LMFDB - Embedded newform 140.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,4,Mod(1,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("140.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	140.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:	$8.26026740080$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +37.0000 q^{9} +O(q^{10})$

$q+8.00000 q^{3} -5.00000 q^{5} +7.00000 q^{7} +37.0000 q^{9} +28.0000 q^{11} +82.0000 q^{13} -40.0000 q^{15} -46.0000 q^{17} +8.00000 q^{19} +56.0000 q^{21} -128.000 q^{23} +25.0000 q^{25} +80.0000 q^{27} +174.000 q^{29} -152.000 q^{31} +224.000 q^{33} -35.0000 q^{35} -290.000 q^{37} +656.000 q^{39} +50.0000 q^{41} +396.000 q^{43} -185.000 q^{45} -296.000 q^{47} +49.0000 q^{49} -368.000 q^{51} -570.000 q^{53} -140.000 q^{55} +64.0000 q^{57} -272.000 q^{59} -662.000 q^{61} +259.000 q^{63} -410.000 q^{65} +876.000 q^{67} -1024.00 q^{69} -880.000 q^{71} -638.000 q^{73} +200.000 q^{75} +196.000 q^{77} -600.000 q^{79} -359.000 q^{81} +624.000 q^{83} +230.000 q^{85} +1392.00 q^{87} +698.000 q^{89} +574.000 q^{91} -1216.00 q^{93} -40.0000 q^{95} +754.000 q^{97} +1036.00 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	82.0000	1.74944	0.874720	−	0.484629i	$-0.161046\pi$
$13$	82.0000	1.74944	0.874720	+	0.484629i	$0.161046\pi$
$14$	0	0
$15$	−40.0000	−0.688530
$16$	0	0
$17$	−46.0000	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$17$	−46.0000	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$18$	0	0
$19$	8.00000	0.0965961	0.0482980	−	0.998833i	$-0.484620\pi$
$19$	8.00000	0.0965961	0.0482980	+	0.998833i	$0.484620\pi$
$20$	0	0
$21$	56.0000	0.581914
$22$	0	0
$23$	−128.000	−1.16043	−0.580214	−	0.814464i	$-0.697031\pi$
$23$	−128.000	−1.16043	−0.580214	+	0.814464i	$0.697031\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	174.000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	174.000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	224.000	1.18162
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−290.000	−1.28853	−0.644266	−	0.764801i	$-0.722837\pi$
$37$	−290.000	−1.28853	−0.644266	+	0.764801i	$0.722837\pi$
$38$	0	0
$39$	656.000	2.69344
$40$	0	0
$41$	50.0000	0.190456	0.0952279	−	0.995455i	$-0.469642\pi$
$41$	50.0000	0.190456	0.0952279	+	0.995455i	$0.469642\pi$
$42$	0	0
$43$	396.000	1.40441	0.702203	−	0.711977i	$-0.252200\pi$
$43$	396.000	1.40441	0.702203	+	0.711977i	$0.252200\pi$
$44$	0	0
$45$	−185.000	−0.612848
$46$	0	0
$47$	−296.000	−0.918639	−0.459320	−	0.888271i	$-0.651907\pi$
$47$	−296.000	−0.918639	−0.459320	+	0.888271i	$0.651907\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−368.000	−1.01040
$52$	0	0
$53$	−570.000	−1.47727	−0.738637	−	0.674103i	$-0.764530\pi$
$53$	−570.000	−1.47727	−0.738637	+	0.674103i	$0.764530\pi$
$54$	0	0
$55$	−140.000	−0.343229
$56$	0	0
$57$	64.0000	0.148719
$58$	0	0
$59$	−272.000	−0.600193	−0.300096	−	0.953909i	$-0.597019\pi$
$59$	−272.000	−0.600193	−0.300096	+	0.953909i	$0.597019\pi$
$60$	0	0
$61$	−662.000	−1.38951	−0.694757	−	0.719244i	$-0.744488\pi$
$61$	−662.000	−1.38951	−0.694757	+	0.719244i	$0.744488\pi$
$62$	0	0
$63$	259.000	0.517951
$64$	0	0
$65$	−410.000	−0.782373
$66$	0	0
$67$	876.000	1.59732	0.798660	−	0.601783i	$-0.205543\pi$
$67$	876.000	1.59732	0.798660	+	0.601783i	$0.205543\pi$
$68$	0	0
$69$	−1024.00	−1.78660
$70$	0	0
$71$	−880.000	−1.47094	−0.735470	−	0.677557i	$-0.763039\pi$
$71$	−880.000	−1.47094	−0.735470	+	0.677557i	$0.763039\pi$
$72$	0	0
$73$	−638.000	−1.02291	−0.511454	−	0.859311i	$-0.670893\pi$
$73$	−638.000	−1.02291	−0.511454	+	0.859311i	$0.670893\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	196.000	0.290081
$78$	0	0
$79$	−600.000	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$79$	−600.000	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	624.000	0.825216	0.412608	−	0.910909i	$-0.364618\pi$
$83$	624.000	0.825216	0.412608	+	0.910909i	$0.364618\pi$
$84$	0	0
$85$	230.000	0.293494
$86$	0	0
$87$	1392.00	1.71538
$88$	0	0
$89$	698.000	0.831324	0.415662	−	0.909519i	$-0.363550\pi$
$89$	698.000	0.831324	0.415662	+	0.909519i	$0.363550\pi$
$90$	0	0
$91$	574.000	0.661226
$92$	0	0
$93$	−1216.00	−1.35584
$94$	0	0
$95$	−40.0000	−0.0431991
$96$	0	0
$97$	754.000	0.789248	0.394624	−	0.918843i	$-0.370875\pi$
$97$	754.000	0.789248	0.394624	+	0.918843i	$0.370875\pi$
$98$	0	0
