LMFDB - Embedded newform 4140.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 16x - 8$ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +4.20905 q^{7} -2.75353 q^{11} -0.753525 q^{13} +4.96257 q^{17} +4.75353 q^{19} -1.00000 q^{23} +1.00000 q^{25} -4.96257 q^{29} -0.209050 q^{31} +4.20905 q^{35} -5.71610 q^{37} +9.38067 q^{41} +12.4181 q^{43} -7.17162 q^{47} +10.7161 q^{49} +9.38067 q^{53} -2.75353 q^{55} +4.96257 q^{59} -5.17162 q^{61} -0.753525 q^{65} -0.209050 q^{67} -9.38067 q^{71} +10.2606 q^{73} -11.5897 q^{77} -2.41810 q^{79} -5.45552 q^{83} +4.96257 q^{85} -4.91105 q^{89} -3.17162 q^{91} +4.75353 q^{95} +10.3432 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{13} + 10 q^{19} - 3 q^{23} + 3 q^{25} + 10 q^{31} + 2 q^{35} + 2 q^{37} - 8 q^{41} + 16 q^{43} + 4 q^{47} + 13 q^{49} - 8 q^{53} - 4 q^{55} + 10 q^{61} + 2 q^{65}+ \cdots - 20 q^{97}+O(q^{100})$ 3 * q + 3 * q^5 + 2 * q^7 - 4 * q^11 + 2 * q^13 + 10 * q^19 - 3 * q^23 + 3 * q^25 + 10 * q^31 + 2 * q^35 + 2 * q^37 - 8 * q^41 + 16 * q^43 + 4 * q^47 + 13 * q^49 - 8 * q^53 - 4 * q^55 + 10 * q^61 + 2 * q^65 + 10 * q^67 + 8 * q^71 + 18 * q^73 + 12 * q^77 + 14 * q^79 - 10 * q^83 - 2 * q^89 + 16 * q^91 + 10 * q^95 - 20 * q^97

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

)

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.20905	1.59087	0.795436	−	0.606038i	$-0.207242\pi$
$7$	4.20905	1.59087	0.795436	+	0.606038i	$0.207242\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.75353	−0.830219	−0.415110	−	0.909771i	$-0.636257\pi$
$11$	−2.75353	−0.830219	−0.415110	+	0.909771i	$0.636257\pi$
$12$	0	0
$13$	−0.753525	−0.208990	−0.104495	−	0.994525i	$-0.533323\pi$
$13$	−0.753525	−0.208990	−0.104495	+	0.994525i	$0.533323\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.96257	1.20360	0.601801	−	0.798646i	$-0.294450\pi$
$17$	4.96257	1.20360	0.601801	+	0.798646i	$0.294450\pi$
$18$	0	0
$19$	4.75353	1.09053	0.545267	−	0.838263i	$-0.316428\pi$
$19$	4.75353	1.09053	0.545267	+	0.838263i	$0.316428\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.96257	−0.921527	−0.460764	−	0.887523i	$-0.652424\pi$
$29$	−4.96257	−0.921527	−0.460764	+	0.887523i	$0.652424\pi$
$30$	0	0
$31$	−0.209050	−0.0375464	−0.0187732	−	0.999824i	$-0.505976\pi$
$31$	−0.209050	−0.0375464	−0.0187732	+	0.999824i	$0.505976\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.20905	0.711459
$36$	0	0
$37$	−5.71610	−0.939721	−0.469861	−	0.882741i	$-0.655696\pi$
$37$	−5.71610	−0.939721	−0.469861	+	0.882741i	$0.655696\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.38067	1.46502	0.732508	−	0.680759i	$-0.238350\pi$
$41$	9.38067	1.46502	0.732508	+	0.680759i	$0.238350\pi$
$42$	0	0
$43$	12.4181	1.89374	0.946871	−	0.321613i	$-0.104225\pi$
$43$	12.4181	1.89374	0.946871	+	0.321613i	$0.104225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.17162	−1.04609	−0.523044	−	0.852305i	$-0.675204\pi$
$47$	−7.17162	−1.04609	−0.523044	+	0.852305i	$0.675204\pi$
$48$	0	0
$49$	10.7161	1.53087
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.38067	1.28853	0.644267	−	0.764800i	$-0.277162\pi$
$53$	9.38067	1.28853	0.644267	+	0.764800i	$0.277162\pi$
$54$	0	0
$55$	−2.75353	−0.371285
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.96257	0.646072	0.323036	−	0.946387i	$-0.395296\pi$
$59$	4.96257	0.646072	0.323036	+	0.946387i	$0.395296\pi$
$60$	0	0
$61$	−5.17162	−0.662159	−0.331079	−	0.943603i	$-0.607413\pi$
$61$	−5.17162	−0.662159	−0.331079	+	0.943603i	$0.607413\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.753525	−0.0934633
$66$	0	0
$67$	−0.209050	−0.0255395	−0.0127697	−	0.999918i	$-0.504065\pi$
$67$	−0.209050	−0.0255395	−0.0127697	+	0.999918i	$0.504065\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.38067	−1.11328	−0.556641	−	0.830753i	$-0.687910\pi$
$71$	−9.38067	−1.11328	−0.556641	+	0.830753i	$0.687910\pi$
$72$	0	0
$73$	10.2606	1.20091	0.600455	−	0.799659i	$-0.294986\pi$
$73$	10.2606	1.20091	0.600455	+	0.799659i	$0.294986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.5897	−1.32077
$78$	0	0
$79$	−2.41810	−0.272057	−0.136029	−	0.990705i	$-0.543434\pi$
$79$	−2.41810	−0.272057	−0.136029	+	0.990705i	$0.543434\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.45552	−0.598822	−0.299411	−	0.954124i	$-0.596790\pi$
