LMFDB - Embedded newform 90.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,4,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("90.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	90.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:	$5.31017190052$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -4.00000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +48.0000 q^{11} +2.00000 q^{13} +8.00000 q^{14} +16.0000 q^{16} +114.000 q^{17} +140.000 q^{19} +20.0000 q^{20} -96.0000 q^{22} -72.0000 q^{23} +25.0000 q^{25} -4.00000 q^{26} -16.0000 q^{28} -210.000 q^{29} +272.000 q^{31} -32.0000 q^{32} -228.000 q^{34} -20.0000 q^{35} -334.000 q^{37} -280.000 q^{38} -40.0000 q^{40} +198.000 q^{41} -268.000 q^{43} +192.000 q^{44} +144.000 q^{46} -216.000 q^{47} -327.000 q^{49} -50.0000 q^{50} +8.00000 q^{52} +78.0000 q^{53} +240.000 q^{55} +32.0000 q^{56} +420.000 q^{58} -240.000 q^{59} +302.000 q^{61} -544.000 q^{62} +64.0000 q^{64} +10.0000 q^{65} +596.000 q^{67} +456.000 q^{68} +40.0000 q^{70} +768.000 q^{71} -478.000 q^{73} +668.000 q^{74} +560.000 q^{76} -192.000 q^{77} -640.000 q^{79} +80.0000 q^{80} -396.000 q^{82} +348.000 q^{83} +570.000 q^{85} +536.000 q^{86} -384.000 q^{88} -210.000 q^{89} -8.00000 q^{91} -288.000 q^{92} +432.000 q^{94} +700.000 q^{95} -1534.00 q^{97} +654.000 q^{98} +O(q^{100})\)

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	8.00000	0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	140.000	1.69043	0.845216	−	0.534425i	$-0.179472\pi$
$19$	140.000	1.69043	0.845216	+	0.534425i	$0.179472\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−96.0000	−0.930330
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−4.00000	−0.0301717
$27$	0	0
$28$	−16.0000	−0.107990
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	272.000	1.57589	0.787946	−	0.615745i	$-0.211145\pi$
$31$	272.000	1.57589	0.787946	+	0.615745i	$0.211145\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−228.000	−1.15005
$35$	−20.0000	−0.0965891
$36$	0	0
$37$	−334.000	−1.48403	−0.742017	−	0.670381i	$-0.766131\pi$
$37$	−334.000	−1.48403	−0.742017	+	0.670381i	$0.766131\pi$
$38$	−280.000	−1.19532
$39$	0	0
$40$	−40.0000	−0.158114
$41$	198.000	0.754205	0.377102	−	0.926172i	$-0.376920\pi$
$41$	198.000	0.754205	0.377102	+	0.926172i	$0.376920\pi$
$42$	0	0
$43$	−268.000	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−268.000	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	192.000	0.657843
$45$	0	0
$46$	144.000	0.461557
$47$	−216.000	−0.670358	−0.335179	−	0.942154i	$-0.608797\pi$
$47$	−216.000	−0.670358	−0.335179	+	0.942154i	$0.608797\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	−50.0000	−0.141421
$51$	0	0
$52$	8.00000	0.0213346
$53$	78.0000	0.202153	0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000	0.202153	0.101077	+	0.994879i	$0.467771\pi$
$54$	0	0
$55$	240.000	0.588393
$56$	32.0000	0.0763604
$57$	0	0
$58$	420.000	0.950840
$59$	−240.000	−0.529582	−0.264791	−	0.964306i	$-0.585303\pi$
$59$	−240.000	−0.529582	−0.264791	+	0.964306i	$0.585303\pi$
$60$	0	0
$61$	302.000	0.633888	0.316944	−	0.948444i	$-0.397343\pi$
$61$	302.000	0.633888	0.316944	+	0.948444i	$0.397343\pi$
$62$	−544.000	−1.11432
$63$	0	0
$64$	64.0000	0.125000
$65$	10.0000	0.0190823
$66$	0	0
$67$	596.000	1.08676	0.543381	−	0.839487i	$-0.317144\pi$
$67$	596.000	1.08676	0.543381	+	0.839487i	$0.317144\pi$
$68$	456.000	0.813208
$69$	0	0
$70$	40.0000	0.0682988
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	−478.000	−0.766379	−0.383190	−	0.923670i	$-0.625174\pi$
$73$	−478.000	−0.766379	−0.383190	+	0.923670i	$0.625174\pi$
$74$	668.000	1.04937
$75$	0	0
$76$	560.000	0.845216
$77$	−192.000	−0.284161
$78$	0	0
$79$	−640.000	−0.911464	−0.455732	−	0.890117i	$-0.650622\pi$
$79$	−640.000	−0.911464	−0.455732	+	0.890117i	$0.650622\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−396.000	−0.533303
$83$	348.000	0.460216	0.230108	−	0.973165i	$-0.426092\pi$
$83$	348.000	0.460216	0.230108	+	0.973165i	$0.426092\pi$
$84$	0	0
$85$	570.000	0.727355
$86$	536.000	0.672074
$87$	0	0
$88$	−384.000	−0.465165
$89$	−210.000	−0.250112	−0.125056	−	0.992150i	$-0.539911\pi$
$89$	−210.000	−0.250112	−0.125056	+	0.992150i	$0.539911\pi$
$90$	0	0
$91$	−8.00000	−0.00921569
$92$	−288.000	−0.326370
$93$	0	0
$94$	432.000	0.474015
$95$	700.000	0.755984
$96$	0	0
$97$	−1534.00	−1.60571	−0.802856	−	0.596173i	$-0.796687\pi$
$97$	−1534.00	−1.60571	−0.802856	+	0.596173i	$0.796687\pi$
$98$	654.000	0.674122
$99$	0	0
$100$	100.000	0.100000
$101$	−1722.00	−1.69649	−0.848245	−	0.529605i	$-0.822340\pi$
$101$	−1722.00	−1.69649	−0.848245	+	0.529605i	$0.822340\pi$
$102$	0	0
