LMFDB - Embedded newform 1470.3.f.d.391.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,3,Mod(391,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1470.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1470.f (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$40.0545988610$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 $ x^16 + 92x^14 - 112x^13 + 5846x^12 - 7728x^11 + 197216x^10 - 298200x^9 + 4836403x^8 - 6808704x^7 + 64376800x^6 - 91953512x^5 + 595763862x^4 - 630430976x^3 + 1087013404x^2 + 294123256x + 101626561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.9
Root			$-2.63284 - 4.56021i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.d.391.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} -0.526446 q^{11} -3.46410i q^{12} +4.22307i q^{13} -3.87298 q^{15} +4.00000 q^{16} -33.3506i q^{17} -4.24264 q^{18} -3.17697i q^{19} -4.47214i q^{20} -0.744507 q^{22} +11.0591 q^{23} -4.89898i q^{24} -5.00000 q^{25} +5.97232i q^{26} +5.19615i q^{27} -56.1302 q^{29} -5.47723 q^{30} -1.89160i q^{31} +5.65685 q^{32} +0.911831i q^{33} -47.1649i q^{34} -6.00000 q^{36} -9.75363 q^{37} -4.49291i q^{38} +7.31457 q^{39} -6.32456i q^{40} -4.07377i q^{41} +46.3519 q^{43} -1.05289 q^{44} +6.70820i q^{45} +15.6399 q^{46} -63.2740i q^{47} -6.92820i q^{48} -7.07107 q^{50} -57.7649 q^{51} +8.44614i q^{52} -46.5211 q^{53} +7.34847i q^{54} +1.17717i q^{55} -5.50267 q^{57} -79.3801 q^{58} -50.5313i q^{59} -7.74597 q^{60} -103.298i q^{61} -2.67513i q^{62} +8.00000 q^{64} +9.44307 q^{65} +1.28952i q^{66} +8.73896 q^{67} -66.7012i q^{68} -19.1549i q^{69} +29.0608 q^{71} -8.48528 q^{72} +16.7329i q^{73} -13.7937 q^{74} +8.66025i q^{75} -6.35393i q^{76} +10.3444 q^{78} -132.295 q^{79} -8.94427i q^{80} +9.00000 q^{81} -5.76119i q^{82} -12.4838i q^{83} -74.5742 q^{85} +65.5515 q^{86} +97.2204i q^{87} -1.48901 q^{88} +68.2544i q^{89} +9.48683i q^{90} +22.1182 q^{92} -3.27635 q^{93} -89.4829i q^{94} -7.10391 q^{95} -9.79796i q^{96} +149.281i q^{97} +1.57934 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 48 * q^9	$ 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	2.00000	0.500000
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	− 2.44949i	− 0.408248i
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	−3.00000	−0.333333
$10$	− 3.16228i	− 0.316228i
$11$	−0.526446	−0.0478587	−0.0239293	−	0.999714i	$-0.507618\pi$
$11$	−0.526446	−0.0478587	−0.0239293	+	0.999714i	$0.507618\pi$
$12$	− 3.46410i	− 0.288675i
$13$	4.22307i	0.324851i	0.986721	+	0.162426i	$0.0519318\pi$
$13$	4.22307i	0.324851i	−0.986721	+	0.162426i	$0.948068\pi$
$14$	0	0
$15$	−3.87298	−0.258199
$16$	4.00000	0.250000
$17$	− 33.3506i	− 1.96180i	−0.194513	−	0.980900i	$-0.562313\pi$
$17$	− 33.3506i	− 1.96180i	0.194513	−	0.980900i	$-0.437687\pi$
$18$	−4.24264	−0.235702
$19$	− 3.17697i	− 0.167209i	−0.996499	−	0.0836044i	$-0.973357\pi$
$19$	− 3.17697i	− 0.167209i	0.996499	−	0.0836044i	$-0.0266432\pi$
$20$	− 4.47214i	− 0.223607i
$21$	0	0
$22$	−0.744507	−0.0338412
$23$	11.0591	0.480829	0.240415	−	0.970670i	$-0.422717\pi$
$23$	11.0591	0.480829	0.240415	+	0.970670i	$0.422717\pi$
$24$	− 4.89898i	− 0.204124i
$25$	−5.00000	−0.200000
$26$	5.97232i	0.229705i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−56.1302	−1.93552	−0.967762	−	0.251866i	$-0.918956\pi$
$29$	−56.1302	−1.93552	−0.967762	+	0.251866i	$0.918956\pi$
$30$	−5.47723	−0.182574
$31$	− 1.89160i	− 0.0610194i	−0.999534	−	0.0305097i	$-0.990287\pi$
$31$	− 1.89160i	− 0.0610194i	0.999534	−	0.0305097i	$-0.00971304\pi$
$32$	5.65685	0.176777
$33$	0.911831i	0.0276312i
$34$	− 47.1649i	− 1.38720i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	−9.75363	−0.263612	−0.131806	−	0.991276i	$-0.542078\pi$
$37$	−9.75363	−0.263612	−0.131806	+	0.991276i	$0.542078\pi$
$38$	− 4.49291i	− 0.118234i
$39$	7.31457	0.187553
$40$	− 6.32456i	− 0.158114i
$41$	− 4.07377i	− 0.0993603i	−0.998765	−	0.0496802i	$-0.984180\pi$
$41$	− 4.07377i	− 0.0993603i	0.998765	−	0.0496802i	$-0.0158202\pi$
$42$	0	0
$43$	46.3519	1.07795	0.538975	−	0.842322i	$-0.318812\pi$
$43$	46.3519	1.07795	0.538975	+	0.842322i	$0.318812\pi$
$44$	−1.05289	−0.0239293
$45$	6.70820i	0.149071i
$46$	15.6399	0.339998
$47$	− 63.2740i	− 1.34625i	−0.739527	−	0.673127i	$-0.764951\pi$
$47$	− 63.2740i	− 1.34625i	0.739527	−	0.673127i	$-0.235049\pi$
$48$	− 6.92820i	− 0.144338i
$49$	0	0
$50$	−7.07107	−0.141421
$51$	−57.7649	−1.13265
$52$	8.44614i	0.162426i
$53$	−46.5211	−0.877757	−0.438878	−	0.898546i	$-0.644624\pi$
$53$	−46.5211	−0.877757	−0.438878	+	0.898546i	$0.644624\pi$
$54$	7.34847i	0.136083i
$55$	1.17717i	0.0214031i
$56$	0	0
$57$	−5.50267	−0.0965380
$58$	−79.3801	−1.36862
$59$	− 50.5313i	− 0.856463i	−0.903669	−	0.428231i	$-0.859137\pi$
$59$	− 50.5313i	− 0.856463i	0.903669	−	0.428231i	$-0.140863\pi$
$60$	−7.74597	−0.129099
$61$	− 103.298i	− 1.69340i	−0.532068	−	0.846702i	$-0.678585\pi$
$61$	− 103.298i	− 1.69340i	0.532068	−	0.846702i	$-0.321415\pi$
$62$	− 2.67513i	− 0.0431472i
$63$	0	0
$64$	8.00000	0.125000
$65$	9.44307	0.145278
$66$	1.28952i	0.0195382i
$67$	8.73896	0.130432	0.0652161	−	0.997871i	$-0.479226\pi$
$67$	8.73896	0.130432	0.0652161	+	0.997871i	$0.479226\pi$
$68$	− 66.7012i	− 0.980900i
$69$	− 19.1549i	− 0.277607i
$70$	0	0
$71$	29.0608	0.409307	0.204653	−	0.978835i	$-0.434393\pi$
$71$	29.0608	0.409307	0.204653	+	0.978835i	$0.434393\pi$
$72$	−8.48528	−0.117851
$73$	16.7329i	0.229218i	0.993411	+	0.114609i	$0.0365616\pi$
$73$	16.7329i	0.229218i	−0.993411	+	0.114609i	$0.963438\pi$
$74$	−13.7937	−0.186402
$75$	8.66025i	0.115470i
$76$	− 6.35393i	− 0.0836044i
