LMFDB - Embedded newform 600.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.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:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -10.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -10.0000 q^{7} +9.00000 q^{9} -46.0000 q^{11} +34.0000 q^{13} -66.0000 q^{17} +104.000 q^{19} +30.0000 q^{21} -164.000 q^{23} -27.0000 q^{27} +224.000 q^{29} -72.0000 q^{31} +138.000 q^{33} +22.0000 q^{37} -102.000 q^{39} +194.000 q^{41} -108.000 q^{43} +480.000 q^{47} -243.000 q^{49} +198.000 q^{51} -286.000 q^{53} -312.000 q^{57} +426.000 q^{59} +698.000 q^{61} -90.0000 q^{63} -328.000 q^{67} +492.000 q^{69} +188.000 q^{71} +740.000 q^{73} +460.000 q^{77} +1168.00 q^{79} +81.0000 q^{81} -412.000 q^{83} -672.000 q^{87} +1206.00 q^{89} -340.000 q^{91} +216.000 q^{93} +1384.00 q^{97} -414.000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−10.0000	−0.539949	−0.269975	−	0.962867i	$-0.587015\pi$
$7$	−10.0000	−0.539949	−0.269975	+	0.962867i	$0.587015\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−46.0000	−1.26087	−0.630433	−	0.776244i	$-0.717123\pi$
$11$	−46.0000	−1.26087	−0.630433	+	0.776244i	$0.717123\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	104.000	1.25575	0.627875	−	0.778314i	$-0.283925\pi$
$19$	104.000	1.25575	0.627875	+	0.778314i	$0.283925\pi$
$20$	0	0
$21$	30.0000	0.311740
$22$	0	0
$23$	−164.000	−1.48680	−0.743399	−	0.668848i	$-0.766788\pi$
$23$	−164.000	−1.48680	−0.743399	+	0.668848i	$0.766788\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	224.000	1.43434	0.717168	−	0.696900i	$-0.245438\pi$
$29$	224.000	1.43434	0.717168	+	0.696900i	$0.245438\pi$
$30$	0	0
$31$	−72.0000	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	0	0
$33$	138.000	0.727961
$34$	0	0
$35$	0	0
$36$	0	0
$37$	22.0000	0.0977507	0.0488754	−	0.998805i	$-0.484436\pi$
$37$	22.0000	0.0977507	0.0488754	+	0.998805i	$0.484436\pi$
$38$	0	0
$39$	−102.000	−0.418797
$40$	0	0
$41$	194.000	0.738969	0.369484	−	0.929237i	$-0.379534\pi$
$41$	194.000	0.738969	0.369484	+	0.929237i	$0.379534\pi$
$42$	0	0
$43$	−108.000	−0.383020	−0.191510	−	0.981491i	$-0.561338\pi$
$43$	−108.000	−0.383020	−0.191510	+	0.981491i	$0.561338\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	480.000	1.48969	0.744843	−	0.667240i	$-0.232525\pi$
$47$	480.000	1.48969	0.744843	+	0.667240i	$0.232525\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	198.000	0.543638
$52$	0	0
$53$	−286.000	−0.741229	−0.370614	−	0.928787i	$-0.620853\pi$
$53$	−286.000	−0.741229	−0.370614	+	0.928787i	$0.620853\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−312.000	−0.725007
$58$	0	0
$59$	426.000	0.940008	0.470004	−	0.882664i	$-0.344252\pi$
$59$	426.000	0.940008	0.470004	+	0.882664i	$0.344252\pi$
$60$	0	0
$61$	698.000	1.46508	0.732539	−	0.680725i	$-0.238335\pi$
$61$	698.000	1.46508	0.732539	+	0.680725i	$0.238335\pi$
$62$	0	0
$63$	−90.0000	−0.179983
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−328.000	−0.598083	−0.299042	−	0.954240i	$-0.596667\pi$
$67$	−328.000	−0.598083	−0.299042	+	0.954240i	$0.596667\pi$
$68$	0	0
$69$	492.000	0.858403
$70$	0	0
$71$	188.000	0.314246	0.157123	−	0.987579i	$-0.449778\pi$
$71$	188.000	0.314246	0.157123	+	0.987579i	$0.449778\pi$
$72$	0	0
$73$	740.000	1.18644	0.593222	−	0.805039i	$-0.297856\pi$
$73$	740.000	1.18644	0.593222	+	0.805039i	$0.297856\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	460.000	0.680803
$78$	0	0
$79$	1168.00	1.66342	0.831711	−	0.555209i	$-0.187362\pi$
$79$	1168.00	1.66342	0.831711	+	0.555209i	$0.187362\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−412.000	−0.544854	−0.272427	−	0.962176i	$-0.587826\pi$
$83$	−412.000	−0.544854	−0.272427	+	0.962176i	$0.587826\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−672.000	−0.828115
$88$	0	0
$89$	1206.00	1.43636	0.718178	−	0.695859i	$-0.244976\pi$
$89$	1206.00	1.43636	0.718178	+	0.695859i	$0.244976\pi$
$90$	0	0
$91$	−340.000	−0.391667
$92$	0	0
$93$	216.000	0.240840
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1384.00	1.44870	0.724350	−	0.689432i	$-0.242140\pi$
$97$	1384.00	1.44870	0.724350	+	0.689432i	$0.242140\pi$
$98$	0	0
$99$	−414.000	−0.420289
$100$	0	0
$101$	−1128.00	−1.11129	−0.555645	−	0.831420i	$-0.687528\pi$
$101$	−1128.00	−1.11129	−0.555645	+	0.831420i	$0.687528\pi$
$102$	0	0