$83$	−5.45552	−0.598822	−0.299411	+	0.954124i	$0.596790\pi$
$84$	0	0
$85$	4.96257	0.538267
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.91105	−0.520570	−0.260285	−	0.965532i	$-0.583817\pi$
$89$	−4.91105	−0.520570	−0.260285	+	0.965532i	$0.583817\pi$
$90$	0	0
$91$	−3.17162	−0.332477
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.75353	0.487701
$96$	0	0
$97$	10.3432	1.05020	0.525099	−	0.851041i	$-0.324028\pi$
$97$	10.3432	1.05020	0.525099	+	0.851041i	$0.324028\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.8877	−1.48138	−0.740692	−	0.671845i	$-0.765502\pi$
$101$	−14.8877	−1.48138	−0.740692	+	0.671845i	$0.765502\pi$
$102$	0	0
$103$	17.9251	1.76622	0.883109	−	0.469168i	$-0.155446\pi$
$103$	17.9251	1.76622	0.883109	+	0.469168i	$0.155446\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4696	1.59218	0.796089	−	0.605179i	$-0.206898\pi$
$107$	16.4696	1.59218	0.796089	+	0.605179i	$0.206898\pi$
$108$	0	0
$109$	−6.26058	−0.599654	−0.299827	−	0.953994i	$-0.596929\pi$
$109$	−6.26058	−0.599654	−0.299827	+	0.953994i	$0.596929\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.38067	0.882460	0.441230	−	0.897394i	$-0.354542\pi$
$113$	9.38067	0.882460	0.441230	+	0.897394i	$0.354542\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.8877	1.91477
$120$	0	0
$121$	−3.41810	−0.310736
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−22.0827	−1.95952	−0.979760	−	0.200175i	$-0.935849\pi$
$127$	−22.0827	−1.95952	−0.979760	+	0.200175i	$0.935849\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.58190	0.138211	0.0691056	−	0.997609i	$-0.477985\pi$
$131$	1.58190	0.138211	0.0691056	+	0.997609i	$0.477985\pi$
$132$	0	0
$133$	20.0078	1.73490
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.91105	0.419579	0.209790	−	0.977747i	$-0.432722\pi$
$137$	4.91105	0.419579	0.209790	+	0.977747i	$0.432722\pi$
$138$	0	0
$139$	14.1342	1.19885	0.599424	−	0.800432i	$-0.295397\pi$
$139$	14.1342	1.19885	0.599424	+	0.800432i	$0.295397\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.07485	0.173508
$144$	0	0
$145$	−4.96257	−0.412119
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.24647	−0.265962	−0.132981	−	0.991119i	$-0.542455\pi$
$149$	−3.24647	−0.265962	−0.132981	+	0.991119i	$0.542455\pi$
$150$	0	0
$151$	0.418100	0.0340245	0.0170122	−	0.999855i	$-0.494585\pi$
$151$	0.418100	0.0340245	0.0170122	+	0.999855i	$0.494585\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.209050	−0.0167913
$156$	0	0
$157$	−21.0452	−1.67959	−0.839797	−	0.542901i	$-0.817326\pi$
$157$	−21.0452	−1.67959	−0.839797	+	0.542901i	$0.817326\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.20905	−0.331720
$162$	0	0
$163$	−7.43220	−0.582135	−0.291067	−	0.956703i	$-0.594010\pi$
$163$	−7.43220	−0.582135	−0.291067	+	0.956703i	$0.594010\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.1716	1.48354	0.741772	−	0.670652i	$-0.233985\pi$
$167$	19.1716	1.48354	0.741772	+	0.670652i	$0.233985\pi$
$168$	0	0
$169$	−12.4322	−0.956323
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	4.20905	0.318174
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.8503	−1.03522	−0.517610	−	0.855617i	$-0.673178\pi$
$179$	−13.8503	−1.03522	−0.517610	+	0.855617i	$0.673178\pi$
$180$	0	0
$181$	12.9110	0.959671	0.479835	−	0.877359i	$-0.340696\pi$
$181$	12.9110	0.959671	0.479835	+	0.877359i	$0.340696\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.71610	−0.420256
$186$	0	0
$187$	−13.6646	−0.999253
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.15752	0.590258	0.295129	−	0.955457i	$-0.404637\pi$
$191$	8.15752	0.590258	0.295129	+	0.955457i	$0.404637\pi$
$192$	0	0
$193$	−16.7613	−1.20651	−0.603254	−	0.797549i	$-0.706130\pi$
$193$	−16.7613	−1.20651	−0.603254	+	0.797549i	$0.706130\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9110	0.777380	0.388690	−	0.921369i	$-0.372928\pi$
$197$	10.9110	0.777380	0.388690	+	0.921369i	$0.372928\pi$
$198$	0	0
$199$	−4.49295	−0.318497	−0.159248	−	0.987239i	$-0.550907\pi$
$199$	−4.49295	−0.318497	−0.159248	+	0.987239i	$0.550907\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.8877	−1.46603
$204$	0	0
$205$	9.38067	0.655175
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.0890	−0.905382
$210$	0	0
$211$	9.71610	0.668884	0.334442	−	0.942416i	$-0.391452\pi$
