LMFDB - Embedded newform 405.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(1,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.23394128186$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	405.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} -3.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +3.00000 q^{28} +1.00000 q^{29} -5.00000 q^{32} +4.00000 q^{34} -3.00000 q^{35} -4.00000 q^{37} +8.00000 q^{38} +3.00000 q^{40} -5.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} +3.00000 q^{46} -7.00000 q^{47} +2.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} -9.00000 q^{56} -1.00000 q^{58} +14.0000 q^{59} +7.00000 q^{61} +7.00000 q^{64} -2.00000 q^{65} -3.00000 q^{67} +4.00000 q^{68} +3.00000 q^{70} -2.00000 q^{71} +4.00000 q^{73} +4.00000 q^{74} +8.00000 q^{76} -6.00000 q^{77} -6.00000 q^{79} -1.00000 q^{80} +5.00000 q^{82} -9.00000 q^{83} -4.00000 q^{85} +8.00000 q^{86} +6.00000 q^{88} +15.0000 q^{89} +6.00000 q^{91} +3.00000 q^{92} +7.00000 q^{94} -8.00000 q^{95} +2.00000 q^{97} -2.00000 q^{98} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−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
$26$	2.00000	0.392232
$27$	0	0
$28$	3.00000	0.566947
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	3.00000	0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−9.00000	−1.20268
$57$	0	0
$58$	−1.00000	−0.131306
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	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
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	3.00000	0.358569
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	8.00000	0.917663
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.00000	0.552158
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	8.00000	0.862662
$87$	0	0
$88$	6.00000	0.639602
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	3.00000	0.312772
$93$	0	0
$94$	7.00000	0.721995
$95$	−8.00000	−0.820783
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−1.00000	−0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	3.00000	0.283473
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−14.0000	−1.28880
$119$	12.0000	1.10004
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−7.00000	−0.633750
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	2.00000	0.175412
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	3.00000	0.259161
$135$	0	0
$136$	−12.0000	−1.02899
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	2.00000	0.167836
$143$	−4.00000	−0.334497
$144$	0	0
$145$	1.00000	0.0830455
$146$	−4.00000	−0.331042
$147$	0	0
$148$	4.00000	0.328798
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−24.0000	−1.94666
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	−5.00000	−0.395285
$161$	9.00000	0.709299
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	9.00000	0.698535
$167$	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
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	0	0
$172$	8.00000	0.609994
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−15.0000	−1.12430
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	−9.00000	−0.663489
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−8.00000	−0.585018
$188$	7.00000	0.510527
$189$	0	0
$190$	8.00000	0.580381
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−18.0000	−1.26648
$203$	−3.00000	−0.210559
$204$	0	0
$205$	−5.00000	−0.349215
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	3.00000	0.205076
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	−5.00000	−0.338643
$219$	0	0
$220$	−2.00000	−0.134840
$221$	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	$0.719481\pi$
$224$	15.0000	1.00223
$225$	0	0
$226$	−8.00000	−0.532152
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−7.00000	−0.456630
$236$	−14.0000	−0.911322
$237$	0	0
$238$	−12.0000	−0.777844
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−7.00000	−0.448129
$245$	2.00000	0.127775
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	5.00000	0.313728
$255$	0	0
$256$	−17.0000	−1.06250