$103$	1052.00	1.00638	0.503188	−	0.864177i	$-0.332160\pi$
$103$	1052.00	1.00638	0.503188	+	0.864177i	$0.332160\pi$
$104$	−16.0000	−0.0150859
$105$	0	0
$106$	−156.000	−0.142944
$107$	564.000	0.509570	0.254785	−	0.966998i	$-0.417995\pi$
$107$	564.000	0.509570	0.254785	+	0.966998i	$0.417995\pi$
$108$	0	0
$109$	−610.000	−0.536031	−0.268016	−	0.963415i	$-0.586368\pi$
$109$	−610.000	−0.536031	−0.268016	+	0.963415i	$0.586368\pi$
$110$	−480.000	−0.416056
$111$	0	0
$112$	−64.0000	−0.0539949
$113$	−1302.00	−1.08391	−0.541955	−	0.840407i	$-0.682316\pi$
$113$	−1302.00	−1.08391	−0.541955	+	0.840407i	$0.682316\pi$
$114$	0	0
$115$	−360.000	−0.291915
$116$	−840.000	−0.672345
$117$	0	0
$118$	480.000	0.374471
$119$	−456.000	−0.351273
$120$	0	0
$121$	973.000	0.731029
$122$	−604.000	−0.448226
$123$	0	0
$124$	1088.00	0.787946
$125$	125.000	0.0894427
$126$	0	0
$127$	−124.000	−0.0866395	−0.0433198	−	0.999061i	$-0.513793\pi$
$127$	−124.000	−0.0866395	−0.0433198	+	0.999061i	$0.513793\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−20.0000	−0.0134932
$131$	−192.000	−0.128054	−0.0640272	−	0.997948i	$-0.520394\pi$
$131$	−192.000	−0.128054	−0.0640272	+	0.997948i	$0.520394\pi$
$132$	0	0
$133$	−560.000	−0.365099
$134$	−1192.00	−0.768456
$135$	0	0
$136$	−912.000	−0.575025
$137$	2514.00	1.56778	0.783889	−	0.620901i	$-0.213233\pi$
$137$	2514.00	1.56778	0.783889	+	0.620901i	$0.213233\pi$
$138$	0	0
$139$	1340.00	0.817679	0.408839	−	0.912606i	$-0.365934\pi$
$139$	1340.00	0.817679	0.408839	+	0.912606i	$0.365934\pi$
$140$	−80.0000	−0.0482945
$141$	0	0
$142$	−1536.00	−0.907734
$143$	96.0000	0.0561393
$144$	0	0
$145$	−1050.00	−0.601364
$146$	956.000	0.541912
$147$	0	0
$148$	−1336.00	−0.742017
$149$	−1410.00	−0.775246	−0.387623	−	0.921818i	$-0.626704\pi$
$149$	−1410.00	−0.775246	−0.387623	+	0.921818i	$0.626704\pi$
$150$	0	0
$151$	−2128.00	−1.14685	−0.573424	−	0.819258i	$-0.694385\pi$
$151$	−2128.00	−1.14685	−0.573424	+	0.819258i	$0.694385\pi$
$152$	−1120.00	−0.597658
$153$	0	0
$154$	384.000	0.200932
$155$	1360.00	0.704760
$156$	0	0
$157$	3026.00	1.53822	0.769112	−	0.639114i	$-0.220699\pi$
$157$	3026.00	1.53822	0.769112	+	0.639114i	$0.220699\pi$
$158$	1280.00	0.644502
$159$	0	0
$160$	−160.000	−0.0790569
$161$	288.000	0.140979
$162$	0	0
$163$	2612.00	1.25514	0.627569	−	0.778561i	$-0.284050\pi$
$163$	2612.00	1.25514	0.627569	+	0.778561i	$0.284050\pi$
$164$	792.000	0.377102
$165$	0	0
$166$	−696.000	−0.325422
$167$	24.0000	0.0111208	0.00556041	−	0.999985i	$-0.498230\pi$
$167$	24.0000	0.0111208	0.00556041	+	0.999985i	$0.498230\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	−1140.00	−0.514318
$171$	0	0
$172$	−1072.00	−0.475228
$173$	−1962.00	−0.862243	−0.431122	−	0.902294i	$-0.641882\pi$
$173$	−1962.00	−0.862243	−0.431122	+	0.902294i	$0.641882\pi$
$174$	0	0
$175$	−100.000	−0.0431959
$176$	768.000	0.328921
$177$	0	0
$178$	420.000	0.176856
$179$	120.000	0.0501074	0.0250537	−	0.999686i	$-0.492024\pi$
$179$	120.000	0.0501074	0.0250537	+	0.999686i	$0.492024\pi$
$180$	0	0
$181$	902.000	0.370415	0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000	0.370415	0.185208	+	0.982699i	$0.440704\pi$
$182$	16.0000	0.00651648
$183$	0	0
$184$	576.000	0.230779
$185$	−1670.00	−0.663680
$186$	0	0
$187$	5472.00	2.13985
$188$	−864.000	−0.335179
$189$	0	0
$190$	−1400.00	−0.534561
$191$	168.000	0.0636443	0.0318221	−	0.999494i	$-0.489869\pi$
$191$	168.000	0.0636443	0.0318221	+	0.999494i	$0.489869\pi$
$192$	0	0
$193$	−1318.00	−0.491563	−0.245782	−	0.969325i	$-0.579045\pi$
$193$	−1318.00	−0.491563	−0.245782	+	0.969325i	$0.579045\pi$
$194$	3068.00	1.13541
$195$	0	0
$196$	−1308.00	−0.476676
$197$	4014.00	1.45170	0.725852	−	0.687851i	$-0.241446\pi$
$197$	4014.00	1.45170	0.725852	+	0.687851i	$0.241446\pi$
$198$	0	0
$199$	2000.00	0.712443	0.356222	−	0.934401i	$-0.384065\pi$
$199$	2000.00	0.712443	0.356222	+	0.934401i	$0.384065\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	3444.00	1.19960
$203$	840.000	0.290426
$204$	0	0
$205$	990.000	0.337291
$206$	−2104.00	−0.711615
$207$	0	0
$208$	32.0000	0.0106673
$209$	6720.00	2.22408
$210$	0	0
$211$	−3868.00	−1.26201	−0.631005	−	0.775779i	$-0.717357\pi$
$211$	−3868.00	−1.26201	−0.631005	+	0.775779i	$0.717357\pi$
$212$	312.000	0.101077
$213$	0	0
$214$	−1128.00	−0.360320
$215$	−1340.00	−0.425057
$216$	0	0
$217$	−1088.00	−0.340361
$218$	1220.00	0.379031
$219$	0	0
$220$	960.000	0.294196
$221$	228.000	0.0693979
$222$	0	0
$223$	−3148.00	−0.945317	−0.472658	−	0.881246i	$-0.656706\pi$
$223$	−3148.00	−0.945317	−0.472658	+	0.881246i	$0.656706\pi$
$224$	128.000	0.0381802
$225$	0	0
$226$	2604.00	0.766440
$227$	−2556.00	−0.747347	−0.373673	−	0.927560i	$-0.621902\pi$