$77$	0	0
$78$	10.3444	0.132620
$79$	−132.295	−1.67462	−0.837308	−	0.546732i	$-0.815872\pi$
$79$	−132.295	−1.67462	−0.837308	+	0.546732i	$0.815872\pi$
$80$	− 8.94427i	− 0.111803i
$81$	9.00000	0.111111
$82$	− 5.76119i	− 0.0702584i
$83$	− 12.4838i	− 0.150407i	−0.997168	−	0.0752033i	$-0.976039\pi$
$83$	− 12.4838i	− 0.150407i	0.997168	−	0.0752033i	$-0.0239606\pi$
$84$	0	0
$85$	−74.5742	−0.877344
$86$	65.5515	0.762226
$87$	97.2204i	1.11748i
$88$	−1.48901	−0.0169206
$89$	68.2544i	0.766904i	0.923561	+	0.383452i	$0.125265\pi$
$89$	68.2544i	0.766904i	−0.923561	+	0.383452i	$0.874735\pi$
$90$	9.48683i	0.105409i
$91$	0	0
$92$	22.1182	0.240415
$93$	−3.27635	−0.0352296
$94$	− 89.4829i	− 0.951946i
$95$	−7.10391	−0.0747780
$96$	− 9.79796i	− 0.102062i
$97$	149.281i	1.53898i	0.638659	+	0.769490i	$0.279490\pi$
$97$	149.281i	1.53898i	−0.638659	+	0.769490i	$0.720510\pi$
$98$	0	0
$99$	1.57934	0.0159529
$100$	−10.0000	−0.100000
$101$	96.7453i	0.957874i	0.877849	+	0.478937i	$0.158978\pi$
$101$	96.7453i	0.957874i	−0.877849	+	0.478937i	$0.841022\pi$
$102$	−81.6919	−0.800901
$103$	− 104.762i	− 1.01711i	−0.861030	−	0.508555i	$-0.830180\pi$
$103$	− 104.762i	− 1.01711i	0.861030	−	0.508555i	$-0.169820\pi$
$104$	11.9446i	0.114852i
$105$	0	0
$106$	−65.7908	−0.620668
$107$	−124.604	−1.16453	−0.582263	−	0.813000i	$-0.697833\pi$
$107$	−124.604	−1.16453	−0.582263	+	0.813000i	$0.697833\pi$
$108$	10.3923i	0.0962250i
$109$	84.6085	0.776224	0.388112	−	0.921612i	$-0.373127\pi$
$109$	84.6085	0.776224	0.388112	+	0.921612i	$0.373127\pi$
$110$	1.66477i	0.0151342i
$111$	16.8938i	0.152196i
$112$	0	0
$113$	−116.241	−1.02868	−0.514340	−	0.857586i	$-0.671963\pi$
$113$	−116.241	−1.02868	−0.514340	+	0.857586i	$0.671963\pi$
$114$	−7.78195	−0.0682627
$115$	− 24.7288i	− 0.215033i
$116$	−112.260	−0.967762
$117$	− 12.6692i	− 0.108284i
$118$	− 71.4621i	− 0.605611i
$119$	0	0
$120$	−10.9545	−0.0912871
$121$	−120.723	−0.997710
$122$	− 146.085i	− 1.19742i
$123$	−7.05598	−0.0573657
$124$	− 3.78320i	− 0.0305097i
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−81.3744	−0.640743	−0.320372	−	0.947292i	$-0.603808\pi$
$127$	−81.3744	−0.640743	−0.320372	+	0.947292i	$0.603808\pi$
$128$	11.3137	0.0883883
$129$	− 80.2838i	− 0.622355i
$130$	13.3545	0.102727
$131$	− 240.626i	− 1.83684i	−0.395604	−	0.918421i	$-0.629465\pi$
$131$	− 240.626i	− 1.83684i	0.395604	−	0.918421i	$-0.370535\pi$
$132$	1.82366i	0.0138156i
$133$	0	0
$134$	12.3588	0.0922295
$135$	11.6190	0.0860663
$136$	− 94.3297i	− 0.693601i
$137$	233.661	1.70556	0.852778	−	0.522273i	$-0.174916\pi$
$137$	233.661	1.70556	0.852778	+	0.522273i	$0.174916\pi$
$138$	− 27.0891i	− 0.196298i
$139$	211.491i	1.52152i	0.649033	+	0.760760i	$0.275174\pi$
$139$	211.491i	1.52152i	−0.649033	+	0.760760i	$0.724826\pi$
$140$	0	0
$141$	−109.594	−0.777261
$142$	41.0981	0.289424
$143$	− 2.22322i	− 0.0155470i
$144$	−12.0000	−0.0833333
$145$	125.511i	0.865593i
$146$	23.6640i	0.162082i
$147$	0	0
$148$	−19.5073	−0.131806
$149$	144.080	0.966982	0.483491	−	0.875349i	$-0.339369\pi$
$149$	144.080	0.966982	0.483491	+	0.875349i	$0.339369\pi$
$150$	12.2474i	0.0816497i
$151$	93.5125	0.619288	0.309644	−	0.950853i	$-0.399790\pi$
$151$	93.5125	0.619288	0.309644	+	0.950853i	$0.399790\pi$
$152$	− 8.98582i	− 0.0591172i
$153$	100.052i	0.653933i
$154$	0	0
$155$	−4.22975	−0.0272887
$156$	14.6291	0.0937765
$157$	− 171.225i	− 1.09061i	−0.838238	−	0.545304i	$-0.816414\pi$
$157$	− 171.225i	− 1.09061i	0.838238	−	0.545304i	$-0.183586\pi$
$158$	−187.093	−1.18413
$159$	80.5769i	0.506773i
$160$	− 12.6491i	− 0.0790569i
$161$	0	0
$162$	12.7279	0.0785674
$163$	93.0129	0.570631	0.285316	−	0.958434i	$-0.407902\pi$
$163$	93.0129	0.570631	0.285316	+	0.958434i	$0.407902\pi$
$164$	− 8.14755i	− 0.0496802i
$165$	2.03892	0.0123571
$166$	− 17.6547i	− 0.106354i
$167$	− 104.991i	− 0.628688i	−0.949309	−	0.314344i	$-0.898216\pi$
$167$	− 104.991i	− 0.628688i	0.949309	−	0.314344i	$-0.101784\pi$
$168$	0	0
$169$	151.166	0.894472
$170$	−105.464	−0.620376
$171$	9.53090i	0.0557362i
$172$	92.7038	0.538975
$173$	− 204.157i	− 1.18010i	−0.807367	−	0.590049i	$-0.799108\pi$
$173$	− 204.157i	− 1.18010i	0.807367	−	0.590049i	$-0.200892\pi$
$174$	137.490i	0.790174i
$175$	0	0
$176$	−2.10578	−0.0119647
$177$	−87.5228	−0.494479
$178$	96.5264i	0.542283i
$179$	−195.899	−1.09441	−0.547204	−	0.836999i	$-0.684308\pi$
$179$	−195.899	−1.09441	−0.547204	+	0.836999i	$0.684308\pi$
$180$	13.4164i	0.0745356i
$181$	− 119.031i	− 0.657632i	−0.944394	−	0.328816i	$-0.893350\pi$
$181$	− 119.031i	− 0.657632i	0.944394	−	0.328816i	$-0.106650\pi$
$182$	0	0
$183$	−178.917	−0.977687
$184$	31.2798	0.169999
$185$	21.8098i	0.117891i
$186$	−4.63346	−0.0249111
$187$	17.5573i	0.0938892i
$188$	− 126.548i	− 0.673127i
$189$	0	0
$190$	−10.0464	−0.0528760
$191$	−65.7314	−0.344143	−0.172072	−	0.985084i	$-0.555046\pi$
$191$	−65.7314	−0.344143	−0.172072	+	0.985084i	$0.555046\pi$
$192$	− 13.8564i	− 0.0721688i
$193$	−96.8700	−0.501917	−0.250959	−	0.967998i	$-0.580746\pi$
$193$	−96.8700	−0.501917	−0.250959	+	0.967998i	$0.580746\pi$
$194$	211.115i	1.08822i
$195$	− 16.3559i	− 0.0838763i
$196$	0	0
$197$	186.672	0.947574	0.473787	−	0.880640i	$-0.342887\pi$
$197$	186.672	0.947574	0.473787	+	0.880640i	$0.342887\pi$
$198$	2.23352	0.0112804
$199$	115.292i	0.579354i	0.957124	+	0.289677i	$0.0935480\pi$
$199$	115.292i	0.579354i	−0.957124	+	0.289677i	$0.906452\pi$
$200$	−14.1421	−0.0707107
$201$	− 15.1363i	− 0.0753051i
$202$	136.818i	0.677319i
$203$	0	0
$204$	−115.530	−0.566323
$205$	−9.10923	−0.0444353
$206$	− 148.156i	− 0.719205i
$207$	−33.1772	−0.160276
