LMFDB - Embedded newform 3525.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} -2.00000 q^{4} -2.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} -2.00000 q^{21} +7.00000 q^{23} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} -6.00000 q^{31} +2.00000 q^{33} -2.00000 q^{36} +4.00000 q^{37} -1.00000 q^{39} -7.00000 q^{43} -4.00000 q^{44} +1.00000 q^{47} +4.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} -6.00000 q^{57} +5.00000 q^{59} +7.00000 q^{61} -2.00000 q^{63} -8.00000 q^{64} -4.00000 q^{67} -4.00000 q^{68} +7.00000 q^{69} -11.0000 q^{71} -7.00000 q^{73} +12.0000 q^{76} -4.00000 q^{77} -13.0000 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{84} -6.00000 q^{87} -7.00000 q^{89} +2.00000 q^{91} -14.0000 q^{92} -6.00000 q^{93} -10.0000 q^{97} +2.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	4.00000	0.577350
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	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
$57$	−6.00000	−0.794719
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.00000	−0.485071
$69$	7.00000	0.842701
$70$	0	0
$71$	−11.0000	−1.30546	−0.652730	−	0.757591i	$-0.726376\pi$
$71$	−11.0000	−1.30546	−0.652730	+	0.757591i	$0.726376\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−14.0000	−1.45960
$93$	−6.00000	−0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−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
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	−8.00000	−0.755929
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	0	0
$116$	12.0000	1.11417
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	12.0000	1.07763
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	−4.00000	−0.348155
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	0	0
$143$	−2.00000	−0.167248
$144$	4.00000	0.333333
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	−8.00000	−0.657596
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	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
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−14.0000	−1.10335
$162$	0	0
$163$	7.00000	0.548282	0.274141	−	0.961689i	$-0.411606\pi$
$163$	7.00000	0.548282	0.274141	+	0.961689i	$0.411606\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−6.00000	−0.458831
$172$	14.0000	1.06749
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	0	0
$176$	8.00000	0.603023
$177$	5.00000	0.375823
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	7.00000	0.517455
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	−2.00000	−0.145865
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	−8.00000	−0.577350
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	0	0
$196$	6.00000	0.428571
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	12.0000	0.842235
$204$	−4.00000	−0.280056
$205$	0	0
$206$	0	0
$207$	7.00000	0.486534
$208$	−4.00000	−0.277350
$209$	−12.0000	−0.830057
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−20.0000	−1.37361
$213$	−11.0000	−0.753708
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	−7.00000	−0.473016
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	12.0000	0.794719
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	−13.0000	−0.844441
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	4.00000	0.251976
$253$	14.0000	0.880172
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−5.00000	−0.311891	−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000	−0.311891	−0.155946	+	0.987766i	$0.549842\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.00000	−0.428393
$268$	8.00000	0.488678
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	8.00000	0.485071
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	−14.0000	−0.842701
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	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
$283$	−8.00000	−0.475551	−0.237775	−	0.971320i	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	22.0000	1.30546
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000	0.819288
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−7.00000	−0.404820
$300$	0	0
$301$	14.0000	0.806947
$302$	0	0
$303$	−3.00000	−0.172345
$304$	−24.0000	−1.37649
$305$	0	0
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	8.00000	0.455842
$309$	−6.00000	−0.341328
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	0	0
$315$	0	0
$316$	26.0000	1.46261
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−9.00000	−0.502331
$322$	0	0
$323$	−12.0000	−0.667698
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	−16.0000	−0.878114
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	−8.00000	−0.436436
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	9.00000	0.488813
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	12.0000	0.643268
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−32.0000	−1.70319	−0.851594	−	0.524202i	$-0.824364\pi$
$353$	−32.0000	−1.70319	−0.851594	+	0.524202i	$0.824364\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	−4.00000	−0.211702
$358$	0	0
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−7.00000	−0.367405
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	28.0000	1.45960
$369$	0	0
$370$	0	0
$371$	−20.0000	−1.03835
$372$	12.0000	0.622171
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.00000	−0.355830
$388$	20.0000	1.01535