$99$	1036.00	1.05174
$100$	0	0
$101$	354.000	0.348756	0.174378	−	0.984679i	$-0.444209\pi$
$101$	354.000	0.348756	0.174378	+	0.984679i	$0.444209\pi$
$102$	0	0
$103$	1904.00	1.82142	0.910712	−	0.413042i	$-0.135534\pi$
$103$	1904.00	1.82142	0.910712	+	0.413042i	$0.135534\pi$
$104$	0	0
$105$	−280.000	−0.260240
$106$	0	0
$107$	1108.00	1.00107	0.500535	−	0.865717i	$-0.333137\pi$
$107$	1108.00	1.00107	0.500535	+	0.865717i	$0.333137\pi$
$108$	0	0
$109$	−146.000	−0.128296	−0.0641480	−	0.997940i	$-0.520433\pi$
$109$	−146.000	−0.128296	−0.0641480	+	0.997940i	$0.520433\pi$
$110$	0	0
$111$	−2320.00	−1.98383
$112$	0	0
$113$	−2078.00	−1.72993	−0.864964	−	0.501834i	$-0.832659\pi$
$113$	−2078.00	−1.72993	−0.864964	+	0.501834i	$0.832659\pi$
$114$	0	0
$115$	640.000	0.518959
$116$	0	0
$117$	3034.00	2.39738
$118$	0	0
$119$	−322.000	−0.248048
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	400.000	0.293226
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−600.000	−0.419224	−0.209612	−	0.977785i	$-0.567220\pi$
$127$	−600.000	−0.419224	−0.209612	+	0.977785i	$0.567220\pi$
$128$	0	0
$129$	3168.00	2.16222
$130$	0	0
$131$	−168.000	−0.112048	−0.0560238	−	0.998429i	$-0.517842\pi$
$131$	−168.000	−0.112048	−0.0560238	+	0.998429i	$0.517842\pi$
$132$	0	0
$133$	56.0000	0.0365099
$134$	0	0
$135$	−400.000	−0.255011
$136$	0	0
$137$	394.000	0.245706	0.122853	−	0.992425i	$-0.460796\pi$
$137$	394.000	0.245706	0.122853	+	0.992425i	$0.460796\pi$
$138$	0	0
$139$	−472.000	−0.288018	−0.144009	−	0.989576i	$-0.545999\pi$
$139$	−472.000	−0.288018	−0.144009	+	0.989576i	$0.545999\pi$
$140$	0	0
$141$	−2368.00	−1.41434
$142$	0	0
$143$	2296.00	1.34267
$144$	0	0
$145$	−870.000	−0.498273
$146$	0	0
$147$	392.000	0.219943
$148$	0	0
$149$	−2658.00	−1.46142	−0.730711	−	0.682687i	$-0.760811\pi$
$149$	−2658.00	−1.46142	−0.730711	+	0.682687i	$0.760811\pi$
$150$	0	0
$151$	1256.00	0.676900	0.338450	−	0.940984i	$-0.390097\pi$
$151$	1256.00	0.676900	0.338450	+	0.940984i	$0.390097\pi$
$152$	0	0
$153$	−1702.00	−0.899337
$154$	0	0
$155$	760.000	0.393837
$156$	0	0
$157$	3266.00	1.66022	0.830112	−	0.557597i	$-0.188276\pi$
$157$	3266.00	1.66022	0.830112	+	0.557597i	$0.188276\pi$
$158$	0	0
$159$	−4560.00	−2.27441
$160$	0	0
$161$	−896.000	−0.438601
$162$	0	0
$163$	1716.00	0.824586	0.412293	−	0.911051i	$-0.364728\pi$
$163$	1716.00	0.824586	0.412293	+	0.911051i	$0.364728\pi$
$164$	0	0
$165$	−1120.00	−0.528436
$166$	0	0
$167$	−2272.00	−1.05277	−0.526385	−	0.850246i	$-0.676453\pi$
$167$	−2272.00	−1.05277	−0.526385	+	0.850246i	$0.676453\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	0	0
$171$	296.000	0.132372
$172$	0	0
$173$	4226.00	1.85721	0.928604	−	0.371073i	$-0.121010\pi$
$173$	4226.00	1.85721	0.928604	+	0.371073i	$0.121010\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−2176.00	−0.924057
$178$	0	0
$179$	988.000	0.412551	0.206275	−	0.978494i	$-0.433866\pi$
$179$	988.000	0.412551	0.206275	+	0.978494i	$0.433866\pi$
$180$	0	0
$181$	2138.00	0.877991	0.438995	−	0.898489i	$-0.355334\pi$
$181$	2138.00	0.877991	0.438995	+	0.898489i	$0.355334\pi$
$182$	0	0
$183$	−5296.00	−2.13930
$184$	0	0
$185$	1450.00	0.576249
$186$	0	0
$187$	−1288.00	−0.503679
$188$	0	0
$189$	560.000	0.215524
$190$	0	0
$191$	−3832.00	−1.45170	−0.725848	−	0.687856i	$-0.758552\pi$
$191$	−3832.00	−1.45170	−0.725848	+	0.687856i	$0.758552\pi$
$192$	0	0
$193$	1378.00	0.513941	0.256970	−	0.966419i	$-0.417276\pi$
$193$	1378.00	0.513941	0.256970	+	0.966419i	$0.417276\pi$
$194$	0	0
$195$	−3280.00	−1.20454
$196$	0	0
$197$	−4266.00	−1.54284	−0.771421	−	0.636325i	$-0.780454\pi$
$197$	−4266.00	−1.54284	−0.771421	+	0.636325i	$0.780454\pi$
$198$	0	0
$199$	5120.00	1.82386	0.911928	−	0.410351i	$-0.134594\pi$
$199$	5120.00	1.82386	0.911928	+	0.410351i	$0.134594\pi$
$200$	0	0
$201$	7008.00	2.45923
$202$	0	0
$203$	1218.00	0.421117
$204$	0	0
$205$	−250.000	−0.0851744
$206$	0	0
$207$	−4736.00	−1.59022
$208$	0	0
$209$	224.000	0.0741359
$210$	0	0
$211$	−1780.00	−0.580759	−0.290380	−	0.956911i	$-0.593782\pi$
$211$	−1780.00	−0.580759	−0.290380	+	0.956911i	$0.593782\pi$
$212$	0	0
$213$	−7040.00	−2.26466
$214$	0	0
$215$	−1980.00	−0.628069
$216$	0	0
$217$	−1064.00	−0.332853
$218$	0	0
$219$	−5104.00	−1.57487
$220$	0	0
$221$	−3772.00	−1.14811
$222$	0	0
$223$	4896.00	1.47023	0.735113	−	0.677945i	$-0.237129\pi$