$208$	16.8923i	0.0812129i
$209$	1.67250i	0.00800239i
$210$	0	0
$211$	139.433	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$211$	139.433	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$212$	−93.0422	−0.438878
$213$	− 50.3347i	− 0.236313i
$214$	−176.217	−0.823445
$215$	− 103.646i	− 0.482074i
$216$	14.6969i	0.0680414i
$217$	0	0
$218$	119.654	0.548874
$219$	28.9823	0.132339
$220$	2.35434i	0.0107015i
$221$	140.842	0.637294
$222$	23.8914i	0.107619i
$223$	258.055i	1.15720i	0.815612	+	0.578599i	$0.196400\pi$
$223$	258.055i	1.15720i	−0.815612	+	0.578599i	$0.803600\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−164.389	−0.727387
$227$	331.032i	1.45829i	0.684358	+	0.729146i	$0.260082\pi$
$227$	331.032i	1.45829i	−0.684358	+	0.729146i	$0.739918\pi$
$228$	−11.0053	−0.0482690
$229$	− 244.238i	− 1.06654i	−0.845944	−	0.533271i	$-0.820963\pi$
$229$	− 244.238i	− 1.06654i	0.845944	−	0.533271i	$-0.179037\pi$
$230$	− 34.9719i	− 0.152052i
$231$	0	0
$232$	−158.760	−0.684311
$233$	211.490	0.907680	0.453840	−	0.891083i	$-0.350054\pi$
$233$	211.490	0.907680	0.453840	+	0.891083i	$0.350054\pi$
$234$	− 17.9170i	− 0.0765682i
$235$	−141.485	−0.602063
$236$	− 101.063i	− 0.428231i
$237$	229.141i	0.966840i
$238$	0	0
$239$	344.134	1.43989	0.719946	−	0.694030i	$-0.244167\pi$
$239$	344.134	1.43989	0.719946	+	0.694030i	$0.244167\pi$
$240$	−15.4919	−0.0645497
$241$	− 171.348i	− 0.710989i	−0.934678	−	0.355495i	$-0.884312\pi$
$241$	− 171.348i	− 0.710989i	0.934678	−	0.355495i	$-0.115688\pi$
$242$	−170.728	−0.705487
$243$	− 15.5885i	− 0.0641500i
$244$	− 206.595i	− 0.846702i
$245$	0	0
$246$	−9.97867	−0.0405637
$247$	13.4165	0.0543180
$248$	− 5.35025i	− 0.0215736i
$249$	−21.6225	−0.0868373
$250$	15.8114i	0.0632456i
$251$	− 327.538i	− 1.30493i	−0.757818	−	0.652467i	$-0.773734\pi$
$251$	− 327.538i	− 1.30493i	0.757818	−	0.652467i	$-0.226266\pi$
$252$	0	0
$253$	−5.82200	−0.0230119
$254$	−115.081	−0.453074
$255$	129.166i	0.506535i
$256$	16.0000	0.0625000
$257$	220.358i	0.857426i	0.903441	+	0.428713i	$0.141033\pi$
$257$	220.358i	0.857426i	−0.903441	+	0.428713i	$0.858967\pi$
$258$	− 113.538i	− 0.440072i
$259$	0	0
$260$	18.8861	0.0726390
$261$	168.391	0.645175
$262$	− 340.297i	− 1.29884i
$263$	133.195	0.506445	0.253223	−	0.967408i	$-0.418509\pi$
$263$	133.195	0.506445	0.253223	+	0.967408i	$0.418509\pi$
$264$	2.57905i	0.00976912i
$265$	104.024i	0.392545i
$266$	0	0
$267$	118.220	0.442772
$268$	17.4779	0.0652161
$269$	− 26.5157i	− 0.0985713i	−0.998785	−	0.0492857i	$-0.984306\pi$
$269$	− 26.5157i	− 0.0985713i	0.998785	−	0.0492857i	$-0.0156945\pi$
$270$	16.4317	0.0608581
$271$	107.015i	0.394889i	0.980314	+	0.197444i	$0.0632642\pi$
$271$	107.015i	0.394889i	−0.980314	+	0.197444i	$0.936736\pi$
$272$	− 133.402i	− 0.490450i
$273$	0	0
$274$	330.447	1.20601
$275$	2.63223	0.00957174
$276$	− 38.3098i	− 0.138803i
$277$	440.381	1.58982	0.794912	−	0.606725i	$-0.207517\pi$
$277$	440.381	1.58982	0.794912	+	0.606725i	$0.207517\pi$
$278$	299.094i	1.07588i
$279$	5.67480i	0.0203398i
$280$	0	0
$281$	322.069	1.14615	0.573076	−	0.819502i	$-0.305750\pi$
$281$	322.069	1.14615	0.573076	+	0.819502i	$0.305750\pi$
$282$	−154.989	−0.549606
$283$	− 68.5331i	− 0.242166i	−0.992642	−	0.121083i	$-0.961363\pi$
$283$	− 68.5331i	− 0.242166i	0.992642	−	0.121083i	$-0.0386368\pi$
$284$	58.1216	0.204653
$285$	12.3043i	0.0431731i
$286$	− 3.14410i	− 0.0109934i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	−823.262	−2.84866
$290$	177.499i	0.612066i
$291$	258.562	0.888531
$292$	33.4659i	0.114609i
$293$	147.510i	0.503448i	0.967799	+	0.251724i	$0.0809975\pi$
$293$	147.510i	0.503448i	−0.967799	+	0.251724i	$0.919003\pi$
$294$	0	0
$295$	−112.991	−0.383022
$296$	−27.5874	−0.0932008
$297$	− 2.73549i	− 0.00921041i
$298$	203.760	0.683760
$299$	46.7032i	0.156198i
$300$	17.3205i	0.0577350i
$301$	0	0
$302$	132.247	0.437903
$303$	167.568	0.553029
$304$	− 12.7079i	− 0.0418022i
$305$	−230.980	−0.757313
$306$	141.495i	0.462401i
$307$	376.010i	1.22479i	0.790553	+	0.612394i	$0.209793\pi$
$307$	376.010i	1.22479i	−0.790553	+	0.612394i	$0.790207\pi$
$308$	0	0
$309$	−181.454	−0.587228
$310$	−5.98177	−0.0192960
$311$	− 389.826i	− 1.25346i	−0.779236	−	0.626730i	$-0.784393\pi$
$311$	− 389.826i	− 1.25346i	0.779236	−	0.626730i	$-0.215607\pi$
$312$	20.6887	0.0663100
$313$	143.056i	0.457049i	0.973538	+	0.228524i	$0.0733901\pi$
$313$	143.056i	0.457049i	−0.973538	+	0.228524i	$0.926610\pi$
$314$	− 242.149i	− 0.771176i
$315$	0	0
$316$	−264.589	−0.837308
$317$	384.444	1.21276	0.606378	−	0.795176i	$-0.292622\pi$
$317$	384.444	1.21276	0.606378	+	0.795176i	$0.292622\pi$
$318$	113.953i	0.358343i
$319$	29.5495	0.0926317
$320$	− 17.8885i	− 0.0559017i
$321$	215.821i	0.672340i
$322$	0	0
$323$	−105.954	−0.328030
$324$	18.0000	0.0555556
$325$	− 21.1153i	− 0.0649703i
$326$	131.540	0.403497
$327$	− 146.546i	− 0.448153i
$328$	− 11.5224i	− 0.0351292i
$329$	0	0
$330$	2.88346	0.00873776
$331$	193.908	0.585824	0.292912	−	0.956139i	$-0.405376\pi$
$331$	193.908	0.585824	0.292912	+	0.956139i	$0.405376\pi$
$332$	− 24.9675i	− 0.0752033i
$333$	29.2609	0.0878706
$334$	− 148.479i	− 0.444549i
$335$	− 19.5409i	− 0.0583310i
$336$	0	0
$337$	125.477	0.372337	0.186168	−	0.982518i	$-0.440393\pi$
$337$	125.477	0.372337	0.186168	+	0.982518i	$0.440393\pi$
$338$	213.781	0.632487
$339$	201.335i	0.593909i
$340$	−149.148	−0.438672
$341$	0.995825i	0.00292031i
$342$	13.4787i	0.0394115i
$343$	0	0
$344$	131.103	0.381113
$345$	−42.8316	−0.124150
$346$	− 288.722i	− 0.834456i
$347$	−503.597	−1.45129	−0.725644	−	0.688071i	$-0.758458\pi$
