LMFDB - Embedded newform 4050.2.a.n.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} +6.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{19} -6.00000 q^{22} -9.00000 q^{23} +2.00000 q^{26} +1.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{37} +4.00000 q^{38} -3.00000 q^{41} -8.00000 q^{43} +6.00000 q^{44} +9.00000 q^{46} +3.00000 q^{47} -6.00000 q^{49} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +13.0000 q^{67} -6.00000 q^{71} +4.00000 q^{73} +8.00000 q^{74} -4.00000 q^{76} +6.00000 q^{77} -10.0000 q^{79} +3.00000 q^{82} +9.00000 q^{83} +8.00000 q^{86} -6.00000 q^{88} +9.00000 q^{89} -2.00000 q^{91} -9.00000 q^{92} -3.00000 q^{94} -2.00000 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	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	0	0
$41$	−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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	9.00000	1.32698
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	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	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	6.00000	0.683763
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	3.00000	0.331295
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−9.00000	−0.938315
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	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	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	13.0000	1.17696
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−13.0000	−1.12303
$135$	0	0
$136$	0	0
$137$	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	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	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
$148$	−8.00000	−0.657596
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−9.00000	−0.698535
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	−9.00000	−0.674579
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	0	0
$188$	3.00000	0.218797
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−6.00000	−0.428571
$197$	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
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	−6.00000	−0.422159
$203$	3.00000	0.210559
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−24.0000	−1.66011
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−3.00000	−0.205076
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	7.00000	0.474100
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	−25.0000	−1.60706
$243$	0	0
$244$	−13.0000	−0.832240
$245$	0	0
$246$	0	0
$247$	8.00000	0.509028
$248$	4.00000	0.254000
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−54.0000	−3.39495
$254$	−7.00000	−0.439219
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	18.0000	1.11204
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	13.0000	0.794101
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	0	0
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	3.00000	0.173785
$299$	18.0000	1.04097
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−14.0000	−0.805609
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	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
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−10.0000	−0.562544
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	9.00000	0.501550
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	3.00000	0.165647
$329$	3.00000	0.165395
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	9.00000	0.493939
$333$	0	0
$334$	−3.00000	−0.164153
$335$	0	0
$336$	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
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	−13.0000	−0.701934
$344$	8.00000	0.431331
$345$	0	0
$346$	24.0000	1.29025
$347$	−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
$349$	23.0000	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$350$	0	0
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−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
$355$	0	0
$356$	9.00000	0.476999
$357$	0	0
$358$	18.0000	0.951330
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−5.00000	−0.262794
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−9.00000	−0.469157
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	0	0
$376$	−3.00000	−0.154713
$377$	−6.00000	−0.309016
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	0	0
$382$	12.0000	0.613973
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	−2.00000	−0.101535
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	0	0
$394$	−12.0000	−0.604551
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	−3.00000	−0.148888
$407$	−48.0000	−2.37927
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	−8.00000	−0.394132
$413$	6.00000	0.295241
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	0	0
$418$	24.0000	1.17388
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−2.00000	−0.0973585
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	−13.0000	−0.629114
$428$	3.00000	0.145010