$223$	4896.00	1.47023	0.735113	+	0.677945i	$0.237129\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	−3504.00	−1.02453	−0.512266	−	0.858827i	$-0.671194\pi$
$227$	−3504.00	−1.02453	−0.512266	+	0.858827i	$0.671194\pi$
$228$	0	0
$229$	826.000	0.238356	0.119178	−	0.992873i	$-0.461974\pi$
$229$	826.000	0.238356	0.119178	+	0.992873i	$0.461974\pi$
$230$	0	0
$231$	1568.00	0.446610
$232$	0	0
$233$	−278.000	−0.0781647	−0.0390824	−	0.999236i	$-0.512443\pi$
$233$	−278.000	−0.0781647	−0.0390824	+	0.999236i	$0.512443\pi$
$234$	0	0
$235$	1480.00	0.410828
$236$	0	0
$237$	−4800.00	−1.31558
$238$	0	0
$239$	4736.00	1.28178	0.640892	−	0.767631i	$-0.278565\pi$
$239$	4736.00	1.28178	0.640892	+	0.767631i	$0.278565\pi$
$240$	0	0
$241$	−3990.00	−1.06647	−0.533233	−	0.845968i	$-0.679023\pi$
$241$	−3990.00	−1.06647	−0.533233	+	0.845968i	$0.679023\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	656.000	0.168989
$248$	0	0
$249$	4992.00	1.27050
$250$	0	0
$251$	2296.00	0.577379	0.288690	−	0.957423i	$-0.406780\pi$
$251$	2296.00	0.577379	0.288690	+	0.957423i	$0.406780\pi$
$252$	0	0
$253$	−3584.00	−0.890609
$254$	0	0
$255$	1840.00	0.451864
$256$	0	0
$257$	4058.00	0.984946	0.492473	−	0.870328i	$-0.336093\pi$
$257$	4058.00	0.984946	0.492473	+	0.870328i	$0.336093\pi$
$258$	0	0
$259$	−2030.00	−0.487020
$260$	0	0
$261$	6438.00	1.52683
$262$	0	0
$263$	5640.00	1.32235	0.661174	−	0.750233i	$-0.270059\pi$
$263$	5640.00	1.32235	0.661174	+	0.750233i	$0.270059\pi$
$264$	0	0
$265$	2850.00	0.660657
$266$	0	0
$267$	5584.00	1.27991
$268$	0	0
$269$	−7758.00	−1.75841	−0.879207	−	0.476439i	$-0.841927\pi$
$269$	−7758.00	−1.75841	−0.879207	+	0.476439i	$0.841927\pi$
$270$	0	0
$271$	3264.00	0.731638	0.365819	−	0.930686i	$-0.380789\pi$
$271$	3264.00	0.731638	0.365819	+	0.930686i	$0.380789\pi$
$272$	0	0
$273$	4592.00	1.01802
$274$	0	0
$275$	700.000	0.153497
$276$	0	0
$277$	566.000	0.122771	0.0613856	−	0.998114i	$-0.480448\pi$
$277$	566.000	0.122771	0.0613856	+	0.998114i	$0.480448\pi$
$278$	0	0
$279$	−5624.00	−1.20681
$280$	0	0
$281$	4234.00	0.898859	0.449429	−	0.893316i	$-0.351627\pi$
$281$	4234.00	0.898859	0.449429	+	0.893316i	$0.351627\pi$
$282$	0	0
$283$	728.000	0.152916	0.0764578	−	0.997073i	$-0.475639\pi$
$283$	728.000	0.152916	0.0764578	+	0.997073i	$0.475639\pi$
$284$	0	0
$285$	−320.000	−0.0665093
$286$	0	0
$287$	350.000	0.0719855
$288$	0	0
$289$	−2797.00	−0.569306
$290$	0	0
$291$	6032.00	1.21513
$292$	0	0
$293$	5490.00	1.09464	0.547319	−	0.836924i	$-0.315648\pi$
$293$	5490.00	1.09464	0.547319	+	0.836924i	$0.315648\pi$
$294$	0	0
$295$	1360.00	0.268414
$296$	0	0
$297$	2240.00	0.437636
$298$	0	0
$299$	−10496.0	−2.03010
$300$	0	0
$301$	2772.00	0.530815
$302$	0	0
$303$	2832.00	0.536944
$304$	0	0
$305$	3310.00	0.621410
$306$	0	0
$307$	1664.00	0.309347	0.154673	−	0.987966i	$-0.450567\pi$
$307$	1664.00	0.309347	0.154673	+	0.987966i	$0.450567\pi$
$308$	0	0
$309$	15232.0	2.80427
$310$	0	0
$311$	1688.00	0.307774	0.153887	−	0.988088i	$-0.450821\pi$
$311$	1688.00	0.307774	0.153887	+	0.988088i	$0.450821\pi$
$312$	0	0
$313$	−2294.00	−0.414264	−0.207132	−	0.978313i	$-0.566413\pi$
$313$	−2294.00	−0.414264	−0.207132	+	0.978313i	$0.566413\pi$
$314$	0	0
$315$	−1295.00	−0.231635
$316$	0	0
$317$	−1426.00	−0.252657	−0.126328	−	0.991988i	$-0.540319\pi$
$317$	−1426.00	−0.252657	−0.126328	+	0.991988i	$0.540319\pi$
$318$	0	0
$319$	4872.00	0.855109
$320$	0	0
$321$	8864.00	1.54125
$322$	0	0
$323$	−368.000	−0.0633934
$324$	0	0
$325$	2050.00	0.349888
$326$	0	0
$327$	−1168.00	−0.197525
$328$	0	0
$329$	−2072.00	−0.347213
$330$	0	0
$331$	−740.000	−0.122882	−0.0614412	−	0.998111i	$-0.519570\pi$
$331$	−740.000	−0.122882	−0.0614412	+	0.998111i	$0.519570\pi$
$332$	0	0
$333$	−10730.0	−1.76577
$334$	0	0
$335$	−4380.00	−0.714343
$336$	0	0
$337$	−11086.0	−1.79197	−0.895984	−	0.444087i	$-0.853528\pi$
$337$	−11086.0	−1.79197	−0.895984	+	0.444087i	$0.853528\pi$
$338$	0	0
$339$	−16624.0	−2.66340
$340$	0	0
$341$	−4256.00	−0.675881
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	5120.00	0.798990
$346$	0	0
$347$	−6420.00	−0.993209	−0.496605	−	0.867977i	$-0.665420\pi$
$347$	−6420.00	−0.993209	−0.496605	+	0.867977i	$0.665420\pi$
$348$	0	0
$349$	−6430.00	−0.986218	−0.493109	−	0.869968i	$-0.664140\pi$
$349$	−6430.00	−0.986218	−0.493109	+	0.869968i	$0.664140\pi$
$350$	0	0
$351$	6560.00	0.997570
$352$	0	0