$347$	−503.597	−1.45129	−0.725644	+	0.688071i	$0.758458\pi$
$348$	194.441i	0.558738i
$349$	− 47.7682i	− 0.136872i	−0.997656	−	0.0684358i	$-0.978199\pi$
$349$	− 47.7682i	− 0.136872i	0.997656	−	0.0684358i	$-0.0218008\pi$
$350$	0	0
$351$	−21.9437	−0.0625177
$352$	−2.97803	−0.00846030
$353$	32.9103i	0.0932304i	0.998913	+	0.0466152i	$0.0148435\pi$
$353$	32.9103i	0.0932304i	−0.998913	+	0.0466152i	$0.985157\pi$
$354$	−123.776	−0.349650
$355$	− 64.9819i	− 0.183048i
$356$	136.509i	0.383452i
$357$	0	0
$358$	−277.043	−0.773863
$359$	267.797	0.745952	0.372976	−	0.927841i	$-0.378337\pi$
$359$	267.797	0.745952	0.372976	+	0.927841i	$0.378337\pi$
$360$	18.9737i	0.0527046i
$361$	350.907	0.972041
$362$	− 168.336i	− 0.465016i
$363$	209.098i	0.576028i
$364$	0	0
$365$	37.4160	0.102510
$366$	−253.026	−0.691329
$367$	− 158.723i	− 0.432487i	−0.976339	−	0.216243i	$-0.930620\pi$
$367$	− 158.723i	− 0.432487i	0.976339	−	0.216243i	$-0.0693805\pi$
$368$	44.2363	0.120207
$369$	12.2213i	0.0331201i
$370$	30.8437i	0.0833614i
$371$	0	0
$372$	−6.55270	−0.0176148
$373$	414.344	1.11084	0.555421	−	0.831570i	$-0.312557\pi$
$373$	414.344	1.11084	0.555421	+	0.831570i	$0.312557\pi$
$374$	24.8297i	0.0663897i
$375$	19.3649	0.0516398
$376$	− 178.966i	− 0.475973i
$377$	− 237.042i	− 0.628758i
$378$	0	0
$379$	−72.8000	−0.192084	−0.0960422	−	0.995377i	$-0.530618\pi$
$379$	−72.8000	−0.192084	−0.0960422	+	0.995377i	$0.530618\pi$
$380$	−14.2078	−0.0373890
$381$	140.945i	0.369933i
$382$	−92.9582	−0.243346
$383$	285.014i	0.744163i	0.928200	+	0.372082i	$0.121356\pi$
$383$	285.014i	0.744163i	−0.928200	+	0.372082i	$0.878644\pi$
$384$	− 19.5959i	− 0.0510310i
$385$	0	0
$386$	−136.995	−0.354909
$387$	−139.056	−0.359317
$388$	298.562i	0.769490i
$389$	−97.1383	−0.249713	−0.124856	−	0.992175i	$-0.539847\pi$
$389$	−97.1383	−0.249713	−0.124856	+	0.992175i	$0.539847\pi$
$390$	− 23.1307i	− 0.0593095i
$391$	− 368.827i	− 0.943291i
$392$	0	0
$393$	−416.777	−1.06050
$394$	263.994	0.670036
$395$	295.820i	0.748911i
$396$	3.15867	0.00797645
$397$	791.704i	1.99422i	0.0759985	+	0.997108i	$0.475786\pi$
$397$	791.704i	1.99422i	−0.0759985	+	0.997108i	$0.524214\pi$
$398$	163.047i	0.409665i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	503.226	1.25493	0.627464	−	0.778645i	$-0.284093\pi$
$401$	503.226	1.25493	0.627464	+	0.778645i	$0.284093\pi$
$402$	− 21.4060i	− 0.0532487i
$403$	7.98836	0.0198222
$404$	193.491i	0.478937i
$405$	− 20.1246i	− 0.0496904i
$406$	0	0
$407$	5.13476	0.0126161
$408$	−163.384	−0.400451
$409$	749.857i	1.83339i	0.399586	+	0.916696i	$0.369154\pi$
$409$	749.857i	1.83339i	−0.399586	+	0.916696i	$0.630846\pi$
$410$	−12.8824	−0.0314205
$411$	− 404.713i	− 0.984704i
$412$	− 209.525i	− 0.508555i
$413$	0	0
$414$	−46.9197	−0.113333
$415$	−27.9145	−0.0672639
$416$	23.8893i	0.0574262i
$417$	366.314	0.878450
$418$	2.36527i	0.00565855i
$419$	465.759i	1.11160i	0.831317	+	0.555799i	$0.187587\pi$
$419$	465.759i	1.11160i	−0.831317	+	0.555799i	$0.812413\pi$
$420$	0	0
$421$	345.980	0.821805	0.410902	−	0.911679i	$-0.365214\pi$
$421$	345.980	0.821805	0.410902	+	0.911679i	$0.365214\pi$
$422$	197.188	0.467269
$423$	189.822i	0.448752i
$424$	−131.582	−0.310334
$425$	166.753i	0.392360i
$426$	− 71.1841i	− 0.167099i
$427$	0	0
$428$	−249.209	−0.582263
$429$	−3.85072	−0.00897605
$430$	− 146.578i	− 0.340878i
$431$	494.600	1.14756	0.573782	−	0.819008i	$-0.305476\pi$
$431$	494.600	1.14756	0.573782	+	0.819008i	$0.305476\pi$
$432$	20.7846i	0.0481125i
$433$	− 730.022i	− 1.68596i	−0.537943	−	0.842981i	$-0.680799\pi$
$433$	− 730.022i	− 1.68596i	0.537943	−	0.842981i	$-0.319201\pi$
$434$	0	0
$435$	217.391	0.499750
$436$	169.217	0.388112
$437$	− 35.1343i	− 0.0803989i
$438$	40.9872	0.0935780
$439$	371.388i	0.845986i	0.906133	+	0.422993i	$0.139021\pi$
$439$	371.388i	0.845986i	−0.906133	+	0.422993i	$0.860979\pi$
$440$	3.32953i	0.00756712i
$441$	0	0
$442$	199.180	0.450635
$443$	408.708	0.922591	0.461295	−	0.887247i	$-0.347385\pi$
$443$	408.708	0.922591	0.461295	+	0.887247i	$0.347385\pi$
$444$	33.7876i	0.0760982i
$445$	152.622	0.342970
$446$	364.945i	0.818263i
$447$	− 249.555i	− 0.558288i
$448$	0	0
$449$	725.469	1.61574	0.807872	−	0.589358i	$-0.200619\pi$
$449$	725.469	1.61574	0.807872	+	0.589358i	$0.200619\pi$
$450$	21.2132	0.0471405
$451$	2.14462i	0.00475526i
$452$	−232.482	−0.514340
$453$	− 161.968i	− 0.357546i
$454$	468.150i	1.03117i
$455$	0	0
$456$	−15.5639	−0.0341313
$457$	401.530	0.878622	0.439311	−	0.898335i	$-0.355223\pi$
$457$	401.530	0.878622	0.439311	+	0.898335i	$0.355223\pi$
$458$	− 345.405i	− 0.754159i
$459$	173.295	0.377549
$460$	− 49.4577i	− 0.107517i
$461$	− 653.050i	− 1.41659i	−0.705915	−	0.708297i	$-0.749464\pi$
$461$	− 653.050i	− 1.41659i	0.705915	−	0.708297i	$-0.250536\pi$
$462$	0	0
$463$	−869.580	−1.87814	−0.939072	−	0.343722i	$-0.888312\pi$
$463$	−869.580	−1.87814	−0.939072	+	0.343722i	$0.888312\pi$
$464$	−224.521	−0.483881
$465$	7.32614i	0.0157551i
$466$	299.091	0.641827
$467$	− 739.202i	− 1.58287i	−0.611251	−	0.791437i	$-0.709333\pi$
$467$	− 739.202i	− 1.58287i	0.611251	−	0.791437i	$-0.290667\pi$
$468$	− 25.3384i	− 0.0541419i
$469$	0	0
$470$	−200.090	−0.425723
$471$	−296.571	−0.629663
$472$	− 142.924i	− 0.302805i
$473$	−24.4017	−0.0515893
$474$	324.054i	0.683659i
$475$	15.8848i	0.0334417i
$476$	0	0
$477$	139.563	0.292586
$478$	486.679	1.01816
$479$	− 606.225i	− 1.26560i	−0.774313	−	0.632802i	$-0.781905\pi$
$479$	− 606.225i	− 1.26560i	0.774313	−	0.632802i	$-0.218095\pi$
$480$	−21.9089	−0.0456435
$481$	− 41.1903i	− 0.0856347i
$482$	− 242.323i	− 0.502745i
$483$	0	0
$484$	−241.446	−0.498855