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−7.00000	−0.335239
$437$	36.0000	1.72211
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−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
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	13.0000	0.607450
$459$	0	0
$460$	0	0
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	13.0000	0.600284
$470$	0	0
$471$	0	0
$472$	−6.00000	−0.276172
$473$	−48.0000	−2.20704
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	−29.0000	−1.32091
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	13.0000	0.588482
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−6.00000	−0.269137
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−27.0000	−1.20387	−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000	−1.20387	−0.601935	+	0.798545i	$0.705603\pi$
$504$	0	0
$505$	0	0
$506$	54.0000	2.40059
$507$	0	0
$508$	7.00000	0.310575
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	18.0000	0.791639
$518$	8.00000	0.351500
$519$	0	0
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	19.0000	0.830812	0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000	0.830812	0.415406	+	0.909636i	$0.363640\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	−4.00000	−0.173422
$533$	6.00000	0.259889
$534$	0	0
$535$	0	0
$536$	−13.0000	−0.561514
$537$	0	0
$538$	−21.0000	−0.905374
$539$	−36.0000	−1.55063
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	43.0000	1.83855	0.919274	−	0.393619i	$-0.128777\pi$
$547$	43.0000	1.83855	0.919274	+	0.393619i	$0.128777\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	−10.0000	−0.425243
$554$	8.00000	0.339887
$555$	0	0
$556$	−16.0000	−0.678551
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	15.0000	0.632737
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	0	0
$566$	−13.0000	−0.546431
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	3.00000	0.125218
$575$	0	0
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	17.0000	0.707107
$579$	0	0
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	−36.0000	−1.49097
$584$	−4.00000	−0.165521
$585$	0	0
$586$	0	0
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	0	0
$598$	−18.0000	−0.736075
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	14.0000	0.569652
$605$	0	0
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	40.0000	1.61558	0.807792	−	0.589467i	$-0.200662\pi$
$613$	40.0000	1.61558	0.807792	+	0.589467i	$0.200662\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	9.00000	0.360577
$624$	0	0
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	−14.0000	−0.558661
$629$	0	0
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	12.0000	0.475457
$638$	−18.0000	−0.712627
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	0	0
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−3.00000	−0.116952
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	−27.0000	−1.04544
$668$	3.00000	0.116073
$669$	0	0
$670$	0	0
$671$	−78.0000	−3.01116
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	24.0000	0.919007
$683$	−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
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−23.0000	−0.870563
$699$	0	0
$700$	0	0
$701$	−45.0000	−1.69963	−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000	−1.69963	−0.849813	+	0.527084i	$0.823285\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	6.00000	0.226134
$705$	0	0
$706$	24.0000	0.903252
$707$	6.00000	0.225653
$708$	0	0
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	0	0
$711$	0	0
$712$	−9.00000	−0.337289
$713$	36.0000	1.34821
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	0	0
$718$	12.0000	0.447836
$719$	−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
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	0	0
$724$	5.00000	0.185824
$725$	0	0
$726$	0	0
$727$	−53.0000	−1.96566	−0.982831	−	0.184510i	$-0.940930\pi$
$727$	−53.0000	−1.96566	−0.982831	+	0.184510i	$0.940930\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	9.00000	0.331744
$737$	78.0000	2.87317
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	0	0
$742$	6.00000	0.220267
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	0	0
$748$	0	0
$749$	3.00000	0.109618
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	22.0000	0.799076
$759$	0	0
$760$	0	0
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−12.0000	−0.433578
$767$	−12.0000	−0.433295
$768$	0	0
$769$	29.0000	1.04577	0.522883	−	0.852404i	$-0.324856\pi$
$769$	29.0000	1.04577	0.522883	+	0.852404i	$0.324856\pi$
$770$	0	0
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	−21.0000	−0.752886
$779$	12.0000	0.429945
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	26.0000	0.923287
$794$	2.00000	0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−6.00000	−0.211867
$803$	24.0000	0.846942
$804$	0	0
$805$	0	0
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−6.00000	−0.211079
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	48.0000	1.68240
$815$	0	0