$211$	9.71610	0.668884	0.334442	+	0.942416i	$0.391452\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.4181	0.846907
$216$	0	0
$217$	−0.879901	−0.0597316
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.73942	−0.251541
$222$	0	0
$223$	−19.4322	−1.30128	−0.650638	−	0.759388i	$-0.725499\pi$
$223$	−19.4322	−1.30128	−0.650638	+	0.759388i	$0.725499\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.985900	0.0654365	0.0327182	−	0.999465i	$-0.489584\pi$
$227$	0.985900	0.0654365	0.0327182	+	0.999465i	$0.489584\pi$
$228$	0	0
$229$	3.08895	0.204124	0.102062	−	0.994778i	$-0.467456\pi$
$229$	3.08895	0.204124	0.102062	+	0.994778i	$0.467456\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.83620	−0.578879	−0.289439	−	0.957196i	$-0.593469\pi$
$233$	−8.83620	−0.578879	−0.289439	+	0.957196i	$0.593469\pi$
$234$	0	0
$235$	−7.17162	−0.467825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.3807	1.38300	0.691500	−	0.722376i	$-0.256950\pi$
$239$	21.3807	1.38300	0.691500	+	0.722376i	$0.256950\pi$
$240$	0	0
$241$	22.2606	1.43393	0.716965	−	0.697109i	$-0.245531\pi$
$241$	22.2606	1.43393	0.716965	+	0.697109i	$0.245531\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10.7161	0.684627
$246$	0	0
$247$	−3.58190	−0.227911
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.59600	0.416336	0.208168	−	0.978093i	$-0.433250\pi$
$251$	6.59600	0.416336	0.208168	+	0.978093i	$0.433250\pi$
$252$	0	0
$253$	2.75353	0.173113
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1857	1.13439	0.567197	−	0.823582i	$-0.308028\pi$
$257$	18.1857	1.13439	0.567197	+	0.823582i	$0.308028\pi$
$258$	0	0
$259$	−24.0593	−1.49498
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.3807	0.948413	0.474207	−	0.880414i	$-0.342735\pi$
$263$	15.3807	0.948413	0.474207	+	0.880414i	$0.342735\pi$
$264$	0	0
$265$	9.38067	0.576250
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.544475	−0.0331972	−0.0165986	−	0.999862i	$-0.505284\pi$
$269$	−0.544475	−0.0331972	−0.0165986	+	0.999862i	$0.505284\pi$
$270$	0	0
$271$	−12.2090	−0.741647	−0.370823	−	0.928703i	$-0.620925\pi$
$271$	−12.2090	−0.741647	−0.370823	+	0.928703i	$0.620925\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.75353	−0.166044
$276$	0	0
$277$	−21.0141	−1.26261	−0.631307	−	0.775533i	$-0.717481\pi$
$277$	−21.0141	−1.26261	−0.631307	+	0.775533i	$0.717481\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.1857	1.44280	0.721400	−	0.692519i	$-0.243499\pi$
$281$	24.1857	1.44280	0.721400	+	0.692519i	$0.243499\pi$
$282$	0	0
$283$	15.1201	0.898797	0.449398	−	0.893331i	$-0.351638\pi$
$283$	15.1201	0.898797	0.449398	+	0.893331i	$0.351638\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	39.4837	2.33065
$288$	0	0
$289$	7.62715	0.448656
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.4696	−0.611642	−0.305821	−	0.952089i	$-0.598931\pi$
$293$	−10.4696	−0.611642	−0.305821	+	0.952089i	$0.598931\pi$
$294$	0	0
$295$	4.96257	0.288932
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.753525	0.0435775
$300$	0	0
$301$	52.2684	3.01270
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.17162	−0.296126
$306$	0	0
$307$	−12.1575	−0.693867	−0.346933	−	0.937890i	$-0.612777\pi$
$307$	−12.1575	−0.693867	−0.346933	+	0.937890i	$0.612777\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	34.4181	1.95167	0.975836	−	0.218506i	$-0.0701182\pi$
$311$	34.4181	1.95167	0.975836	+	0.218506i	$0.0701182\pi$
$312$	0	0
$313$	29.5664	1.67119	0.835596	−	0.549345i	$-0.185123\pi$
$313$	29.5664	1.67119	0.835596	+	0.549345i	$0.185123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.66457	−0.0934918	−0.0467459	−	0.998907i	$-0.514885\pi$
$317$	−1.66457	−0.0934918	−0.0467459	+	0.998907i	$0.514885\pi$
$318$	0	0
$319$	13.6646	0.765069
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.5897	1.31257
$324$	0	0
$325$	−0.753525	−0.0417981
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−30.1857	−1.66419
$330$	0	0
$331$	3.12010	0.171496	0.0857481	−	0.996317i	$-0.472672\pi$
$331$	3.12010	0.171496	0.0857481	+	0.996317i	$0.472672\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.209050	−0.0114216
$336$	0	0
$337$	−11.5819	−0.630906	−0.315453	−	0.948941i	$-0.602157\pi$
$337$	−11.5819	−0.630906	−0.315453	+	0.948941i	$0.602157\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.575624	0.0311718