$227$	−2556.00	−0.747347	−0.373673	+	0.927560i	$0.621902\pi$
$228$	0	0
$229$	−610.000	−0.176026	−0.0880130	−	0.996119i	$-0.528052\pi$
$229$	−610.000	−0.176026	−0.0880130	+	0.996119i	$0.528052\pi$
$230$	720.000	0.206415
$231$	0	0
$232$	1680.00	0.475420
$233$	2058.00	0.578644	0.289322	−	0.957232i	$-0.406570\pi$
$233$	2058.00	0.578644	0.289322	+	0.957232i	$0.406570\pi$
$234$	0	0
$235$	−1080.00	−0.299793
$236$	−960.000	−0.264791
$237$	0	0
$238$	912.000	0.248387
$239$	−4920.00	−1.33158	−0.665792	−	0.746138i	$-0.731906\pi$
$239$	−4920.00	−1.33158	−0.665792	+	0.746138i	$0.731906\pi$
$240$	0	0
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	−1946.00	−0.516916
$243$	0	0
$244$	1208.00	0.316944
$245$	−1635.00	−0.426352
$246$	0	0
$247$	280.000	0.0721294
$248$	−2176.00	−0.557162
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−792.000	−0.199166	−0.0995829	−	0.995029i	$-0.531751\pi$
$251$	−792.000	−0.199166	−0.0995829	+	0.995029i	$0.531751\pi$
$252$	0	0
$253$	−3456.00	−0.858802
$254$	248.000	0.0612634
$255$	0	0
$256$	256.000	0.0625000
$257$	−2166.00	−0.525725	−0.262863	−	0.964833i	$-0.584667\pi$
$257$	−2166.00	−0.525725	−0.262863	+	0.964833i	$0.584667\pi$
$258$	0	0
$259$	1336.00	0.320521
$260$	40.0000	0.00954113
$261$	0	0
$262$	384.000	0.0905481
$263$	−3192.00	−0.748392	−0.374196	−	0.927350i	$-0.622081\pi$
$263$	−3192.00	−0.748392	−0.374196	+	0.927350i	$0.622081\pi$
$264$	0	0
$265$	390.000	0.0904057
$266$	1120.00	0.258164
$267$	0	0
$268$	2384.00	0.543381
$269$	−5490.00	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$269$	−5490.00	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$270$	0	0
$271$	−6328.00	−1.41845	−0.709223	−	0.704985i	$-0.750954\pi$
$271$	−6328.00	−1.41845	−0.709223	+	0.704985i	$0.750954\pi$
$272$	1824.00	0.406604
$273$	0	0
$274$	−5028.00	−1.10859
$275$	1200.00	0.263137
$276$	0	0
$277$	−574.000	−0.124507	−0.0622533	−	0.998060i	$-0.519829\pi$
$277$	−574.000	−0.124507	−0.0622533	+	0.998060i	$0.519829\pi$
$278$	−2680.00	−0.578186
$279$	0	0
$280$	160.000	0.0341494
$281$	−4242.00	−0.900557	−0.450278	−	0.892888i	$-0.648675\pi$
$281$	−4242.00	−0.900557	−0.450278	+	0.892888i	$0.648675\pi$
$282$	0	0
$283$	−628.000	−0.131911	−0.0659553	−	0.997823i	$-0.521009\pi$
$283$	−628.000	−0.131911	−0.0659553	+	0.997823i	$0.521009\pi$
$284$	3072.00	0.641865
$285$	0	0
$286$	−192.000	−0.0396965
$287$	−792.000	−0.162893
$288$	0	0
$289$	8083.00	1.64523
$290$	2100.00	0.425228
$291$	0	0
$292$	−1912.00	−0.383190
$293$	558.000	0.111258	0.0556292	−	0.998451i	$-0.482284\pi$
$293$	558.000	0.111258	0.0556292	+	0.998451i	$0.482284\pi$
$294$	0	0
$295$	−1200.00	−0.236836
$296$	2672.00	0.524685
$297$	0	0
$298$	2820.00	0.548182
$299$	−144.000	−0.0278520
$300$	0	0
$301$	1072.00	0.205279
$302$	4256.00	0.810945
$303$	0	0
$304$	2240.00	0.422608
$305$	1510.00	0.283483
$306$	0	0
$307$	−6964.00	−1.29465	−0.647323	−	0.762216i	$-0.724112\pi$
$307$	−6964.00	−1.29465	−0.647323	+	0.762216i	$0.724112\pi$
$308$	−768.000	−0.142081
$309$	0	0
$310$	−2720.00	−0.498341
$311$	−2832.00	−0.516360	−0.258180	−	0.966097i	$-0.583123\pi$
$311$	−2832.00	−0.516360	−0.258180	+	0.966097i	$0.583123\pi$
$312$	0	0
$313$	8642.00	1.56062	0.780311	−	0.625392i	$-0.215061\pi$
$313$	8642.00	1.56062	0.780311	+	0.625392i	$0.215061\pi$
$314$	−6052.00	−1.08769
$315$	0	0
$316$	−2560.00	−0.455732
$317$	2214.00	0.392273	0.196137	−	0.980577i	$-0.437160\pi$
$317$	2214.00	0.392273	0.196137	+	0.980577i	$0.437160\pi$
$318$	0	0
$319$	−10080.0	−1.76919
$320$	320.000	0.0559017
$321$	0	0
$322$	−576.000	−0.0996870
$323$	15960.0	2.74934
$324$	0	0
$325$	50.0000	0.00853385
$326$	−5224.00	−0.887517
$327$	0	0
$328$	−1584.00	−0.266652
$329$	864.000	0.144784
$330$	0	0
$331$	10772.0	1.78877	0.894385	−	0.447299i	$-0.147614\pi$
$331$	10772.0	1.78877	0.894385	+	0.447299i	$0.147614\pi$
$332$	1392.00	0.230108
$333$	0	0
$334$	−48.0000	−0.00786360
$335$	2980.00	0.486014
$336$	0	0
$337$	−1654.00	−0.267356	−0.133678	−	0.991025i	$-0.542679\pi$
$337$	−1654.00	−0.267356	−0.133678	+	0.991025i	$0.542679\pi$
$338$	4386.00	0.705819
$339$	0	0
$340$	2280.00	0.363678
$341$	13056.0	2.07338
$342$	0	0
$343$	2680.00	0.421885
$344$	2144.00	0.336037
$345$	0	0
$346$	3924.00	0.609698
$347$	−2196.00	−0.339733	−0.169867	−	0.985467i	$-0.554334\pi$
$347$	−2196.00	−0.339733	−0.169867	+	0.985467i	$0.554334\pi$
$348$	0	0
$349$	8270.00	1.26843	0.634216	−	0.773156i	$-0.281323\pi$
$349$	8270.00	1.26843	0.634216	+	0.773156i	$0.281323\pi$
$350$	200.000	0.0305441
$351$	0	0
$352$	−1536.00	−0.232583
$353$	−10302.0	−1.55331	−0.776657	−	0.629923i	$-0.783086\pi$
$353$	−10302.0	−1.55331	−0.776657	+	0.629923i	$0.783086\pi$
$354$	0	0
$355$	3840.00	0.574102