$103$	758.000	0.725126	0.362563	−	0.931959i	$-0.381902\pi$
$103$	758.000	0.725126	0.362563	+	0.931959i	$0.381902\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1324.00	−1.19622	−0.598112	−	0.801413i	$-0.704082\pi$
$107$	−1324.00	−1.19622	−0.598112	+	0.801413i	$0.704082\pi$
$108$	0	0
$109$	1602.00	1.40774	0.703871	−	0.710328i	$-0.251454\pi$
$109$	1602.00	1.40774	0.703871	+	0.710328i	$0.251454\pi$
$110$	0	0
$111$	−66.0000	−0.0564364
$112$	0	0
$113$	2074.00	1.72660	0.863299	−	0.504693i	$-0.168394\pi$
$113$	2074.00	1.72660	0.863299	+	0.504693i	$0.168394\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	306.000	0.241792
$118$	0	0
$119$	660.000	0.508421
$120$	0	0
$121$	785.000	0.589782
$122$	0	0
$123$	−582.000	−0.426644
$124$	0	0
$125$	0	0
$126$	0	0
$127$	534.000	0.373109	0.186554	−	0.982445i	$-0.440268\pi$
$127$	534.000	0.373109	0.186554	+	0.982445i	$0.440268\pi$
$128$	0	0
$129$	324.000	0.221137
$130$	0	0
$131$	1806.00	1.20451	0.602256	−	0.798303i	$-0.294269\pi$
$131$	1806.00	1.20451	0.602256	+	0.798303i	$0.294269\pi$
$132$	0	0
$133$	−1040.00	−0.678041
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1822.00	−1.13623	−0.568117	−	0.822948i	$-0.692328\pi$
$137$	−1822.00	−1.13623	−0.568117	+	0.822948i	$0.692328\pi$
$138$	0	0
$139$	532.000	0.324631	0.162315	−	0.986739i	$-0.448104\pi$
$139$	532.000	0.324631	0.162315	+	0.986739i	$0.448104\pi$
$140$	0	0
$141$	−1440.00	−0.860070
$142$	0	0
$143$	−1564.00	−0.914603
$144$	0	0
$145$	0	0
$146$	0	0
$147$	729.000	0.409027
$148$	0	0
$149$	−1284.00	−0.705969	−0.352984	−	0.935629i	$-0.614833\pi$
$149$	−1284.00	−0.705969	−0.352984	+	0.935629i	$0.614833\pi$
$150$	0	0
$151$	184.000	0.0991636	0.0495818	−	0.998770i	$-0.484211\pi$
$151$	184.000	0.0991636	0.0495818	+	0.998770i	$0.484211\pi$
$152$	0	0
$153$	−594.000	−0.313870
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3746.00	1.90423	0.952113	−	0.305748i	$-0.0989064\pi$
$157$	3746.00	1.90423	0.952113	+	0.305748i	$0.0989064\pi$
$158$	0	0
$159$	858.000	0.427949
$160$	0	0
$161$	1640.00	0.802796
$162$	0	0
$163$	−1504.00	−0.722714	−0.361357	−	0.932428i	$-0.617686\pi$
$163$	−1504.00	−0.722714	−0.361357	+	0.932428i	$0.617686\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3012.00	−1.39566	−0.697831	−	0.716262i	$-0.745851\pi$
$167$	−3012.00	−1.39566	−0.697831	+	0.716262i	$0.745851\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	936.000	0.418583
$172$	0	0
$173$	438.000	0.192489	0.0962443	−	0.995358i	$-0.469317\pi$
$173$	438.000	0.192489	0.0962443	+	0.995358i	$0.469317\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1278.00	−0.542714
$178$	0	0
$179$	−1462.00	−0.610475	−0.305237	−	0.952276i	$-0.598736\pi$
$179$	−1462.00	−0.610475	−0.305237	+	0.952276i	$0.598736\pi$
$180$	0	0
$181$	586.000	0.240647	0.120323	−	0.992735i	$-0.461607\pi$
$181$	586.000	0.240647	0.120323	+	0.992735i	$0.461607\pi$
$182$	0	0
$183$	−2094.00	−0.845863
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3036.00	1.18724
$188$	0	0
$189$	270.000	0.103913
$190$	0	0
$191$	60.0000	0.0227301	0.0113650	−	0.999935i	$-0.496382\pi$
$191$	60.0000	0.0227301	0.0113650	+	0.999935i	$0.496382\pi$
$192$	0	0
$193$	4676.00	1.74397	0.871984	−	0.489534i	$-0.162833\pi$
$193$	4676.00	1.74397	0.871984	+	0.489534i	$0.162833\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2286.00	−0.826755	−0.413378	−	0.910560i	$-0.635651\pi$
$197$	−2286.00	−0.826755	−0.413378	+	0.910560i	$0.635651\pi$
$198$	0	0
$199$	−3536.00	−1.25960	−0.629800	−	0.776757i	$-0.716863\pi$
$199$	−3536.00	−1.25960	−0.629800	+	0.776757i	$0.716863\pi$
$200$	0	0
$201$	984.000	0.345304
$202$	0	0
$203$	−2240.00	−0.774469
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1476.00	−0.495599
$208$	0	0
$209$	−4784.00	−1.58333
$210$	0	0
$211$	−3500.00	−1.14194	−0.570971	−	0.820970i	$-0.693433\pi$
$211$	−3500.00	−1.14194	−0.570971	+	0.820970i	$0.693433\pi$
$212$	0	0
$213$	−564.000	−0.181430
$214$	0	0
$215$	0	0
$216$	0	0
$217$	720.000	0.225239
$218$	0	0
$219$	−2220.00	−0.684994
$220$	0	0
$221$	−2244.00	−0.683022
$222$	0	0
$223$	−5874.00	−1.76391	−0.881955	−	0.471333i	$-0.843773\pi$
$223$	−5874.00	−1.76391	−0.881955	+	0.471333i	$0.843773\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	124.000	0.0362563	0.0181281	−	0.999836i	$-0.494229\pi$
$227$	124.000	0.0362563	0.0181281	+	0.999836i	$0.494229\pi$
$228$	0	0
