LMFDB - Embedded newform 810.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(1,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("810.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.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:	$6.46788256372$
Analytic rank:	$0$
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	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +5.00000 q^{19} -1.00000 q^{20} -3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{26} -1.00000 q^{28} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{35} +2.00000 q^{37} +5.00000 q^{38} -1.00000 q^{40} -3.00000 q^{41} -4.00000 q^{43} -3.00000 q^{46} -9.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} +5.00000 q^{52} +9.00000 q^{53} -1.00000 q^{56} -15.0000 q^{59} -4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -5.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{70} -6.00000 q^{71} +14.0000 q^{73} +2.00000 q^{74} +5.00000 q^{76} +14.0000 q^{79} -1.00000 q^{80} -3.00000 q^{82} +6.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -18.0000 q^{89} -5.00000 q^{91} -3.00000 q^{92} -9.00000 q^{94} -5.00000 q^{95} -16.0000 q^{97} -6.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−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
$25$	1.00000	0.200000
$26$	5.00000	0.980581
$27$	0	0
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	5.00000	0.693375
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.00000	−0.620174
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	1.00000	0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−3.00000	−0.331295
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−9.00000	−0.928279
$95$	−5.00000	−0.512989
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	9.00000	0.874157
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	−15.0000	−1.38086
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−4.00000	−0.362143
$123$	0	0
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.00000	−0.438529
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−22.0000	−1.79033	−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000	−1.79033	−0.895167	+	0.445730i	$0.852944\pi$
$152$	5.00000	0.405554
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	14.0000	1.11378
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	3.00000	0.236433
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	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
$169$	12.0000	0.923077
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−4.00000	−0.304997
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0	0
$178$	−18.0000	−1.34916
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−5.00000	−0.370625
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−2.00000	−0.147043
$186$	0	0
$187$	0	0
$188$	−9.00000	−0.656392
$189$	0	0
$190$	−5.00000	−0.362738
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.00000	0.209529
$206$	17.0000	1.18445
$207$	0	0
$208$	5.00000	0.346688
$209$	0	0
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	−6.00000	−0.410152
$215$	4.00000	0.272798
$216$	0	0
$217$	−8.00000	−0.543075
$218$	8.00000	0.541828
$219$	0	0
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	3.00000	0.197814
$231$	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
$235$	9.00000	0.587095
$236$	−15.0000	−0.976417
$237$	0	0
$238$	−6.00000	−0.388922
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−4.00000	−0.256074
$245$	6.00000	0.383326
$246$	0	0
$247$	25.0000	1.59071
$248$	8.00000	0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	0	0
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−5.00000	−0.310087
$261$	0	0
$262$	15.0000	0.926703
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	−5.00000	−0.306570
$267$	0	0
$268$	−4.00000	−0.244339
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	1.00000	0.0597614
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	14.0000	0.819288
$293$	−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
$295$	15.0000	0.873334
$296$	2.00000	0.116248
$297$	0	0
$298$	6.00000	0.347571
$299$	−15.0000	−0.867472
$300$	0	0
$301$	4.00000	0.230556
$302$	−22.0000	−1.26596
$303$	0	0
$304$	5.00000	0.286770
$305$	4.00000	0.229039
