LMFDB - Embedded newform 4050.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(1,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, 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("4050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.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:	$32.3394128186$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} -2.00000 q^{22} +1.00000 q^{23} -6.00000 q^{26} -1.00000 q^{28} +9.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{37} +6.00000 q^{38} -11.0000 q^{41} -4.00000 q^{43} -2.00000 q^{44} +1.00000 q^{46} -7.00000 q^{47} -6.00000 q^{49} -6.00000 q^{52} -1.00000 q^{56} +9.00000 q^{58} -4.00000 q^{59} -7.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -11.0000 q^{67} -2.00000 q^{68} -6.00000 q^{71} -4.00000 q^{73} +2.00000 q^{74} +6.00000 q^{76} +2.00000 q^{77} -12.0000 q^{79} -11.0000 q^{82} -11.0000 q^{83} -4.00000 q^{86} -2.00000 q^{88} +1.00000 q^{89} +6.00000 q^{91} +1.00000 q^{92} -7.00000 q^{94} -8.00000 q^{97} -6.00000 q^{98} +O(q^{100})$

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	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
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−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
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	0	0
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	0	0
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	−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
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−6.00000	−0.832050
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	9.00000	1.18176
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−11.0000	−1.34386	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	−11.0000	−1.34386	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	−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
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	2.00000	0.227921
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	−11.0000	−1.21475
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−2.00000	−0.213201
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	1.00000	0.104257
$93$	0	0
$94$	−7.00000	−0.721995
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	0	0
$116$	9.00000	0.835629
$117$	0	0
$118$	−4.00000	−0.368230
$119$	2.00000	0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−7.00000	−0.633750
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−2.00000	−0.171499
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	2.00000	0.164399
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−11.0000	−0.858956
$165$	0	0
$166$	−11.0000	−0.853766
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	1.00000	0.0749532
$179$	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
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	1.00000	0.0737210
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	−7.00000	−0.510527
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−6.00000	−0.428571
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	2.00000	0.140720
$203$	−9.00000	−0.631676
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−12.0000	−0.830057
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	0	0
$214$	3.00000	0.205076
$215$	0	0
$216$	0	0
$217$	2.00000	0.135769
$218$	7.00000	0.474100
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	12.0000	0.798228
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	9.00000	0.590879
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	2.00000	0.129641
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−7.00000	−0.448129
$245$	0	0
$246$	0	0
$247$	−36.0000	−2.29063
$248$	−2.00000	−0.127000
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	−19.0000	−1.19217
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	12.0000	0.741362
$263$	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
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	−11.0000	−0.671932
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	11.0000	0.649309
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−4.00000	−0.234082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.00000	−0.346989
$300$	0	0
$301$	4.00000	0.230556
$302$	10.0000	0.575435