$353$	−5910.00	−0.891098	−0.445549	−	0.895258i	$-0.646991\pi$
$353$	−5910.00	−0.891098	−0.445549	+	0.895258i	$0.646991\pi$
$354$	0	0
$355$	4400.00	0.657825
$356$	0	0
$357$	−2576.00	−0.381895
$358$	0	0
$359$	−5720.00	−0.840919	−0.420460	−	0.907311i	$-0.638131\pi$
$359$	−5720.00	−0.840919	−0.420460	+	0.907311i	$0.638131\pi$
$360$	0	0
$361$	−6795.00	−0.990669
$362$	0	0
$363$	−4376.00	−0.632728
$364$	0	0
$365$	3190.00	0.457458
$366$	0	0
$367$	8320.00	1.18338	0.591690	−	0.806166i	$-0.298461\pi$
$367$	8320.00	1.18338	0.591690	+	0.806166i	$0.298461\pi$
$368$	0	0
$369$	1850.00	0.260995
$370$	0	0
$371$	−3990.00	−0.558357
$372$	0	0
$373$	1718.00	0.238484	0.119242	−	0.992865i	$-0.461954\pi$
$373$	1718.00	0.238484	0.119242	+	0.992865i	$0.461954\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	14268.0	1.94918
$378$	0	0
$379$	364.000	0.0493336	0.0246668	−	0.999696i	$-0.492148\pi$
$379$	364.000	0.0493336	0.0246668	+	0.999696i	$0.492148\pi$
$380$	0	0
$381$	−4800.00	−0.645437
$382$	0	0
$383$	2088.00	0.278569	0.139284	−	0.990252i	$-0.455520\pi$
$383$	2088.00	0.278569	0.139284	+	0.990252i	$0.455520\pi$
$384$	0	0
$385$	−980.000	−0.129728
$386$	0	0
$387$	14652.0	1.92456
$388$	0	0
$389$	1446.00	0.188471	0.0942354	−	0.995550i	$-0.469959\pi$
$389$	1446.00	0.188471	0.0942354	+	0.995550i	$0.469959\pi$
$390$	0	0
$391$	5888.00	0.761557
$392$	0	0
$393$	−1344.00	−0.172508
$394$	0	0
$395$	3000.00	0.382143
$396$	0	0
$397$	−10910.0	−1.37924	−0.689619	−	0.724173i	$-0.742222\pi$
$397$	−10910.0	−1.37924	−0.689619	+	0.724173i	$0.742222\pi$
$398$	0	0
$399$	448.000	0.0562107
$400$	0	0
$401$	3090.00	0.384806	0.192403	−	0.981316i	$-0.438372\pi$
$401$	3090.00	0.384806	0.192403	+	0.981316i	$0.438372\pi$
$402$	0	0
$403$	−12464.0	−1.54064
$404$	0	0
$405$	1795.00	0.220233
$406$	0	0
$407$	−8120.00	−0.988927
$408$	0	0
$409$	2194.00	0.265248	0.132624	−	0.991166i	$-0.457660\pi$
$409$	2194.00	0.265248	0.132624	+	0.991166i	$0.457660\pi$
$410$	0	0
$411$	3152.00	0.378289
$412$	0	0
$413$	−1904.00	−0.226852
$414$	0	0
$415$	−3120.00	−0.369048
$416$	0	0
$417$	−3776.00	−0.443433
$418$	0	0
$419$	3744.00	0.436531	0.218265	−	0.975889i	$-0.429960\pi$
$419$	3744.00	0.436531	0.218265	+	0.975889i	$0.429960\pi$
$420$	0	0
$421$	−13106.0	−1.51721	−0.758607	−	0.651548i	$-0.774120\pi$
$421$	−13106.0	−1.51721	−0.758607	+	0.651548i	$0.774120\pi$
$422$	0	0
$423$	−10952.0	−1.25888
$424$	0	0
$425$	−1150.00	−0.131255
$426$	0	0
$427$	−4634.00	−0.525187
$428$	0	0
$429$	18368.0	2.06717
$430$	0	0
$431$	12240.0	1.36794	0.683968	−	0.729512i	$-0.260253\pi$
$431$	12240.0	1.36794	0.683968	+	0.729512i	$0.260253\pi$
$432$	0	0
$433$	12402.0	1.37645	0.688224	−	0.725498i	$-0.258391\pi$
$433$	12402.0	1.37645	0.688224	+	0.725498i	$0.258391\pi$
$434$	0	0
$435$	−6960.00	−0.767141
$436$	0	0
$437$	−1024.00	−0.112093
$438$	0	0
$439$	−12280.0	−1.33506	−0.667531	−	0.744582i	$-0.732649\pi$
$439$	−12280.0	−1.33506	−0.667531	+	0.744582i	$0.732649\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	0	0
$443$	3092.00	0.331615	0.165807	−	0.986158i	$-0.446977\pi$
$443$	3092.00	0.331615	0.165807	+	0.986158i	$0.446977\pi$
$444$	0	0
$445$	−3490.00	−0.371779
$446$	0	0
$447$	−21264.0	−2.25001
$448$	0	0
$449$	−14750.0	−1.55032	−0.775162	−	0.631762i	$-0.782332\pi$
$449$	−14750.0	−1.55032	−0.775162	+	0.631762i	$0.782332\pi$
$450$	0	0
$451$	1400.00	0.146172
$452$	0	0
$453$	10048.0	1.04216
$454$	0	0
$455$	−2870.00	−0.295709
$456$	0	0
$457$	6394.00	0.654483	0.327241	−	0.944941i	$-0.393881\pi$
$457$	6394.00	0.654483	0.327241	+	0.944941i	$0.393881\pi$
$458$	0	0
$459$	−3680.00	−0.374222
$460$	0	0
$461$	−14590.0	−1.47402	−0.737011	−	0.675881i	$-0.763763\pi$
$461$	−14590.0	−1.47402	−0.737011	+	0.675881i	$0.763763\pi$
$462$	0	0
$463$	13664.0	1.37153	0.685767	−	0.727821i	$-0.259467\pi$
$463$	13664.0	1.37153	0.685767	+	0.727821i	$0.259467\pi$
$464$	0	0
$465$	6080.00	0.606351
$466$	0	0
$467$	4960.00	0.491481	0.245740	−	0.969336i	$-0.420969\pi$
$467$	4960.00	0.491481	0.245740	+	0.969336i	$0.420969\pi$
$468$	0	0
$469$	6132.00	0.603730
$470$	0	0
$471$	26128.0	2.55608
$472$	0	0
$473$	11088.0	1.07786
$474$	0	0
$475$	200.000	0.0193192
$476$	0	0
$477$	−21090.0	−2.02441
$478$	0	0
$479$	5304.00	0.505941	0.252971	−	0.967474i	$-0.418592\pi$
$479$	5304.00	0.505941	0.252971	+	0.967474i	$0.418592\pi$
$480$	0	0
$481$	−23780.0	−2.25421
$482$	0	0