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	14.0000	0.787562
$317$	33.0000	1.85346	0.926732	−	0.375722i	$-0.122605\pi$
$317$	33.0000	1.85346	0.926732	+	0.375722i	$0.122605\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.00000	0.167183
$323$	30.0000	1.66924
$324$	0	0
$325$	5.00000	0.277350
$326$	−16.0000	−0.886158
$327$	0	0
$328$	−3.00000	−0.165647
$329$	9.00000	0.496186
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−6.00000	−0.325396
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−4.00000	−0.215666
$345$	0	0
$346$	9.00000	0.483843
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−18.0000	−0.953998
$357$	0	0
$358$	−21.0000	−1.10988
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	2.00000	0.105118
$363$	0	0
$364$	−5.00000	−0.262071
$365$	−14.0000	−0.732793
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−2.00000	−0.103975
$371$	−9.00000	−0.467257
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	−9.00000	−0.464140
$377$	0	0
$378$	0	0
$379$	−25.0000	−1.28416	−0.642082	−	0.766636i	$-0.721929\pi$
$379$	−25.0000	−1.28416	−0.642082	+	0.766636i	$0.721929\pi$
$380$	−5.00000	−0.256495
$381$	0	0
$382$	12.0000	0.613973
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	−16.0000	−0.812277
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	−6.00000	−0.303046
$393$	0	0
$394$	−9.00000	−0.453413
$395$	−14.0000	−0.704416
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	40.0000	1.99254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	3.00000	0.148159
$411$	0	0
$412$	17.0000	0.837530
$413$	15.0000	0.738102
$414$	0	0
$415$	−6.00000	−0.294528
$416$	5.00000	0.245145
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	9.00000	0.437079
$425$	6.00000	0.291043
$426$	0	0
$427$	4.00000	0.193574
$428$	−6.00000	−0.290021
$429$	0	0
$430$	4.00000	0.192897
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	8.00000	0.383131
$437$	−15.0000	−0.717547
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	30.0000	1.42695
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	−16.0000	−0.757622
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	−12.0000	−0.563188
$455$	5.00000	0.234404
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−28.0000	−1.30835
$459$	0	0
$460$	3.00000	0.139876
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	0	0
$465$	0	0
$466$	24.0000	1.11178
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	9.00000	0.415139
$471$	0	0
$472$	−15.0000	−0.690431
$473$	0	0
$474$	0	0
$475$	5.00000	0.229416
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−6.00000	−0.274434
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	16.0000	0.726523
$486$	0	0
$487$	29.0000	1.31412	0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000	1.31412	0.657058	+	0.753840i	$0.271801\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	6.00000	0.271052
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	0	0
$494$	25.0000	1.12480
$495$	0	0
$496$	8.00000	0.359211
$497$	6.00000	0.269137
$498$	0	0
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−21.0000	−0.937276
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−7.00000	−0.310575
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−17.0000	−0.749110
$516$	0	0
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	−5.00000	−0.219265
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	3.00000	0.130806
$527$	48.0000	2.09091
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−9.00000	−0.390935
$531$	0	0
$532$	−5.00000	−0.216777
$533$	−15.0000	−0.649722
$534$	0	0
$535$	6.00000	0.259403
$536$	−4.00000	−0.172774
$537$	0	0
$538$	6.00000	0.258678
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	−22.0000	−0.944981
$543$	0	0
$544$	6.00000	0.257248
$545$	−8.00000	−0.342682
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−14.0000	−0.595341
$554$	11.0000	0.467345
$555$	0	0
$556$	−13.0000	−0.551323
$557$	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
$559$	−20.0000	−0.845910
$560$	1.00000	0.0422577
$561$	0	0
$562$	15.0000	0.632737
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	8.00000	0.336265
$567$	0	0
$568$	−6.00000	−0.251754
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	3.00000	0.125218