$342$	0	0
$343$	15.6412	0.844548
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.3432	−1.41418	−0.707090	−	0.707124i	$-0.749992\pi$
$347$	−26.3432	−1.41418	−0.707090	+	0.707124i	$0.749992\pi$
$348$	0	0
$349$	9.22315	0.493704	0.246852	−	0.969053i	$-0.420604\pi$
$349$	9.22315	0.493704	0.246852	+	0.969053i	$0.420604\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.1857	0.967928	0.483964	−	0.875088i	$-0.339196\pi$
$353$	18.1857	0.967928	0.483964	+	0.875088i	$0.339196\pi$
$354$	0	0
$355$	−9.38067	−0.497875
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.17162	−0.378504	−0.189252	−	0.981929i	$-0.560606\pi$
$359$	−7.17162	−0.378504	−0.189252	+	0.981929i	$0.560606\pi$
$360$	0	0
$361$	3.59600	0.189263
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.2606	0.537063
$366$	0	0
$367$	17.5664	0.916959	0.458479	−	0.888705i	$-0.348394\pi$
$367$	17.5664	0.916959	0.458479	+	0.888705i	$0.348394\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	39.4837	2.04989
$372$	0	0
$373$	−33.6724	−1.74349	−0.871745	−	0.489959i	$-0.837012\pi$
$373$	−33.6724	−1.74349	−0.871745	+	0.489959i	$0.837012\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.73942	0.192590
$378$	0	0
$379$	18.4181	0.946074	0.473037	−	0.881043i	$-0.343158\pi$
$379$	18.4181	0.946074	0.473037	+	0.881043i	$0.343158\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.7847	1.06205	0.531024	−	0.847357i	$-0.321808\pi$
$383$	20.7847	1.06205	0.531024	+	0.847357i	$0.321808\pi$
$384$	0	0
$385$	−11.5897	−0.590667
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.5212	−1.14187	−0.570934	−	0.820996i	$-0.693419\pi$
$389$	−22.5212	−1.14187	−0.570934	+	0.820996i	$0.693419\pi$
$390$	0	0
$391$	−4.96257	−0.250968
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.41810	−0.121668
$396$	0	0
$397$	−15.4040	−0.773105	−0.386552	−	0.922267i	$-0.626334\pi$
$397$	−15.4040	−0.773105	−0.386552	+	0.922267i	$0.626334\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.5212	0.525401	0.262701	−	0.964877i	$-0.415387\pi$
$401$	10.5212	0.525401	0.262701	+	0.964877i	$0.415387\pi$
$402$	0	0
$403$	0.157524	0.00784684
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.7394	0.780174
$408$	0	0
$409$	−26.0593	−1.28855	−0.644276	−	0.764793i	$-0.722841\pi$
$409$	−26.0593	−1.28855	−0.644276	+	0.764793i	$0.722841\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20.8877	1.02782
$414$	0	0
$415$	−5.45552	−0.267801
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.73942	0.182683	0.0913414	−	0.995820i	$-0.470885\pi$
$419$	3.73942	0.182683	0.0913414	+	0.995820i	$0.470885\pi$
$420$	0	0
$421$	−3.09677	−0.150928	−0.0754638	−	0.997149i	$-0.524044\pi$
$421$	−3.09677	−0.150928	−0.0754638	+	0.997149i	$0.524044\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.96257	0.240720
$426$	0	0
$427$	−21.7676	−1.05341
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.1653	1.74202	0.871012	−	0.491262i	$-0.163464\pi$
$431$	36.1653	1.74202	0.871012	+	0.491262i	$0.163464\pi$
$432$	0	0
$433$	−16.6271	−0.799050	−0.399525	−	0.916722i	$-0.630825\pi$
$433$	−16.6271	−0.799050	−0.399525	+	0.916722i	$0.630825\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.75353	−0.227392
$438$	0	0
$439$	38.7613	1.84998	0.924989	−	0.379994i	$-0.124074\pi$
$439$	38.7613	1.84998	0.924989	+	0.379994i	$0.124074\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−33.5149	−1.59234	−0.796170	−	0.605073i	$-0.793144\pi$
$443$	−33.5149	−1.59234	−0.796170	+	0.605073i	$0.793144\pi$
$444$	0	0
$445$	−4.91105	−0.232806
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.9018	0.656068	0.328034	−	0.944666i	$-0.393614\pi$
$449$	13.9018	0.656068	0.328034	+	0.944666i	$0.393614\pi$
$450$	0	0
$451$	−25.8299	−1.21628
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.17162	−0.148688
$456$	0	0
$457$	−2.28390	−0.106836	−0.0534182	−	0.998572i	$-0.517012\pi$
$457$	−2.28390	−0.106836	−0.0534182	+	0.998572i	$0.517012\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.34325	−0.388584	−0.194292	−	0.980944i	$-0.562241\pi$
$461$	−8.34325	−0.388584	−0.194292	+	0.980944i	$0.562241\pi$
$462$	0	0
$463$	−27.6928	−1.28699	−0.643496	−	0.765449i	$-0.722517\pi$
$463$	−27.6928	−1.28699	−0.643496	+	0.765449i	$0.722517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.53037	−0.348464	−0.174232	−	0.984705i	$-0.555744\pi$