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	−9.00000	−0.513657	−0.256829	−	0.966457i	$-0.582678\pi$
$307$	−9.00000	−0.513657	−0.256829	+	0.966457i	$0.582678\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	−12.0000	−0.667698
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	−11.0000	−0.607373
$329$	7.00000	0.385922
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−11.0000	−0.603703
$333$	0	0
$334$	−3.00000	−0.164153
$335$	0	0
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	13.0000	0.701934
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−4.00000	−0.215041
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	0	0
$355$	0	0
$356$	1.00000	0.0529999
$357$	0	0
$358$	2.00000	0.105703
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−13.0000	−0.683265
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−7.00000	−0.360997
$377$	−54.0000	−2.78114
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	6.00000	0.306987
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−8.00000	−0.406138
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−6.00000	−0.303046
$393$	0	0
$394$	8.00000	0.403034
$395$	0	0
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	2.00000	0.0995037
$405$	0	0
$406$	−9.00000	−0.446663
$407$	−4.00000	−0.198273
$408$	0	0
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	−6.00000	−0.294174
$417$	0	0
$418$	−12.0000	−0.586939
$419$	34.0000	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	18.0000	0.876226
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.00000	0.338754
$428$	3.00000	0.145010
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	7.00000	0.335239
$437$	6.00000	0.287019
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	0	0
$446$	23.0000	1.08908
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	22.0000	1.03594
$452$	12.0000	0.564433
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−7.00000	−0.327089
$459$	0	0
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−10.0000	−0.463241
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	11.0000	0.507933
$470$	0	0
$471$	0	0
$472$	−4.00000	−0.184115
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	28.0000	1.28069
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	1.00000	0.0455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−7.00000	−0.316875
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	6.00000	0.269137
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	0	0
$502$	18.0000	0.803379
$503$	27.0000	1.20387	0.601935	−	0.798545i	$-0.294397\pi$
$503$	27.0000	1.20387	0.601935	+	0.798545i	$0.294397\pi$
$504$	0	0
$505$	0	0
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−19.0000	−0.842989
$509$	−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
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	14.0000	0.615719
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	0	0
$521$	−37.0000	−1.62100	−0.810500	−	0.585739i	$-0.800804\pi$
$521$	−37.0000	−1.62100	−0.810500	+	0.585739i	$0.800804\pi$
$522$	0	0
$523$	−29.0000	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	16.0000	0.697633
$527$	4.00000	0.174243
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	0	0
$532$	−6.00000	−0.260133
$533$	66.0000	2.85878
$534$	0	0
$535$	0	0
$536$	−11.0000	−0.475128
$537$	0	0
$538$	3.00000	0.129339
$539$	12.0000	0.516877
$540$	0	0
$541$	−17.0000	−0.730887	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000	−0.730887	−0.365444	+	0.930834i	$0.619083\pi$
$542$	−14.0000	−0.601351
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	54.0000	2.30048
$552$	0	0
$553$	12.0000	0.510292
$554$	−22.0000	−0.934690
$555$	0	0
$556$	16.0000	0.678551
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	−3.00000	−0.126547
$563$	−37.0000	−1.55936	−0.779682	−	0.626176i	$-0.784619\pi$
$563$	−37.0000	−1.55936	−0.779682	+	0.626176i	$0.784619\pi$
$564$	0	0
$565$	0	0
$566$	1.00000	0.0420331
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	11.0000	0.459131
$575$	0	0
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	0	0
$581$	11.0000	0.456357
$582$	0	0
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	0	0
$611$	42.0000	1.69914
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−9.00000	−0.363210
$615$	0	0
$616$	2.00000	0.0805823