$483$	−7168.00	−0.675270
$484$	0	0
$485$	−3770.00	−0.352963
$486$	0	0
$487$	−21120.0	−1.96517	−0.982586	−	0.185810i	$-0.940509\pi$
$487$	−21120.0	−1.96517	−0.982586	+	0.185810i	$0.940509\pi$
$488$	0	0
$489$	13728.0	1.26953
$490$	0	0
$491$	10500.0	0.965088	0.482544	−	0.875872i	$-0.339713\pi$
$491$	10500.0	0.965088	0.482544	+	0.875872i	$0.339713\pi$
$492$	0	0
$493$	−8004.00	−0.731201
$494$	0	0
$495$	−5180.00	−0.470351
$496$	0	0
$497$	−6160.00	−0.555963
$498$	0	0
$499$	−3620.00	−0.324756	−0.162378	−	0.986729i	$-0.551916\pi$
$499$	−3620.00	−0.324756	−0.162378	+	0.986729i	$0.551916\pi$
$500$	0	0
$501$	−18176.0	−1.62085
$502$	0	0
$503$	11176.0	0.990682	0.495341	−	0.868699i	$-0.335043\pi$
$503$	11176.0	0.990682	0.495341	+	0.868699i	$0.335043\pi$
$504$	0	0
$505$	−1770.00	−0.155968
$506$	0	0
$507$	36216.0	3.17240
$508$	0	0
$509$	3362.00	0.292766	0.146383	−	0.989228i	$-0.453237\pi$
$509$	3362.00	0.292766	0.146383	+	0.989228i	$0.453237\pi$
$510$	0	0
$511$	−4466.00	−0.386623
$512$	0	0
$513$	640.000	0.0550813
$514$	0	0
$515$	−9520.00	−0.814566
$516$	0	0
$517$	−8288.00	−0.705040
$518$	0	0
$519$	33808.0	2.85936
$520$	0	0
$521$	16530.0	1.39000	0.695002	−	0.719007i	$-0.255403\pi$
$521$	16530.0	1.39000	0.695002	+	0.719007i	$0.255403\pi$
$522$	0	0
$523$	6440.00	0.538435	0.269218	−	0.963079i	$-0.413235\pi$
$523$	6440.00	0.538435	0.269218	+	0.963079i	$0.413235\pi$
$524$	0	0
$525$	1400.00	0.116383
$526$	0	0
$527$	6992.00	0.577944
$528$	0	0
$529$	4217.00	0.346593
$530$	0	0
$531$	−10064.0	−0.822487
$532$	0	0
$533$	4100.00	0.333191
$534$	0	0
$535$	−5540.00	−0.447692
$536$	0	0
$537$	7904.00	0.635163
$538$	0	0
$539$	1372.00	0.109640
$540$	0	0
$541$	−7210.00	−0.572980	−0.286490	−	0.958083i	$-0.592488\pi$
$541$	−7210.00	−0.572980	−0.286490	+	0.958083i	$0.592488\pi$
$542$	0	0
$543$	17104.0	1.35175
$544$	0	0
$545$	730.000	0.0573757
$546$	0	0
$547$	21796.0	1.70371	0.851855	−	0.523777i	$-0.175478\pi$
$547$	21796.0	1.70371	0.851855	+	0.523777i	$0.175478\pi$
$548$	0	0
$549$	−24494.0	−1.90415
$550$	0	0
$551$	1392.00	0.107625
$552$	0	0
$553$	−4200.00	−0.322970
$554$	0	0
$555$	11600.0	0.887194
$556$	0	0
$557$	10030.0	0.762989	0.381494	−	0.924371i	$-0.375410\pi$
$557$	10030.0	0.762989	0.381494	+	0.924371i	$0.375410\pi$
$558$	0	0
$559$	32472.0	2.45692
$560$	0	0
$561$	−10304.0	−0.775464
$562$	0	0
$563$	9144.00	0.684500	0.342250	−	0.939609i	$-0.388811\pi$
$563$	9144.00	0.684500	0.342250	+	0.939609i	$0.388811\pi$
$564$	0	0
$565$	10390.0	0.773647
$566$	0	0
$567$	−2513.00	−0.186131
$568$	0	0
$569$	11946.0	0.880145	0.440072	−	0.897962i	$-0.354953\pi$
$569$	11946.0	0.880145	0.440072	+	0.897962i	$0.354953\pi$
$570$	0	0
$571$	11596.0	0.849873	0.424937	−	0.905223i	$-0.360296\pi$
$571$	11596.0	0.849873	0.424937	+	0.905223i	$0.360296\pi$
$572$	0	0
$573$	−30656.0	−2.23503
$574$	0	0
$575$	−3200.00	−0.232086
$576$	0	0
$577$	1754.00	0.126551	0.0632755	−	0.997996i	$-0.479845\pi$
$577$	1754.00	0.126551	0.0632755	+	0.997996i	$0.479845\pi$
$578$	0	0
$579$	11024.0	0.791264
$580$	0	0
$581$	4368.00	0.311902
$582$	0	0
$583$	−15960.0	−1.13378
$584$	0	0
$585$	−15170.0	−1.07214
$586$	0	0
$587$	13152.0	0.924772	0.462386	−	0.886679i	$-0.346993\pi$
$587$	13152.0	0.924772	0.462386	+	0.886679i	$0.346993\pi$
$588$	0	0
$589$	−1216.00	−0.0850669
$590$	0	0
$591$	−34128.0	−2.37536
$592$	0	0
$593$	−13014.0	−0.901216	−0.450608	−	0.892722i	$-0.648793\pi$
$593$	−13014.0	−0.901216	−0.450608	+	0.892722i	$0.648793\pi$
$594$	0	0
$595$	1610.00	0.110930
$596$	0	0
$597$	40960.0	2.80801
$598$	0	0
$599$	−19936.0	−1.35987	−0.679936	−	0.733272i	$-0.737992\pi$
$599$	−19936.0	−1.35987	−0.679936	+	0.733272i	$0.737992\pi$
$600$	0	0
$601$	15578.0	1.05730	0.528652	−	0.848839i	$-0.322698\pi$
$601$	15578.0	1.05730	0.528652	+	0.848839i	$0.322698\pi$
$602$	0	0
$603$	32412.0	2.18892
$604$	0	0
$605$	2735.00	0.183791
$606$	0	0
$607$	−3792.00	−0.253563	−0.126781	−	0.991931i	$-0.540465\pi$
$607$	−3792.00	−0.253563	−0.126781	+	0.991931i	$0.540465\pi$
$608$	0	0
$609$	9744.00	0.648353
$610$	0	0
$611$	−24272.0	−1.60710
$612$	0	0
$613$	−5074.00	−0.334318	−0.167159	−	0.985930i	$-0.553459\pi$
$613$	−5074.00	−0.334318	−0.167159	+	0.985930i	$0.553459\pi$
$614$	0	0
$615$	−2000.00	−0.131135
$616$	0	0
$617$	−13670.0	−0.891951	−0.445975	−	0.895045i	$-0.647143\pi$
$617$	−13670.0	−0.891951	−0.445975	+	0.895045i	$0.647143\pi$