$229$	−1362.00	−0.393028	−0.196514	−	0.980501i	$-0.562962\pi$
$229$	−1362.00	−0.393028	−0.196514	+	0.980501i	$0.562962\pi$
$230$	0	0
$231$	−1380.00	−0.393062
$232$	0	0
$233$	3870.00	1.08812	0.544060	−	0.839046i	$-0.316886\pi$
$233$	3870.00	1.08812	0.544060	+	0.839046i	$0.316886\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3504.00	−0.960377
$238$	0	0
$239$	6116.00	1.65528	0.827638	−	0.561262i	$-0.189684\pi$
$239$	6116.00	1.65528	0.827638	+	0.561262i	$0.189684\pi$
$240$	0	0
$241$	−5962.00	−1.59355	−0.796776	−	0.604274i	$-0.793463\pi$
$241$	−5962.00	−1.59355	−0.796776	+	0.604274i	$0.793463\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3536.00	0.910892
$248$	0	0
$249$	1236.00	0.314572
$250$	0	0
$251$	−1490.00	−0.374693	−0.187347	−	0.982294i	$-0.559989\pi$
$251$	−1490.00	−0.374693	−0.187347	+	0.982294i	$0.559989\pi$
$252$	0	0
$253$	7544.00	1.87465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5394.00	1.30922	0.654608	−	0.755969i	$-0.272834\pi$
$257$	5394.00	1.30922	0.654608	+	0.755969i	$0.272834\pi$
$258$	0	0
$259$	−220.000	−0.0527804
$260$	0	0
$261$	2016.00	0.478112
$262$	0	0
$263$	636.000	0.149116	0.0745579	−	0.997217i	$-0.476245\pi$
$263$	636.000	0.149116	0.0745579	+	0.997217i	$0.476245\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3618.00	−0.829281
$268$	0	0
$269$	−3360.00	−0.761572	−0.380786	−	0.924663i	$-0.624347\pi$
$269$	−3360.00	−0.761572	−0.380786	+	0.924663i	$0.624347\pi$
$270$	0	0
$271$	5768.00	1.29292	0.646459	−	0.762948i	$-0.276249\pi$
$271$	5768.00	1.29292	0.646459	+	0.762948i	$0.276249\pi$
$272$	0	0
$273$	1020.00	0.226129
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1398.00	0.303241	0.151620	−	0.988439i	$-0.451551\pi$
$277$	1398.00	0.303241	0.151620	+	0.988439i	$0.451551\pi$
$278$	0	0
$279$	−648.000	−0.139049
$280$	0	0
$281$	−4194.00	−0.890367	−0.445183	−	0.895439i	$-0.646862\pi$
$281$	−4194.00	−0.890367	−0.445183	+	0.895439i	$0.646862\pi$
$282$	0	0
$283$	8256.00	1.73416	0.867082	−	0.498166i	$-0.165993\pi$
$283$	8256.00	1.73416	0.867082	+	0.498166i	$0.165993\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1940.00	−0.399006
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	−4152.00	−0.836407
$292$	0	0
$293$	−5534.00	−1.10341	−0.551706	−	0.834039i	$-0.686023\pi$
$293$	−5534.00	−1.10341	−0.551706	+	0.834039i	$0.686023\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1242.00	0.242654
$298$	0	0
$299$	−5576.00	−1.07849
$300$	0	0
$301$	1080.00	0.206811
$302$	0	0
$303$	3384.00	0.641603
$304$	0	0
$305$	0	0
$306$	0	0
$307$	484.000	0.0899783	0.0449892	−	0.998987i	$-0.485675\pi$
$307$	484.000	0.0899783	0.0449892	+	0.998987i	$0.485675\pi$
$308$	0	0
$309$	−2274.00	−0.418652
$310$	0	0
$311$	−2724.00	−0.496668	−0.248334	−	0.968674i	$-0.579883\pi$
$311$	−2724.00	−0.496668	−0.248334	+	0.968674i	$0.579883\pi$
$312$	0	0
$313$	−5308.00	−0.958549	−0.479275	−	0.877665i	$-0.659100\pi$
$313$	−5308.00	−0.958549	−0.479275	+	0.877665i	$0.659100\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4218.00	−0.747339	−0.373670	−	0.927562i	$-0.621901\pi$
$317$	−4218.00	−0.747339	−0.373670	+	0.927562i	$0.621901\pi$
$318$	0	0
$319$	−10304.0	−1.80851
$320$	0	0
$321$	3972.00	0.690640
$322$	0	0
$323$	−6864.00	−1.18242
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4806.00	−0.812760
$328$	0	0
$329$	−4800.00	−0.804354
$330$	0	0
$331$	−4640.00	−0.770506	−0.385253	−	0.922811i	$-0.625886\pi$
$331$	−4640.00	−0.770506	−0.385253	+	0.922811i	$0.625886\pi$
$332$	0	0
$333$	198.000	0.0325836
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8156.00	1.31835	0.659177	−	0.751987i	$-0.270905\pi$
$337$	8156.00	1.31835	0.659177	+	0.751987i	$0.270905\pi$
$338$	0	0
$339$	−6222.00	−0.996851
$340$	0	0
$341$	3312.00	0.525967
$342$	0	0
$343$	5860.00	0.922479
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4124.00	−0.638006	−0.319003	−	0.947754i	$-0.603348\pi$
$347$	−4124.00	−0.638006	−0.319003	+	0.947754i	$0.603348\pi$
$348$	0	0
$349$	3650.00	0.559828	0.279914	−	0.960025i	$-0.409694\pi$
$349$	3650.00	0.559828	0.279914	+	0.960025i	$0.409694\pi$
$350$	0	0
$351$	−918.000	−0.139599
$352$	0	0
$353$	−6834.00	−1.03042	−0.515208	−	0.857065i	$-0.672285\pi$
$353$	−6834.00	−1.03042	−0.515208	+	0.857065i	$0.672285\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1980.00	−0.293537
$358$	0	0