$257$	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
$259$	12.0000	0.745644
$260$	2.00000	0.124035
$261$	0	0
$262$	−6.00000	−0.370681
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	−24.0000	−1.47153
$267$	0	0
$268$	3.00000	0.183254
$269$	−25.0000	−1.52428	−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000	−1.52428	−0.762138	+	0.647414i	$0.775850\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	2.00000	0.120605
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	−9.00000	−0.537853
$281$	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
$283$	21.0000	1.24832	0.624160	−	0.781296i	$-0.285441\pi$
$283$	21.0000	1.24832	0.624160	+	0.781296i	$0.285441\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	4.00000	0.236525
$287$	15.0000	0.885422
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−1.00000	−0.0587220
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	14.0000	0.815112
$296$	−12.0000	−0.697486
$297$	0	0
$298$	17.0000	0.984784
$299$	6.00000	0.346989
$300$	0	0
$301$	24.0000	1.38334
$302$	2.00000	0.115087
$303$	0	0
$304$	8.00000	0.458831
$305$	7.00000	0.400819
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	6.00000	0.341882
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	6.00000	0.337526
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	7.00000	0.391312
$321$	0	0
$322$	−9.00000	−0.501550
$323$	32.0000	1.78053
$324$	0	0
$325$	−2.00000	−0.110940
$326$	4.00000	0.221540
$327$	0	0
$328$	−15.0000	−0.828236
$329$	21.0000	1.15777
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	9.00000	0.493939
$333$	0	0
$334$	−9.00000	−0.492458
$335$	−3.00000	−0.163908
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	4.00000	0.216930
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	−24.0000	−1.29399
$345$	0	0
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	3.00000	0.160357
$351$	0	0
$352$	−10.0000	−0.533002
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	−15.0000	−0.794998
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	7.00000	0.367912
$363$	0	0
$364$	−6.00000	−0.314485
$365$	4.00000	0.209370
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	4.00000	0.207950
$371$	−6.00000	−0.311504
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−21.0000	−1.08299
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	8.00000	0.409316
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	10.0000	0.508987
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	6.00000	0.303046
$393$	0	0
$394$	−12.0000	−0.604551
$395$	−6.00000	−0.301893
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	0	0
$404$	−18.0000	−0.895533
$405$	0	0
$406$	3.00000	0.148888
$407$	−8.00000	−0.396545
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	5.00000	0.246932
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−42.0000	−2.06668
$414$	0	0
$415$	−9.00000	−0.441793
$416$	10.0000	0.490290
$417$	0	0
$418$	16.0000	0.782586
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	6.00000	0.291386
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−21.0000	−1.01626
$428$	3.00000	0.145010
$429$	0	0
$430$	8.00000	0.385794
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	0	0
$436$	−5.00000	−0.239457
$437$	24.0000	1.14808
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	−8.00000	−0.380521
$443$	−15.0000	−0.712672	−0.356336	−	0.934358i	$-0.615974\pi$
$443$	−15.0000	−0.712672	−0.356336	+	0.934358i	$0.615974\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	19.0000	0.899676
$447$	0	0
$448$	−21.0000	−0.992157
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	−8.00000	−0.376288
$453$	0	0
$454$	−4.00000	−0.187729
$455$	6.00000	0.281284
$456$	0	0
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	−15.0000	−0.700904
$459$	0	0
$460$	3.00000	0.139876
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	24.0000	1.11178
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	7.00000	0.322886
$471$	0	0
$472$	42.0000	1.93321
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−8.00000	−0.367065
$476$	−12.0000	−0.550019
$477$	0	0
$478$	−8.00000	−0.365911