$467$	−7.53037	−0.348464	−0.174232	+	0.984705i	$0.555744\pi$
$468$	0	0
$469$	−0.879901	−0.0406300
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−34.1935	−1.57222
$474$	0	0
$475$	4.75353	0.218107
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.1857	0.830927	0.415463	−	0.909610i	$-0.363619\pi$
$479$	18.1857	0.830927	0.415463	+	0.909610i	$0.363619\pi$
$480$	0	0
$481$	4.30722	0.196393
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.3432	0.469663
$486$	0	0
$487$	−24.2606	−1.09935	−0.549676	−	0.835378i	$-0.685249\pi$
$487$	−24.2606	−1.09935	−0.549676	+	0.835378i	$0.685249\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.8877	−0.671874	−0.335937	−	0.941885i	$-0.609053\pi$
$491$	−14.8877	−0.671874	−0.335937	+	0.941885i	$0.609053\pi$
$492$	0	0
$493$	−24.6271	−1.10915
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.4837	−1.77109
$498$	0	0
$499$	−0.312101	−0.0139716	−0.00698578	−	0.999976i	$-0.502224\pi$
$499$	−0.312101	−0.0139716	−0.00698578	+	0.999976i	$0.502224\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.8877	−0.931338	−0.465669	−	0.884959i	$-0.654186\pi$
$503$	−20.8877	−0.931338	−0.465669	+	0.884959i	$0.654186\pi$
$504$	0	0
$505$	−14.8877	−0.662495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.9251	−1.23776	−0.618880	−	0.785485i	$-0.712413\pi$
$509$	−27.9251	−1.23776	−0.618880	+	0.785485i	$0.712413\pi$
$510$	0	0
$511$	43.1873	1.91049
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.9251	0.789876
$516$	0	0
$517$	19.7472	0.868483
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.0360	−1.40352	−0.701762	−	0.712412i	$-0.747603\pi$
$521$	−32.0360	−1.40352	−0.701762	+	0.712412i	$0.747603\pi$
$522$	0	0
$523$	38.8644	1.69942	0.849711	−	0.527249i	$-0.176777\pi$
$523$	38.8644	1.69942	0.849711	+	0.527249i	$0.176777\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.03743	−0.0451909
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.06857	−0.306174
$534$	0	0
$535$	16.4696	0.712044
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−29.5071	−1.27096
$540$	0	0
$541$	4.07485	0.175191	0.0875957	−	0.996156i	$-0.472082\pi$
$541$	4.07485	0.175191	0.0875957	+	0.996156i	$0.472082\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.26058	−0.268174
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23.5897	−1.00496
$552$	0	0
$553$	−10.1779	−0.432808
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.63343	−0.0692105	−0.0346052	−	0.999401i	$-0.511017\pi$
$557$	−1.63343	−0.0692105	−0.0346052	+	0.999401i	$0.511017\pi$
$558$	0	0
$559$	−9.35735	−0.395774
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.2310	0.979069	0.489534	−	0.871984i	$-0.337167\pi$
$563$	23.2310	0.979069	0.489534	+	0.871984i	$0.337167\pi$
$564$	0	0
$565$	9.38067	0.394648
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.3291	−0.894164	−0.447082	−	0.894493i	$-0.647537\pi$
$569$	−21.3291	−0.894164	−0.447082	+	0.894493i	$0.647537\pi$
$570$	0	0
$571$	−4.18573	−0.175167	−0.0875836	−	0.996157i	$-0.527914\pi$
$571$	−4.18573	−0.175167	−0.0875836	+	0.996157i	$0.527914\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	10.5678	0.439943	0.219972	−	0.975506i	$-0.429404\pi$
$577$	10.5678	0.439943	0.219972	+	0.975506i	$0.429404\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.9626	−0.952648
$582$	0	0
$583$	−25.8299	−1.06977
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.4181	−1.17294	−0.586470	−	0.809971i	$-0.699483\pi$
$587$	−28.4181	−1.17294	−0.586470	+	0.809971i	$0.699483\pi$
$588$	0	0
$589$	−0.993723	−0.0409457
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.9937	−1.19063	−0.595315	−	0.803493i	$-0.702973\pi$
$593$	−28.9937	−1.19063	−0.595315	+	0.803493i	$0.702973\pi$
$594$	0	0
$595$	20.8877	0.856313
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.10305	0.249364	0.124682	−	0.992197i	$-0.460209\pi$
$599$	6.10305	0.249364	0.124682	+	0.992197i	$0.460209\pi$
$600$	0	0
$601$	−29.3885	−1.19878	−0.599391	−	0.800456i	$-0.704590\pi$
$601$	−29.3885	−1.19878	−0.599391	+	0.800456i	$0.704590\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.41810	−0.138966
$606$	0	0
$607$	−3.58972	−0.145702	−0.0728512	−	0.997343i	$-0.523210\pi$
$607$	−3.58972	−0.145702	−0.0728512	+	0.997343i	$0.523210\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.40400	0.218622