$618$	0	0
$619$	−18560.0	−1.20515	−0.602576	−	0.798061i	$-0.705859\pi$
$619$	−18560.0	−1.20515	−0.602576	+	0.798061i	$0.705859\pi$
$620$	0	0
$621$	−10240.0	−0.661702
$622$	0	0
$623$	4886.00	0.314211
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1792.00	0.114140
$628$	0	0
$629$	13340.0	0.845629
$630$	0	0
$631$	−8400.00	−0.529950	−0.264975	−	0.964255i	$-0.585364\pi$
$631$	−8400.00	−0.529950	−0.264975	+	0.964255i	$0.585364\pi$
$632$	0	0
$633$	−14240.0	−0.894138
$634$	0	0
$635$	3000.00	0.187482
$636$	0	0
$637$	4018.00	0.249920
$638$	0	0
$639$	−32560.0	−2.01573
$640$	0	0
$641$	16770.0	1.03335	0.516673	−	0.856183i	$-0.327170\pi$
$641$	16770.0	1.03335	0.516673	+	0.856183i	$0.327170\pi$
$642$	0	0
$643$	−12152.0	−0.745300	−0.372650	−	0.927972i	$-0.621551\pi$
$643$	−12152.0	−0.745300	−0.372650	+	0.927972i	$0.621551\pi$
$644$	0	0
$645$	−15840.0	−0.966976
$646$	0	0
$647$	14896.0	0.905135	0.452567	−	0.891730i	$-0.350508\pi$
$647$	14896.0	0.905135	0.452567	+	0.891730i	$0.350508\pi$
$648$	0	0
$649$	−7616.00	−0.460638
$650$	0	0
$651$	−8512.00	−0.512460
$652$	0	0
$653$	11942.0	0.715661	0.357830	−	0.933787i	$-0.383517\pi$
$653$	11942.0	0.715661	0.357830	+	0.933787i	$0.383517\pi$
$654$	0	0
$655$	840.000	0.0501092
$656$	0	0
$657$	−23606.0	−1.40176
$658$	0	0
$659$	−31772.0	−1.87809	−0.939045	−	0.343794i	$-0.888288\pi$
$659$	−31772.0	−1.87809	−0.939045	+	0.343794i	$0.888288\pi$
$660$	0	0
$661$	−9502.00	−0.559130	−0.279565	−	0.960127i	$-0.590190\pi$
$661$	−9502.00	−0.559130	−0.279565	+	0.960127i	$0.590190\pi$
$662$	0	0
$663$	−30176.0	−1.76763
$664$	0	0
$665$	−280.000	−0.0163277
$666$	0	0
$667$	−22272.0	−1.29292
$668$	0	0
$669$	39168.0	2.26356
$670$	0	0
$671$	−18536.0	−1.06643
$672$	0	0
$673$	−4750.00	−0.272064	−0.136032	−	0.990704i	$-0.543435\pi$
$673$	−4750.00	−0.272064	−0.136032	+	0.990704i	$0.543435\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	23610.0	1.34033	0.670167	−	0.742210i	$-0.266223\pi$
$677$	23610.0	1.34033	0.670167	+	0.742210i	$0.266223\pi$
$678$	0	0
$679$	5278.00	0.298308
$680$	0	0
$681$	−28032.0	−1.57737
$682$	0	0
$683$	26852.0	1.50434	0.752169	−	0.658970i	$-0.229008\pi$
$683$	26852.0	1.50434	0.752169	+	0.658970i	$0.229008\pi$
$684$	0	0
$685$	−1970.00	−0.109883
$686$	0	0
$687$	6608.00	0.366974
$688$	0	0
$689$	−46740.0	−2.58440
$690$	0	0
$691$	−2648.00	−0.145781	−0.0728905	−	0.997340i	$-0.523222\pi$
$691$	−2648.00	−0.145781	−0.0728905	+	0.997340i	$0.523222\pi$
$692$	0	0
$693$	7252.00	0.397519
$694$	0	0
$695$	2360.00	0.128806
$696$	0	0
$697$	−2300.00	−0.124991
$698$	0	0
$699$	−2224.00	−0.120342
$700$	0	0
$701$	29118.0	1.56886	0.784431	−	0.620217i	$-0.212955\pi$
$701$	29118.0	1.56886	0.784431	+	0.620217i	$0.212955\pi$
$702$	0	0
$703$	−2320.00	−0.124467
$704$	0	0
$705$	11840.0	0.632511
$706$	0	0
$707$	2478.00	0.131817
$708$	0	0
$709$	−14474.0	−0.766689	−0.383344	−	0.923605i	$-0.625228\pi$
$709$	−14474.0	−0.766689	−0.383344	+	0.923605i	$0.625228\pi$
$710$	0	0
$711$	−22200.0	−1.17098
$712$	0	0
$713$	19456.0	1.02193
$714$	0	0
$715$	−11480.0	−0.600458
$716$	0	0
$717$	37888.0	1.97344
$718$	0	0
$719$	10968.0	0.568898	0.284449	−	0.958691i	$-0.408189\pi$
$719$	10968.0	0.568898	0.284449	+	0.958691i	$0.408189\pi$
$720$	0	0
$721$	13328.0	0.688434
$722$	0	0
$723$	−31920.0	−1.64193
$724$	0	0
$725$	4350.00	0.222834
$726$	0	0
$727$	18624.0	0.950104	0.475052	−	0.879958i	$-0.342429\pi$
$727$	18624.0	0.950104	0.475052	+	0.879958i	$0.342429\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−18216.0	−0.921673
$732$	0	0
$733$	−37014.0	−1.86513	−0.932567	−	0.360997i	$-0.882437\pi$
$733$	−37014.0	−1.86513	−0.932567	+	0.360997i	$0.882437\pi$
$734$	0	0
$735$	−1960.00	−0.0983615
$736$	0	0
$737$	24528.0	1.22592
$738$	0	0
$739$	1284.00	0.0639143	0.0319572	−	0.999489i	$-0.489826\pi$
$739$	1284.00	0.0639143	0.0319572	+	0.999489i	$0.489826\pi$
$740$	0	0
$741$	5248.00	0.260176
$742$	0	0
$743$	25584.0	1.26324	0.631619	−	0.775279i	$-0.282391\pi$
$743$	25584.0	1.26324	0.631619	+	0.775279i	$0.282391\pi$
$744$	0	0
$745$	13290.0	0.653568
$746$	0	0
$747$	23088.0	1.13085
$748$	0	0
$749$	7756.00	0.378369
$750$	0	0
$751$	−3872.00	−0.188138	−0.0940688	−	0.995566i	$-0.529987\pi$
$751$	−3872.00	−0.188138	−0.0940688	+	0.995566i	$0.529987\pi$
$752$	0	0
$753$	18368.0	0.888934
$754$	0	0
$755$	−6280.00	−0.302719
$756$	0	0
$757$	22238.0	1.06771	0.533853	−	0.845577i	$-0.320743\pi$