$359$	3904.00	0.573942	0.286971	−	0.957939i	$-0.407352\pi$
$359$	3904.00	0.573942	0.286971	+	0.957939i	$0.407352\pi$
$360$	0	0
$361$	3957.00	0.576906
$362$	0	0
$363$	−2355.00	−0.340511
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13174.0	1.87378	0.936890	−	0.349624i	$-0.113691\pi$
$367$	13174.0	1.87378	0.936890	+	0.349624i	$0.113691\pi$
$368$	0	0
$369$	1746.00	0.246323
$370$	0	0
$371$	2860.00	0.400226
$372$	0	0
$373$	1090.00	0.151308	0.0756542	−	0.997134i	$-0.475895\pi$
$373$	1090.00	0.151308	0.0756542	+	0.997134i	$0.475895\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7616.00	1.04043
$378$	0	0
$379$	−9220.00	−1.24960	−0.624802	−	0.780784i	$-0.714820\pi$
$379$	−9220.00	−1.24960	−0.624802	+	0.780784i	$0.714820\pi$
$380$	0	0
$381$	−1602.00	−0.215415
$382$	0	0
$383$	3960.00	0.528320	0.264160	−	0.964479i	$-0.414905\pi$
$383$	3960.00	0.528320	0.264160	+	0.964479i	$0.414905\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−972.000	−0.127673
$388$	0	0
$389$	−1788.00	−0.233047	−0.116523	−	0.993188i	$-0.537175\pi$
$389$	−1788.00	−0.233047	−0.116523	+	0.993188i	$0.537175\pi$
$390$	0	0
$391$	10824.0	1.39998
$392$	0	0
$393$	−5418.00	−0.695425
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9642.00	−1.21894	−0.609469	−	0.792810i	$-0.708617\pi$
$397$	−9642.00	−1.21894	−0.609469	+	0.792810i	$0.708617\pi$
$398$	0	0
$399$	3120.00	0.391467
$400$	0	0
$401$	−410.000	−0.0510584	−0.0255292	−	0.999674i	$-0.508127\pi$
$401$	−410.000	−0.0510584	−0.0255292	+	0.999674i	$0.508127\pi$
$402$	0	0
$403$	−2448.00	−0.302589
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1012.00	−0.123251
$408$	0	0
$409$	13766.0	1.66427	0.832133	−	0.554576i	$-0.187120\pi$
$409$	13766.0	1.66427	0.832133	+	0.554576i	$0.187120\pi$
$410$	0	0
$411$	5466.00	0.656005
$412$	0	0
$413$	−4260.00	−0.507557
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1596.00	−0.187426
$418$	0	0
$419$	16998.0	1.98188	0.990939	−	0.134315i	$-0.0428833\pi$
$419$	16998.0	1.98188	0.990939	+	0.134315i	$0.0428833\pi$
$420$	0	0
$421$	−2450.00	−0.283624	−0.141812	−	0.989894i	$-0.545293\pi$
$421$	−2450.00	−0.283624	−0.141812	+	0.989894i	$0.545293\pi$
$422$	0	0
$423$	4320.00	0.496562
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6980.00	−0.791068
$428$	0	0
$429$	4692.00	0.528046
$430$	0	0
$431$	−9248.00	−1.03355	−0.516776	−	0.856121i	$-0.672868\pi$
$431$	−9248.00	−1.03355	−0.516776	+	0.856121i	$0.672868\pi$
$432$	0	0
$433$	5028.00	0.558038	0.279019	−	0.960286i	$-0.409991\pi$
$433$	5028.00	0.558038	0.279019	+	0.960286i	$0.409991\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17056.0	−1.86705
$438$	0	0
$439$	3120.00	0.339202	0.169601	−	0.985513i	$-0.445752\pi$
$439$	3120.00	0.339202	0.169601	+	0.985513i	$0.445752\pi$
$440$	0	0
$441$	−2187.00	−0.236152
$442$	0	0
$443$	8220.00	0.881589	0.440795	−	0.897608i	$-0.354697\pi$
$443$	8220.00	0.881589	0.440795	+	0.897608i	$0.354697\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3852.00	0.407591
$448$	0	0
$449$	−5826.00	−0.612352	−0.306176	−	0.951975i	$-0.599050\pi$
$449$	−5826.00	−0.612352	−0.306176	+	0.951975i	$0.599050\pi$
$450$	0	0
$451$	−8924.00	−0.931740
$452$	0	0
$453$	−552.000	−0.0572521
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16896.0	1.72946	0.864728	−	0.502240i	$-0.167491\pi$
$457$	16896.0	1.72946	0.864728	+	0.502240i	$0.167491\pi$
$458$	0	0
$459$	1782.00	0.181213
$460$	0	0
$461$	−996.000	−0.100625	−0.0503127	−	0.998734i	$-0.516022\pi$
$461$	−996.000	−0.100625	−0.0503127	+	0.998734i	$0.516022\pi$
$462$	0	0
$463$	−10046.0	−1.00837	−0.504187	−	0.863594i	$-0.668208\pi$
$463$	−10046.0	−1.00837	−0.504187	+	0.863594i	$0.668208\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8388.00	0.831157	0.415579	−	0.909557i	$-0.363579\pi$
$467$	8388.00	0.831157	0.415579	+	0.909557i	$0.363579\pi$
$468$	0	0
$469$	3280.00	0.322935
$470$	0	0
$471$	−11238.0	−1.09940
$472$	0	0
$473$	4968.00	0.482936
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2574.00	−0.247076
$478$	0	0
$479$	11396.0	1.08705	0.543525	−	0.839393i	$-0.317089\pi$
$479$	11396.0	1.08705	0.543525	+	0.839393i	$0.317089\pi$
$480$	0	0
$481$	748.000	0.0709062
$482$	0	0
$483$	−4920.00	−0.463494
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6454.00	−0.600531	−0.300266	−	0.953856i	$-0.597075\pi$
$487$	−6454.00	−0.600531	−0.300266	+	0.953856i	$0.597075\pi$