$816$	0	0
$817$	32.0000	1.11954
$818$	10.0000	0.349642
$819$	0	0
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	0	0
$823$	25.0000	0.871445	0.435723	−	0.900081i	$-0.356493\pi$
$823$	25.0000	0.871445	0.435723	+	0.900081i	$0.356493\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	30.0000	1.03633
$839$	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
$841$	−20.0000	−0.689655
$842$	22.0000	0.758170
$843$	0	0
$844$	2.00000	0.0688428
$845$	0	0
$846$	0	0
$847$	25.0000	0.859010
$848$	−6.00000	−0.206041
$849$	0	0
$850$	0	0
$851$	72.0000	2.46813
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	13.0000	0.444851
$855$	0	0
$856$	−3.00000	−0.102538
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	0	0
$861$	0	0
$862$	−6.00000	−0.204361
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	0	0
$865$	0	0
$866$	−16.0000	−0.543702
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−60.0000	−2.03536
$870$	0	0
$871$	−26.0000	−0.880976
$872$	7.00000	0.237050
$873$	0	0
$874$	−36.0000	−1.21772
$875$	0	0
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	0	0
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	0	0
$883$	31.0000	1.04323	0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000	1.04323	0.521617	+	0.853180i	$0.325329\pi$
$884$	0	0
$885$	0	0
$886$	9.00000	0.302361
$887$	−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
$889$	7.00000	0.234772
$890$	0	0
$891$	0	0
$892$	1.00000	0.0334825
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−12.0000	−0.400222
$900$	0	0
$901$	0	0
$902$	18.0000	0.599334
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	54.0000	1.78714
$914$	8.00000	0.264616
$915$	0	0
$916$	−13.0000	−0.429532
$917$	−18.0000	−0.594412
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	0	0
$922$	27.0000	0.889198
$923$	12.0000	0.394985
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	12.0000	0.393073
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	−56.0000	−1.82944	−0.914720	−	0.404088i	$-0.867589\pi$
$937$	−56.0000	−1.82944	−0.914720	+	0.404088i	$0.867589\pi$
$938$	−13.0000	−0.424465
$939$	0	0
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	27.0000	0.879241
$944$	6.00000	0.195283
$945$	0	0
$946$	48.0000	1.56061
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	30.0000	0.969256
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−16.0000	−0.515861
$963$	0	0
$964$	29.0000	0.934027
$965$	0	0
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	8.00000	0.256337
$975$	0	0
$976$	−13.0000	−0.416120
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	72.0000	2.28947
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−32.0000	−1.01294
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.n.1.1		1
3.2	odd	2		4050.2.a.ba.1.1		1
5.2	odd	4		4050.2.c.t.649.1		2
5.3	odd	4		4050.2.c.t.649.2		2
5.4	even	2		810.2.a.g.1.1		1
9.2	odd	6		1350.2.e.b.901.1		2
9.4	even	3		450.2.e.e.151.1		2
9.5	odd	6		1350.2.e.b.451.1		2
9.7	even	3		450.2.e.e.301.1		2
15.2	even	4		4050.2.c.a.649.2		2
15.8	even	4		4050.2.c.a.649.1		2
15.14	odd	2		810.2.a.b.1.1		1
20.19	odd	2		6480.2.a.g.1.1		1
45.2	even	12		1350.2.j.e.199.2		4
45.4	even	6		90.2.e.a.61.1	yes	2
45.7	odd	12		450.2.j.c.49.1		4
45.13	odd	12		450.2.j.c.349.1		4
45.14	odd	6		270.2.e.b.181.1		2
45.22	odd	12		450.2.j.c.349.2		4
45.23	even	12		1350.2.j.e.1099.2		4
45.29	odd	6		270.2.e.b.91.1		2
45.32	even	12		1350.2.j.e.1099.1		4
45.34	even	6		90.2.e.a.31.1	✓	2
45.38	even	12		1350.2.j.e.199.1		4
45.43	odd	12		450.2.j.c.49.2		4
60.59	even	2		6480.2.a.v.1.1		1
180.59	even	6		2160.2.q.b.721.1		2
180.79	odd	6		720.2.q.b.481.1		2
180.119	even	6		2160.2.q.b.1441.1		2
180.139	odd	6		720.2.q.b.241.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.a.31.1	✓	2	45.34	even	6
90.2.e.a.61.1	yes	2	45.4	even	6
270.2.e.b.91.1		2	45.29	odd	6
270.2.e.b.181.1		2	45.14	odd	6
450.2.e.e.151.1		2	9.4	even	3
450.2.e.e.301.1		2	9.7	even	3
450.2.j.c.49.1		4	45.7	odd	12
450.2.j.c.49.2		4	45.43	odd	12
450.2.j.c.349.1		4	45.13	odd	12
450.2.j.c.349.2		4	45.22	odd	12
720.2.q.b.241.1		2	180.139	odd	6
720.2.q.b.481.1		2	180.79	odd	6
810.2.a.b.1.1		1	15.14	odd	2
810.2.a.g.1.1		1	5.4	even	2
1350.2.e.b.451.1		2	9.5	odd	6
1350.2.e.b.901.1		2	9.2	odd	6
1350.2.j.e.199.1		4	45.38	even	12
1350.2.j.e.199.2		4	45.2	even	12
1350.2.j.e.1099.1		4	45.32	even	12
1350.2.j.e.1099.2		4	45.23	even	12
2160.2.q.b.721.1		2	180.59	even	6
2160.2.q.b.1441.1		2	180.119	even	6
4050.2.a.n.1.1		1	1.1	even	1	trivial
4050.2.a.ba.1.1		1	3.2	odd	2
4050.2.c.a.649.1		2	15.8	even	4
4050.2.c.a.649.2		2	15.2	even	4
4050.2.c.t.649.1		2	5.2	odd	4
4050.2.c.t.649.2		2	5.3	odd	4
6480.2.a.g.1.1		1	20.19	odd	2
6480.2.a.v.1.1		1	60.59	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 $\F_{q}$
Belyi maps

Introduction

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4050.2.a.n.1.1

Level	$4050$
Weight	$2$
Character	4050.1
Self dual	yes
Analytic conductor	$32.339$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$