LMFDB - Embedded newform 2475.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{2} +2.00000 q^{4} +2.00000 q^{7} -1.00000 q^{11} -4.00000 q^{13} -4.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} +2.00000 q^{22} -1.00000 q^{23} +8.00000 q^{26} +4.00000 q^{28} +7.00000 q^{31} +8.00000 q^{32} +4.00000 q^{34} -3.00000 q^{37} +8.00000 q^{41} +6.00000 q^{43} -2.00000 q^{44} +2.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} -8.00000 q^{52} -6.00000 q^{53} -5.00000 q^{59} +12.0000 q^{61} -14.0000 q^{62} -8.00000 q^{64} +7.00000 q^{67} -4.00000 q^{68} +3.00000 q^{71} -4.00000 q^{73} +6.00000 q^{74} -2.00000 q^{77} -10.0000 q^{79} -16.0000 q^{82} -6.00000 q^{83} -12.0000 q^{86} -15.0000 q^{89} -8.00000 q^{91} -2.00000 q^{92} -16.0000 q^{94} +7.00000 q^{97} +6.00000 q^{98} +O(q^{100})$

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	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	0	0
$26$	8.00000	1.56893
$27$	0	0
$28$	4.00000	0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	2.00000	0.294884
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−8.00000	−1.10940
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	−14.0000	−1.77800
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	−16.0000	−1.76690
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	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
$91$	−8.00000	−0.838628
$92$	−2.00000	−0.208514
$93$	0	0
$94$	−16.0000	−1.65027
$95$	0	0
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	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
$106$	12.0000	1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	−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
$116$	0	0
$117$	0	0
$118$	10.0000	0.920575
$119$	−4.00000	−0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	−24.0000	−2.17286
$123$	0	0
$124$	14.0000	1.25724
$125$	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
$129$	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
$133$	0	0
$134$	−14.0000	−1.20942
$135$	0	0
$136$	0	0
$137$	−7.00000	−0.598050	−0.299025	−	0.954245i	$-0.596661\pi$
$137$	−7.00000	−0.598050	−0.299025	+	0.954245i	$0.596661\pi$
$138$	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
$141$	0	0
$142$	−6.00000	−0.503509
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	8.00000	0.662085
$147$	0	0
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	20.0000	1.59111
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	16.0000	1.24939
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	12.0000	0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	30.0000	2.24860
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	16.0000	1.18600
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.00000	0.146254
$188$	16.0000	1.16692
$189$	0	0
$190$	0	0
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	4.00000	0.281439
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−32.0000	−2.22955
$207$	0	0
$208$	16.0000	1.10940
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−36.0000	−2.46091
$215$	0	0
$216$	0	0
$217$	14.0000	0.950382
$218$	−20.0000	−1.35457
$219$	0	0
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	16.0000	1.06904
$225$	0	0
$226$	−18.0000	−1.19734
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	8.00000	0.518563
$239$	30.0000	1.94054	0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000	1.94054	0.970269	+	0.242028i	$0.0778125\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	24.0000	1.53644
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	16.0000	1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	−36.0000	−2.22409
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	14.0000	0.855186
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	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
$273$	0	0
$274$	14.0000	0.845771
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−8.00000	−0.473050
$287$	16.0000	0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−8.00000	−0.468165
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−20.0000	−1.15857
$299$	4.00000	0.231326
$300$	0	0
$301$	12.0000	0.691669
$302$	−4.00000	−0.230174
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−20.0000	−1.12509
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	4.00000	0.222911
$323$	0	0
$324$	0	0
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−6.00000	−0.326357
$339$	0	0
$340$	0	0
$341$	−7.00000	−0.379071