$488$	0	0
$489$	4512.00	0.417259
$490$	0	0
$491$	18638.0	1.71308	0.856539	−	0.516083i	$-0.172610\pi$
$491$	18638.0	1.71308	0.856539	+	0.516083i	$0.172610\pi$
$492$	0	0
$493$	−14784.0	−1.35058
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1880.00	−0.169677
$498$	0	0
$499$	17768.0	1.59400	0.796999	−	0.603981i	$-0.206420\pi$
$499$	17768.0	1.59400	0.796999	+	0.603981i	$0.206420\pi$
$500$	0	0
$501$	9036.00	0.805786
$502$	0	0
$503$	7952.00	0.704895	0.352447	−	0.935832i	$-0.385350\pi$
$503$	7952.00	0.704895	0.352447	+	0.935832i	$0.385350\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3123.00	0.273565
$508$	0	0
$509$	−12896.0	−1.12300	−0.561498	−	0.827478i	$-0.689775\pi$
$509$	−12896.0	−1.12300	−0.561498	+	0.827478i	$0.689775\pi$
$510$	0	0
$511$	−7400.00	−0.640620
$512$	0	0
$513$	−2808.00	−0.241669
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22080.0	−1.87829
$518$	0	0
$519$	−1314.00	−0.111133
$520$	0	0
$521$	−2714.00	−0.228220	−0.114110	−	0.993468i	$-0.536402\pi$
$521$	−2714.00	−0.228220	−0.114110	+	0.993468i	$0.536402\pi$
$522$	0	0
$523$	13792.0	1.15312	0.576560	−	0.817055i	$-0.304395\pi$
$523$	13792.0	1.15312	0.576560	+	0.817055i	$0.304395\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4752.00	0.392790
$528$	0	0
$529$	14729.0	1.21057
$530$	0	0
$531$	3834.00	0.313336
$532$	0	0
$533$	6596.00	0.536031
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4386.00	0.352458
$538$	0	0
$539$	11178.0	0.893266
$540$	0	0
$541$	6802.00	0.540556	0.270278	−	0.962782i	$-0.412884\pi$
$541$	6802.00	0.540556	0.270278	+	0.962782i	$0.412884\pi$
$542$	0	0
$543$	−1758.00	−0.138937
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18188.0	1.42169	0.710843	−	0.703350i	$-0.248313\pi$
$547$	18188.0	1.42169	0.710843	+	0.703350i	$0.248313\pi$
$548$	0	0
$549$	6282.00	0.488359
$550$	0	0
$551$	23296.0	1.80117
$552$	0	0
$553$	−11680.0	−0.898163
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21462.0	−1.63263	−0.816314	−	0.577608i	$-0.803986\pi$
$557$	−21462.0	−1.63263	−0.816314	+	0.577608i	$0.803986\pi$
$558$	0	0
$559$	−3672.00	−0.277834
$560$	0	0
$561$	−9108.00	−0.685455
$562$	0	0
$563$	17244.0	1.29085	0.645424	−	0.763824i	$-0.276681\pi$
$563$	17244.0	1.29085	0.645424	+	0.763824i	$0.276681\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−810.000	−0.0599944
$568$	0	0
$569$	8790.00	0.647620	0.323810	−	0.946122i	$-0.395036\pi$
$569$	8790.00	0.647620	0.323810	+	0.946122i	$0.395036\pi$
$570$	0	0
$571$	−5984.00	−0.438568	−0.219284	−	0.975661i	$-0.570372\pi$
$571$	−5984.00	−0.438568	−0.219284	+	0.975661i	$0.570372\pi$
$572$	0	0
$573$	−180.000	−0.0131232
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9344.00	0.674170	0.337085	−	0.941474i	$-0.390559\pi$
$577$	9344.00	0.674170	0.337085	+	0.941474i	$0.390559\pi$
$578$	0	0
$579$	−14028.0	−1.00688
$580$	0	0
$581$	4120.00	0.294193
$582$	0	0
$583$	13156.0	0.934590
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6932.00	0.487418	0.243709	−	0.969848i	$-0.421636\pi$
$587$	6932.00	0.487418	0.243709	+	0.969848i	$0.421636\pi$
$588$	0	0
$589$	−7488.00	−0.523833
$590$	0	0
$591$	6858.00	0.477327
$592$	0	0
$593$	−9382.00	−0.649701	−0.324850	−	0.945765i	$-0.605314\pi$
$593$	−9382.00	−0.649701	−0.324850	+	0.945765i	$0.605314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10608.0	0.727230
$598$	0	0
$599$	4096.00	0.279396	0.139698	−	0.990194i	$-0.455387\pi$
$599$	4096.00	0.279396	0.139698	+	0.990194i	$0.455387\pi$
$600$	0	0
$601$	22962.0	1.55847	0.779234	−	0.626733i	$-0.215608\pi$
$601$	22962.0	1.55847	0.779234	+	0.626733i	$0.215608\pi$
$602$	0	0
$603$	−2952.00	−0.199361
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3490.00	0.233369	0.116684	−	0.993169i	$-0.462773\pi$
$607$	3490.00	0.233369	0.116684	+	0.993169i	$0.462773\pi$
$608$	0	0
$609$	6720.00	0.447140
$610$	0	0
$611$	16320.0	1.08058
$612$	0	0
$613$	−6386.00	−0.420764	−0.210382	−	0.977619i	$-0.567471\pi$
$613$	−6386.00	−0.420764	−0.210382	+	0.977619i	$0.567471\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19534.0	1.27457	0.637285	−	0.770629i	$-0.280058\pi$
$617$	19534.0	1.27457	0.637285	+	0.770629i	$0.280058\pi$
$618$	0	0
$619$	8764.00	0.569071	0.284535	−	0.958666i	$-0.408161\pi$
$619$	8764.00	0.569071	0.284535	+	0.958666i	$0.408161\pi$
$620$	0	0
$621$	4428.00	0.286134
$622$	0	0