$485$	333.803	0.688253
$486$	− 22.0454i	− 0.0453609i
$487$	649.576	1.33383	0.666916	−	0.745133i	$-0.267614\pi$
$487$	649.576	1.33383	0.666916	+	0.745133i	$0.267614\pi$
$488$	− 292.170i	− 0.598708i
$489$	− 161.103i	− 0.329454i
$490$	0	0
$491$	266.629	0.543033	0.271516	−	0.962434i	$-0.412475\pi$
$491$	266.629	0.543033	0.271516	+	0.962434i	$0.412475\pi$
$492$	−14.1120	−0.0286829
$493$	1871.98i	3.79711i
$494$	18.9739	0.0384086
$495$	− 3.53150i	− 0.00713435i
$496$	− 7.56640i	− 0.0152548i
$497$	0	0
$498$	−30.5788	−0.0614033
$499$	−681.106	−1.36494	−0.682471	−	0.730913i	$-0.739095\pi$
$499$	−681.106	−1.36494	−0.682471	+	0.730913i	$0.739095\pi$
$500$	22.3607i	0.0447214i
$501$	−181.849	−0.362973
$502$	− 463.209i	− 0.922727i
$503$	− 453.326i	− 0.901245i	−0.892715	−	0.450622i	$-0.851202\pi$
$503$	− 453.326i	− 0.901245i	0.892715	−	0.450622i	$-0.148798\pi$
$504$	0	0
$505$	216.329	0.428374
$506$	−8.23356	−0.0162718
$507$	− 261.827i	− 0.516423i
$508$	−162.749	−0.320372
$509$	50.2641i	0.0987507i	0.998780	+	0.0493754i	$0.0157231\pi$
$509$	50.2641i	0.0987507i	−0.998780	+	0.0493754i	$0.984277\pi$
$510$	182.669i	0.358174i
$511$	0	0
$512$	22.6274	0.0441942
$513$	16.5080	0.0321793
$514$	311.634i	0.606291i
$515$	−234.256	−0.454865
$516$	− 160.568i	− 0.311178i
$517$	33.3103i	0.0644300i
$518$	0	0
$519$	−353.610	−0.681330
$520$	26.7090	0.0513635
$521$	− 103.111i	− 0.197911i	−0.995092	−	0.0989553i	$-0.968450\pi$
$521$	− 103.111i	− 0.197911i	0.995092	−	0.0989553i	$-0.0315501\pi$
$522$	238.140	0.456207
$523$	366.867i	0.701467i	0.936475	+	0.350733i	$0.114068\pi$
$523$	366.867i	0.701467i	−0.936475	+	0.350733i	$0.885932\pi$
$524$	− 481.253i	− 0.918421i
$525$	0	0
$526$	188.366	0.358111
$527$	−63.0860	−0.119708
$528$	3.64732i	0.00690781i
$529$	−406.697	−0.768803
$530$	147.113i	0.277571i
$531$	151.594i	0.285488i
$532$	0	0
$533$	17.2038	0.0322773
$534$	167.189	0.313087
$535$	278.624i	0.520792i
$536$	24.7175	0.0461147
$537$	339.307i	0.631856i
$538$	− 37.4988i	− 0.0697005i
$539$	0	0
$540$	23.2379	0.0430331
$541$	−533.117	−0.985429	−0.492714	−	0.870191i	$-0.663995\pi$
$541$	−533.117	−0.985429	−0.492714	+	0.870191i	$0.663995\pi$
$542$	151.342i	0.279228i
$543$	−206.168	−0.379684
$544$	− 188.659i	− 0.346800i
$545$	− 189.190i	− 0.347138i
$546$	0	0
$547$	69.6218	0.127279	0.0636396	−	0.997973i	$-0.479729\pi$
$547$	69.6218	0.127279	0.0636396	+	0.997973i	$0.479729\pi$
$548$	467.323	0.852778
$549$	309.893i	0.564468i
$550$	3.72253	0.00676824
$551$	178.324i	0.323637i
$552$	− 54.1782i	− 0.0981489i
$553$	0	0
$554$	622.793	1.12418
$555$	37.7757	0.0680643
$556$	422.983i	0.760760i
$557$	16.8624	0.0302737	0.0151368	−	0.999885i	$-0.495182\pi$
$557$	16.8624	0.0302737	0.0151368	+	0.999885i	$0.495182\pi$
$558$	8.02538i	0.0143824i
$559$	195.747i	0.350174i
$560$	0	0
$561$	30.4101	0.0542069
$562$	455.474	0.810452
$563$	915.784i	1.62662i	0.581834	+	0.813308i	$0.302335\pi$
$563$	915.784i	1.62662i	−0.581834	+	0.813308i	$0.697665\pi$
$564$	−219.187	−0.388630
$565$	259.922i	0.460040i
$566$	− 96.9204i	− 0.171237i
$567$	0	0
$568$	82.1963	0.144712
$569$	407.655	0.716441	0.358221	−	0.933637i	$-0.383384\pi$
$569$	407.655	0.716441	0.358221	+	0.933637i	$0.383384\pi$
$570$	17.4010i	0.0305280i
$571$	−895.820	−1.56886	−0.784431	−	0.620216i	$-0.787045\pi$
$571$	−895.820	−1.56886	−0.784431	+	0.620216i	$0.787045\pi$
$572$	− 4.44643i	− 0.00777348i
$573$	113.850i	0.198691i
$574$	0	0
$575$	−55.2954	−0.0961659
$576$	−24.0000	−0.0416667
$577$	681.406i	1.18095i	0.807057	+	0.590473i	$0.201059\pi$
$577$	681.406i	1.18095i	−0.807057	+	0.590473i	$0.798941\pi$
$578$	−1164.27	−2.01431
$579$	167.784i	0.289782i
$580$	251.022i	0.432796i
$581$	0	0
$582$	365.662	0.628286
$583$	24.4908	0.0420083
$584$	47.3279i	0.0810410i
$585$	−28.3292	−0.0484260
$586$	208.611i	0.355991i
$587$	− 833.001i	− 1.41908i	−0.704665	−	0.709541i	$-0.748903\pi$
$587$	− 833.001i	− 1.41908i	0.704665	−	0.709541i	$-0.251097\pi$
$588$	0	0
$589$	−6.00955	−0.0102030
$590$	−159.794	−0.270837
$591$	− 323.325i	− 0.547082i
$592$	−39.0145	−0.0659029
$593$	− 312.741i	− 0.527388i	−0.964606	−	0.263694i	$-0.915059\pi$
$593$	− 312.741i	− 0.527388i	0.964606	−	0.263694i	$-0.0849409\pi$
$594$	− 3.86857i	− 0.00651274i
$595$	0	0
$596$	288.161	0.483491
$597$	199.691	0.334490
$598$	66.0484i	0.110449i
$599$	214.243	0.357667	0.178834	−	0.983879i	$-0.442768\pi$
$599$	214.243	0.357667	0.178834	+	0.983879i	$0.442768\pi$
$600$	24.4949i	0.0408248i
$601$	176.849i	0.294257i	0.989117	+	0.147129i	$0.0470031\pi$
$601$	176.849i	0.294257i	−0.989117	+	0.147129i	$0.952997\pi$
$602$	0	0
$603$	−26.2169	−0.0434774
$604$	187.025	0.309644
$605$	269.945i	0.446189i
$606$	236.977	0.391050
$607$	− 37.1010i	− 0.0611218i	−0.999533	−	0.0305609i	$-0.990271\pi$
$607$	− 37.1010i	− 0.0611218i	0.999533	−	0.0305609i	$-0.00972936\pi$
$608$	− 17.9716i	− 0.0295586i
$609$	0	0
$610$	−326.656	−0.535501
$611$	267.210	0.437333
$612$	200.104i	0.326967i
$613$	−899.719	−1.46773	−0.733866	−	0.679295i	$-0.762286\pi$
$613$	−899.719	−1.46773	−0.733866	+	0.679295i	$0.762286\pi$
$614$	531.758i	0.866056i
$615$	15.7777i	0.0256547i
$616$	0	0
$617$	−626.244	−1.01498	−0.507491	−	0.861657i	$-0.669427\pi$
$617$	−626.244	−1.01498	−0.507491	+	0.861657i	$0.669427\pi$
$618$	−256.614	−0.415233
$619$	− 896.480i	− 1.44827i	−0.689657	−	0.724136i	$-0.742239\pi$
$619$	− 896.480i	− 1.44827i	0.689657	−	0.724136i	$-0.257761\pi$
$620$	−8.45950	−0.0136443
$621$	57.4646i	0.0925357i
$622$	− 551.298i	− 0.886331i
$623$	0	0
$624$	29.2583	0.0468883
$625$	25.0000	0.0400000
$626$	202.312i	0.323182i
$627$	2.89685	0.00462018
