LMFDB - Embedded newform 197.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [197,2,Mod(1,197)] 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("197.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(197, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	197.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.57305291982$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	197.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{7} -3.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -8.00000 q^{17} +6.00000 q^{18} -3.00000 q^{19} -8.00000 q^{22} -3.00000 q^{23} -5.00000 q^{25} +4.00000 q^{26} -6.00000 q^{28} +7.00000 q^{29} -10.0000 q^{31} +8.00000 q^{32} +16.0000 q^{34} -6.00000 q^{36} +7.00000 q^{37} +6.00000 q^{38} +9.00000 q^{41} +1.00000 q^{43} +8.00000 q^{44} +6.00000 q^{46} -11.0000 q^{47} +2.00000 q^{49} +10.0000 q^{50} -4.00000 q^{52} +10.0000 q^{53} -14.0000 q^{58} +5.00000 q^{61} +20.0000 q^{62} +9.00000 q^{63} -8.00000 q^{64} -10.0000 q^{67} -16.0000 q^{68} +8.00000 q^{71} +6.00000 q^{73} -14.0000 q^{74} -6.00000 q^{76} -12.0000 q^{77} +2.00000 q^{79} +9.00000 q^{81} -18.0000 q^{82} -7.00000 q^{83} -2.00000 q^{86} -8.00000 q^{89} +6.00000 q^{91} -6.00000 q^{92} +22.0000 q^{94} -2.00000 q^{97} -4.00000 q^{98} -12.0000 q^{99} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	2.00000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−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
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	6.00000	1.41421
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	−8.00000	−1.70561
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	0	0
$28$	−6.00000	−1.13389
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	16.0000	2.74398
$35$	0	0
$36$	−6.00000	−1.00000
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	8.00000	1.20605
$45$	0	0
$46$	6.00000	0.884652
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	10.0000	1.41421
$51$	0	0
$52$	−4.00000	−0.554700
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−14.0000	−1.83829
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	20.0000	2.54000
$63$	9.00000	1.13389
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−16.0000	−1.94029
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−14.0000	−1.62747
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−12.0000	−1.36753
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−18.0000	−1.98777
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	−6.00000	−0.625543
$93$	0	0
$94$	22.0000	2.26913
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−4.00000	−0.404061
$99$	−12.0000	−1.20605
$100$	−10.0000	−1.00000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	−20.0000	−1.94257
$107$	−9.00000	−0.870063	−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000	−0.870063	−0.435031	+	0.900415i	$0.643263\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	0	0
$112$	12.0000	1.13389
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	0	0
$116$	14.0000	1.29987
$117$	6.00000	0.554700
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−20.0000	−1.79605
$125$	0	0
$126$	−18.0000	−1.60357
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	9.00000	0.780399
$134$	20.0000	1.72774
$135$	0	0
$136$	0	0
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−8.00000	−0.668994
$144$	12.0000	1.00000
$145$	0	0
$146$	−12.0000	−0.993127
$147$	0	0
$148$	14.0000	1.15079
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	24.0000	1.93398
$155$	0	0
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	0	0
$161$	9.00000	0.709299
$162$	−18.0000	−1.41421
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	18.0000	1.40556
$165$	0	0
$166$	14.0000	1.08661
$167$	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
$169$	−9.00000	−0.692308
$170$	0	0
$171$	9.00000	0.688247
$172$	2.00000	0.152499
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	15.0000	1.13389
$176$	−16.0000	−1.20605
$177$	0	0
$178$	16.0000	1.19925
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−12.0000	−0.889499
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−32.0000	−2.34007