$623$	−12060.0	−0.775560
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14352.0	0.914137
$628$	0	0
$629$	−1452.00	−0.0920430
$630$	0	0
$631$	7856.00	0.495630	0.247815	−	0.968807i	$-0.420288\pi$
$631$	7856.00	0.495630	0.247815	+	0.968807i	$0.420288\pi$
$632$	0	0
$633$	10500.0	0.659301
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8262.00	−0.513897
$638$	0	0
$639$	1692.00	0.104749
$640$	0	0
$641$	−22974.0	−1.41563	−0.707815	−	0.706398i	$-0.750319\pi$
$641$	−22974.0	−1.41563	−0.707815	+	0.706398i	$0.750319\pi$
$642$	0	0
$643$	6216.00	0.381237	0.190618	−	0.981664i	$-0.438951\pi$
$643$	6216.00	0.381237	0.190618	+	0.981664i	$0.438951\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13384.0	−0.813260	−0.406630	−	0.913593i	$-0.633296\pi$
$647$	−13384.0	−0.813260	−0.406630	+	0.913593i	$0.633296\pi$
$648$	0	0
$649$	−19596.0	−1.18522
$650$	0	0
$651$	−2160.00	−0.130042
$652$	0	0
$653$	−12882.0	−0.771993	−0.385997	−	0.922500i	$-0.626142\pi$
$653$	−12882.0	−0.771993	−0.385997	+	0.922500i	$0.626142\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6660.00	0.395482
$658$	0	0
$659$	−2082.00	−0.123070	−0.0615351	−	0.998105i	$-0.519600\pi$
$659$	−2082.00	−0.123070	−0.0615351	+	0.998105i	$0.519600\pi$
$660$	0	0
$661$	−9430.00	−0.554893	−0.277447	−	0.960741i	$-0.589488\pi$
$661$	−9430.00	−0.554893	−0.277447	+	0.960741i	$0.589488\pi$
$662$	0	0
$663$	6732.00	0.394343
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36736.0	−2.13257
$668$	0	0
$669$	17622.0	1.01839
$670$	0	0
$671$	−32108.0	−1.84727
$672$	0	0
$673$	−3268.00	−0.187180	−0.0935900	−	0.995611i	$-0.529834\pi$
$673$	−3268.00	−0.187180	−0.0935900	+	0.995611i	$0.529834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15606.0	0.885949	0.442974	−	0.896534i	$-0.353923\pi$
$677$	15606.0	0.885949	0.442974	+	0.896534i	$0.353923\pi$
$678$	0	0
$679$	−13840.0	−0.782225
$680$	0	0
$681$	−372.000	−0.0209326
$682$	0	0
$683$	−428.000	−0.0239780	−0.0119890	−	0.999928i	$-0.503816\pi$
$683$	−428.000	−0.0239780	−0.0119890	+	0.999928i	$0.503816\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4086.00	0.226915
$688$	0	0
$689$	−9724.00	−0.537670
$690$	0	0
$691$	−6384.00	−0.351460	−0.175730	−	0.984438i	$-0.556229\pi$
$691$	−6384.00	−0.351460	−0.175730	+	0.984438i	$0.556229\pi$
$692$	0	0
$693$	4140.00	0.226934
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12804.0	−0.695819
$698$	0	0
$699$	−11610.0	−0.628227
$700$	0	0
$701$	−12224.0	−0.658622	−0.329311	−	0.944221i	$-0.606816\pi$
$701$	−12224.0	−0.658622	−0.329311	+	0.944221i	$0.606816\pi$
$702$	0	0
$703$	2288.00	0.122750
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11280.0	0.600040
$708$	0	0
$709$	19510.0	1.03345	0.516723	−	0.856153i	$-0.327152\pi$
$709$	19510.0	1.03345	0.516723	+	0.856153i	$0.327152\pi$
$710$	0	0
$711$	10512.0	0.554474
$712$	0	0
$713$	11808.0	0.620215
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18348.0	−0.955674
$718$	0	0
$719$	−3368.00	−0.174694	−0.0873472	−	0.996178i	$-0.527839\pi$
$719$	−3368.00	−0.174694	−0.0873472	+	0.996178i	$0.527839\pi$
$720$	0	0
$721$	−7580.00	−0.391531
$722$	0	0
$723$	17886.0	0.920038
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−22134.0	−1.12917	−0.564584	−	0.825376i	$-0.690963\pi$
$727$	−22134.0	−1.12917	−0.564584	+	0.825376i	$0.690963\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7128.00	0.360655
$732$	0	0
$733$	−32298.0	−1.62750	−0.813748	−	0.581219i	$-0.802576\pi$
$733$	−32298.0	−1.62750	−0.813748	+	0.581219i	$0.802576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15088.0	0.754103
$738$	0	0
$739$	25104.0	1.24962	0.624808	−	0.780779i	$-0.285177\pi$
$739$	25104.0	1.24962	0.624808	+	0.780779i	$0.285177\pi$
$740$	0	0
$741$	−10608.0	−0.525904
$742$	0	0
$743$	27696.0	1.36752	0.683760	−	0.729707i	$-0.260343\pi$
$743$	27696.0	1.36752	0.683760	+	0.729707i	$0.260343\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3708.00	−0.181618
$748$	0	0
$749$	13240.0	0.645900
$750$	0	0
$751$	−2176.00	−0.105730	−0.0528651	−	0.998602i	$-0.516835\pi$
$751$	−2176.00	−0.105730	−0.0528651	+	0.998602i	$0.516835\pi$
$752$	0	0
$753$	4470.00	0.216329
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25514.0	−1.22500	−0.612498	−	0.790472i	$-0.709835\pi$
$757$	−25514.0	−1.22500	−0.612498	+	0.790472i	$0.709835\pi$
$758$	0	0
$759$	−22632.0	−1.08233
$760$	0	0