$628$	− 342.451i	− 0.545304i
$629$	325.290i	0.517153i
$630$	0	0
$631$	−500.730	−0.793550	−0.396775	−	0.917916i	$-0.629871\pi$
$631$	−500.730	−0.793550	−0.396775	+	0.917916i	$0.629871\pi$
$632$	−374.186	−0.592066
$633$	− 241.505i	− 0.381524i
$634$	543.686	0.857549
$635$	181.959i	0.286549i
$636$	161.154i	0.253387i
$637$	0	0
$638$	41.7893	0.0655005
$639$	−87.1823	−0.136436
$640$	− 25.2982i	− 0.0395285i
$641$	620.578	0.968140	0.484070	−	0.875029i	$-0.339158\pi$
$641$	620.578	0.968140	0.484070	+	0.875029i	$0.339158\pi$
$642$	305.217i	0.475416i
$643$	1127.93i	1.75417i	0.480334	+	0.877086i	$0.340516\pi$
$643$	1127.93i	1.75417i	−0.480334	+	0.877086i	$0.659484\pi$
$644$	0	0
$645$	−179.520	−0.278326
$646$	−149.841	−0.231952
$647$	− 135.250i	− 0.209041i	−0.994523	−	0.104521i	$-0.966669\pi$
$647$	− 135.250i	− 0.209041i	0.994523	−	0.104521i	$-0.0333308\pi$
$648$	25.4558	0.0392837
$649$	26.6020i	0.0409892i
$650$	− 29.8616i	− 0.0459409i
$651$	0	0
$652$	186.026	0.285316
$653$	−390.290	−0.597687	−0.298843	−	0.954302i	$-0.596601\pi$
$653$	−390.290	−0.597687	−0.298843	+	0.954302i	$0.596601\pi$
$654$	− 207.248i	− 0.316892i
$655$	−538.057	−0.821461
$656$	− 16.2951i	− 0.0248401i
$657$	− 50.1988i	− 0.0764061i
$658$	0	0
$659$	−864.853	−1.31237	−0.656186	−	0.754599i	$-0.727831\pi$
$659$	−864.853	−1.31237	−0.656186	+	0.754599i	$0.727831\pi$
$660$	4.07783	0.00617853
$661$	− 1008.21i	− 1.52528i	−0.646825	−	0.762638i	$-0.723904\pi$
$661$	− 1008.21i	− 1.52528i	0.646825	−	0.762638i	$-0.276096\pi$
$662$	274.227	0.414240
$663$	− 243.945i	− 0.367942i
$664$	− 35.3094i	− 0.0531768i
$665$	0	0
$666$	41.3812	0.0621339
$667$	−620.748	−0.930657
$668$	− 209.982i	− 0.314344i
$669$	446.965	0.668109
$670$	− 27.6350i	− 0.0412463i
$671$	54.3806i	0.0810441i
$672$	0	0
$673$	−109.959	−0.163386	−0.0816928	−	0.996658i	$-0.526033\pi$
$673$	−109.959	−0.163386	−0.0816928	+	0.996658i	$0.526033\pi$
$674$	177.452	0.263282
$675$	− 25.9808i	− 0.0384900i
$676$	302.331	0.447236
$677$	− 168.833i	− 0.249383i	−0.992196	−	0.124692i	$-0.960206\pi$
$677$	− 168.833i	− 0.249383i	0.992196	−	0.124692i	$-0.0397942\pi$
$678$	284.731i	0.419957i
$679$	0	0
$680$	−210.928	−0.310188
$681$	573.365	0.841945
$682$	1.40831i	0.00206497i
$683$	−436.130	−0.638551	−0.319276	−	0.947662i	$-0.603440\pi$
$683$	−436.130	−0.638551	−0.319276	+	0.947662i	$0.603440\pi$
$684$	19.0618i	0.0278681i
$685$	− 522.482i	− 0.762748i
$686$	0	0
$687$	−423.033	−0.615768
$688$	185.408	0.269488
$689$	− 196.462i	− 0.285141i
$690$	−60.5731	−0.0877870
$691$	68.4471i	0.0990552i	0.998773	+	0.0495276i	$0.0157716\pi$
$691$	68.4471i	0.0990552i	−0.998773	+	0.0495276i	$0.984228\pi$
$692$	− 408.314i	− 0.590049i
$693$	0	0
$694$	−712.193	−1.02622
$695$	472.909	0.680444
$696$	274.981i	0.395087i
$697$	−135.863	−0.194925
$698$	− 67.5544i	− 0.0967829i
$699$	− 366.311i	− 0.524050i
$700$	0	0
$701$	1283.41	1.83083	0.915414	−	0.402514i	$-0.131864\pi$
$701$	1283.41	1.83083	0.915414	+	0.402514i	$0.131864\pi$
$702$	−31.0331	−0.0442067
$703$	30.9870i	0.0440782i
$704$	−4.21157	−0.00598234
$705$	245.059i	0.347602i
$706$	46.5422i	0.0659238i
$707$	0	0
$708$	−175.046	−0.247240
$709$	172.400	0.243159	0.121580	−	0.992582i	$-0.461204\pi$
$709$	172.400	0.243159	0.121580	+	0.992582i	$0.461204\pi$
$710$	− 91.8983i	− 0.129434i
$711$	396.884	0.558205
$712$	193.053i	0.271141i
$713$	− 20.9194i	− 0.0293399i
$714$	0	0
$715$	−4.97126	−0.00695282
$716$	−391.798	−0.547204
$717$	− 596.058i	− 0.831322i
$718$	378.722	0.527468
$719$	502.312i	0.698626i	0.937006	+	0.349313i	$0.113585\pi$
$719$	502.312i	0.698626i	−0.937006	+	0.349313i	$0.886415\pi$
$720$	26.8328i	0.0372678i
$721$	0	0
$722$	496.257	0.687337
$723$	−296.784	−0.410490
$724$	− 238.063i	− 0.328816i
$725$	280.651	0.387105
$726$	295.709i	0.407313i
$727$	748.693i	1.02984i	0.857238	+	0.514920i	$0.172178\pi$
$727$	748.693i	1.02984i	−0.857238	+	0.514920i	$0.827822\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	52.9142	0.0724852
$731$	− 1545.86i	− 2.11472i
$732$	−357.833	−0.488843
$733$	938.496i	1.28035i	0.768229	+	0.640175i	$0.221138\pi$
$733$	938.496i	1.28035i	−0.768229	+	0.640175i	$0.778862\pi$
$734$	− 224.468i	− 0.305814i
$735$	0	0
$736$	62.5596	0.0849994
$737$	−4.60059	−0.00624231
$738$	17.2836i	0.0234195i
$739$	213.641	0.289095	0.144547	−	0.989498i	$-0.453827\pi$
$739$	213.641	0.289095	0.144547	+	0.989498i	$0.453827\pi$
$740$	43.6196i	0.0589454i
$741$	− 23.2381i	− 0.0313605i
$742$	0	0
$743$	544.013	0.732184	0.366092	−	0.930579i	$-0.380696\pi$
$743$	544.013	0.732184	0.366092	+	0.930579i	$0.380696\pi$
$744$	−9.26691	−0.0124555
$745$	− 322.173i	− 0.432448i
$746$	585.971	0.785483
$747$	37.4513i	0.0501356i
$748$	35.1146i	0.0469446i
$749$	0	0
$750$	27.3861	0.0365148
$751$	589.409	0.784832	0.392416	−	0.919788i	$-0.371639\pi$
$751$	589.409	0.784832	0.392416	+	0.919788i	$0.371639\pi$
$752$	− 253.096i	− 0.336564i
$753$	−567.313	−0.753403
$754$	− 335.228i	− 0.444599i
$755$	− 209.100i	− 0.276954i
$756$	0	0
$757$	−448.997	−0.593127	−0.296564	−	0.955013i	$-0.595841\pi$
$757$	−448.997	−0.593127	−0.296564	+	0.955013i	$0.595841\pi$
$758$	−102.955	−0.135824
$759$	10.0840i	0.0132859i
$760$	−20.0929	−0.0264380
$761$	− 429.449i	− 0.564322i	−0.959367	−	0.282161i	$-0.908949\pi$
$761$	− 429.449i	− 0.564322i	0.959367	−	0.282161i	$-0.0910513\pi$
$762$	199.326i	0.261582i
$763$	0	0
$764$	−131.463	−0.172072
$765$	223.723	0.292448
$766$	403.071i	0.526203i
$767$	213.397	0.278223
$768$	− 27.7128i	− 0.0360844i
$769$	− 32.5790i	− 0.0423655i	−0.999776	−	0.0211827i	$-0.993257\pi$
$769$	− 32.5790i	− 0.0423655i	0.999776	−	0.0211827i	$-0.00674318\pi$