$617$	−32.0000	−1.28827	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−32.0000	−1.28827	−0.644136	+	0.764911i	$0.722783\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	6.00000	0.240578
$623$	−1.00000	−0.0400642
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	4.00000	0.159617
$629$	−4.00000	−0.159490
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	0	0
$637$	36.0000	1.42637
$638$	−18.0000	−0.712627
$639$	0	0
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−12.0000	−0.472134
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	0	0
$656$	−11.0000	−0.429478
$657$	0	0
$658$	7.00000	0.272888
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−11.0000	−0.426883
$665$	0	0
$666$	0	0
$667$	9.00000	0.348481
$668$	−3.00000	−0.116073
$669$	0	0
$670$	0	0
$671$	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
$676$	23.0000	0.884615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	4.00000	0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	22.0000	0.833309
$698$	−11.0000	−0.416356
$699$	0	0
$700$	0	0
$701$	13.0000	0.491003	0.245502	−	0.969396i	$-0.421047\pi$
$701$	13.0000	0.491003	0.245502	+	0.969396i	$0.421047\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−16.0000	−0.602168
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	0	0
$711$	0	0
$712$	1.00000	0.0374766
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	0	0
$716$	2.00000	0.0747435
$717$	0	0
$718$	30.0000	1.11959
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	17.0000	0.632674
$723$	0	0
$724$	−13.0000	−0.483141
$725$	0	0
$726$	0	0
$727$	21.0000	0.778847	0.389423	−	0.921059i	$-0.372674\pi$
$727$	21.0000	0.778847	0.389423	+	0.921059i	$0.372674\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	1.00000	0.0368605
$737$	22.0000	0.810380
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	0	0
$746$	12.0000	0.439351
$747$	0	0
$748$	4.00000	0.146254
$749$	−3.00000	−0.109618
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−7.00000	−0.255264
$753$	0	0
$754$	−54.0000	−1.96656
$755$	0	0
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	6.00000	0.217072
$765$	0	0
$766$	32.0000	1.15621
$767$	24.0000	0.866590
$768$	0	0
$769$	−15.0000	−0.540914	−0.270457	−	0.962732i	$-0.587175\pi$
$769$	−15.0000	−0.540914	−0.270457	+	0.962732i	$0.587175\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	19.0000	0.681183
$779$	−66.0000	−2.36470
$780$	0	0
$781$	12.0000	0.429394
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	44.0000	1.56843	0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000	1.56843	0.784215	+	0.620489i	$0.213066\pi$
$788$	8.00000	0.284988
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	42.0000	1.49146
$794$	−4.00000	−0.141955
$795$	0	0
$796$	−18.0000	−0.637993
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	0	0
$802$	−10.0000	−0.353112
$803$	8.00000	0.282314
$804$	0	0
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	2.00000	0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−9.00000	−0.315838
$813$	0	0
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	−24.0000	−0.839654
$818$	38.0000	1.32864
$819$	0	0
$820$	0	0
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	15.0000	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$823$	15.0000	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	4.00000	0.139178
$827$	−35.0000	−1.21707	−0.608535	−	0.793527i	$-0.708242\pi$
$827$	−35.0000	−1.21707	−0.608535	+	0.793527i	$0.708242\pi$
$828$	0	0
$829$	15.0000	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$830$	0	0
$831$	0	0
$832$	−6.00000	−0.208013
$833$	12.0000	0.415775
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	34.0000	1.17451
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	22.0000	0.758170
$843$	0	0
$844$	18.0000	0.619586
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	7.00000	0.239535
$855$	0	0
$856$	3.00000	0.102538
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	16.0000	0.544962
$863$	−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
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	2.00000	0.0678844
$869$	24.0000	0.814144
$870$	0	0
$871$	66.0000	2.23632
$872$	7.00000	0.237050
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	3.00000	0.100958	0.0504790	−	0.998725i	$-0.483925\pi$