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	−8.00000	−0.426401
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	0	0
$356$	−30.0000	−1.59000
$357$	0	0
$358$	−30.0000	−1.58555
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−14.0000	−0.735824
$363$	0	0
$364$	−16.0000	−0.838628
$365$	0	0
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	0	0
$382$	34.0000	1.73959
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	0	0
$386$	8.00000	0.407189
$387$	0	0
$388$	14.0000	0.710742
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−28.0000	−1.39478
$404$	−4.00000	−0.199007
$405$	0	0
$406$	0	0
$407$	3.00000	0.148704
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	0	0
$412$	32.0000	1.57653
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	−32.0000	−1.56893
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−24.0000	−1.16830
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.0000	1.16144
$428$	36.0000	1.74013
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	−28.0000	−1.34404
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	0	0
$442$	−16.0000	−0.761042
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	0	0
$445$	0	0
$446$	38.0000	1.79935
$447$	0	0
$448$	−16.0000	−0.755929
$449$	−35.0000	−1.65175	−0.825876	−	0.563852i	$-0.809319\pi$
$449$	−35.0000	−1.65175	−0.825876	+	0.563852i	$0.809319\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	18.0000	0.846649
$453$	0	0
$454$	−36.0000	−1.68956
$455$	0	0
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	−30.0000	−1.40181
$459$	0	0
$460$	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
$463$	11.0000	0.511213	0.255607	−	0.966781i	$-0.417725\pi$
$463$	11.0000	0.511213	0.255607	+	0.966781i	$0.417725\pi$
$464$	0	0
$465$	0	0
$466$	−48.0000	−2.22356
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	−60.0000	−2.74434
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	16.0000	0.728780
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−28.0000	−1.25724
$497$	6.00000	0.269137
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	−46.0000	−2.05308
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−32.0000	−1.41421
$513$	0	0
$514$	4.00000	0.176432
$515$	0	0
$516$	0	0
$517$	−8.00000	−0.351840
$518$	12.0000	0.527250
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	−28.0000	−1.22086
$527$	−14.0000	−0.609850
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	20.0000	0.862261
$539$	3.00000	0.129219
$540$	0	0
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	56.0000	2.40541
$543$	0	0
$544$	−16.0000	−0.685994
$545$	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	$0.554709\pi$
$548$	−14.0000	−0.598050
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	−4.00000	−0.169944
$555$	0	0
$556$	20.0000	0.848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	−36.0000	−1.51857
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	−32.0000	−1.33565
$575$	0	0
$576$	0	0
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	26.0000	1.08146
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	0	0
$586$	−48.0000	−1.98286
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	12.0000	0.493197
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	0	0
$595$	0	0
$596$	20.0000	0.819232
$597$	0	0
$598$	−8.00000	−0.327144
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−24.0000	−0.978167
$603$	0	0
$604$	4.00000	0.162758
$605$	0	0
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.0000	−1.29458
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	−30.0000	−1.20192
$624$	0	0
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	14.0000	0.558661
$629$	6.00000	0.239236
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	0	0
$634$	−26.0000	−1.03259
$635$	0	0
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	−29.0000	−1.14365	−0.571824	−	0.820376i	$-0.693764\pi$
$643$	−29.0000	−1.14365	−0.571824	+	0.820376i	$0.693764\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	0	0
$647$	−7.00000	−0.275198	−0.137599	−	0.990488i	$-0.543939\pi$
$647$	−7.00000	−0.275198	−0.137599	+	0.990488i	$0.543939\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−41.0000	−1.60445	−0.802227	−	0.597019i	$-0.796352\pi$
$653$	−41.0000	−1.60445	−0.802227	+	0.597019i	$0.796352\pi$
$654$	0	0
$655$	0	0
$656$	−32.0000	−1.24939
$657$	0	0
$658$	−32.0000	−1.24749
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0	0