$188$	−22.0000	−1.60451
$189$	0	0
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	4.00000	0.285714
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	24.0000	1.70561
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	30.0000	2.11079
$203$	−21.0000	−1.47391
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	9.00000	0.625543
$208$	8.00000	0.554700
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	20.0000	1.37361
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	30.0000	2.03653
$218$	−26.0000	−1.76094
$219$	0	0
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	−24.0000	−1.60357
$225$	15.0000	1.00000
$226$	8.00000	0.532152
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−48.0000	−3.11138
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	18.0000	1.13389
$253$	−12.0000	−0.754434
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	−1.00000	−0.0623783	−0.0311891	−	0.999514i	$-0.509929\pi$
$257$	−1.00000	−0.0623783	−0.0311891	+	0.999514i	$0.509929\pi$
$258$	0	0
$259$	−21.0000	−1.30488
$260$	0	0
$261$	−21.0000	−1.29987
$262$	−36.0000	−2.22409
$263$	26.0000	1.60323	0.801614	−	0.597841i	$-0.203975\pi$
$263$	26.0000	1.60323	0.801614	+	0.597841i	$0.203975\pi$
$264$	0	0
$265$	0	0
$266$	−18.0000	−1.10365
$267$	0	0
$268$	−20.0000	−1.22169
$269$	28.0000	1.70719	0.853595	−	0.520937i	$-0.174417\pi$
$269$	28.0000	1.70719	0.853595	+	0.520937i	$0.174417\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	32.0000	1.94029
$273$	0	0
$274$	42.0000	2.53731
$275$	−20.0000	−1.20605
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	20.0000	1.19952
$279$	30.0000	1.79605
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	16.0000	0.946100
$287$	−27.0000	−1.59376
$288$	−24.0000	−1.41421
$289$	47.0000	2.76471
$290$	0	0
$291$	0	0
$292$	12.0000	0.702247
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−24.0000	−1.39028
$299$	6.00000	0.346989
$300$	0	0
$301$	−3.00000	−0.172917
$302$	20.0000	1.15087
$303$	0	0
$304$	12.0000	0.688247
$305$	0	0
$306$	−48.0000	−2.74398
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−24.0000	−1.36753
$309$	0	0
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	17.0000	0.960897	0.480448	−	0.877023i	$-0.340474\pi$
$313$	17.0000	0.960897	0.480448	+	0.877023i	$0.340474\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	4.00000	0.225018
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	28.0000	1.56770
$320$	0	0
$321$	0	0
$322$	−18.0000	−1.00310
$323$	24.0000	1.33540
$324$	18.0000	1.00000
$325$	10.0000	0.554700
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	33.0000	1.81935
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	−14.0000	−0.768350
$333$	−21.0000	−1.15079
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	18.0000	0.979071
$339$	0	0
$340$	0	0
$341$	−40.0000	−2.16612
$342$	−18.0000	−0.973329
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	4.00000	0.215041
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	−30.0000	−1.60357
$351$	0	0
$352$	32.0000	1.70561
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	0	0
$356$	−16.0000	−0.847998
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−14.0000	−0.738892	−0.369446	−	0.929252i	$-0.620452\pi$
$359$	−14.0000	−0.738892	−0.369446	+	0.929252i	$0.620452\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−4.00000	−0.210235
$363$	0	0
$364$	12.0000	0.628971
$365$	0	0
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	12.0000	0.625543
$369$	−27.0000	−1.40556
$370$	0	0
$371$	−30.0000	−1.55752
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	64.0000	3.30936
$375$	0	0
$376$	0	0
$377$	−14.0000	−0.721037
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	26.0000	1.33028
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	−3.00000	−0.152499
$388$	−4.00000	−0.203069
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	−24.0000	−1.20605
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	20.0000	1.00000
$401$	−33.0000	−1.64794	−0.823971	−	0.566632i	$-0.808246\pi$
$401$	−33.0000	−1.64794	−0.823971	+	0.566632i	$0.808246\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	−30.0000	−1.49256