$883$	3.00000	0.100958	0.0504790	+	0.998725i	$0.483925\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−9.00000	−0.302361
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	19.0000	0.637240
$890$	0	0
$891$	0	0
$892$	23.0000	0.770097
$893$	−42.0000	−1.40548
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	18.0000	0.600668
$899$	−18.0000	−0.600334
$900$	0	0
$901$	0	0
$902$	22.0000	0.732520
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	0	0
$907$	33.0000	1.09575	0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000	1.09575	0.547874	+	0.836561i	$0.315438\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	22.0000	0.728094
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−7.00000	−0.231287
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	0	0
$921$	0	0
$922$	−21.0000	−0.691598
$923$	36.0000	1.18495
$924$	0	0
$925$	0	0
$926$	36.0000	1.18303
$927$	0	0
$928$	9.00000	0.295439
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−10.0000	−0.327561
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	11.0000	0.359163
$939$	0	0
$940$	0	0
$941$	−41.0000	−1.33656	−0.668281	−	0.743909i	$-0.732970\pi$
$941$	−41.0000	−1.33656	−0.668281	+	0.743909i	$0.732970\pi$
$942$	0	0
$943$	−11.0000	−0.358209
$944$	−4.00000	−0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	−51.0000	−1.65728	−0.828639	−	0.559784i	$-0.810884\pi$
$947$	−51.0000	−1.65728	−0.828639	+	0.559784i	$0.810884\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	2.00000	0.0648204
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	0	0
$956$	28.0000	0.905585
$957$	0	0
$958$	28.0000	0.904639
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−12.0000	−0.386896
$963$	0	0
$964$	1.00000	0.0322078
$965$	0	0
$966$	0	0
$967$	−17.0000	−0.546683	−0.273342	−	0.961917i	$-0.588129\pi$
$967$	−17.0000	−0.546683	−0.273342	+	0.961917i	$0.588129\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	−12.0000	−0.384505
$975$	0	0
$976$	−7.00000	−0.224065
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	−36.0000	−1.14531
$989$	−4.00000	−0.127193
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.x.1.1		1
3.2	odd	2		4050.2.a.g.1.1		1
5.2	odd	4		810.2.c.b.649.2		2
5.3	odd	4		810.2.c.b.649.1		2
5.4	even	2		4050.2.a.j.1.1		1
9.2	odd	6		1350.2.e.i.901.1		2
9.4	even	3		450.2.e.b.151.1		2
9.5	odd	6		1350.2.e.i.451.1		2
9.7	even	3		450.2.e.b.301.1		2
15.2	even	4		810.2.c.c.649.1		2
15.8	even	4		810.2.c.c.649.2		2
15.14	odd	2		4050.2.a.be.1.1		1
45.2	even	12		270.2.i.a.199.1		4
45.4	even	6		450.2.e.g.151.1		2
45.7	odd	12		90.2.i.a.49.2	yes	4
45.13	odd	12		90.2.i.a.79.2	yes	4
45.14	odd	6		1350.2.e.a.451.1		2
45.22	odd	12		90.2.i.a.79.1	yes	4
45.23	even	12		270.2.i.a.19.1		4
45.29	odd	6		1350.2.e.a.901.1		2
45.32	even	12		270.2.i.a.19.2		4
45.34	even	6		450.2.e.g.301.1		2
45.38	even	12		270.2.i.a.199.2		4
45.43	odd	12		90.2.i.a.49.1	✓	4
180.7	even	12		720.2.by.a.49.2		4
180.23	odd	12		2160.2.by.b.289.1		4
180.43	even	12		720.2.by.a.49.1		4
180.47	odd	12		2160.2.by.b.1009.1		4
180.67	even	12		720.2.by.a.529.1		4
180.83	odd	12		2160.2.by.b.1009.2		4
180.103	even	12		720.2.by.a.529.2		4
180.167	odd	12		2160.2.by.b.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.a.49.1	✓	4	45.43	odd	12
90.2.i.a.49.2	yes	4	45.7	odd	12
90.2.i.a.79.1	yes	4	45.22	odd	12
90.2.i.a.79.2	yes	4	45.13	odd	12
270.2.i.a.19.1		4	45.23	even	12
270.2.i.a.19.2		4	45.32	even	12
270.2.i.a.199.1		4	45.2	even	12
270.2.i.a.199.2		4	45.38	even	12
450.2.e.b.151.1		2	9.4	even	3
450.2.e.b.301.1		2	9.7	even	3
450.2.e.g.151.1		2	45.4	even	6
450.2.e.g.301.1		2	45.34	even	6
720.2.by.a.49.1		4	180.43	even	12
720.2.by.a.49.2		4	180.7	even	12
720.2.by.a.529.1		4	180.67	even	12
720.2.by.a.529.2		4	180.103	even	12
810.2.c.b.649.1		2	5.3	odd	4
810.2.c.b.649.2		2	5.2	odd	4
810.2.c.c.649.1		2	15.2	even	4
810.2.c.c.649.2		2	15.8	even	4
1350.2.e.a.451.1		2	45.14	odd	6
1350.2.e.a.901.1		2	45.29	odd	6
1350.2.e.i.451.1		2	9.5	odd	6
1350.2.e.i.901.1		2	9.2	odd	6
2160.2.by.b.289.1		4	180.23	odd	12
2160.2.by.b.289.2		4	180.167	odd	12
2160.2.by.b.1009.1		4	180.47	odd	12
2160.2.by.b.1009.2		4	180.83	odd	12
4050.2.a.g.1.1		1	3.2	odd	2
4050.2.a.j.1.1		1	5.4	even	2
4050.2.a.x.1.1		1	1.1	even	1	trivial
4050.2.a.be.1.1		1	15.14	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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more