$757$	22238.0	1.06771	0.533853	+	0.845577i	$0.320743\pi$
$758$	0	0
$759$	−28672.0	−1.37118
$760$	0	0
$761$	10178.0	0.484826	0.242413	−	0.970173i	$-0.422061\pi$
$761$	10178.0	0.484826	0.242413	+	0.970173i	$0.422061\pi$
$762$	0	0
$763$	−1022.00	−0.0484913
$764$	0	0
$765$	8510.00	0.402196
$766$	0	0
$767$	−22304.0	−1.05000
$768$	0	0
$769$	31498.0	1.47704	0.738522	−	0.674229i	$-0.235524\pi$
$769$	31498.0	1.47704	0.738522	+	0.674229i	$0.235524\pi$
$770$	0	0
$771$	32464.0	1.51642
$772$	0	0
$773$	19090.0	0.888253	0.444127	−	0.895964i	$-0.353514\pi$
$773$	19090.0	0.888253	0.444127	+	0.895964i	$0.353514\pi$
$774$	0	0
$775$	−3800.00	−0.176129
$776$	0	0
$777$	−16240.0	−0.749816
$778$	0	0
$779$	400.000	0.0183973
$780$	0	0
$781$	−24640.0	−1.12892
$782$	0	0
$783$	13920.0	0.635326
$784$	0	0
$785$	−16330.0	−0.742475
$786$	0	0
$787$	23360.0	1.05806	0.529031	−	0.848603i	$-0.322556\pi$
$787$	23360.0	1.05806	0.529031	+	0.848603i	$0.322556\pi$
$788$	0	0
$789$	45120.0	2.03589
$790$	0	0
$791$	−14546.0	−0.653851
$792$	0	0
$793$	−54284.0	−2.43087
$794$	0	0
$795$	22800.0	1.01715
$796$	0	0
$797$	−28686.0	−1.27492	−0.637459	−	0.770484i	$-0.720015\pi$
$797$	−28686.0	−1.27492	−0.637459	+	0.770484i	$0.720015\pi$
$798$	0	0
$799$	13616.0	0.602878
$800$	0	0
$801$	25826.0	1.13922
$802$	0	0
$803$	−17864.0	−0.785065
$804$	0	0
$805$	4480.00	0.196148
$806$	0	0
$807$	−62064.0	−2.70726
$808$	0	0
$809$	33226.0	1.44396	0.721980	−	0.691914i	$-0.243232\pi$
$809$	33226.0	1.44396	0.721980	+	0.691914i	$0.243232\pi$
$810$	0	0
$811$	−33024.0	−1.42988	−0.714938	−	0.699188i	$-0.753545\pi$
$811$	−33024.0	−1.42988	−0.714938	+	0.699188i	$0.753545\pi$
$812$	0	0
$813$	26112.0	1.12643
$814$	0	0
$815$	−8580.00	−0.368766
$816$	0	0
$817$	3168.00	0.135660
$818$	0	0
$819$	21238.0	0.906124
$820$	0	0
$821$	−17090.0	−0.726486	−0.363243	−	0.931694i	$-0.618331\pi$
$821$	−17090.0	−0.726486	−0.363243	+	0.931694i	$0.618331\pi$
$822$	0	0
$823$	−11720.0	−0.496396	−0.248198	−	0.968709i	$-0.579838\pi$
$823$	−11720.0	−0.496396	−0.248198	+	0.968709i	$0.579838\pi$
$824$	0	0
$825$	5600.00	0.236324
$826$	0	0
$827$	−34764.0	−1.46174	−0.730872	−	0.682514i	$-0.760886\pi$
$827$	−34764.0	−1.46174	−0.730872	+	0.682514i	$0.760886\pi$
$828$	0	0
$829$	650.000	0.0272321	0.0136161	−	0.999907i	$-0.495666\pi$
$829$	650.000	0.0272321	0.0136161	+	0.999907i	$0.495666\pi$
$830$	0	0
$831$	4528.00	0.189019
$832$	0	0
$833$	−2254.00	−0.0937533
$834$	0	0
$835$	11360.0	0.470813
$836$	0	0
$837$	−12160.0	−0.502164
$838$	0	0
$839$	−33888.0	−1.39445	−0.697225	−	0.716852i	$-0.745582\pi$
$839$	−33888.0	−1.39445	−0.697225	+	0.716852i	$0.745582\pi$
$840$	0	0
$841$	5887.00	0.241379
$842$	0	0
$843$	33872.0	1.38388
$844$	0	0
$845$	−22635.0	−0.921500
$846$	0	0
$847$	−3829.00	−0.155332
$848$	0	0
$849$	5824.00	0.235429
$850$	0	0
$851$	37120.0	1.49525
$852$	0	0
$853$	−14246.0	−0.571833	−0.285917	−	0.958255i	$-0.592298\pi$
$853$	−14246.0	−0.571833	−0.285917	+	0.958255i	$0.592298\pi$
$854$	0	0
$855$	−1480.00	−0.0591988
$856$	0	0
$857$	362.000	0.0144290	0.00721452	−	0.999974i	$-0.497704\pi$
$857$	362.000	0.0144290	0.00721452	+	0.999974i	$0.497704\pi$
$858$	0	0
$859$	25544.0	1.01461	0.507305	−	0.861767i	$-0.330642\pi$
$859$	25544.0	1.01461	0.507305	+	0.861767i	$0.330642\pi$
$860$	0	0
$861$	2800.00	0.110829
$862$	0	0
$863$	18576.0	0.732717	0.366358	−	0.930474i	$-0.380604\pi$
$863$	18576.0	0.732717	0.366358	+	0.930474i	$0.380604\pi$
$864$	0	0
$865$	−21130.0	−0.830568
$866$	0	0
$867$	−22376.0	−0.876504
$868$	0	0
$869$	−16800.0	−0.655812
$870$	0	0
$871$	71832.0	2.79441
$872$	0	0
$873$	27898.0	1.08156
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	29606.0	1.13994	0.569968	−	0.821667i	$-0.306956\pi$
$877$	29606.0	1.13994	0.569968	+	0.821667i	$0.306956\pi$
$878$	0	0
$879$	43920.0	1.68531
$880$	0	0
$881$	45378.0	1.73533	0.867664	−	0.497151i	$-0.165621\pi$
$881$	45378.0	1.73533	0.867664	+	0.497151i	$0.165621\pi$
$882$	0	0
$883$	22540.0	0.859039	0.429519	−	0.903058i	$-0.358683\pi$
$883$	22540.0	0.859039	0.429519	+	0.903058i	$0.358683\pi$
$884$	0	0
$885$	10880.0	0.413251
$886$	0	0
$887$	−20336.0	−0.769804	−0.384902	−	0.922957i	$-0.625765\pi$
$887$	−20336.0	−0.769804	−0.384902	+	0.922957i	$0.625765\pi$
$888$	0	0
$889$	−4200.00	−0.158452
$890$	0	0
$891$	−10052.0	−0.377951
$892$	0	0
$893$	−2368.00	−0.0887370