$761$	18238.0	0.868761	0.434380	−	0.900730i	$-0.356967\pi$
$761$	18238.0	0.868761	0.434380	+	0.900730i	$0.356967\pi$
$762$	0	0
$763$	−16020.0	−0.760109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14484.0	0.681860
$768$	0	0
$769$	−14462.0	−0.678170	−0.339085	−	0.940756i	$-0.610118\pi$
$769$	−14462.0	−0.678170	−0.339085	+	0.940756i	$0.610118\pi$
$770$	0	0
$771$	−16182.0	−0.755876
$772$	0	0
$773$	−34034.0	−1.58359	−0.791797	−	0.610785i	$-0.790854\pi$
$773$	−34034.0	−1.58359	−0.791797	+	0.610785i	$0.790854\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	660.000	0.0304728
$778$	0	0
$779$	20176.0	0.927959
$780$	0	0
$781$	−8648.00	−0.396222
$782$	0	0
$783$	−6048.00	−0.276038
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−22064.0	−0.999360	−0.499680	−	0.866210i	$-0.666549\pi$
$787$	−22064.0	−0.999360	−0.499680	+	0.866210i	$0.666549\pi$
$788$	0	0
$789$	−1908.00	−0.0860920
$790$	0	0
$791$	−20740.0	−0.932275
$792$	0	0
$793$	23732.0	1.06273
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23334.0	1.03705	0.518527	−	0.855061i	$-0.326480\pi$
$797$	23334.0	1.03705	0.518527	+	0.855061i	$0.326480\pi$
$798$	0	0
$799$	−31680.0	−1.40270
$800$	0	0
$801$	10854.0	0.478786
$802$	0	0
$803$	−34040.0	−1.49595
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10080.0	0.439694
$808$	0	0
$809$	−7566.00	−0.328809	−0.164404	−	0.986393i	$-0.552570\pi$
$809$	−7566.00	−0.328809	−0.164404	+	0.986393i	$0.552570\pi$
$810$	0	0
$811$	5964.00	0.258230	0.129115	−	0.991630i	$-0.458786\pi$
$811$	5964.00	0.258230	0.129115	+	0.991630i	$0.458786\pi$
$812$	0	0
$813$	−17304.0	−0.746467
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11232.0	−0.480977
$818$	0	0
$819$	−3060.00	−0.130556
$820$	0	0
$821$	−11880.0	−0.505012	−0.252506	−	0.967595i	$-0.581255\pi$
$821$	−11880.0	−0.505012	−0.252506	+	0.967595i	$0.581255\pi$
$822$	0	0
$823$	1762.00	0.0746287	0.0373144	−	0.999304i	$-0.488120\pi$
$823$	1762.00	0.0746287	0.0373144	+	0.999304i	$0.488120\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14124.0	0.593881	0.296941	−	0.954896i	$-0.404034\pi$
$827$	14124.0	0.593881	0.296941	+	0.954896i	$0.404034\pi$
$828$	0	0
$829$	−21350.0	−0.894471	−0.447235	−	0.894416i	$-0.647591\pi$
$829$	−21350.0	−0.894471	−0.447235	+	0.894416i	$0.647591\pi$
$830$	0	0
$831$	−4194.00	−0.175076
$832$	0	0
$833$	16038.0	0.667087
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1944.00	0.0802801
$838$	0	0
$839$	−26136.0	−1.07546	−0.537732	−	0.843116i	$-0.680719\pi$
$839$	−26136.0	−1.07546	−0.537732	+	0.843116i	$0.680719\pi$
$840$	0	0
$841$	25787.0	1.05732
$842$	0	0
$843$	12582.0	0.514053
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7850.00	−0.318452
$848$	0	0
$849$	−24768.0	−1.00122
$850$	0	0
$851$	−3608.00	−0.145336
$852$	0	0
$853$	7030.00	0.282184	0.141092	−	0.989997i	$-0.454939\pi$
$853$	7030.00	0.282184	0.141092	+	0.989997i	$0.454939\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9574.00	0.381612	0.190806	−	0.981628i	$-0.438890\pi$
$857$	9574.00	0.381612	0.190806	+	0.981628i	$0.438890\pi$
$858$	0	0
$859$	−43748.0	−1.73767	−0.868837	−	0.495098i	$-0.835132\pi$
$859$	−43748.0	−1.73767	−0.868837	+	0.495098i	$0.835132\pi$
$860$	0	0
$861$	5820.00	0.230366
$862$	0	0
$863$	−41436.0	−1.63441	−0.817206	−	0.576345i	$-0.804478\pi$
$863$	−41436.0	−1.63441	−0.817206	+	0.576345i	$0.804478\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1671.00	0.0654558
$868$	0	0
$869$	−53728.0	−2.09735
$870$	0	0
$871$	−11152.0	−0.433836
$872$	0	0
$873$	12456.0	0.482900
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4606.00	−0.177347	−0.0886736	−	0.996061i	$-0.528263\pi$
$877$	−4606.00	−0.177347	−0.0886736	+	0.996061i	$0.528263\pi$
$878$	0	0
$879$	16602.0	0.637055
$880$	0	0
$881$	−4610.00	−0.176294	−0.0881469	−	0.996107i	$-0.528094\pi$
$881$	−4610.00	−0.176294	−0.0881469	+	0.996107i	$0.528094\pi$
$882$	0	0
$883$	−23512.0	−0.896084	−0.448042	−	0.894013i	$-0.647878\pi$
$883$	−23512.0	−0.896084	−0.448042	+	0.894013i	$0.647878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30396.0	1.15062	0.575309	−	0.817936i	$-0.304882\pi$
$887$	30396.0	1.15062	0.575309	+	0.817936i	$0.304882\pi$
$888$	0	0
$889$	−5340.00	−0.201460
$890$	0	0
$891$	−3726.00	−0.140096
$892$	0	0
$893$	49920.0	1.87067
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16728.0	0.622666