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	11.0000	0.501036
$483$	0	0
$484$	7.00000	0.318182
$485$	2.00000	0.0908153
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	21.0000	0.950625
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	−16.0000	−0.719874
$495$	0	0
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	7.00000	0.312115	0.156057	−	0.987748i	$-0.450122\pi$
$503$	7.00000	0.312115	0.156057	+	0.987748i	$0.450122\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	6.00000	0.266733
$507$	0	0
$508$	5.00000	0.221839
$509$	−43.0000	−1.90594	−0.952971	−	0.303062i	$-0.901991\pi$
$509$	−43.0000	−1.90594	−0.952971	+	0.303062i	$0.901991\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	11.0000	0.486136
$513$	0	0
$514$	−6.00000	−0.264649
$515$	8.00000	0.352522
$516$	0	0
$517$	−14.0000	−0.615719
$518$	−12.0000	−0.527250
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	0	0
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−24.0000	−1.04053
$533$	10.0000	0.433148
$534$	0	0
$535$	−3.00000	−0.129701
$536$	−9.00000	−0.388741
$537$	0	0
$538$	25.0000	1.07783
$539$	4.00000	0.172292
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	20.0000	0.857493
$545$	5.00000	0.214176
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	−8.00000	−0.340811
$552$	0	0
$553$	18.0000	0.765438
$554$	−12.0000	−0.509831
$555$	0	0
$556$	16.0000	0.678551
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	3.00000	0.126773
$561$	0	0
$562$	−15.0000	−0.632737
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	8.00000	0.336563
$566$	−21.0000	−0.882696
$567$	0	0
$568$	−6.00000	−0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−15.0000	−0.626088
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−1.00000	−0.0415227
$581$	27.0000	1.12015
$582$	0	0
$583$	4.00000	0.165663
$584$	12.0000	0.496564
$585$	0	0
$586$	12.0000	0.495715
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	0	0
$590$	−14.0000	−0.576371
$591$	0	0
$592$	4.00000	0.164399
$593$	−20.0000	−0.821302	−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000	−0.821302	−0.410651	+	0.911793i	$0.634698\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	17.0000	0.696347
$597$	0	0
$598$	−6.00000	−0.245358
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	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
$604$	2.00000	0.0813788
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−41.0000	−1.66414	−0.832069	−	0.554672i	$-0.812844\pi$
$607$	−41.0000	−1.66414	−0.832069	+	0.554672i	$0.812844\pi$
$608$	40.0000	1.62221
$609$	0	0
$610$	−7.00000	−0.283422
$611$	14.0000	0.566379
$612$	0	0
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−18.0000	−0.725241
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−45.0000	−1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−14.0000	−0.558661
$629$	16.0000	0.637962
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−18.0000	−0.716002
$633$	0	0
$634$	−34.0000	−1.35031
$635$	−5.00000	−0.198419
$636$	0	0
$637$	−4.00000	−0.158486
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	3.00000	0.118585
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	−32.0000	−1.25902
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	0	0
$649$	28.0000	1.09910
$650$	2.00000	0.0784465
$651$	0	0
$652$	4.00000	0.156652
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	5.00000	0.195217
$657$	0	0
$658$	−21.0000	−0.818665
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	6.00000	0.233197
$663$	0	0
$664$	−27.0000	−1.04780
$665$	24.0000	0.930680
$666$	0	0
$667$	−3.00000	−0.116160
$668$	−9.00000	−0.348220
$669$	0	0
$670$	3.00000	0.115900
$671$	14.0000	0.540464
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	9.00000	0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	−12.0000	−0.460179
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	−15.0000	−0.572703
$687$	0	0
$688$	8.00000	0.304997
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	0	0
$694$	4.00000	0.151838
$695$	−16.0000	−0.606915
$696$	0	0
$697$	20.0000	0.757554
$698$	5.00000	0.189253