$612$	0	0
$613$	−21.4040	−0.864499	−0.432250	−	0.901754i	$-0.642280\pi$
$613$	−21.4040	−0.864499	−0.432250	+	0.901754i	$0.642280\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.2917	−1.30002	−0.650008	−	0.759927i	$-0.725234\pi$
$617$	−32.2917	−1.30002	−0.650008	+	0.759927i	$0.725234\pi$
$618$	0	0
$619$	47.1046	1.89329	0.946647	−	0.322273i	$-0.104447\pi$
$619$	47.1046	1.89329	0.946647	+	0.322273i	$0.104447\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20.6709	−0.828160
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.3666	−1.13105
$630$	0	0
$631$	23.5149	0.936112	0.468056	−	0.883699i	$-0.344954\pi$
$631$	23.5149	0.936112	0.468056	+	0.883699i	$0.344954\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.0827	−0.876324
$636$	0	0
$637$	−8.07485	−0.319937
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.67867	0.263792	0.131896	−	0.991264i	$-0.457894\pi$
$641$	6.67867	0.263792	0.131896	+	0.991264i	$0.457894\pi$
$642$	0	0
$643$	27.1201	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$643$	27.1201	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.7394	1.09055	0.545275	−	0.838257i	$-0.316425\pi$
$647$	27.7394	1.09055	0.545275	+	0.838257i	$0.316425\pi$
$648$	0	0
$649$	−13.6646	−0.536381
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.75353	0.107754	0.0538769	−	0.998548i	$-0.482842\pi$
$653$	2.75353	0.107754	0.0538769	+	0.998548i	$0.482842\pi$
$654$	0	0
$655$	1.58190	0.0618100
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.5897	−0.451472	−0.225736	−	0.974189i	$-0.572479\pi$
$659$	−11.5897	−0.451472	−0.225736	+	0.974189i	$0.572479\pi$
$660$	0	0
$661$	28.3432	1.10242	0.551212	−	0.834365i	$-0.314165\pi$
$661$	28.3432	1.10242	0.551212	+	0.834365i	$0.314165\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.0078	0.775870
$666$	0	0
$667$	4.96257	0.192152
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.2402	0.549737
$672$	0	0
$673$	−48.0360	−1.85165	−0.925826	−	0.377949i	$-0.876629\pi$
$673$	−48.0360	−1.85165	−0.925826	+	0.377949i	$0.876629\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.51627	−0.0967083	−0.0483541	−	0.998830i	$-0.515398\pi$
$677$	−2.51627	−0.0967083	−0.0483541	+	0.998830i	$0.515398\pi$
$678$	0	0
$679$	43.5353	1.67073
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.84248	0.147028	0.0735141	−	0.997294i	$-0.476579\pi$
$683$	3.84248	0.147028	0.0735141	+	0.997294i	$0.476579\pi$
$684$	0	0
$685$	4.91105	0.187642
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.06857	−0.269291
$690$	0	0
$691$	2.59600	0.0987565	0.0493783	−	0.998780i	$-0.484276\pi$
$691$	2.59600	0.0987565	0.0493783	+	0.998780i	$0.484276\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.1342	0.536141
$696$	0	0
$697$	46.5523	1.76329
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−41.5897	−1.57082	−0.785411	−	0.618974i	$-0.787548\pi$
$701$	−41.5897	−1.57082	−0.785411	+	0.618974i	$0.787548\pi$
$702$	0	0
$703$	−27.1716	−1.02480
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−62.6632	−2.35669
$708$	0	0
$709$	−53.1716	−1.99690	−0.998451	−	0.0556356i	$-0.982281\pi$
$709$	−53.1716	−1.99690	−0.998451	+	0.0556356i	$0.982281\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.209050	0.00782898
$714$	0	0
$715$	2.07485	0.0775950
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−45.3807	−1.69241	−0.846207	−	0.532855i	$-0.821119\pi$
$719$	−45.3807	−1.69241	−0.846207	+	0.532855i	$0.821119\pi$
$720$	0	0
$721$	75.4478	2.80982
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.96257	−0.184305
$726$	0	0
$727$	−46.4026	−1.72098	−0.860489	−	0.509470i	$-0.829842\pi$
$727$	−46.4026	−1.72098	−0.860489	+	0.509470i	$0.829842\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	61.6257	2.27931
$732$	0	0
$733$	22.8051	0.842324	0.421162	−	0.906985i	$-0.361622\pi$
$733$	22.8051	0.842324	0.421162	+	0.906985i	$0.361622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.575624	0.0212034
$738$	0	0
$739$	−16.5241	−0.607849	−0.303924	−	0.952696i	$-0.598297\pi$
$739$	−16.5241	−0.607849	−0.303924	+	0.952696i	$0.598297\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.8503	−1.60871	−0.804356	−	0.594148i	$-0.797489\pi$
$743$	−43.8503	−1.60871	−0.804356	+	0.594148i	$0.797489\pi$
$744$	0	0