$898$	0	0
$899$	−16128.0	−0.598330
$900$	0	0
$901$	18876.0	0.697948
$902$	0	0
$903$	−3240.00	−0.119402
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3452.00	−0.126375	−0.0631873	−	0.998002i	$-0.520127\pi$
$907$	−3452.00	−0.126375	−0.0631873	+	0.998002i	$0.520127\pi$
$908$	0	0
$909$	−10152.0	−0.370430
$910$	0	0
$911$	14256.0	0.518466	0.259233	−	0.965815i	$-0.416530\pi$
$911$	14256.0	0.518466	0.259233	+	0.965815i	$0.416530\pi$
$912$	0	0
$913$	18952.0	0.686988
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18060.0	−0.650375
$918$	0	0
$919$	8064.00	0.289452	0.144726	−	0.989472i	$-0.453770\pi$
$919$	8064.00	0.289452	0.144726	+	0.989472i	$0.453770\pi$
$920$	0	0
$921$	−1452.00	−0.0519490
$922$	0	0
$923$	6392.00	0.227947
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6822.00	0.241709
$928$	0	0
$929$	−40930.0	−1.44550	−0.722750	−	0.691109i	$-0.757122\pi$
$929$	−40930.0	−1.44550	−0.722750	+	0.691109i	$0.757122\pi$
$930$	0	0
$931$	−25272.0	−0.889642
$932$	0	0
$933$	8172.00	0.286752
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9016.00	−0.314344	−0.157172	−	0.987571i	$-0.550238\pi$
$937$	−9016.00	−0.314344	−0.157172	+	0.987571i	$0.550238\pi$
$938$	0	0
$939$	15924.0	0.553419
$940$	0	0
$941$	−10444.0	−0.361812	−0.180906	−	0.983500i	$-0.557903\pi$
$941$	−10444.0	−0.361812	−0.180906	+	0.983500i	$0.557903\pi$
$942$	0	0
$943$	−31816.0	−1.09870
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21516.0	−0.738306	−0.369153	−	0.929369i	$-0.620352\pi$
$947$	−21516.0	−0.738306	−0.369153	+	0.929369i	$0.620352\pi$
$948$	0	0
$949$	25160.0	0.860620
$950$	0	0
$951$	12654.0	0.431476
$952$	0	0
$953$	28098.0	0.955072	0.477536	−	0.878612i	$-0.341530\pi$
$953$	28098.0	0.955072	0.477536	+	0.878612i	$0.341530\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	30912.0	1.04414
$958$	0	0
$959$	18220.0	0.613508
$960$	0	0
$961$	−24607.0	−0.825988
$962$	0	0
$963$	−11916.0	−0.398741
$964$	0	0
$965$	0	0
$966$	0	0
$967$	14558.0	0.484130	0.242065	−	0.970260i	$-0.422175\pi$
$967$	14558.0	0.484130	0.242065	+	0.970260i	$0.422175\pi$
$968$	0	0
$969$	20592.0	0.682673
$970$	0	0
$971$	24846.0	0.821160	0.410580	−	0.911825i	$-0.365326\pi$
$971$	24846.0	0.821160	0.410580	+	0.911825i	$0.365326\pi$
$972$	0	0
$973$	−5320.00	−0.175284
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12950.0	0.424061	0.212030	−	0.977263i	$-0.431992\pi$
$977$	12950.0	0.424061	0.212030	+	0.977263i	$0.431992\pi$
$978$	0	0
$979$	−55476.0	−1.81105
$980$	0	0
$981$	14418.0	0.469247
$982$	0	0
$983$	26728.0	0.867234	0.433617	−	0.901097i	$-0.357237\pi$
$983$	26728.0	0.867234	0.433617	+	0.901097i	$0.357237\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14400.0	0.464394
$988$	0	0
$989$	17712.0	0.569473
$990$	0	0
$991$	3880.00	0.124372	0.0621858	−	0.998065i	$-0.480193\pi$
$991$	3880.00	0.124372	0.0621858	+	0.998065i	$0.480193\pi$
$992$	0	0
$993$	13920.0	0.444852
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18582.0	−0.590269	−0.295134	−	0.955456i	$-0.595364\pi$
$997$	−18582.0	−0.590269	−0.295134	+	0.955456i	$0.595364\pi$
$998$	0	0
$999$	−594.000	−0.0188121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.a.b.1.1		1
3.2	odd	2		1800.4.a.j.1.1		1
4.3	odd	2		1200.4.a.bh.1.1		1
5.2	odd	4		120.4.f.a.49.2	yes	2
5.3	odd	4		120.4.f.a.49.1	✓	2
5.4	even	2		600.4.a.o.1.1		1
15.2	even	4		360.4.f.c.289.1		2
15.8	even	4		360.4.f.c.289.2		2
15.14	odd	2		1800.4.a.y.1.1		1
20.3	even	4		240.4.f.b.49.2		2
20.7	even	4		240.4.f.b.49.1		2
20.19	odd	2		1200.4.a.f.1.1		1
40.3	even	4		960.4.f.g.769.1		2
40.13	odd	4		960.4.f.l.769.2		2
40.27	even	4		960.4.f.g.769.2		2
40.37	odd	4		960.4.f.l.769.1		2
60.23	odd	4		720.4.f.g.289.2		2
60.47	odd	4		720.4.f.g.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.a.49.1	✓	2	5.3	odd	4
120.4.f.a.49.2	yes	2	5.2	odd	4
240.4.f.b.49.1		2	20.7	even	4
240.4.f.b.49.2		2	20.3	even	4
360.4.f.c.289.1		2	15.2	even	4
360.4.f.c.289.2		2	15.8	even	4
600.4.a.b.1.1		1	1.1	even	1	trivial
600.4.a.o.1.1		1	5.4	even	2
720.4.f.g.289.1		2	60.47	odd	4
720.4.f.g.289.2		2	60.23	odd	4
960.4.f.g.769.1		2	40.3	even	4
960.4.f.g.769.2		2	40.27	even	4
960.4.f.l.769.1		2	40.37	odd	4
960.4.f.l.769.2		2	40.13	odd	4
1200.4.a.f.1.1		1	20.19	odd	2
1200.4.a.bh.1.1		1	4.3	odd	2
1800.4.a.j.1.1		1	3.2	odd	2
1800.4.a.y.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more