000	0.0961756
$974$	−14.0000	−0.448589
$975$	0	0
$976$	−52.0000	−1.66448
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	−26.0000	−0.831388
$979$	15.0000	0.479402
$980$	0	0
$981$	−16.0000	−0.510841
$982$	−56.0000	−1.78703
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	0	0
$986$	4.00000	0.127386
$987$	−30.0000	−0.954911
$988$	−4.00000	−0.127257
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	−48.0000	−1.52400
$993$	0	0
$994$	−30.0000	−0.951542
$995$	0	0
$996$	24.0000	0.760469
$997$	36.0000	1.14013	0.570066	−	0.821599i	$-0.306918\pi$
$997$	36.0000	1.14013	0.570066	+	0.821599i	$0.306918\pi$
$998$	28.0000	0.886325
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.c.1.1		1
3.2	odd	2		2925.2.a.q.1.1		1
5.2	odd	4		975.2.c.a.274.1		2
5.3	odd	4		975.2.c.a.274.2		2
5.4	even	2		195.2.a.b.1.1	✓	1
15.2	even	4		2925.2.c.c.2224.2		2
15.8	even	4		2925.2.c.c.2224.1		2
15.14	odd	2		585.2.a.b.1.1		1
20.19	odd	2		3120.2.a.u.1.1		1
35.34	odd	2		9555.2.a.v.1.1		1
60.59	even	2		9360.2.a.d.1.1		1
65.64	even	2		2535.2.a.a.1.1		1
195.194	odd	2		7605.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.b.1.1	✓	1	5.4	even	2
585.2.a.b.1.1		1	15.14	odd	2
975.2.a.c.1.1		1	1.1	even	1	trivial
975.2.c.a.274.1		2	5.2	odd	4
975.2.c.a.274.2		2	5.3	odd	4
2535.2.a.a.1.1		1	65.64	even	2
2925.2.a.q.1.1		1	3.2	odd	2
2925.2.c.c.2224.1		2	15.8	even	4
2925.2.c.c.2224.2		2	15.2	even	4
3120.2.a.u.1.1		1	20.19	odd	2
7605.2.a.u.1.1		1	195.194	odd	2
9360.2.a.d.1.1		1	60.59	even	2
9555.2.a.v.1.1		1	35.34	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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