$770$	0	0
$771$	381.672	0.495035
$772$	−193.740	−0.250959
$773$	1243.54i	1.60871i	0.594146	+	0.804357i	$0.297490\pi$
$773$	1243.54i	1.60871i	−0.594146	+	0.804357i	$0.702510\pi$
$774$	−196.654	−0.254075
$775$	9.45800i	0.0122039i
$776$	422.231i	0.544112i
$777$	0	0
$778$	−137.374	−0.176574
$779$	−12.9422	−0.0166139
$780$	− 32.7118i	− 0.0419381i
$781$	−15.2989	−0.0195889
$782$	− 521.600i	− 0.667008i
$783$	− 291.661i	− 0.372492i
$784$	0	0
$785$	−382.872	−0.487735
$786$	−589.412	−0.749888
$787$	190.628i	0.242220i	0.992639	+	0.121110i	$0.0386455\pi$
$787$	190.628i	0.242220i	−0.992639	+	0.121110i	$0.961355\pi$
$788$	373.344	0.473787
$789$	− 230.701i	− 0.292396i
$790$	418.352i	0.529560i
$791$	0	0
$792$	4.46704	0.00564020
$793$	436.233	0.550105
$794$	1119.64i	1.41012i
$795$	180.175	0.226636
$796$	230.583i	0.289677i
$797$	− 387.796i	− 0.486570i	−0.969955	−	0.243285i	$-0.921775\pi$
$797$	− 387.796i	− 0.486570i	0.969955	−	0.243285i	$-0.0782250\pi$
$798$	0	0
$799$	−2110.23	−2.64108
$800$	−28.2843	−0.0353553
$801$	− 204.763i	− 0.255635i
$802$	711.670	0.887369
$803$	− 8.80899i	− 0.0109701i
$804$	− 30.2726i	− 0.0376525i
$805$	0	0
$806$	11.2972	0.0140164
$807$	−45.9265	−0.0569102
$808$	273.637i	0.338660i
$809$	365.226	0.451453	0.225727	−	0.974191i	$-0.427524\pi$
$809$	365.226	0.451453	0.225727	+	0.974191i	$0.427524\pi$
$810$	− 28.4605i	− 0.0351364i
$811$	944.189i	1.16423i	0.813107	+	0.582114i	$0.197774\pi$
$811$	944.189i	1.16423i	−0.813107	+	0.582114i	$0.802226\pi$
$812$	0	0
$813$	185.355	0.227989
$814$	7.26164	0.00892094
$815$	− 207.983i	− 0.255194i
$816$	−231.060	−0.283161
$817$	− 147.258i	− 0.180243i
$818$	1060.46i	1.29640i
$819$	0	0
$820$	−18.2185	−0.0222176
$821$	−684.675	−0.833952	−0.416976	−	0.908917i	$-0.636910\pi$
$821$	−684.675	−0.833952	−0.416976	+	0.908917i	$0.636910\pi$
$822$	− 572.351i	− 0.696291i
$823$	−1045.67	−1.27056	−0.635280	−	0.772282i	$-0.719115\pi$
$823$	−1045.67	−1.27056	−0.635280	+	0.772282i	$0.719115\pi$
$824$	− 296.312i	− 0.359602i
$825$	− 4.55915i	− 0.00552625i
$826$	0	0
$827$	830.505	1.00424	0.502119	−	0.864798i	$-0.332554\pi$
$827$	830.505	1.00424	0.502119	+	0.864798i	$0.332554\pi$
$828$	−66.3545	−0.0801382
$829$	− 718.204i	− 0.866349i	−0.901310	−	0.433175i	$-0.857393\pi$
$829$	− 718.204i	− 0.866349i	0.901310	−	0.433175i	$-0.142607\pi$
$830$	−39.4771	−0.0475628
$831$	− 762.763i	− 0.917885i
$832$	33.7846i	0.0406064i
$833$	0	0
$834$	518.046	0.621158
$835$	−234.767	−0.281158
$836$	3.34500i	0.00400120i
$837$	9.82905	0.0117432
$838$	658.683i	0.786018i
$839$	− 55.2900i	− 0.0658999i	−0.999457	−	0.0329499i	$-0.989510\pi$
$839$	− 55.2900i	− 0.0658999i	0.999457	−	0.0329499i	$-0.0104902\pi$
$840$	0	0
$841$	2309.60	2.74625
$842$	489.289	0.581104
$843$	− 557.839i	− 0.661731i
$844$	278.866	0.330409
$845$	− 338.017i	− 0.400020i
$846$	268.449i	0.317315i
$847$	0	0
$848$	−186.084	−0.219439
$849$	−118.703	−0.139815
$850$	235.824i	0.277440i
$851$	−107.866	−0.126752
$852$	− 100.669i	− 0.118157i
$853$	− 21.9601i	− 0.0257445i	−0.999917	−	0.0128723i	$-0.995903\pi$
$853$	− 21.9601i	− 0.0257445i	0.999917	−	0.0128723i	$-0.00409748\pi$
$854$	0	0
$855$	21.3117	0.0249260
$856$	−352.434	−0.411722
$857$	− 1615.81i	− 1.88542i	−0.333610	−	0.942711i	$-0.608267\pi$
$857$	− 1615.81i	− 1.88542i	0.333610	−	0.942711i	$-0.391733\pi$
$858$	−5.44575	−0.00634702
$859$	− 891.601i	− 1.03795i	−0.854789	−	0.518976i	$-0.826313\pi$
$859$	− 891.601i	− 1.03795i	0.854789	−	0.518976i	$-0.173687\pi$
$860$	− 207.292i	− 0.241037i
$861$	0	0
$862$	699.470	0.811450
$863$	−896.346	−1.03864	−0.519320	−	0.854580i	$-0.673815\pi$
$863$	−896.346	−1.03864	−0.519320	+	0.854580i	$0.673815\pi$
$864$	29.3939i	0.0340207i
$865$	−456.509	−0.527756
$866$	− 1032.41i	− 1.19216i
$867$	1425.93i	1.64467i
$868$	0	0
$869$	69.6459	0.0801449
$870$	307.438	0.353377
$871$	36.9052i	0.0423711i
$872$	239.309	0.274437
$873$	− 447.843i	− 0.512993i
$874$	− 49.6874i	− 0.0568506i
$875$	0	0
$876$	57.9646	0.0661697
$877$	−337.770	−0.385143	−0.192571	−	0.981283i	$-0.561683\pi$
$877$	−337.770	−0.385143	−0.192571	+	0.981283i	$0.561683\pi$
$878$	525.222i	0.598203i
$879$	255.495	0.290666
$880$	4.70867i	0.00535076i
$881$	301.377i	0.342085i	0.985264	+	0.171042i	$0.0547135\pi$
$881$	301.377i	0.342085i	−0.985264	+	0.171042i	$0.945287\pi$
$882$	0	0
$883$	38.3679	0.0434518	0.0217259	−	0.999764i	$-0.493084\pi$
$883$	38.3679	0.0434518	0.0217259	+	0.999764i	$0.493084\pi$
$884$	281.684	0.318647
$885$	195.707i	0.221138i
$886$	578.000	0.652370
$887$	− 661.452i	− 0.745718i	−0.927888	−	0.372859i	$-0.878377\pi$
$887$	− 661.452i	− 0.745718i	0.927888	−	0.372859i	$-0.121623\pi$
$888$	47.7829i	0.0538095i
$889$	0	0
$890$	215.840	0.242516
$891$	−4.73801	−0.00531763
$892$	516.110i	0.578599i
$893$	−201.019	−0.225106
$894$	− 352.923i	− 0.394769i
$895$	438.043i	0.489434i
$896$	0	0
$897$	80.8924	0.0901810
$898$	1025.97	1.14250
$899$	106.176i	0.118104i
$900$	30.0000	0.0333333
$901$	1551.51i	1.72198i
$902$	3.03295i	0.00336247i
$903$	0	0
$904$	−328.779	−0.363693
$905$	−266.162	−0.294102
$906$	− 229.058i	− 0.252823i
$907$	−968.304	−1.06759	−0.533795	−	0.845614i	$-0.679235\pi$
$907$	−968.304	−1.06759	−0.533795	+	0.845614i	$0.679235\pi$
$908$	662.065i	0.729146i
$909$	− 290.236i	− 0.319291i
$910$	0	0
$911$	220.674	0.242233	0.121116	−	0.992638i	$-0.461353\pi$
$911$	220.674	0.242233	0.121116	+	0.992638i	$0.461353\pi$
$912$	−22.0107	−0.0241345
$913$	6.57202i	0.00719827i
$914$	567.850	0.621280
$915$	400.070i	0.437235i
$916$	− 488.476i	− 0.533271i
$917$	0	0
$918$	245.076	0.266967