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	−14.0000	−0.544125
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−24.0000	−0.928588
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−44.0000	−1.69482
$675$	0	0
$676$	6.00000	0.230769
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	0	0
$682$	14.0000	0.536088
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	0	0
$686$	40.0000	1.52721
$687$	0	0
$688$	−24.0000	−0.914991
$689$	24.0000	0.914327
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−56.0000	−2.12573
$695$	0	0
$696$	0	0
$697$	−16.0000	−0.606043
$698$	−60.0000	−2.27103
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	0	0
$704$	8.00000	0.301511
$705$	0	0
$706$	42.0000	1.58069
$707$	−4.00000	−0.150435
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.00000	−0.262152
$714$	0	0
$715$	0	0
$716$	30.0000	1.12115
$717$	0	0
$718$	−40.0000	−1.49279
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	38.0000	1.41421
$723$	0	0
$724$	14.0000	0.520306
$725$	0	0
$726$	0	0
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	−8.00000	−0.294884
$737$	−7.00000	−0.257848
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	0	0
$746$	−52.0000	−1.90386
$747$	0	0
$748$	4.00000	0.146254
$749$	36.0000	1.31541
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−32.0000	−1.16692
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	−34.0000	−1.23008
$765$	0	0
$766$	2.00000	0.0722629
$767$	20.0000	0.722158
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	0	0
$772$	−8.00000	−0.287926
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−30.0000	−1.07555
$779$	0	0
$780$	0	0
$781$	−3.00000	−0.107348
$782$	−4.00000	−0.143040
$783$	0	0
$784$	12.0000	0.428571
$785$	0	0
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−4.00000	−0.142494
$789$	0	0
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−48.0000	−1.70453
$794$	−4.00000	−0.141955
$795$	0	0
$796$	0	0
$797$	53.0000	1.87736	0.938678	−	0.344795i	$-0.112051\pi$
$797$	53.0000	1.87736	0.938678	+	0.344795i	$0.112051\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	0	0
$802$	4.00000	0.141245
$803$	4.00000	0.141157
$804$	0	0
$805$	0	0
$806$	56.0000	1.97252
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	0	0
$814$	−6.00000	−0.210300
$815$	0	0
$816$	0	0
$817$	0	0
$818$	60.0000	2.09785
$819$	0	0
$820$	0	0
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	0	0
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	0	0
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	0	0
$832$	32.0000	1.10940
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	40.0000	1.38178
$839$	5.00000	0.172619	0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000	0.172619	0.0863096	+	0.996268i	$0.472493\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−44.0000	−1.51634
$843$	0	0
$844$	24.0000	0.826114
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	24.0000	0.824163
$849$	0	0
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−48.0000	−1.64253
$855$	0	0
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	−15.0000	−0.511793	−0.255897	−	0.966704i	$-0.582371\pi$
$859$	−15.0000	−0.511793	−0.255897	+	0.966704i	$0.582371\pi$
$860$	0	0
$861$	0	0
$862$	−36.0000	−1.22616
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	28.0000	0.950382
$869$	10.0000	0.339227
$870$	0	0
$871$	−28.0000	−0.948744
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.0000	0.405211	0.202606	−	0.979260i	$-0.435059\pi$
$877$	12.0000	0.405211	0.202606	+	0.979260i	$0.435059\pi$
$878$	−80.0000	−2.69987
$879$	0	0
$880$	0	0
$881$	43.0000	1.44871	0.724353	−	0.689429i	$-0.242138\pi$
$881$	43.0000	1.44871	0.724353	+	0.689429i	$0.242138\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	22.0000	0.739104
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	−38.0000	−1.27233
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	70.0000	2.33593
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	16.0000	0.532742
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	36.0000	1.19470
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−24.0000	−0.793849
$915$	0	0
$916$	30.0000	0.991228
$917$	36.0000	1.18882
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	0	0
$922$	24.0000	0.790398
$923$	−12.0000	−0.394985
$924$	0	0
$925$	0	0
$926$	−22.0000	−0.722965
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	48.0000	1.57229
$933$	0	0
$934$	54.0000	1.76693
$935$	0	0