$356$	−840.000	−0.125056
$357$	0	0
$358$	−240.000	−0.0354313
$359$	2280.00	0.335192	0.167596	−	0.985856i	$-0.446400\pi$
$359$	2280.00	0.335192	0.167596	+	0.985856i	$0.446400\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	−1804.00	−0.261923
$363$	0	0
$364$	−32.0000	−0.00460785
$365$	−2390.00	−0.342735
$366$	0	0
$367$	−8764.00	−1.24653	−0.623266	−	0.782010i	$-0.714195\pi$
$367$	−8764.00	−1.24653	−0.623266	+	0.782010i	$0.714195\pi$
$368$	−1152.00	−0.163185
$369$	0	0
$370$	3340.00	0.469293
$371$	−312.000	−0.0436610
$372$	0	0
$373$	−1318.00	−0.182958	−0.0914792	−	0.995807i	$-0.529159\pi$
$373$	−1318.00	−0.182958	−0.0914792	+	0.995807i	$0.529159\pi$
$374$	−10944.0	−1.51310
$375$	0	0
$376$	1728.00	0.237007
$377$	−420.000	−0.0573769
$378$	0	0
$379$	1100.00	0.149085	0.0745425	−	0.997218i	$-0.476250\pi$
$379$	1100.00	0.149085	0.0745425	+	0.997218i	$0.476250\pi$
$380$	2800.00	0.377992
$381$	0	0
$382$	−336.000	−0.0450033
$383$	3528.00	0.470685	0.235343	−	0.971912i	$-0.424379\pi$
$383$	3528.00	0.470685	0.235343	+	0.971912i	$0.424379\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	2636.00	0.347588
$387$	0	0
$388$	−6136.00	−0.802856
$389$	9630.00	1.25517	0.627584	−	0.778549i	$-0.284044\pi$
$389$	9630.00	1.25517	0.627584	+	0.778549i	$0.284044\pi$
$390$	0	0
$391$	−8208.00	−1.06163
$392$	2616.00	0.337061
$393$	0	0
$394$	−8028.00	−1.02651
$395$	−3200.00	−0.407619
$396$	0	0
$397$	−3094.00	−0.391142	−0.195571	−	0.980690i	$-0.562656\pi$
$397$	−3094.00	−0.391142	−0.195571	+	0.980690i	$0.562656\pi$
$398$	−4000.00	−0.503774
$399$	0	0
$400$	400.000	0.0500000
$401$	1638.00	0.203985	0.101992	−	0.994785i	$-0.467478\pi$
$401$	1638.00	0.203985	0.101992	+	0.994785i	$0.467478\pi$
$402$	0	0
$403$	544.000	0.0672421
$404$	−6888.00	−0.848245
$405$	0	0
$406$	−1680.00	−0.205362
$407$	−16032.0	−1.95252
$408$	0	0
$409$	−13750.0	−1.66233	−0.831166	−	0.556024i	$-0.812326\pi$
$409$	−13750.0	−1.66233	−0.831166	+	0.556024i	$0.812326\pi$
$410$	−1980.00	−0.238501
$411$	0	0
$412$	4208.00	0.503188
$413$	960.000	0.114379
$414$	0	0
$415$	1740.00	0.205815
$416$	−64.0000	−0.00754293
$417$	0	0
$418$	−13440.0	−1.57266
$419$	12480.0	1.45510	0.727551	−	0.686053i	$-0.240658\pi$
$419$	12480.0	1.45510	0.727551	+	0.686053i	$0.240658\pi$
$420$	0	0
$421$	7262.00	0.840685	0.420342	−	0.907366i	$-0.361910\pi$
$421$	7262.00	0.840685	0.420342	+	0.907366i	$0.361910\pi$
$422$	7736.00	0.892376
$423$	0	0
$424$	−624.000	−0.0714720
$425$	2850.00	0.325283
$426$	0	0
$427$	−1208.00	−0.136907
$428$	2256.00	0.254785
$429$	0	0
$430$	2680.00	0.300561
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	1802.00	0.199997	0.0999984	−	0.994988i	$-0.468116\pi$
$433$	1802.00	0.199997	0.0999984	+	0.994988i	$0.468116\pi$
$434$	2176.00	0.240671
$435$	0	0
$436$	−2440.00	−0.268016
$437$	−10080.0	−1.10341
$438$	0	0
$439$	−2320.00	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$439$	−2320.00	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$440$	−1920.00	−0.208028
$441$	0	0
$442$	−456.000	−0.0490717
$443$	−11172.0	−1.19819	−0.599095	−	0.800678i	$-0.704473\pi$
$443$	−11172.0	−1.19819	−0.599095	+	0.800678i	$0.704473\pi$
$444$	0	0
$445$	−1050.00	−0.111853
$446$	6296.00	0.668440
$447$	0	0
$448$	−256.000	−0.0269975
$449$	−6810.00	−0.715777	−0.357888	−	0.933764i	$-0.616503\pi$
$449$	−6810.00	−0.715777	−0.357888	+	0.933764i	$0.616503\pi$
$450$	0	0
$451$	9504.00	0.992297
$452$	−5208.00	−0.541955
$453$	0	0
$454$	5112.00	0.528454
$455$	−40.0000	−0.00412138
$456$	0	0
$457$	17066.0	1.74686	0.873429	−	0.486952i	$-0.161891\pi$
$457$	17066.0	1.74686	0.873429	+	0.486952i	$0.161891\pi$
$458$	1220.00	0.124469
$459$	0	0
$460$	−1440.00	−0.145957
$461$	18918.0	1.91128	0.955639	−	0.294541i	$-0.0951667\pi$
$461$	18918.0	1.91128	0.955639	+	0.294541i	$0.0951667\pi$
$462$	0	0
$463$	1052.00	0.105595	0.0527976	−	0.998605i	$-0.483186\pi$
$463$	1052.00	0.105595	0.0527976	+	0.998605i	$0.483186\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	−4116.00	−0.409163
$467$	−11076.0	−1.09751	−0.548754	−	0.835984i	$-0.684898\pi$
$467$	−11076.0	−1.09751	−0.548754	+	0.835984i	$0.684898\pi$
$468$	0	0
$469$	−2384.00	−0.234718
$470$	2160.00	0.211986
$471$	0	0
$472$	1920.00	0.187236
$473$	−12864.0	−1.25050
$474$	0	0
$475$	3500.00	0.338086
$476$	−1824.00	−0.175636
$477$	0	0
$478$	9840.00	0.941571
$479$	9000.00	0.858498	0.429249	−	0.903186i	$-0.358778\pi$
$479$	9000.00	0.858498	0.429249	+	0.903186i	$0.358778\pi$
$480$	0	0
$481$	−668.000	−0.0633226
$482$	2876.00	0.271781
$483$	0	0
$484$	3892.00	0.365515
$485$	−7670.00	−0.718096
$486$	0	0
$487$	−8764.00	−0.815472	−0.407736	−	0.913100i	$-0.633682\pi$
$487$	−8764.00	−0.815472	−0.407736	+	0.913100i	$0.633682\pi$