$699$	0	0
$700$	3.00000	0.113389
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	14.0000	0.527645
$705$	0	0
$706$	24.0000	0.903252
$707$	−54.0000	−2.03088
$708$	0	0
$709$	−41.0000	−1.53979	−0.769894	−	0.638172i	$-0.779691\pi$
$709$	−41.0000	−1.53979	−0.769894	+	0.638172i	$0.779691\pi$
$710$	2.00000	0.0750587
$711$	0	0
$712$	45.0000	1.68645
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−2.00000	−0.0747435
$717$	0	0
$718$	24.0000	0.895672
$719$	−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
$721$	−24.0000	−0.893807
$722$	−45.0000	−1.67473
$723$	0	0
$724$	7.00000	0.260153
$725$	1.00000	0.0371391
$726$	0	0
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	18.0000	0.667124
$729$	0	0
$730$	−4.00000	−0.148047
$731$	32.0000	1.18356
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	15.0000	0.552907
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	6.00000	0.220267
$743$	29.0000	1.06391	0.531953	−	0.846774i	$-0.321458\pi$
$743$	29.0000	1.06391	0.531953	+	0.846774i	$0.321458\pi$
$744$	0	0
$745$	−17.0000	−0.622832
$746$	−10.0000	−0.366126
$747$	0	0
$748$	8.00000	0.292509
$749$	9.00000	0.328853
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	7.00000	0.255264
$753$	0	0
$754$	2.00000	0.0728357
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	26.0000	0.944363
$759$	0	0
$760$	−24.0000	−0.870572
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	−15.0000	−0.543036
$764$	8.00000	0.289430
$765$	0	0
$766$	36.0000	1.30073
$767$	−28.0000	−1.01102
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	6.00000	0.216225
$771$	0	0
$772$	10.0000	0.359908
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	33.0000	1.18311
$779$	40.0000	1.43315
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	14.0000	0.499681
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−12.0000	−0.427482
$789$	0	0
$790$	6.00000	0.213470
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−14.0000	−0.497155
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	28.0000	0.990569
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−18.0000	−0.635602
$803$	8.00000	0.282314
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	0	0
$808$	54.0000	1.89971
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	8.00000	0.280400
$815$	−4.00000	−0.140114
$816$	0	0
$817$	64.0000	2.23908
$818$	−14.0000	−0.489499
$819$	0	0
$820$	5.00000	0.174608
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	53.0000	1.84746	0.923732	−	0.383040i	$-0.125123\pi$
$823$	53.0000	1.84746	0.923732	+	0.383040i	$0.125123\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	42.0000	1.46137
$827$	−37.0000	−1.28662	−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000	−1.28662	−0.643308	+	0.765607i	$0.722439\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	9.00000	0.312395
$831$	0	0
$832$	−14.0000	−0.485363
$833$	−8.00000	−0.277184
$834$	0	0
$835$	9.00000	0.311458
$836$	16.0000	0.553372
$837$	0	0
$838$	26.0000	0.898155
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−34.0000	−1.17172
$843$	0	0
$844$	22.0000	0.757271
$845$	−9.00000	−0.309609
$846$	0	0
$847$	21.0000	0.721569
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	4.00000	0.137199
$851$	12.0000	0.411355
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	21.0000	0.718605
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−17.0000	−0.578687	−0.289343	−	0.957225i	$-0.593437\pi$
$863$	−17.0000	−0.578687	−0.289343	+	0.957225i	$0.593437\pi$
$864$	0	0
$865$	0	0
$866$	−28.0000	−0.951479
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	6.00000	0.203302
$872$	15.0000	0.507964
$873$	0	0
$874$	−24.0000	−0.811812
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	0	0
$883$	23.0000	0.774012	0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000	0.774012	0.387006	+	0.922077i	$0.373509\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	15.0000	0.503935
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	15.0000	0.503084
$890$	−15.0000	−0.502801
$891$	0	0
$892$	19.0000	0.636167
$893$	56.0000	1.87397
$894$	0	0