$405$	0	0
$406$	42.0000	2.08443
$407$	28.0000	1.38791
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−18.0000	−0.884652
$415$	0	0
$416$	−16.0000	−0.784465
$417$	0	0
$418$	24.0000	1.17388
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	40.0000	1.94717
$423$	33.0000	1.60451
$424$	0	0
$425$	40.0000	1.94029
$426$	0	0
$427$	−15.0000	−0.725901
$428$	−18.0000	−0.870063
$429$	0	0
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	−60.0000	−2.88009
$435$	0	0
$436$	26.0000	1.24517
$437$	9.00000	0.430528
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−32.0000	−1.52208
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	0	0
$445$	0	0
$446$	18.0000	0.852325
$447$	0	0
$448$	24.0000	1.13389
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	−30.0000	−1.41421
$451$	36.0000	1.69517
$452$	−8.00000	−0.376288
$453$	0	0
$454$	−16.0000	−0.750917
$455$	0	0
$456$	0	0
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	48.0000	2.24289
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−28.0000	−1.29987
$465$	0	0
$466$	6.00000	0.277945
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	12.0000	0.554700
$469$	30.0000	1.38527
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	15.0000	0.688247
$476$	48.0000	2.20008
$477$	−30.0000	−1.37361
$478$	42.0000	1.92104
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	24.0000	1.09317
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	0	0
$487$	−5.00000	−0.226572	−0.113286	−	0.993562i	$-0.536138\pi$
$487$	−5.00000	−0.226572	−0.113286	+	0.993562i	$0.536138\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	0	0
$493$	−56.0000	−2.52211
$494$	−12.0000	−0.539906
$495$	0	0
$496$	40.0000	1.79605
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	0	0
$501$	0	0
$502$	−10.0000	−0.446322
$503$	−7.00000	−0.312115	−0.156057	−	0.987748i	$-0.549878\pi$
$503$	−7.00000	−0.312115	−0.156057	+	0.987748i	$0.549878\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	0	0
$508$	16.0000	0.709885
$509$	4.00000	0.177297	0.0886484	−	0.996063i	$-0.471745\pi$
$509$	4.00000	0.177297	0.0886484	+	0.996063i	$0.471745\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	−32.0000	−1.41421
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	−44.0000	−1.93512
$518$	42.0000	1.84537
$519$	0	0
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	42.0000	1.83829
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	−52.0000	−2.26731
$527$	80.0000	3.48485
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	18.0000	0.780399
$533$	−18.0000	−0.779667
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−56.0000	−2.41433
$539$	8.00000	0.344584
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	40.0000	1.71815
$543$	0	0
$544$	−64.0000	−2.74398
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−42.0000	−1.79415
$549$	−15.0000	−0.640184
$550$	40.0000	1.70561
$551$	−21.0000	−0.894630
$552$	0	0
$553$	−6.00000	−0.255146
$554$	−16.0000	−0.679775
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	−60.0000	−2.54000
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	0	0
$562$	−20.0000	−0.843649
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	−27.0000	−1.13389
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−14.0000	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$572$	−16.0000	−0.668994
$573$	0	0
$574$	54.0000	2.25392
$575$	15.0000	0.625543
$576$	24.0000	1.00000
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−94.0000	−3.90988
$579$	0	0
$580$	0	0
$581$	21.0000	0.871227
$582$	0	0
$583$	40.0000	1.65663
$584$	0	0
$585$	0	0
$586$	18.0000	0.743573
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	0	0
$589$	30.0000	1.23613
$590$	0	0
$591$	0	0
$592$	−28.0000	−1.15079
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	0	0
$596$	24.0000	0.983078
$597$	0	0
$598$	−12.0000	−0.490716
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	6.00000	0.244542
$603$	30.0000	1.22169
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	25.0000	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$607$	25.0000	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$608$	−24.0000	−0.973329