$488$	−2416.00	−0.224113
$489$	0	0
$490$	3270.00	0.301477
$491$	−5592.00	−0.513978	−0.256989	−	0.966414i	$-0.582730\pi$
$491$	−5592.00	−0.513978	−0.256989	+	0.966414i	$0.582730\pi$
$492$	0	0
$493$	−23940.0	−2.18703
$494$	−560.000	−0.0510032
$495$	0	0
$496$	4352.00	0.393973
$497$	−3072.00	−0.277260
$498$	0	0
$499$	4700.00	0.421645	0.210823	−	0.977524i	$-0.432386\pi$
$499$	4700.00	0.421645	0.210823	+	0.977524i	$0.432386\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	1584.00	0.140831
$503$	11808.0	1.04671	0.523353	−	0.852116i	$-0.324681\pi$
$503$	11808.0	1.04671	0.523353	+	0.852116i	$0.324681\pi$
$504$	0	0
$505$	−8610.00	−0.758693
$506$	6912.00	0.607265
$507$	0	0
$508$	−496.000	−0.0433198
$509$	−1170.00	−0.101885	−0.0509424	−	0.998702i	$-0.516222\pi$
$509$	−1170.00	−0.101885	−0.0509424	+	0.998702i	$0.516222\pi$
$510$	0	0
$511$	1912.00	0.165522
$512$	−512.000	−0.0441942
$513$	0	0
$514$	4332.00	0.371744
$515$	5260.00	0.450065
$516$	0	0
$517$	−10368.0	−0.881981
$518$	−2672.00	−0.226643
$519$	0	0
$520$	−80.0000	−0.00674660
$521$	16638.0	1.39909	0.699543	−	0.714590i	$-0.253387\pi$
$521$	16638.0	1.39909	0.699543	+	0.714590i	$0.253387\pi$
$522$	0	0
$523$	15692.0	1.31198	0.655988	−	0.754771i	$-0.272252\pi$
$523$	15692.0	1.31198	0.655988	+	0.754771i	$0.272252\pi$
$524$	−768.000	−0.0640272
$525$	0	0
$526$	6384.00	0.529193
$527$	31008.0	2.56305
$528$	0	0
$529$	−6983.00	−0.573929
$530$	−780.000	−0.0639265
$531$	0	0
$532$	−2240.00	−0.182549
$533$	396.000	0.0321814
$534$	0	0
$535$	2820.00	0.227886
$536$	−4768.00	−0.384228
$537$	0	0
$538$	10980.0	0.879891
$539$	−15696.0	−1.25431
$540$	0	0
$541$	−22018.0	−1.74977	−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0	−1.74977	−0.874887	+	0.484327i	$0.839064\pi$
$542$	12656.0	1.00299
$543$	0	0
$544$	−3648.00	−0.287512
$545$	−3050.00	−0.239720
$546$	0	0
$547$	−4564.00	−0.356751	−0.178375	−	0.983963i	$-0.557084\pi$
$547$	−4564.00	−0.356751	−0.178375	+	0.983963i	$0.557084\pi$
$548$	10056.0	0.783889
$549$	0	0
$550$	−2400.00	−0.186066
$551$	−29400.0	−2.27311
$552$	0	0
$553$	2560.00	0.196858
$554$	1148.00	0.0880394
$555$	0	0
$556$	5360.00	0.408839
$557$	7734.00	0.588331	0.294165	−	0.955755i	$-0.404958\pi$
$557$	7734.00	0.588331	0.294165	+	0.955755i	$0.404958\pi$
$558$	0	0
$559$	−536.000	−0.0405552
$560$	−320.000	−0.0241473
$561$	0	0
$562$	8484.00	0.636790
$563$	20148.0	1.50824	0.754118	−	0.656739i	$-0.228065\pi$
$563$	20148.0	1.50824	0.754118	+	0.656739i	$0.228065\pi$
$564$	0	0
$565$	−6510.00	−0.484739
$566$	1256.00	0.0932749
$567$	0	0
$568$	−6144.00	−0.453867
$569$	24030.0	1.77046	0.885228	−	0.465156i	$-0.154002\pi$
$569$	24030.0	1.77046	0.885228	+	0.465156i	$0.154002\pi$
$570$	0	0
$571$	2372.00	0.173844	0.0869222	−	0.996215i	$-0.472297\pi$
$571$	2372.00	0.173844	0.0869222	+	0.996215i	$0.472297\pi$
$572$	384.000	0.0280697
$573$	0	0
$574$	1584.00	0.115183
$575$	−1800.00	−0.130548
$576$	0	0
$577$	8546.00	0.616594	0.308297	−	0.951290i	$-0.400241\pi$
$577$	8546.00	0.616594	0.308297	+	0.951290i	$0.400241\pi$
$578$	−16166.0	−1.16335
$579$	0	0
$580$	−4200.00	−0.300682
$581$	−1392.00	−0.0993974
$582$	0	0
$583$	3744.00	0.265970
$584$	3824.00	0.270956
$585$	0	0
$586$	−1116.00	−0.0786716
$587$	15444.0	1.08593	0.542966	−	0.839755i	$-0.317301\pi$
$587$	15444.0	1.08593	0.542966	+	0.839755i	$0.317301\pi$
$588$	0	0
$589$	38080.0	2.66394
$590$	2400.00	0.167469
$591$	0	0
$592$	−5344.00	−0.371009
$593$	−18342.0	−1.27018	−0.635089	−	0.772439i	$-0.719037\pi$
$593$	−18342.0	−1.27018	−0.635089	+	0.772439i	$0.719037\pi$
$594$	0	0
$595$	−2280.00	−0.157094
$596$	−5640.00	−0.387623
$597$	0	0
$598$	288.000	0.0196943
$599$	−24600.0	−1.67801	−0.839006	−	0.544123i	$-0.816863\pi$
$599$	−24600.0	−1.67801	−0.839006	+	0.544123i	$0.816863\pi$
$600$	0	0
$601$	−8998.00	−0.610709	−0.305354	−	0.952239i	$-0.598775\pi$
$601$	−8998.00	−0.610709	−0.305354	+	0.952239i	$0.598775\pi$
$602$	−2144.00	−0.145154
$603$	0	0
$604$	−8512.00	−0.573424
$605$	4865.00	0.326926
$606$	0	0
$607$	4076.00	0.272553	0.136277	−	0.990671i	$-0.456486\pi$
$607$	4076.00	0.272553	0.136277	+	0.990671i	$0.456486\pi$
$608$	−4480.00	−0.298829
$609$	0	0
$610$	−3020.00	−0.200453
$611$	−432.000	−0.0286037
$612$	0	0
$613$	−4078.00	−0.268693	−0.134347	−	0.990934i	$-0.542894\pi$
$613$	−4078.00	−0.268693	−0.134347	+	0.990934i	$0.542894\pi$
$614$	13928.0	0.915453
$615$	0	0
$616$	1536.00	0.100466
$617$	−10086.0	−0.658099	−0.329049	−	0.944313i	$-0.606728\pi$
$617$	−10086.0	−0.658099	−0.329049	+	0.944313i	$0.606728\pi$
$618$	0	0
$619$	8780.00	0.570110	0.285055	−	0.958511i	$-0.407988\pi$
$619$	8780.00	0.570110	0.285055	+	0.958511i	$0.407988\pi$