$895$	2.00000	0.0668526
$896$	−9.00000	−0.300669
$897$	0	0
$898$	−26.0000	−0.867631
$899$	0	0
$900$	0	0
$901$	−8.00000	−0.266519
$902$	10.0000	0.332964
$903$	0	0
$904$	24.0000	0.798228
$905$	−7.00000	−0.232688
$906$	0	0
$907$	−51.0000	−1.69343	−0.846714	−	0.532049i	$-0.821422\pi$
$907$	−51.0000	−1.69343	−0.846714	+	0.532049i	$0.821422\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−6.00000	−0.198898
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	20.0000	0.661541
$915$	0	0
$916$	−15.0000	−0.495614
$917$	−18.0000	−0.594412
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−9.00000	−0.296721
$921$	0	0
$922$	9.00000	0.296399
$923$	4.00000	0.131662
$924$	0	0
$925$	−4.00000	−0.131519
$926$	36.0000	1.18303
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	24.0000	0.786146
$933$	0	0
$934$	20.0000	0.654420
$935$	−8.00000	−0.261628
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	−9.00000	−0.293860
$939$	0	0
$940$	7.00000	0.228315
$941$	7.00000	0.228193	0.114097	−	0.993470i	$-0.463603\pi$
$941$	7.00000	0.228193	0.114097	+	0.993470i	$0.463603\pi$
$942$	0	0
$943$	15.0000	0.488467
$944$	−14.0000	−0.455661
$945$	0	0
$946$	16.0000	0.520205
$947$	−57.0000	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−57.0000	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	8.00000	0.259554
$951$	0	0
$952$	36.0000	1.16677
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−18.0000	−0.581554
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	0	0
$964$	11.0000	0.354286
$965$	−10.0000	−0.321911
$966$	0	0
$967$	41.0000	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	48.0000	1.53881
$974$	16.0000	0.512673
$975$	0	0
$976$	−7.00000	−0.224065
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	20.0000	0.638226
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	4.00000	0.127386
$987$	0	0
$988$	−16.0000	−0.509028
$989$	24.0000	0.763156
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	0	0
$994$	−6.00000	−0.190308
$995$	4.00000	0.126809
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.b.1.1		1
3.2	odd	2		405.2.a.e.1.1		1
4.3	odd	2		6480.2.a.x.1.1		1
5.2	odd	4		2025.2.b.d.649.1		2
5.3	odd	4		2025.2.b.d.649.2		2
5.4	even	2		2025.2.a.e.1.1		1
9.2	odd	6		45.2.e.a.31.1	yes	2
9.4	even	3		135.2.e.a.46.1		2
9.5	odd	6		45.2.e.a.16.1	✓	2
9.7	even	3		135.2.e.a.91.1		2
12.11	even	2		6480.2.a.k.1.1		1
15.2	even	4		2025.2.b.c.649.2		2
15.8	even	4		2025.2.b.c.649.1		2
15.14	odd	2		2025.2.a.b.1.1		1
36.7	odd	6		2160.2.q.a.1441.1		2
36.11	even	6		720.2.q.d.481.1		2
36.23	even	6		720.2.q.d.241.1		2
36.31	odd	6		2160.2.q.a.721.1		2
45.2	even	12		225.2.k.a.49.2		4
45.4	even	6		675.2.e.a.451.1		2
45.7	odd	12		675.2.k.a.199.1		4
45.13	odd	12		675.2.k.a.424.1		4
45.14	odd	6		225.2.e.a.151.1		2
45.22	odd	12		675.2.k.a.424.2		4
45.23	even	12		225.2.k.a.124.2		4
45.29	odd	6		225.2.e.a.76.1		2
45.32	even	12		225.2.k.a.124.1		4
45.34	even	6		675.2.e.a.226.1		2
45.38	even	12		225.2.k.a.49.1		4
45.43	odd	12		675.2.k.a.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.a.16.1	✓	2	9.5	odd	6
45.2.e.a.31.1	yes	2	9.2	odd	6
135.2.e.a.46.1		2	9.4	even	3
135.2.e.a.91.1		2	9.7	even	3
225.2.e.a.76.1		2	45.29	odd	6
225.2.e.a.151.1		2	45.14	odd	6
225.2.k.a.49.1		4	45.38	even	12
225.2.k.a.49.2		4	45.2	even	12
225.2.k.a.124.1		4	45.32	even	12
225.2.k.a.124.2		4	45.23	even	12
405.2.a.b.1.1		1	1.1	even	1	trivial
405.2.a.e.1.1		1	3.2	odd	2
675.2.e.a.226.1		2	45.34	even	6
675.2.e.a.451.1		2	45.4	even	6
675.2.k.a.199.1		4	45.7	odd	12
675.2.k.a.199.2		4	45.43	odd	12
675.2.k.a.424.1		4	45.13	odd	12
675.2.k.a.424.2		4	45.22	odd	12
720.2.q.d.241.1		2	36.23	even	6
720.2.q.d.481.1		2	36.11	even	6
2025.2.a.b.1.1		1	15.14	odd	2
2025.2.a.e.1.1		1	5.4	even	2
2025.2.b.c.649.1		2	15.8	even	4
2025.2.b.c.649.2		2	15.2	even	4
2025.2.b.d.649.1		2	5.2	odd	4
2025.2.b.d.649.2		2	5.3	odd	4
2160.2.q.a.721.1		2	36.31	odd	6
2160.2.q.a.1441.1		2	36.7	odd	6
6480.2.a.k.1.1		1	12.11	even	2
6480.2.a.x.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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