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	14.0000	0.708010
$392$	0	0
$393$	−21.0000	−1.05931
$394$	0	0
$395$	0	0
$396$	−4.00000	−0.201008
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	11.0000	0.549314	0.274657	−	0.961542i	$-0.411436\pi$
$401$	11.0000	0.549314	0.274657	+	0.961542i	$0.411436\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	12.0000	0.591198
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.0000	0.489702
$418$	0	0
$419$	−22.0000	−1.07477	−0.537385	−	0.843337i	$-0.680588\pi$
$419$	−22.0000	−1.07477	−0.537385	+	0.843337i	$0.680588\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	1.00000	0.0486217
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.0000	−0.677507
$428$	18.0000	0.870063
$429$	−2.00000	−0.0965609
$430$	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
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−42.0000	−2.00913
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	0	0
$447$	−15.0000	−0.709476
$448$	16.0000	0.755929
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−10.0000	−0.469841
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.0000	1.12267	0.561336	−	0.827588i	$-0.310287\pi$
$457$	24.0000	1.12267	0.561336	+	0.827588i	$0.310287\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	15.0000	0.697109	0.348555	−	0.937288i	$-0.386673\pi$
$463$	15.0000	0.697109	0.348555	+	0.937288i	$0.386673\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	2.00000	0.0924500
$469$	8.00000	0.369406
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	−14.0000	−0.643721
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	10.0000	0.457869
$478$	0	0
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−14.0000	−0.637022
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	42.0000	1.90320	0.951601	−	0.307337i	$-0.0994378\pi$
$487$	42.0000	1.90320	0.951601	+	0.307337i	$0.0994378\pi$
$488$	0	0
$489$	7.00000	0.316551
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	−24.0000	−1.07763
$497$	22.0000	0.986835
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	9.00000	0.402090
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	16.0000	0.709885
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	0	0
$516$	14.0000	0.616316
$517$	2.00000	0.0879599
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	−18.0000	−0.787085	−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000	−0.787085	−0.393543	+	0.919306i	$0.628751\pi$
$524$	42.0000	1.83478
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	8.00000	0.348155
$529$	26.0000	1.13043
$530$	0	0
$531$	5.00000	0.216982
$532$	−24.0000	−1.04053
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−10.0000	−0.427179
$549$	7.00000	0.298753
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	26.0000	1.10563
$554$	0	0
$555$	0	0
$556$	−20.0000	−0.848189
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	0	0
$559$	7.00000	0.296068
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	−2.00000	−0.0842152
$565$	0	0
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	35.0000	1.46470	0.732352	−	0.680926i	$-0.238422\pi$
$571$	35.0000	1.46470	0.732352	+	0.680926i	$0.238422\pi$
$572$	4.00000	0.167248
$573$	−9.00000	−0.375980
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	−26.0000	−1.08052
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.0000	−1.77480	−0.887400	−	0.461000i	$-0.847491\pi$
$587$	−43.0000	−1.77480	−0.887400	+	0.461000i	$0.847491\pi$
$588$	6.00000	0.247436
$589$	36.0000	1.48335
$590$	0	0
$591$	−12.0000	−0.493614
$592$	16.0000	0.657596
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	30.0000	1.22885
$597$	−18.0000	−0.736691
$598$	0	0
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−1.00000	−0.0404557
$612$	−4.00000	−0.161690
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	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	$0.179345\pi$
$618$	0	0
$619$	−29.0000	−1.16561	−0.582804	−	0.812613i	$-0.698045\pi$
$619$	−29.0000	−1.16561	−0.582804	+	0.812613i	$0.698045\pi$
$620$	0	0
$621$	7.00000	0.280900
$622$	0	0
$623$	14.0000	0.560898
$624$	−4.00000	−0.160128
$625$	0	0
$626$	0	0
$627$	−12.0000	−0.479234
$628$	28.0000	1.11732
$629$	8.00000	0.318981
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	−20.0000	−0.793052
$637$	3.00000	0.118864
$638$	0	0
$639$	−11.0000	−0.435153
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	12.0000	0.470317
$652$	−14.0000	−0.548282
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.00000	−0.273096
$658$	0	0
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.0000	−1.62625
$668$	−18.0000	−0.696441
$669$	1.00000	0.0386622
$670$	0	0
$671$	14.0000	0.540464
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	0	0
$675$	0	0
$676$	24.0000	0.923077
$677$	3.00000	0.115299	0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000	0.115299	0.0576497	+	0.998337i	$0.481639\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−32.0000	−1.22445	−0.612223	−	0.790685i	$-0.709725\pi$
$683$	−32.0000	−1.22445	−0.612223	+	0.790685i	$0.709725\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	0	0
$687$	16.0000	0.610438
$688$	−28.0000	−1.06749
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	44.0000	1.67263
$693$	−4.00000	−0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−16.0000	−0.603023
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	−10.0000	−0.375823