$620$	5440.00	0.352380
$621$	0	0
$622$	5664.00	0.365122
$623$	840.000	0.0540191
$624$	0	0
$625$	625.000	0.0400000
$626$	−17284.0	−1.10353
$627$	0	0
$628$	12104.0	0.769112
$629$	−38076.0	−2.41366
$630$	0	0
$631$	2792.00	0.176145	0.0880727	−	0.996114i	$-0.471929\pi$
$631$	2792.00	0.176145	0.0880727	+	0.996114i	$0.471929\pi$
$632$	5120.00	0.322251
$633$	0	0
$634$	−4428.00	−0.277379
$635$	−620.000	−0.0387464
$636$	0	0
$637$	−654.000	−0.0406788
$638$	20160.0	1.25101
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−7602.00	−0.468426	−0.234213	−	0.972185i	$-0.575251\pi$
$641$	−7602.00	−0.468426	−0.234213	+	0.972185i	$0.575251\pi$
$642$	0	0
$643$	24212.0	1.48496	0.742479	−	0.669869i	$-0.233650\pi$
$643$	24212.0	1.48496	0.742479	+	0.669869i	$0.233650\pi$
$644$	1152.00	0.0704894
$645$	0	0
$646$	−31920.0	−1.94408
$647$	−9456.00	−0.574581	−0.287290	−	0.957844i	$-0.592754\pi$
$647$	−9456.00	−0.574581	−0.287290	+	0.957844i	$0.592754\pi$
$648$	0	0
$649$	−11520.0	−0.696764
$650$	−100.000	−0.00603434
$651$	0	0
$652$	10448.0	0.627569
$653$	9558.00	0.572792	0.286396	−	0.958111i	$-0.407543\pi$
$653$	9558.00	0.572792	0.286396	+	0.958111i	$0.407543\pi$
$654$	0	0
$655$	−960.000	−0.0572676
$656$	3168.00	0.188551
$657$	0	0
$658$	−1728.00	−0.102378
$659$	29280.0	1.73078	0.865392	−	0.501095i	$-0.167069\pi$
$659$	29280.0	1.73078	0.865392	+	0.501095i	$0.167069\pi$
$660$	0	0
$661$	−29098.0	−1.71223	−0.856113	−	0.516789i	$-0.827127\pi$
$661$	−29098.0	−1.71223	−0.856113	+	0.516789i	$0.827127\pi$
$662$	−21544.0	−1.26485
$663$	0	0
$664$	−2784.00	−0.162711
$665$	−2800.00	−0.163277
$666$	0	0
$667$	15120.0	0.877734
$668$	96.0000	0.00556041
$669$	0	0
$670$	−5960.00	−0.343664
$671$	14496.0	0.833997
$672$	0	0
$673$	−11638.0	−0.666585	−0.333293	−	0.942823i	$-0.608160\pi$
$673$	−11638.0	−0.666585	−0.333293	+	0.942823i	$0.608160\pi$
$674$	3308.00	0.189050
$675$	0	0
$676$	−8772.00	−0.499090
$677$	−3426.00	−0.194493	−0.0972466	−	0.995260i	$-0.531004\pi$
$677$	−3426.00	−0.194493	−0.0972466	+	0.995260i	$0.531004\pi$
$678$	0	0
$679$	6136.00	0.346801
$680$	−4560.00	−0.257159
$681$	0	0
$682$	−26112.0	−1.46610
$683$	20148.0	1.12876	0.564379	−	0.825516i	$-0.309116\pi$
$683$	20148.0	1.12876	0.564379	+	0.825516i	$0.309116\pi$
$684$	0	0
$685$	12570.0	0.701131
$686$	−5360.00	−0.298317
$687$	0	0
$688$	−4288.00	−0.237614
$689$	156.000	0.00862573
$690$	0	0
$691$	−29428.0	−1.62011	−0.810053	−	0.586356i	$-0.800562\pi$
$691$	−29428.0	−1.62011	−0.810053	+	0.586356i	$0.800562\pi$
$692$	−7848.00	−0.431122
$693$	0	0
$694$	4392.00	0.240228
$695$	6700.00	0.365677
$696$	0	0
$697$	22572.0	1.22665
$698$	−16540.0	−0.896917
$699$	0	0
$700$	−400.000	−0.0215980
$701$	−16242.0	−0.875110	−0.437555	−	0.899192i	$-0.644155\pi$
$701$	−16242.0	−0.875110	−0.437555	+	0.899192i	$0.644155\pi$
$702$	0	0
$703$	−46760.0	−2.50866
$704$	3072.00	0.164461
$705$	0	0
$706$	20604.0	1.09836
$707$	6888.00	0.366407
$708$	0	0
$709$	2030.00	0.107529	0.0537646	−	0.998554i	$-0.482878\pi$
$709$	2030.00	0.107529	0.0537646	+	0.998554i	$0.482878\pi$
$710$	−7680.00	−0.405951
$711$	0	0
$712$	1680.00	0.0884279
$713$	−19584.0	−1.02865
$714$	0	0
$715$	480.000	0.0251063
$716$	480.000	0.0250537
$717$	0	0
$718$	−4560.00	−0.237016
$719$	−6960.00	−0.361007	−0.180504	−	0.983574i	$-0.557773\pi$
$719$	−6960.00	−0.361007	−0.180504	+	0.983574i	$0.557773\pi$
$720$	0	0
$721$	−4208.00	−0.217357
$722$	−25482.0	−1.31349
$723$	0	0
$724$	3608.00	0.185208
$725$	−5250.00	−0.268938
$726$	0	0
$727$	18596.0	0.948676	0.474338	−	0.880343i	$-0.342687\pi$
$727$	18596.0	0.948676	0.474338	+	0.880343i	$0.342687\pi$
$728$	64.0000	0.00325824
$729$	0	0
$730$	4780.00	0.242350
$731$	−30552.0	−1.54584
$732$	0	0
$733$	21242.0	1.07038	0.535192	−	0.844731i	$-0.320239\pi$
$733$	21242.0	1.07038	0.535192	+	0.844731i	$0.320239\pi$
$734$	17528.0	0.881431
$735$	0	0
$736$	2304.00	0.115389
$737$	28608.0	1.42984
$738$	0	0
$739$	−340.000	−0.0169244	−0.00846218	−	0.999964i	$-0.502694\pi$
$739$	−340.000	−0.0169244	−0.00846218	+	0.999964i	$0.502694\pi$
$740$	−6680.00	−0.331840
$741$	0	0
$742$	624.000	0.0308730
$743$	21888.0	1.08074	0.540372	−	0.841426i	$-0.318284\pi$
$743$	21888.0	1.08074	0.540372	+	0.841426i	$0.318284\pi$
$744$	0	0
$745$	−7050.00	−0.346701
$746$	2636.00	0.129371
$747$	0	0
$748$	21888.0	1.06993
$749$	−2256.00	−0.110057
$750$	0	0
$751$	17792.0	0.864500	0.432250	−	0.901754i	$-0.357720\pi$
$751$	17792.0	0.864500	0.432250	+	0.901754i	$0.357720\pi$
$752$	−3456.00	−0.167590
$753$	0	0
$754$	840.000	0.0405716
$755$	−10640.0	−0.512886
$756$	0	0
$757$	37346.0	1.79308	0.896541	−	0.442960i	$-0.146072\pi$
$757$	37346.0	1.79308	0.896541	+	0.442960i	$0.146072\pi$
$758$	−2200.00	−0.105419