$745$	−3.24647	−0.118942
$746$	0	0
$747$	0	0
$748$	0	0
$749$	69.3215	2.53295
$750$	0	0
$751$	5.73942	0.209435	0.104717	−	0.994502i	$-0.466606\pi$
$751$	5.73942	0.209435	0.104717	+	0.994502i	$0.466606\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.418100	0.0152162
$756$	0	0
$757$	−9.86580	−0.358579	−0.179289	−	0.983796i	$-0.557380\pi$
$757$	−9.86580	−0.358579	−0.179289	+	0.983796i	$0.557380\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−4.69418	−0.170164	−0.0850819	−	0.996374i	$-0.527115\pi$
$761$	−4.69418	−0.170164	−0.0850819	+	0.996374i	$0.527115\pi$
$762$	0	0
$763$	−26.3511	−0.953973
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.73942	−0.135023
$768$	0	0
$769$	45.5431	1.64233	0.821163	−	0.570694i	$-0.193326\pi$
$769$	45.5431	1.64233	0.821163	+	0.570694i	$0.193326\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.4929	−0.449340	−0.224670	−	0.974435i	$-0.572130\pi$
$773$	−12.4929	−0.449340	−0.224670	+	0.974435i	$0.572130\pi$
$774$	0	0
$775$	−0.209050	−0.00750929
$776$	0	0
$777$	0	0
$778$	0	0
$779$	44.5913	1.59765
$780$	0	0
$781$	25.8299	0.924267
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−21.0452	−0.751137
$786$	0	0
$787$	−5.71610	−0.203757	−0.101878	−	0.994797i	$-0.532485\pi$
$787$	−5.71610	−0.203757	−0.101878	+	0.994797i	$0.532485\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	39.4837	1.40388
$792$	0	0
$793$	3.89695	0.138385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.48373	0.335931	0.167965	−	0.985793i	$-0.446280\pi$
$797$	9.48373	0.335931	0.167965	+	0.985793i	$0.446280\pi$
$798$	0	0
$799$	−35.5897	−1.25907
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.2528	−0.997018
$804$	0	0
$805$	−4.20905	−0.148350
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−50.0672	−1.76027	−0.880134	−	0.474725i	$-0.842547\pi$
$809$	−50.0672	−1.76027	−0.880134	+	0.474725i	$0.842547\pi$
$810$	0	0
$811$	−9.14830	−0.321240	−0.160620	−	0.987016i	$-0.551349\pi$
$811$	−9.14830	−0.321240	−0.160620	+	0.987016i	$0.551349\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.43220	−0.260339
$816$	0	0
$817$	59.0297	2.06519
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.91105	0.171397	0.0856984	−	0.996321i	$-0.472688\pi$
$821$	4.91105	0.171397	0.0856984	+	0.996321i	$0.472688\pi$
$822$	0	0
$823$	23.7006	0.826151	0.413075	−	0.910697i	$-0.364455\pi$
$823$	23.7006	0.826151	0.413075	+	0.910697i	$0.364455\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.1405	−0.456939	−0.228470	−	0.973551i	$-0.573372\pi$
$827$	−13.1405	−0.456939	−0.228470	+	0.973551i	$0.573372\pi$
$828$	0	0
$829$	−45.6412	−1.58519	−0.792593	−	0.609751i	$-0.791269\pi$
$829$	−45.6412	−1.58519	−0.792593	+	0.609751i	$0.791269\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	53.1794	1.84256
$834$	0	0
$835$	19.1716	0.663461
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.40400	0.186567	0.0932834	−	0.995640i	$-0.470264\pi$
$839$	5.40400	0.186567	0.0932834	+	0.995640i	$0.470264\pi$
$840$	0	0
$841$	−4.37285	−0.150788
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.4322	−0.427681
$846$	0	0
$847$	−14.3870	−0.494341
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.71610	0.195945
$852$	0	0
$853$	−53.0297	−1.81570	−0.907852	−	0.419291i	$-0.862279\pi$
$853$	−53.0297	−1.81570	−0.907852	+	0.419291i	$0.862279\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.4463	0.493476	0.246738	−	0.969082i	$-0.420641\pi$
$857$	14.4463	0.493476	0.246738	+	0.969082i	$0.420641\pi$
$858$	0	0
$859$	13.8658	0.473095	0.236548	−	0.971620i	$-0.423984\pi$
$859$	13.8658	0.473095	0.236548	+	0.971620i	$0.423984\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.5834	−1.78996	−0.894981	−	0.446105i	$-0.852811\pi$
$863$	−52.5834	−1.78996	−0.894981	+	0.446105i	$0.852811\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.65830	0.225867
$870$	0	0
$871$	0.157524	0.00533751
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.20905	0.142292
$876$	0	0
$877$	−25.3291	−0.855305	−0.427652	−	0.903943i	$-0.640659\pi$
$877$	−25.3291	−0.855305	−0.427652	+	0.903943i	$0.640659\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.2606	−0.480451	−0.240225	−	0.970717i	$-0.577221\pi$
$881$	−14.2606	−0.480451	−0.240225	+	0.970717i	$0.577221\pi$
$882$	0	0
$883$	40.6320	1.36738	0.683688	−	0.729774i	$-0.260375\pi$