$575$	−3.00000	−0.125109
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	14.0000	0.579324
$585$	0	0
$586$	−9.00000	−0.371787
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	40.0000	1.64817
$590$	15.0000	0.617540
$591$	0	0
$592$	2.00000	0.0821995
$593$	−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
$595$	6.00000	0.245976
$596$	6.00000	0.245770
$597$	0	0
$598$	−15.0000	−0.613396
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	−22.0000	−0.895167
$605$	11.0000	0.447214
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	4.00000	0.161955
$611$	−45.0000	−1.82051
$612$	0	0
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−18.0000	−0.721734
$623$	18.0000	0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	0	0
$628$	−7.00000	−0.279330
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	14.0000	0.556890
$633$	0	0
$634$	33.0000	1.31060
$635$	7.00000	0.277787
$636$	0	0
$637$	−30.0000	−1.18864
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−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
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	30.0000	1.18033
$647$	−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
$650$	5.00000	0.196116
$651$	0	0
$652$	−16.0000	−0.626608
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	−15.0000	−0.586098
$656$	−3.00000	−0.117130
$657$	0	0
$658$	9.00000	0.350857
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	32.0000	1.24372
$663$	0	0
$664$	6.00000	0.232845
$665$	5.00000	0.193892
$666$	0	0
$667$	0	0
$668$	12.0000	0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	0	0
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	12.0000	0.461538
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	−6.00000	−0.230089
$681$	0	0
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	13.0000	0.496342
$687$	0	0
$688$	−4.00000	−0.152499
$689$	45.0000	1.71436
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	−6.00000	−0.227757
$695$	13.0000	0.493118
$696$	0	0
$697$	−18.0000	−0.681799
$698$	26.0000	0.984115
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	−18.0000	−0.674579
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	−21.0000	−0.784807
$717$	0	0
$718$	18.0000	0.671754
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−17.0000	−0.633113
$722$	6.00000	0.223297
$723$	0	0
$724$	2.00000	0.0743294
$725$	0	0
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	−5.00000	−0.185312
$729$	0	0
$730$	−14.0000	−0.518163
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−9.00000	−0.330400
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−10.0000	−0.366126
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−9.00000	−0.328196
$753$	0	0
$754$	0	0
$755$	22.0000	0.800662
$756$	0	0
$757$	−37.0000	−1.34479	−0.672394	−	0.740193i	$-0.734734\pi$
$757$	−37.0000	−1.34479	−0.672394	+	0.740193i	$0.734734\pi$
$758$	−25.0000	−0.908041
$759$	0	0
$760$	−5.00000	−0.181369
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	12.0000	0.434145
$765$	0	0
$766$	−15.0000	−0.541972
$767$	−75.0000	−2.70809
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	20.0000	0.719816
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−36.0000	−1.29066
$779$	−15.0000	−0.537431
$780$	0	0
$781$	0	0
$782$	−18.0000	−0.643679
$783$	0	0
$784$	−6.00000	−0.214286
$785$	7.00000	0.249841
$786$	0	0
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	−9.00000	−0.320612
$789$	0	0
$790$	−14.0000	−0.498098
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−20.0000	−0.710221
$794$	14.0000	0.496841
$795$	0	0
$796$	−16.0000	−0.567105
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−54.0000	−1.91038
$800$	1.00000	0.0353553
$801$	0	0
$802$	15.0000	0.529668
$803$	0	0
$804$	0	0
$805$	−3.00000	−0.105736
$806$	40.0000	1.40894
$807$	0	0
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	−20.0000	−0.699711
$818$	5.00000	0.174821
$819$	0	0
$820$	3.00000	0.104765
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	17.0000	0.592223
$825$	0	0
$826$	15.0000	0.521917
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	8.00000	0.277851	0.138926	−	0.990303i	$-0.455635\pi$
$829$	8.00000	0.277851	0.138926	+	0.990303i	$0.455635\pi$
$830$	−6.00000	−0.208263
$831$	0	0