$609$	0	0
$610$	0	0
$611$	22.0000	0.890025
$612$	48.0000	1.94029
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	54.0000	2.16520
$623$	24.0000	0.961540
$624$	0	0
$625$	25.0000	1.00000
$626$	−34.0000	−1.35891
$627$	0	0
$628$	22.0000	0.877896
$629$	−56.0000	−2.23287
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−56.0000	−2.21706
$639$	−24.0000	−0.949425
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	−48.0000	−1.88853
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	−20.0000	−0.784465
$651$	0	0
$652$	−8.00000	−0.313304
$653$	39.0000	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$653$	39.0000	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$654$	0	0
$655$	0	0
$656$	−36.0000	−1.40556
$657$	−18.0000	−0.702247
$658$	−66.0000	−2.57295
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	22.0000	0.855054
$663$	0	0
$664$	0	0
$665$	0	0
$666$	42.0000	1.62747
$667$	−21.0000	−0.813123
$668$	24.0000	0.928588
$669$	0	0
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	28.0000	1.07852
$675$	0	0
$676$	−18.0000	−0.692308
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	0	0
$681$	0	0
$682$	80.0000	3.06336
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	18.0000	0.688247
$685$	0	0
$686$	−30.0000	−1.14541
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−20.0000	−0.761939
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	−4.00000	−0.152057
$693$	36.0000	1.36753
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	−72.0000	−2.72719
$698$	40.0000	1.51402
$699$	0	0
$700$	30.0000	1.13389
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−21.0000	−0.792030
$704$	−32.0000	−1.20605
$705$	0	0
$706$	−18.0000	−0.677439
$707$	45.0000	1.69240
$708$	0	0
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	30.0000	1.12351
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	0	0
$718$	28.0000	1.04495
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	20.0000	0.744323
$723$	0	0
$724$	4.00000	0.148659
$725$	−35.0000	−1.29987
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	42.0000	1.55131	0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000	1.55131	0.775653	+	0.631160i	$0.217421\pi$
$734$	20.0000	0.738213
$735$	0	0
$736$	−24.0000	−0.884652
$737$	−40.0000	−1.47342
$738$	54.0000	1.98777
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	60.0000	2.20267
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	0	0
$746$	−12.0000	−0.439351
$747$	21.0000	0.768350
$748$	−64.0000	−2.34007
$749$	27.0000	0.986559
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	44.0000	1.60451
$753$	0	0
$754$	28.0000	1.01970
$755$	0	0
$756$	0	0
$757$	48.0000	1.74459	0.872295	−	0.488980i	$-0.162631\pi$
$757$	48.0000	1.74459	0.872295	+	0.488980i	$0.162631\pi$
$758$	56.0000	2.03401
$759$	0	0
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	0	0
$763$	−39.0000	−1.41189
$764$	−26.0000	−0.940647
$765$	0	0
$766$	4.00000	0.144526
$767$	0	0
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	−20.0000	−0.719816
$773$	17.0000	0.611448	0.305724	−	0.952120i	$-0.401102\pi$
$773$	17.0000	0.611448	0.305724	+	0.952120i	$0.401102\pi$
$774$	6.00000	0.215666
$775$	50.0000	1.79605
$776$	0	0
$777$	0	0
$778$	8.00000	0.286814
$779$	−27.0000	−0.967375
$780$	0	0
$781$	32.0000	1.14505
$782$	−48.0000	−1.71648
$783$	0	0
$784$	−8.00000	−0.285714
$785$	0	0
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−32.0000	−1.13564
$795$	0	0
$796$	4.00000	0.141776
$797$	5.00000	0.177109	0.0885545	−	0.996071i	$-0.471775\pi$
$797$	5.00000	0.177109	0.0885545	+	0.996071i	$0.471775\pi$
$798$	0	0
$799$	88.0000	3.11322
$800$	−40.0000	−1.41421
$801$	24.0000	0.847998
$802$	66.0000	2.33054
$803$	24.0000	0.846942
$804$	0	0
$805$	0	0
$806$	−40.0000	−1.40894
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	−42.0000	−1.47391
$813$	0	0
$814$	−56.0000	−1.96280
$815$	0	0
$816$	0	0
$817$	−3.00000	−0.104957
$818$	50.0000	1.74821
$819$	−18.0000	−0.628971
$820$	0	0
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	18.0000	0.625543