$759$	0	0
$760$	−5600.00	−0.267281
$761$	11358.0	0.541034	0.270517	−	0.962715i	$-0.412805\pi$
$761$	11358.0	0.541034	0.270517	+	0.962715i	$0.412805\pi$
$762$	0	0
$763$	2440.00	0.115772
$764$	672.000	0.0318221
$765$	0	0
$766$	−7056.00	−0.332825
$767$	−480.000	−0.0225969
$768$	0	0
$769$	−34270.0	−1.60703	−0.803516	−	0.595283i	$-0.797040\pi$
$769$	−34270.0	−1.60703	−0.803516	+	0.595283i	$0.797040\pi$
$770$	1920.00	0.0898597
$771$	0	0
$772$	−5272.00	−0.245782
$773$	13278.0	0.617822	0.308911	−	0.951091i	$-0.400035\pi$
$773$	13278.0	0.617822	0.308911	+	0.951091i	$0.400035\pi$
$774$	0	0
$775$	6800.00	0.315178
$776$	12272.0	0.567705
$777$	0	0
$778$	−19260.0	−0.887538
$779$	27720.0	1.27493
$780$	0	0
$781$	36864.0	1.68899
$782$	16416.0	0.750684
$783$	0	0
$784$	−5232.00	−0.238338
$785$	15130.0	0.687914
$786$	0	0
$787$	−11164.0	−0.505659	−0.252829	−	0.967511i	$-0.581361\pi$
$787$	−11164.0	−0.505659	−0.252829	+	0.967511i	$0.581361\pi$
$788$	16056.0	0.725852
$789$	0	0
$790$	6400.00	0.288230
$791$	5208.00	0.234103
$792$	0	0
$793$	604.000	0.0270475
$794$	6188.00	0.276579
$795$	0	0
$796$	8000.00	0.356222
$797$	5094.00	0.226397	0.113199	−	0.993572i	$-0.463890\pi$
$797$	5094.00	0.226397	0.113199	+	0.993572i	$0.463890\pi$
$798$	0	0
$799$	−24624.0	−1.09028
$800$	−800.000	−0.0353553
$801$	0	0
$802$	−3276.00	−0.144239
$803$	−22944.0	−1.00831
$804$	0	0
$805$	1440.00	0.0630476
$806$	−1088.00	−0.0475474
$807$	0	0
$808$	13776.0	0.599799
$809$	8790.00	0.382002	0.191001	−	0.981590i	$-0.438827\pi$
$809$	8790.00	0.382002	0.191001	+	0.981590i	$0.438827\pi$
$810$	0	0
$811$	5852.00	0.253380	0.126690	−	0.991942i	$-0.459565\pi$
$811$	5852.00	0.253380	0.126690	+	0.991942i	$0.459565\pi$
$812$	3360.00	0.145213
$813$	0	0
$814$	32064.0	1.38064
$815$	13060.0	0.561315
$816$	0	0
$817$	−37520.0	−1.60668
$818$	27500.0	1.17545
$819$	0	0
$820$	3960.00	0.168645
$821$	29478.0	1.25309	0.626546	−	0.779384i	$-0.284468\pi$
$821$	29478.0	1.25309	0.626546	+	0.779384i	$0.284468\pi$
$822$	0	0
$823$	39332.0	1.66589	0.832945	−	0.553356i	$-0.186653\pi$
$823$	39332.0	1.66589	0.832945	+	0.553356i	$0.186653\pi$
$824$	−8416.00	−0.355807
$825$	0	0
$826$	−1920.00	−0.0808781
$827$	−6756.00	−0.284074	−0.142037	−	0.989861i	$-0.545365\pi$
$827$	−6756.00	−0.284074	−0.142037	+	0.989861i	$0.545365\pi$
$828$	0	0
$829$	3950.00	0.165488	0.0827438	−	0.996571i	$-0.473632\pi$
$829$	3950.00	0.165488	0.0827438	+	0.996571i	$0.473632\pi$
$830$	−3480.00	−0.145533
$831$	0	0
$832$	128.000	0.00533366
$833$	−37278.0	−1.55055
$834$	0	0
$835$	120.000	0.00497338
$836$	26880.0	1.11204
$837$	0	0
$838$	−24960.0	−1.02891
$839$	−12360.0	−0.508599	−0.254300	−	0.967126i	$-0.581845\pi$
$839$	−12360.0	−0.508599	−0.254300	+	0.967126i	$0.581845\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	−14524.0	−0.594454
$843$	0	0
$844$	−15472.0	−0.631005
$845$	−10965.0	−0.446399
$846$	0	0
$847$	−3892.00	−0.157887
$848$	1248.00	0.0505383
$849$	0	0
$850$	−5700.00	−0.230010
$851$	24048.0	0.968690
$852$	0	0
$853$	−35998.0	−1.44496	−0.722478	−	0.691394i	$-0.756997\pi$
$853$	−35998.0	−1.44496	−0.722478	+	0.691394i	$0.756997\pi$
$854$	2416.00	0.0968077
$855$	0	0
$856$	−4512.00	−0.180160
$857$	21594.0	0.860720	0.430360	−	0.902657i	$-0.358387\pi$
$857$	21594.0	0.860720	0.430360	+	0.902657i	$0.358387\pi$
$858$	0	0
$859$	9260.00	0.367808	0.183904	−	0.982944i	$-0.441126\pi$
$859$	9260.00	0.367808	0.183904	+	0.982944i	$0.441126\pi$
$860$	−5360.00	−0.212528
$861$	0	0
$862$	19584.0	0.773821
$863$	−31632.0	−1.24770	−0.623850	−	0.781544i	$-0.714433\pi$
$863$	−31632.0	−1.24770	−0.623850	+	0.781544i	$0.714433\pi$
$864$	0	0
$865$	−9810.00	−0.385607
$866$	−3604.00	−0.141419
$867$	0	0
$868$	−4352.00	−0.170180
$869$	−30720.0	−1.19920
$870$	0	0
$871$	1192.00	0.0463713
$872$	4880.00	0.189516
$873$	0	0
$874$	20160.0	0.780231
$875$	−500.000	−0.0193178
$876$	0	0
$877$	−39694.0	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$877$	−39694.0	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$878$	4640.00	0.178351
$879$	0	0
$880$	3840.00	0.147098
$881$	−1242.00	−0.0474961	−0.0237480	−	0.999718i	$-0.507560\pi$
$881$	−1242.00	−0.0474961	−0.0237480	+	0.999718i	$0.507560\pi$
$882$	0	0
$883$	−2668.00	−0.101682	−0.0508411	−	0.998707i	$-0.516190\pi$
$883$	−2668.00	−0.101682	−0.0508411	+	0.998707i	$0.516190\pi$
$884$	912.000	0.0346990
$885$	0	0
$886$	22344.0	0.847248
$887$	4344.00	0.164439	0.0822194	−	0.996614i	$-0.473799\pi$
$887$	4344.00	0.164439	0.0822194	+	0.996614i	$0.473799\pi$
$888$	0	0
$889$	496.000	0.0187124
$890$	2100.00	0.0790923
$891$	0	0
$892$	−12592.0	−0.472658
$893$	−30240.0	−1.13319
$894$	0	0
$895$	600.000	0.0224087
$896$	512.000	0.0190901