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	0	0
$711$	−13.0000	−0.487538
$712$	0	0
$713$	−42.0000	−1.57291
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	23.0000	0.855379
$724$	−44.0000	−1.63525
$725$	0	0
$726$	0	0
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.0000	−0.517809
$732$	−14.0000	−0.517455
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−23.0000	−0.846069	−0.423034	−	0.906114i	$-0.639035\pi$
$739$	−23.0000	−0.846069	−0.423034	+	0.906114i	$0.639035\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.00000	0.292705
$748$	−8.00000	−0.292509
$749$	18.0000	0.657706
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	4.00000	0.145865
$753$	−8.00000	−0.291536
$754$	0	0
$755$	0	0
$756$	4.00000	0.145479
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	14.0000	0.508168
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	0	0
$763$	0	0
$764$	18.0000	0.651217
$765$	0	0
$766$	0	0
$767$	−5.00000	−0.180540
$768$	16.0000	0.577350
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	0	0
$771$	−5.00000	−0.180071
$772$	52.0000	1.87152
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−22.0000	−0.787222
$782$	0	0
$783$	−6.00000	−0.214423
$784$	−12.0000	−0.428571
$785$	0	0
$786$	0	0
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	24.0000	0.854965
$789$	2.00000	0.0712019
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−7.00000	−0.248577
$794$	0	0
$795$	0	0
$796$	36.0000	1.27599
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	0	0
$801$	−7.00000	−0.247333
$802$	0	0
$803$	−14.0000	−0.494049
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	27.0000	0.950445
$808$	0	0
$809$	48.0000	1.68759	0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000	1.68759	0.843795	+	0.536666i	$0.180316\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−24.0000	−0.842235
$813$	−28.0000	−0.982003
$814$	0	0
$815$	0	0
$816$	8.00000	0.280056
$817$	42.0000	1.46939
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	−14.0000	−0.486534
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	8.00000	0.277350
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	−6.00000	−0.207390
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000	0.481046
$848$	40.0000	1.37361
$849$	−8.00000	−0.274559
$850$	0	0
$851$	28.0000	0.959828
$852$	22.0000	0.753708
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26.0000	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13.0000	−0.441503
$868$	−24.0000	−0.814613
$869$	−26.0000	−0.881990
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	14.0000	0.473016
$877$	50.0000	1.68838	0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000	1.68838	0.844190	+	0.536044i	$0.180082\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	0	0
$887$	43.0000	1.44380	0.721899	−	0.691998i	$-0.243269\pi$
$887$	43.0000	1.44380	0.721899	+	0.691998i	$0.243269\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	2.00000	0.0670025
$892$	−2.00000	−0.0669650
$893$	−6.00000	−0.200782
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.00000	−0.233723
$898$	0	0
$899$	36.0000	1.20067
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	14.0000	0.465891
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−8.00000	−0.265489
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	−24.0000	−0.794719
$913$	16.0000	0.529523
$914$	0	0
$915$	0	0
$916$	−32.0000	−1.05731
$917$	42.0000	1.38696
$918$	0	0
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	11.0000	0.362069
$924$	8.00000	0.263181
$925$	0	0
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−9.00000	−0.295280	−0.147640	−	0.989041i	$-0.547168\pi$
$929$	−9.00000	−0.295280	−0.147640	+	0.989041i	$0.547168\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	12.0000	0.393073
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	34.0000	1.10955
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	20.0000	0.650945
$945$	0	0
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	26.0000	0.844441
$949$	7.00000	0.227230
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	41.0000	1.32812	0.664060	−	0.747679i	$-0.268832\pi$
$953$	41.0000	1.32812	0.664060	+	0.747679i	$0.268832\pi$
$954$	0	0
$955$	0	0
$956$	−48.0000	−1.55243
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−9.00000	−0.290021
$964$	−46.0000	−1.48156
$965$	0	0
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	−2.00000	−0.0641500
$973$	−20.0000	−0.641171
$974$	0	0
$975$	0	0
$976$	28.0000	0.896258
$977$	40.0000	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$977$	40.0000	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$978$	0	0
$979$	−14.0000	−0.447442
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.0000	0.861166	0.430583	−	0.902551i	$-0.358308\pi$
$983$	27.0000	0.861166	0.430583	+	0.902551i	$0.358308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	−12.0000	−0.381771
$989$	−49.0000	−1.55811
$990$	0	0
$991$	−35.0000	−1.11181	−0.555906	−	0.831245i	$-0.687628\pi$
$991$	−35.0000	−1.11181	−0.555906	+	0.831245i	$0.687628\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.g.1.1		1
5.4	even	2		705.2.a.c.1.1	✓	1
15.14	odd	2		2115.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.c.1.1	✓	1	5.4	even	2
2115.2.a.e.1.1		1	15.14	odd	2
3525.2.a.g.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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