$919$	−1148.04	−1.24923	−0.624613	−	0.780934i	$-0.714743\pi$
$919$	−1148.04	−1.24923	−0.624613	+	0.780934i	$0.714743\pi$
$920$	− 69.9437i	− 0.0760258i
$921$	651.268	0.707132
$922$	− 923.552i	− 1.00168i
$923$	122.726i	0.132964i
$924$	0	0
$925$	48.7682	0.0527223
$926$	−1229.77	−1.32805
$927$	314.287i	0.339036i
$928$	−317.520	−0.342156
$929$	305.300i	0.328633i	0.986408	+	0.164317i	$0.0525419\pi$
$929$	305.300i	0.328633i	−0.986408	+	0.164317i	$0.947458\pi$
$930$	10.3607i	0.0111406i
$931$	0	0
$932$	422.979	0.453840
$933$	−675.199	−0.723686
$934$	− 1045.39i	− 1.11926i
$935$	39.2593	0.0419885
$936$	− 35.8339i	− 0.0382841i
$937$	− 12.4049i	− 0.0132390i	−0.999978	−	0.00661948i	$-0.997893\pi$
$937$	− 12.4049i	− 0.0132390i	0.999978	−	0.00661948i	$-0.00210706\pi$
$938$	0	0
$939$	247.781	0.263877
$940$	−282.970	−0.301032
$941$	− 289.147i	− 0.307276i	−0.988127	−	0.153638i	$-0.950901\pi$
$941$	− 289.147i	− 0.307276i	0.988127	−	0.153638i	$-0.0490990\pi$
$942$	−419.415	−0.445239
$943$	− 45.0522i	− 0.0477754i
$944$	− 202.125i	− 0.214116i
$945$	0	0
$946$	−34.5093	−0.0364792
$947$	−1426.96	−1.50682	−0.753409	−	0.657552i	$-0.771592\pi$
$947$	−1426.96	−1.50682	−0.753409	+	0.657552i	$0.771592\pi$
$948$	458.282i	0.483420i
$949$	−70.6644	−0.0744619
$950$	22.4645i	0.0236469i
$951$	− 665.876i	− 0.700186i
$952$	0	0
$953$	−668.525	−0.701495	−0.350747	−	0.936470i	$-0.614072\pi$
$953$	−668.525	−0.701495	−0.350747	+	0.936470i	$0.614072\pi$
$954$	197.372	0.206889
$955$	146.980i	0.153906i
$956$	688.268	0.719946
$957$	− 51.1812i	− 0.0534809i
$958$	− 857.331i	− 0.894918i
$959$	0	0
$960$	−30.9839	−0.0322749
$961$	957.422	0.996277
$962$	− 58.2518i	− 0.0605528i
$963$	373.813	0.388176
$964$	− 342.697i	− 0.355495i
$965$	216.608i	0.224464i
$966$	0	0
$967$	1647.14	1.70335	0.851676	−	0.524069i	$-0.175586\pi$
$967$	1647.14	1.70335	0.851676	+	0.524069i	$0.175586\pi$
$968$	−341.456	−0.352744
$969$	183.517i	0.189388i
$970$	472.068	0.486668
$971$	− 1314.93i	− 1.35420i	−0.735889	−	0.677102i	$-0.763236\pi$
$971$	− 1314.93i	− 1.35420i	0.735889	−	0.677102i	$-0.236764\pi$
$972$	− 31.1769i	− 0.0320750i
$973$	0	0
$974$	918.639	0.943161
$975$	−36.5729	−0.0375106
$976$	− 413.190i	− 0.423351i
$977$	1556.04	1.59267	0.796337	−	0.604854i	$-0.206768\pi$
$977$	1556.04	1.59267	0.796337	+	0.604854i	$0.206768\pi$
$978$	− 227.834i	− 0.232959i
$979$	− 35.9323i	− 0.0367030i
$980$	0	0
$981$	−253.825	−0.258741
$982$	377.070	0.383982
$983$	− 1372.79i	− 1.39654i	−0.715837	−	0.698268i	$-0.753955\pi$
$983$	− 1372.79i	− 1.39654i	0.715837	−	0.698268i	$-0.246045\pi$
$984$	−19.9573	−0.0202818
$985$	− 417.411i	− 0.423768i
$986$	2647.37i	2.68496i
$987$	0	0
$988$	26.8331	0.0271590
$989$	512.609	0.518310
$990$	− 4.99430i	− 0.00504475i
$991$	643.624	0.649470	0.324735	−	0.945805i	$-0.394725\pi$
$991$	643.624	0.649470	0.324735	+	0.945805i	$0.394725\pi$
$992$	− 10.7005i	− 0.0107868i
$993$	− 335.858i	− 0.338226i
$994$	0	0
$995$	257.800	0.259095
$996$	−43.2450	−0.0434187
$997$	704.865i	0.706986i	0.935437	+	0.353493i	$0.115006\pi$
$997$	704.865i	0.706986i	−0.935437	+	0.353493i	$0.884994\pi$
$998$	−963.229	−0.965159
$999$	− 50.6814i	− 0.0507321i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.d.391.9		16
7.4	even	3		210.3.o.b.61.4	yes	16
7.5	odd	6		210.3.o.b.31.4	✓	16
7.6	odd	2	inner	1470.3.f.d.391.15		16
21.5	even	6		630.3.v.c.451.6		16
21.11	odd	6		630.3.v.c.271.6		16
35.4	even	6		1050.3.p.i.901.6		16
35.12	even	12		1050.3.q.e.199.12		32
35.18	odd	12		1050.3.q.e.649.12		32
35.19	odd	6		1050.3.p.i.451.6		16
35.32	odd	12		1050.3.q.e.649.4		32
35.33	even	12		1050.3.q.e.199.4		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.b.31.4	✓	16	7.5	odd	6
210.3.o.b.61.4	yes	16	7.4	even	3
630.3.v.c.271.6		16	21.11	odd	6
630.3.v.c.451.6		16	21.5	even	6
1050.3.p.i.451.6		16	35.19	odd	6
1050.3.p.i.901.6		16	35.4	even	6
1050.3.q.e.199.4		32	35.33	even	12
1050.3.q.e.199.12		32	35.12	even	12
1050.3.q.e.649.4		32	35.32	odd	12
1050.3.q.e.649.12		32	35.18	odd	12
1470.3.f.d.391.9		16	1.1	even	1	trivial
1470.3.f.d.391.15		16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1470.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} -0.526446 q^{11} -3.46410i q^{12} +4.22307i q^{13} -3.87298 q^{15} +4.00000 q^{16} -33.3506i q^{17} -4.24264 q^{18} -3.17697i q^{19} -4.47214i q^{20} -0.744507 q^{22} +11.0591 q^{23} -4.89898i q^{24} -5.00000 q^{25} +5.97232i q^{26} +5.19615i q^{27} -56.1302 q^{29} -5.47723 q^{30} -1.89160i q^{31} +5.65685 q^{32} +0.911831i q^{33} -47.1649i q^{34} -6.00000 q^{36} -9.75363 q^{37} -4.49291i q^{38} +7.31457 q^{39} -6.32456i q^{40} -4.07377i q^{41} +46.3519 q^{43} -1.05289 q^{44} +6.70820i q^{45} +15.6399 q^{46} -63.2740i q^{47} -6.92820i q^{48} -7.07107 q^{50} -57.7649 q^{51} +8.44614i q^{52} -46.5211 q^{53} +7.34847i q^{54} +1.17717i q^{55} -5.50267 q^{57} -79.3801 q^{58} -50.5313i q^{59} -7.74597 q^{60} -103.298i q^{61} -2.67513i q^{62} +8.00000 q^{64} +9.44307 q^{65} +1.28952i q^{66} +8.73896 q^{67} -66.7012i q^{68} -19.1549i q^{69} +29.0608 q^{71} -8.48528 q^{72} +16.7329i q^{73} -13.7937 q^{74} +8.66025i q^{75} -6.35393i q^{76} +10.3444 q^{78} -132.295 q^{79} -8.94427i q^{80} +9.00000 q^{81} -5.76119i q^{82} -12.4838i q^{83} -74.5742 q^{85} +65.5515 q^{86} +97.2204i q^{87} -1.48901 q^{88} +68.2544i q^{89} +9.48683i q^{90} +22.1182 q^{92} -3.27635 q^{93} -89.4829i q^{94} -7.10391 q^{95} -9.79796i q^{96} +149.281i q^{97} +1.57934 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 48 * q^9	\( 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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