$894$	0	0
$895$	−4940.00	−0.184498
$896$	0	0
$897$	−83968.0	−3.12554
$898$	0	0
$899$	−26448.0	−0.981190
$900$	0	0
$901$	26220.0	0.969495
$902$	0	0
$903$	22176.0	0.817244
$904$	0	0
$905$	−10690.0	−0.392649
$906$	0	0
$907$	−34764.0	−1.27268	−0.636339	−	0.771409i	$-0.719552\pi$
$907$	−34764.0	−1.27268	−0.636339	+	0.771409i	$0.719552\pi$
$908$	0	0
$909$	13098.0	0.477924
$910$	0	0
$911$	−240.000	−0.00872838	−0.00436419	−	0.999990i	$-0.501389\pi$
$911$	−240.000	−0.00872838	−0.00436419	+	0.999990i	$0.501389\pi$
$912$	0	0
$913$	17472.0	0.633339
$914$	0	0
$915$	26480.0	0.956723
$916$	0	0
$917$	−1176.00	−0.0423500
$918$	0	0
$919$	37264.0	1.33757	0.668785	−	0.743456i	$-0.266815\pi$
$919$	37264.0	1.33757	0.668785	+	0.743456i	$0.266815\pi$
$920$	0	0
$921$	13312.0	0.476271
$922$	0	0
$923$	−72160.0	−2.57332
$924$	0	0
$925$	−7250.00	−0.257707
$926$	0	0
$927$	70448.0	2.49603
$928$	0	0
$929$	−13286.0	−0.469214	−0.234607	−	0.972090i	$-0.575380\pi$
$929$	−13286.0	−0.469214	−0.234607	+	0.972090i	$0.575380\pi$
$930$	0	0
$931$	392.000	0.0137994
$932$	0	0
$933$	13504.0	0.473849
$934$	0	0
$935$	6440.00	0.225252
$936$	0	0
$937$	−13614.0	−0.474653	−0.237327	−	0.971430i	$-0.576271\pi$
$937$	−13614.0	−0.474653	−0.237327	+	0.971430i	$0.576271\pi$
$938$	0	0
$939$	−18352.0	−0.637801
$940$	0	0
$941$	36570.0	1.26689	0.633447	−	0.773786i	$-0.281639\pi$
$941$	36570.0	1.26689	0.633447	+	0.773786i	$0.281639\pi$
$942$	0	0
$943$	−6400.00	−0.221010
$944$	0	0
$945$	−2800.00	−0.0963852
$946$	0	0
$947$	8772.00	0.301005	0.150502	−	0.988610i	$-0.451911\pi$
$947$	8772.00	0.301005	0.150502	+	0.988610i	$0.451911\pi$
$948$	0	0
$949$	−52316.0	−1.78951
$950$	0	0
$951$	−11408.0	−0.388990
$952$	0	0
$953$	−8742.00	−0.297147	−0.148574	−	0.988901i	$-0.547468\pi$
$953$	−8742.00	−0.297147	−0.148574	+	0.988901i	$0.547468\pi$
$954$	0	0
$955$	19160.0	0.649218
$956$	0	0
$957$	38976.0	1.31653
$958$	0	0
$959$	2758.00	0.0928681
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	40996.0	1.37184
$964$	0	0
$965$	−6890.00	−0.229841
$966$	0	0
$967$	−51248.0	−1.70427	−0.852133	−	0.523326i	$-0.824691\pi$
$967$	−51248.0	−1.70427	−0.852133	+	0.523326i	$0.824691\pi$
$968$	0	0
$969$	−2944.00	−0.0976005
$970$	0	0
$971$	41376.0	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$971$	41376.0	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$972$	0	0
$973$	−3304.00	−0.108861
$974$	0	0
$975$	16400.0	0.538688
$976$	0	0
$977$	15234.0	0.498852	0.249426	−	0.968394i	$-0.419758\pi$
$977$	15234.0	0.498852	0.249426	+	0.968394i	$0.419758\pi$
$978$	0	0
$979$	19544.0	0.638028
$980$	0	0
$981$	−5402.00	−0.175813
$982$	0	0
$983$	−22384.0	−0.726286	−0.363143	−	0.931733i	$-0.618296\pi$
$983$	−22384.0	−0.726286	−0.363143	+	0.931733i	$0.618296\pi$
$984$	0	0
$985$	21330.0	0.689980
$986$	0	0
$987$	−16576.0	−0.534569
$988$	0	0
$989$	−50688.0	−1.62971
$990$	0	0
$991$	−3656.00	−0.117191	−0.0585957	−	0.998282i	$-0.518662\pi$
$991$	−3656.00	−0.117191	−0.0585957	+	0.998282i	$0.518662\pi$
$992$	0	0
$993$	−5920.00	−0.189190
$994$	0	0
$995$	−25600.0	−0.815653
$996$	0	0
$997$	−15406.0	−0.489381	−0.244691	−	0.969601i	$-0.578686\pi$
$997$	−15406.0	−0.489381	−0.244691	+	0.969601i	$0.578686\pi$
$998$	0	0
$999$	−23200.0	−0.734750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.4.a.e.1.1	✓	1
3.2	odd	2		1260.4.a.j.1.1		1
4.3	odd	2		560.4.a.b.1.1		1
5.2	odd	4		700.4.e.c.449.1		2
5.3	odd	4		700.4.e.c.449.2		2
5.4	even	2		700.4.a.b.1.1		1
7.2	even	3		980.4.i.b.361.1		2
7.3	odd	6		980.4.i.q.961.1		2
7.4	even	3		980.4.i.b.961.1		2
7.5	odd	6		980.4.i.q.361.1		2
7.6	odd	2		980.4.a.b.1.1		1
8.3	odd	2		2240.4.a.bj.1.1		1
8.5	even	2		2240.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.4.a.e.1.1	✓	1	1.1	even	1	trivial
560.4.a.b.1.1		1	4.3	odd	2
700.4.a.b.1.1		1	5.4	even	2
700.4.e.c.449.1		2	5.2	odd	4
700.4.e.c.449.2		2	5.3	odd	4
980.4.a.b.1.1		1	7.6	odd	2
980.4.i.b.361.1		2	7.2	even	3
980.4.i.b.961.1		2	7.4	even	3
980.4.i.q.361.1		2	7.5	odd	6
980.4.i.q.961.1		2	7.3	odd	6
1260.4.a.j.1.1		1	3.2	odd	2
2240.4.a.c.1.1		1	8.5	even	2
2240.4.a.bj.1.1		1	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

Introduction

L-functions

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Representations

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Database

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