$897$	0	0
$898$	13620.0	0.506131
$899$	−57120.0	−2.11909
$900$	0	0
$901$	8892.00	0.328785
$902$	−19008.0	−0.701660
$903$	0	0
$904$	10416.0	0.383220
$905$	4510.00	0.165655
$906$	0	0
$907$	4436.00	0.162398	0.0811990	−	0.996698i	$-0.474125\pi$
$907$	4436.00	0.162398	0.0811990	+	0.996698i	$0.474125\pi$
$908$	−10224.0	−0.373673
$909$	0	0
$910$	80.0000	0.00291426
$911$	−22752.0	−0.827450	−0.413725	−	0.910402i	$-0.635773\pi$
$911$	−22752.0	−0.827450	−0.413725	+	0.910402i	$0.635773\pi$
$912$	0	0
$913$	16704.0	0.605500
$914$	−34132.0	−1.23521
$915$	0	0
$916$	−2440.00	−0.0880130
$917$	768.000	0.0276571
$918$	0	0
$919$	−27160.0	−0.974892	−0.487446	−	0.873153i	$-0.662071\pi$
$919$	−27160.0	−0.974892	−0.487446	+	0.873153i	$0.662071\pi$
$920$	2880.00	0.103207
$921$	0	0
$922$	−37836.0	−1.35148
$923$	1536.00	0.0547758
$924$	0	0
$925$	−8350.00	−0.296807
$926$	−2104.00	−0.0746671
$927$	0	0
$928$	6720.00	0.237710
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	0	0
$931$	−45780.0	−1.61158
$932$	8232.00	0.289322
$933$	0	0
$934$	22152.0	0.776055
$935$	27360.0	0.956971
$936$	0	0
$937$	−29974.0	−1.04505	−0.522523	−	0.852625i	$-0.675009\pi$
$937$	−29974.0	−1.04505	−0.522523	+	0.852625i	$0.675009\pi$
$938$	4768.00	0.165971
$939$	0	0
$940$	−4320.00	−0.149897
$941$	−13962.0	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$941$	−13962.0	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$942$	0	0
$943$	−14256.0	−0.492300
$944$	−3840.00	−0.132396
$945$	0	0
$946$	25728.0	0.884238
$947$	−35196.0	−1.20773	−0.603863	−	0.797088i	$-0.706373\pi$
$947$	−35196.0	−1.20773	−0.603863	+	0.797088i	$0.706373\pi$
$948$	0	0
$949$	−956.000	−0.0327008
$950$	−7000.00	−0.239063
$951$	0	0
$952$	3648.00	0.124194
$953$	28338.0	0.963230	0.481615	−	0.876383i	$-0.340050\pi$
$953$	28338.0	0.963230	0.481615	+	0.876383i	$0.340050\pi$
$954$	0	0
$955$	840.000	0.0284626
$956$	−19680.0	−0.665792
$957$	0	0
$958$	−18000.0	−0.607050
$959$	−10056.0	−0.338608
$960$	0	0
$961$	44193.0	1.48343
$962$	1336.00	0.0447759
$963$	0	0
$964$	−5752.00	−0.192178
$965$	−6590.00	−0.219834
$966$	0	0
$967$	−17524.0	−0.582765	−0.291383	−	0.956607i	$-0.594115\pi$
$967$	−17524.0	−0.582765	−0.291383	+	0.956607i	$0.594115\pi$
$968$	−7784.00	−0.258458
$969$	0	0
$970$	15340.0	0.507771
$971$	26808.0	0.886004	0.443002	−	0.896521i	$-0.353913\pi$
$971$	26808.0	0.886004	0.443002	+	0.896521i	$0.353913\pi$
$972$	0	0
$973$	−5360.00	−0.176602
$974$	17528.0	0.576626
$975$	0	0
$976$	4832.00	0.158472
$977$	10914.0	0.357390	0.178695	−	0.983905i	$-0.442813\pi$
$977$	10914.0	0.357390	0.178695	+	0.983905i	$0.442813\pi$
$978$	0	0
$979$	−10080.0	−0.329069
$980$	−6540.00	−0.213176
$981$	0	0
$982$	11184.0	0.363438
$983$	−22272.0	−0.722652	−0.361326	−	0.932440i	$-0.617676\pi$
$983$	−22272.0	−0.722652	−0.361326	+	0.932440i	$0.617676\pi$
$984$	0	0
$985$	20070.0	0.649222
$986$	47880.0	1.54646
$987$	0	0
$988$	1120.00	0.0360647
$989$	19296.0	0.620402
$990$	0	0
$991$	14072.0	0.451071	0.225536	−	0.974235i	$-0.427587\pi$
$991$	14072.0	0.451071	0.225536	+	0.974235i	$0.427587\pi$
$992$	−8704.00	−0.278581
$993$	0	0
$994$	6144.00	0.196052
$995$	10000.0	0.318614
$996$	0	0
$997$	4826.00	0.153301	0.0766504	−	0.997058i	$-0.475577\pi$
$997$	4826.00	0.153301	0.0766504	+	0.997058i	$0.475577\pi$
$998$	−9400.00	−0.298148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.4.a.c.1.1		1
3.2	odd	2		30.4.a.b.1.1	✓	1
4.3	odd	2		720.4.a.y.1.1		1
5.2	odd	4		450.4.c.j.199.1		2
5.3	odd	4		450.4.c.j.199.2		2
5.4	even	2		450.4.a.r.1.1		1
9.2	odd	6		810.4.e.i.271.1		2
9.4	even	3		810.4.e.p.541.1		2
9.5	odd	6		810.4.e.i.541.1		2
9.7	even	3		810.4.e.p.271.1		2
12.11	even	2		240.4.a.b.1.1		1
15.2	even	4		150.4.c.c.49.2		2
15.8	even	4		150.4.c.c.49.1		2
15.14	odd	2		150.4.a.b.1.1		1
21.20	even	2		1470.4.a.r.1.1		1
24.5	odd	2		960.4.a.n.1.1		1
24.11	even	2		960.4.a.bg.1.1		1
60.23	odd	4		1200.4.f.r.49.1		2
60.47	odd	4		1200.4.f.r.49.2		2
60.59	even	2		1200.4.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	3.2	odd	2
90.4.a.c.1.1		1	1.1	even	1	trivial
150.4.a.b.1.1		1	15.14	odd	2
150.4.c.c.49.1		2	15.8	even	4
150.4.c.c.49.2		2	15.2	even	4
240.4.a.b.1.1		1	12.11	even	2
450.4.a.r.1.1		1	5.4	even	2
450.4.c.j.199.1		2	5.2	odd	4
450.4.c.j.199.2		2	5.3	odd	4
720.4.a.y.1.1		1	4.3	odd	2
810.4.e.i.271.1		2	9.2	odd	6
810.4.e.i.541.1		2	9.5	odd	6
810.4.e.p.271.1		2	9.7	even	3
810.4.e.p.541.1		2	9.4	even	3
960.4.a.n.1.1		1	24.5	odd	2
960.4.a.bg.1.1		1	24.11	even	2
1200.4.a.ba.1.1		1	60.59	even	2
1200.4.f.r.49.1		2	60.23	odd	4
1200.4.f.r.49.2		2	60.47	odd	4
1470.4.a.r.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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