$883$	40.6320	1.36738	0.683688	+	0.729774i	$0.260375\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.9534	0.669968	0.334984	−	0.942224i	$-0.391269\pi$
$887$	19.9534	0.669968	0.334984	+	0.942224i	$0.391269\pi$
$888$	0	0
$889$	−92.9471	−3.11734
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−34.0905	−1.14080
$894$	0	0
$895$	−13.8503	−0.462964
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.03743	0.0346001
$900$	0	0
$901$	46.5523	1.55088
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.9110	0.429178
$906$	0	0
$907$	−29.8814	−0.992197	−0.496099	−	0.868266i	$-0.665235\pi$
$907$	−29.8814	−0.992197	−0.496099	+	0.868266i	$0.665235\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.8644	−0.625005	−0.312503	−	0.949917i	$-0.601167\pi$
$911$	−18.8644	−0.625005	−0.312503	+	0.949917i	$0.601167\pi$
$912$	0	0
$913$	15.0219	0.497153
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.65830	0.219876
$918$	0	0
$919$	16.2402	0.535715	0.267857	−	0.963459i	$-0.413684\pi$
$919$	16.2402	0.535715	0.267857	+	0.963459i	$0.413684\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.06857	0.232665
$924$	0	0
$925$	−5.71610	−0.187944
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.3340	−0.962418	−0.481209	−	0.876606i	$-0.659802\pi$
$929$	−29.3340	−0.962418	−0.481209	+	0.876606i	$0.659802\pi$
$930$	0	0
$931$	50.9393	1.66947
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.6646	−0.446879
$936$	0	0
$937$	−41.3573	−1.35109	−0.675543	−	0.737321i	$-0.736091\pi$
$937$	−41.3573	−1.35109	−0.675543	+	0.737321i	$0.736091\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50.4259	−1.64384	−0.821919	−	0.569604i	$-0.807096\pi$
$941$	−50.4259	−1.64384	−0.821919	+	0.569604i	$0.807096\pi$
$942$	0	0
$943$	−9.38067	−0.305477
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.50705	−0.178955	−0.0894775	−	0.995989i	$-0.528520\pi$
$947$	−5.50705	−0.178955	−0.0894775	+	0.995989i	$0.528520\pi$
$948$	0	0
$949$	−7.73160	−0.250978
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.864400	−0.0280007	−0.0140003	−	0.999902i	$-0.504457\pi$
$953$	−0.864400	−0.0280007	−0.0140003	+	0.999902i	$0.504457\pi$
$954$	0	0
$955$	8.15752	0.263971
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.6709	0.667497
$960$	0	0
$961$	−30.9563	−0.998590
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.7613	−0.539567
$966$	0	0
$967$	10.7535	0.345810	0.172905	−	0.984939i	$-0.444685\pi$
$967$	10.7535	0.345810	0.172905	+	0.984939i	$0.444685\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.1794	−1.12896	−0.564481	−	0.825446i	$-0.690924\pi$
$971$	−35.1794	−1.12896	−0.564481	+	0.825446i	$0.690924\pi$
$972$	0	0
$973$	59.4915	1.90721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.9767	−0.511139	−0.255570	−	0.966791i	$-0.582263\pi$
$977$	−15.9767	−0.511139	−0.255570	+	0.966791i	$0.582263\pi$
$978$	0	0
$979$	13.5227	0.432187
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.2592	1.82628	0.913142	−	0.407642i	$-0.133649\pi$
$983$	57.2592	1.82628	0.913142	+	0.407642i	$0.133649\pi$
$984$	0	0
$985$	10.9110	0.347655
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.4181	−0.394873
$990$	0	0
$991$	12.8799	0.409144	0.204572	−	0.978852i	$-0.434420\pi$
$991$	12.8799	0.409144	0.204572	+	0.978852i	$0.434420\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.49295	−0.142436
$996$	0	0
$997$	−39.7754	−1.25970	−0.629851	−	0.776716i	$-0.716884\pi$
$997$	−39.7754	−1.25970	−0.629851	+	0.776716i	$0.716884\pi$
$998$	0	0
$999$	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	4140.2.a.s.1.3		3
3.2	odd	2		1380.2.a.j.1.3	✓	3
12.11	even	2		5520.2.a.bv.1.1		3
15.2	even	4		6900.2.f.r.6349.3		6
15.8	even	4		6900.2.f.r.6349.4		6
15.14	odd	2		6900.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.3	✓	3	3.2	odd	2
4140.2.a.s.1.3		3	1.1	even	1	trivial
5520.2.a.bv.1.1		3	12.11	even	2
6900.2.a.x.1.1		3	15.14	odd	2
6900.2.f.r.6349.3		6	15.2	even	4
6900.2.f.r.6349.4		6	15.8	even	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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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	4140.2.a.s.1.3

Level	$4140$
Weight	$2$
Character	4140.1
Self dual	yes
Analytic conductor	$33.058$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$