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	−28.0000	−0.914232
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	20.0000	0.650945
$945$	0	0
$946$	12.0000	0.390154
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	0	0
$956$	60.0000	1.94054
$957$	0	0
$958$	40.0000	1.29234
$959$	−14.0000	−0.452084
$960$	0	0
$961$	18.0000	0.580645
$962$	−24.0000	−0.773791
$963$	0	0
$964$	−16.0000	−0.515325
$965$	0	0
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.0000	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$971$	−47.0000	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	46.0000	1.47394
$975$	0	0
$976$	−48.0000	−1.53644
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	0	0
$981$	0	0
$982$	−16.0000	−0.510581
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	56.0000	1.77800
$993$	0	0
$994$	−12.0000	−0.380617
$995$	0	0
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−40.0000	−1.26618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.a.1.1		1
3.2	odd	2		275.2.a.b.1.1		1
5.2	odd	4		2475.2.c.a.199.1		2
5.3	odd	4		2475.2.c.a.199.2		2
5.4	even	2		99.2.a.d.1.1		1
12.11	even	2		4400.2.a.i.1.1		1
15.2	even	4		275.2.b.a.199.2		2
15.8	even	4		275.2.b.a.199.1		2
15.14	odd	2		11.2.a.a.1.1	✓	1
20.19	odd	2		1584.2.a.g.1.1		1
33.32	even	2		3025.2.a.a.1.1		1
35.34	odd	2		4851.2.a.t.1.1		1
40.19	odd	2		6336.2.a.bu.1.1		1
40.29	even	2		6336.2.a.br.1.1		1
45.4	even	6		891.2.e.b.298.1		2
45.14	odd	6		891.2.e.k.298.1		2
45.29	odd	6		891.2.e.k.595.1		2
45.34	even	6		891.2.e.b.595.1		2
55.54	odd	2		1089.2.a.b.1.1		1
60.23	odd	4		4400.2.b.h.4049.1		2
60.47	odd	4		4400.2.b.h.4049.2		2
60.59	even	2		176.2.a.b.1.1		1
105.44	odd	6		539.2.e.h.67.1		2
105.59	even	6		539.2.e.g.177.1		2
105.74	odd	6		539.2.e.h.177.1		2
105.89	even	6		539.2.e.g.67.1		2
105.104	even	2		539.2.a.a.1.1		1
120.29	odd	2		704.2.a.h.1.1		1
120.59	even	2		704.2.a.c.1.1		1
165.14	odd	10		121.2.c.e.9.1		4
165.29	even	10		121.2.c.a.27.1		4
165.59	odd	10		121.2.c.e.27.1		4
165.74	even	10		121.2.c.a.9.1		4
165.104	odd	10		121.2.c.e.3.1		4
165.119	odd	10		121.2.c.e.81.1		4
165.134	even	10		121.2.c.a.81.1		4
165.149	even	10		121.2.c.a.3.1		4
165.164	even	2		121.2.a.d.1.1		1
195.194	odd	2		1859.2.a.b.1.1		1
240.29	odd	4		2816.2.c.j.1409.1		2
240.59	even	4		2816.2.c.f.1409.1		2
240.149	odd	4		2816.2.c.j.1409.2		2
240.179	even	4		2816.2.c.f.1409.2		2
255.254	odd	2		3179.2.a.a.1.1		1
285.284	even	2		3971.2.a.b.1.1		1
345.344	even	2		5819.2.a.a.1.1		1
420.419	odd	2		8624.2.a.j.1.1		1
435.434	odd	2		9251.2.a.d.1.1		1
660.659	odd	2		1936.2.a.i.1.1		1
1155.1154	odd	2		5929.2.a.h.1.1		1
1320.659	odd	2		7744.2.a.k.1.1		1
1320.989	even	2		7744.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	15.14	odd	2
99.2.a.d.1.1		1	5.4	even	2
121.2.a.d.1.1		1	165.164	even	2
121.2.c.a.3.1		4	165.149	even	10
121.2.c.a.9.1		4	165.74	even	10
121.2.c.a.27.1		4	165.29	even	10
121.2.c.a.81.1		4	165.134	even	10
121.2.c.e.3.1		4	165.104	odd	10
121.2.c.e.9.1		4	165.14	odd	10
121.2.c.e.27.1		4	165.59	odd	10
121.2.c.e.81.1		4	165.119	odd	10
176.2.a.b.1.1		1	60.59	even	2
275.2.a.b.1.1		1	3.2	odd	2
275.2.b.a.199.1		2	15.8	even	4
275.2.b.a.199.2		2	15.2	even	4
539.2.a.a.1.1		1	105.104	even	2
539.2.e.g.67.1		2	105.89	even	6
539.2.e.g.177.1		2	105.59	even	6
539.2.e.h.67.1		2	105.44	odd	6
539.2.e.h.177.1		2	105.74	odd	6
704.2.a.c.1.1		1	120.59	even	2
704.2.a.h.1.1		1	120.29	odd	2
891.2.e.b.298.1		2	45.4	even	6
891.2.e.b.595.1		2	45.34	even	6
891.2.e.k.298.1		2	45.14	odd	6
891.2.e.k.595.1		2	45.29	odd	6
1089.2.a.b.1.1		1	55.54	odd	2
1584.2.a.g.1.1		1	20.19	odd	2
1859.2.a.b.1.1		1	195.194	odd	2
1936.2.a.i.1.1		1	660.659	odd	2
2475.2.a.a.1.1		1	1.1	even	1	trivial
2475.2.c.a.199.1		2	5.2	odd	4
2475.2.c.a.199.2		2	5.3	odd	4
2816.2.c.f.1409.1		2	240.59	even	4
2816.2.c.f.1409.2		2	240.179	even	4
2816.2.c.j.1409.1		2	240.29	odd	4
2816.2.c.j.1409.2		2	240.149	odd	4
3025.2.a.a.1.1		1	33.32	even	2
3179.2.a.a.1.1		1	255.254	odd	2
3971.2.a.b.1.1		1	285.284	even	2
4400.2.a.i.1.1		1	12.11	even	2
4400.2.b.h.4049.1		2	60.23	odd	4
4400.2.b.h.4049.2		2	60.47	odd	4
4851.2.a.t.1.1		1	35.34	odd	2
5819.2.a.a.1.1		1	345.344	even	2
5929.2.a.h.1.1		1	1155.1154	odd	2
6336.2.a.br.1.1		1	40.29	even	2
6336.2.a.bu.1.1		1	40.19	odd	2
7744.2.a.k.1.1		1	1320.659	odd	2
7744.2.a.x.1.1		1	1320.989	even	2
8624.2.a.j.1.1		1	420.419	odd	2
9251.2.a.d.1.1		1	435.434	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more