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	0	0
$832$	16.0000	0.554700
$833$	−16.0000	−0.554367
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	−18.0000	−0.621800
$839$	−13.0000	−0.448810	−0.224405	−	0.974496i	$-0.572044\pi$
$839$	−13.0000	−0.448810	−0.224405	+	0.974496i	$0.572044\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	20.0000	0.689246
$843$	0	0
$844$	−40.0000	−1.37686
$845$	0	0
$846$	−66.0000	−2.26913
$847$	−15.0000	−0.515406
$848$	−40.0000	−1.37361
$849$	0	0
$850$	−80.0000	−2.74398
$851$	−21.0000	−0.719871
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	30.0000	1.02658
$855$	0	0
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	−66.0000	−2.24797
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	60.0000	2.03653
$869$	8.00000	0.271381
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	6.00000	0.203069
$874$	−18.0000	−0.608859
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−40.0000	−1.34993
$879$	0	0
$880$	0	0
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	12.0000	0.404061
$883$	46.0000	1.54802	0.774012	−	0.633171i	$-0.218247\pi$
$883$	46.0000	1.54802	0.774012	+	0.633171i	$0.218247\pi$
$884$	32.0000	1.07628
$885$	0	0
$886$	78.0000	2.62046
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	36.0000	1.20605
$892$	−18.0000	−0.602685
$893$	33.0000	1.10430
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	−70.0000	−2.33463
$900$	30.0000	1.00000
$901$	−80.0000	−2.66519
$902$	−72.0000	−2.39734
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34.0000	1.12895	0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000	1.12895	0.564476	+	0.825450i	$0.309078\pi$
$908$	16.0000	0.530979
$909$	45.0000	1.49256
$910$	0	0
$911$	34.0000	1.12647	0.563235	−	0.826297i	$-0.309557\pi$
$911$	34.0000	1.12647	0.563235	+	0.826297i	$0.309557\pi$
$912$	0	0
$913$	−28.0000	−0.926665
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−48.0000	−1.58596
$917$	−54.0000	−1.78324
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	−36.0000	−1.18560
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−35.0000	−1.15079
$926$	−12.0000	−0.394344
$927$	12.0000	0.394132
$928$	56.0000	1.83829
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−6.00000	−0.196537
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	−60.0000	−1.95907
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	0	0
$945$	0	0
$946$	−8.00000	−0.260102
$947$	58.0000	1.88475	0.942373	−	0.334563i	$-0.108589\pi$
$947$	58.0000	1.88475	0.942373	+	0.334563i	$0.108589\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	−30.0000	−0.973329
$951$	0	0
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	60.0000	1.94257
$955$	0	0
$956$	−42.0000	−1.35838
$957$	0	0
$958$	8.00000	0.258468
$959$	63.0000	2.03438
$960$	0	0
$961$	69.0000	2.22581
$962$	28.0000	0.902756
$963$	27.0000	0.870063
$964$	−24.0000	−0.772988
$965$	0	0
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	30.0000	0.961756
$974$	10.0000	0.320421
$975$	0	0
$976$	−20.0000	−0.640184
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	−39.0000	−1.24517
$982$	−10.0000	−0.319113
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	112.000	3.56681
$987$	0	0
$988$	12.0000	0.381771
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−15.0000	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$991$	−15.0000	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$992$	−80.0000	−2.54000
$993$	0	0
$994$	48.0000	1.52247
$995$	0	0
$996$	0	0
$997$	−6.00000	−0.190022	−0.0950110	−	0.995476i	$-0.530289\pi$
$997$	−6.00000	−0.190022	−0.0950110	+	0.995476i	$0.530289\pi$
$998$	88.0000	2.78559
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	197.2.a.a.1.1	✓	1
3.2	odd	2		1773.2.a.b.1.1		1
4.3	odd	2		3152.2.a.a.1.1		1
5.4	even	2		4925.2.a.d.1.1		1
7.6	odd	2		9653.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
197.2.a.a.1.1	✓	1	1.1	even	1	trivial
1773.2.a.b.1.1		1	3.2	odd	2
3152.2.a.a.1.1		1	4.3	odd	2
4925.2.a.d.1.1		1	5.4	even	2
9653.2.a.a.1.1		1	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	197.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more