$832$	5.00000	0.173344
$833$	−36.0000	−1.24733
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	−24.0000	−0.829066
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−10.0000	−0.344623
$843$	0	0
$844$	5.00000	0.172107
$845$	−12.0000	−0.412813
$846$	0	0
$847$	11.0000	0.377964
$848$	9.00000	0.309061
$849$	0	0
$850$	6.00000	0.205798
$851$	−6.00000	−0.205677
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−6.00000	−0.205076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	24.0000	0.817443
$863$	−45.0000	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$863$	−45.0000	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$864$	0	0
$865$	−9.00000	−0.306009
$866$	8.00000	0.271851
$867$	0	0
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	−20.0000	−0.677674
$872$	8.00000	0.270914
$873$	0	0
$874$	−15.0000	−0.507383
$875$	1.00000	0.0338062
$876$	0	0
$877$	−31.0000	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−30.0000	−1.00787
$887$	33.0000	1.10803	0.554016	−	0.832506i	$-0.313095\pi$
$887$	33.0000	1.10803	0.554016	+	0.832506i	$0.313095\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	18.0000	0.603361
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−45.0000	−1.50587
$894$	0	0
$895$	21.0000	0.701953
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	27.0000	0.901002
$899$	0	0
$900$	0	0
$901$	54.0000	1.79900
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	50.0000	1.66022	0.830111	−	0.557598i	$-0.188277\pi$
$907$	50.0000	1.66022	0.830111	+	0.557598i	$0.188277\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	5.00000	0.165748
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	0	0
$914$	8.00000	0.264616
$915$	0	0
$916$	−28.0000	−0.925146
$917$	−15.0000	−0.495344
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	0	0
$923$	−30.0000	−0.987462
$924$	0	0
$925$	2.00000	0.0657596
$926$	29.0000	0.952999
$927$	0	0
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	−30.0000	−0.983210
$932$	24.0000	0.786146
$933$	0	0
$934$	−18.0000	−0.588978
$935$	0	0
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	9.00000	0.293548
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	−15.0000	−0.488208
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	70.0000	2.27230
$950$	5.00000	0.162221
$951$	0	0
$952$	−6.00000	−0.194461
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	−6.00000	−0.194054
$957$	0	0
$958$	30.0000	0.969256
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	10.0000	0.322413
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−20.0000	−0.643823
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	16.0000	0.513729
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	29.0000	0.929220
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	6.00000	0.191663
$981$	0	0
$982$	−3.00000	−0.0957338
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	0	0
$988$	25.0000	0.795356
$989$	12.0000	0.381578
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	6.00000	0.190308
$995$	16.0000	0.507234
$996$	0	0
$997$	−55.0000	−1.74187	−0.870934	−	0.491400i	$-0.836485\pi$
$997$	−55.0000	−1.74187	−0.870934	+	0.491400i	$0.836485\pi$
$998$	−13.0000	−0.411508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.a.f.1.1	yes	1
3.2	odd	2		810.2.a.c.1.1	✓	1
4.3	odd	2		6480.2.a.i.1.1		1
5.2	odd	4		4050.2.c.k.649.2		2
5.3	odd	4		4050.2.c.k.649.1		2
5.4	even	2		4050.2.a.k.1.1		1
9.2	odd	6		810.2.e.j.271.1		2
9.4	even	3		810.2.e.f.541.1		2
9.5	odd	6		810.2.e.j.541.1		2
9.7	even	3		810.2.e.f.271.1		2
12.11	even	2		6480.2.a.u.1.1		1
15.2	even	4		4050.2.c.i.649.1		2
15.8	even	4		4050.2.c.i.649.2		2
15.14	odd	2		4050.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.a.c.1.1	✓	1	3.2	odd	2
810.2.a.f.1.1	yes	1	1.1	even	1	trivial
810.2.e.f.271.1		2	9.7	even	3
810.2.e.f.541.1		2	9.4	even	3
810.2.e.j.271.1		2	9.2	odd	6
810.2.e.j.541.1		2	9.5	odd	6
4050.2.a.k.1.1		1	5.4	even	2
4050.2.a.bc.1.1		1	15.14	odd	2
4050.2.c.i.649.1		2	15.2	even	4
4050.2.c.i.649.2		2	15.8	even	4
4050.2.c.k.649.1		2	5.3	odd	4
4050.2.c.k.649.2		2	5.2	odd	4
6480.2.a.i.1.1		1	4.3	odd	2
6480.2.a.u.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

Introduction

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