LMFDB - Embedded newform 1470.3.f.a.391.3

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:	$8$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.3
Root			$-1.72286 - 0.178197i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.a.391.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -1.83883 q^{11} +3.46410i q^{12} -5.40765i q^{13} +3.87298 q^{15} +4.00000 q^{16} +10.0585i q^{17} +4.24264 q^{18} +9.19238i q^{19} -4.47214i q^{20} +2.60049 q^{22} -0.921288 q^{23} -4.89898i q^{24} -5.00000 q^{25} +7.64757i q^{26} -5.19615i q^{27} +12.5573 q^{29} -5.47723 q^{30} -41.7856i q^{31} -5.65685 q^{32} -3.18494i q^{33} -14.2249i q^{34} -6.00000 q^{36} +7.28388 q^{37} -13.0000i q^{38} +9.36632 q^{39} +6.32456i q^{40} +52.3877i q^{41} -8.12312 q^{43} -3.67765 q^{44} +6.70820i q^{45} +1.30290 q^{46} -33.9617i q^{47} +6.92820i q^{48} +7.07107 q^{50} -17.4219 q^{51} -10.8153i q^{52} -104.079 q^{53} +7.34847i q^{54} +4.11174i q^{55} -15.9217 q^{57} -17.7588 q^{58} +14.5398i q^{59} +7.74597 q^{60} -23.7768i q^{61} +59.0937i q^{62} +8.00000 q^{64} -12.0919 q^{65} +4.50419i q^{66} -14.9268 q^{67} +20.1170i q^{68} -1.59572i q^{69} -17.9620 q^{71} +8.48528 q^{72} -124.367i q^{73} -10.3010 q^{74} -8.66025i q^{75} +18.3848i q^{76} -13.2460 q^{78} -80.9365 q^{79} -8.94427i q^{80} +9.00000 q^{81} -74.0875i q^{82} +154.716i q^{83} +22.4915 q^{85} +11.4878 q^{86} +21.7500i q^{87} +5.20099 q^{88} -68.1223i q^{89} -9.48683i q^{90} -1.84258 q^{92} +72.3747 q^{93} +48.0291i q^{94} +20.5548 q^{95} -9.79796i q^{96} +88.1736i q^{97} +5.51648 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) $ 8 * q + 16 * q^4 - 24 * q^9	$ 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) $ 8 * q + 16 * q^4 - 24 * q^9 + 8 * q^11 + 32 * q^16 - 48 * q^22 - 24 * q^23 - 40 * q^25 + 72 * q^29 - 48 * q^36 + 192 * q^37 - 48 * q^39 - 112 * q^43 + 16 * q^44 - 16 * q^46 - 168 * q^51 - 64 * q^53 + 216 * q^57 - 208 * q^58 + 64 * q^64 - 40 * q^65 + 240 * q^67 + 8 * q^71 + 32 * q^74 - 192 * q^78 - 24 * q^79 + 72 * q^81 + 120 * q^85 + 80 * q^86 - 96 * q^88 - 48 * q^92 + 264 * q^93 + 80 * 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
$11$	−1.83883	−0.167166	−0.0835831	−	0.996501i	$-0.526636\pi$
$11$	−1.83883	−0.167166	−0.0835831	+	0.996501i	$0.526636\pi$
$12$	3.46410i	0.288675i
$13$	− 5.40765i	− 0.415973i	−0.978132	−	0.207986i	$-0.933309\pi$
$13$	− 5.40765i	− 0.415973i	0.978132	−	0.207986i	$-0.0666910\pi$
$14$	0	0
$15$	3.87298	0.258199
$16$	4.00000	0.250000
$17$	10.0585i	0.591677i	0.955238	+	0.295839i	$0.0955991\pi$
$17$	10.0585i	0.591677i	−0.955238	+	0.295839i	$0.904401\pi$
$18$	4.24264	0.235702
$19$	9.19238i	0.483810i	0.970300	+	0.241905i	$0.0777722\pi$
$19$	9.19238i	0.483810i	−0.970300	+	0.241905i	$0.922228\pi$
$20$	− 4.47214i	− 0.223607i
$21$	0	0
$22$	2.60049	0.118204
$23$	−0.921288	−0.0400560	−0.0200280	−	0.999799i	$-0.506376\pi$
$23$	−0.921288	−0.0400560	−0.0200280	+	0.999799i	$0.506376\pi$
$24$	− 4.89898i	− 0.204124i
$25$	−5.00000	−0.200000
$26$	7.64757i	0.294137i
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	12.5573	0.433012	0.216506	−	0.976281i	$-0.430534\pi$
$29$	12.5573	0.433012	0.216506	+	0.976281i	$0.430534\pi$
$30$	−5.47723	−0.182574
$31$	− 41.7856i	− 1.34792i	−0.738767	−	0.673961i	$-0.764592\pi$
$31$	− 41.7856i	− 1.34792i	0.738767	−	0.673961i	$-0.235408\pi$
$32$	−5.65685	−0.176777
$33$	− 3.18494i	− 0.0965134i
$34$	− 14.2249i	− 0.418379i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	7.28388	0.196862	0.0984308	−	0.995144i	$-0.468618\pi$
$37$	7.28388	0.196862	0.0984308	+	0.995144i	$0.468618\pi$
$38$	− 13.0000i	− 0.342105i
$39$	9.36632	0.240162
$40$	6.32456i	0.158114i
$41$	52.3877i	1.27775i	0.769311	+	0.638875i	$0.220600\pi$
$41$	52.3877i	1.27775i	−0.769311	+	0.638875i	$0.779400\pi$
$42$	0	0
$43$	−8.12312	−0.188910	−0.0944549	−	0.995529i	$-0.530111\pi$
$43$	−8.12312	−0.188910	−0.0944549	+	0.995529i	$0.530111\pi$
$44$	−3.67765	−0.0835831
$45$	6.70820i	0.149071i
$46$	1.30290	0.0283239
$47$	− 33.9617i	− 0.722589i	−0.932452	−	0.361295i	$-0.882335\pi$
$47$	− 33.9617i	− 0.722589i	0.932452	−	0.361295i	$-0.117665\pi$
$48$	6.92820i	0.144338i
$49$	0	0
$50$	7.07107	0.141421
$51$	−17.4219	−0.341605
$52$	− 10.8153i	− 0.207986i
$53$	−104.079	−1.96376	−0.981880	−	0.189503i	$-0.939312\pi$
$53$	−104.079	−1.96376	−0.981880	+	0.189503i	$0.939312\pi$
$54$	7.34847i	0.136083i
$55$	4.11174i	0.0747590i
$56$	0	0
$57$	−15.9217	−0.279328
$58$	−17.7588	−0.306186
$59$	14.5398i	0.246437i	0.992380	+	0.123218i	$0.0393216\pi$
$59$	14.5398i	0.246437i	−0.992380	+	0.123218i	$0.960678\pi$
$60$	7.74597	0.129099
$61$	− 23.7768i	− 0.389783i	−0.980825	−	0.194892i	$-0.937564\pi$
$61$	− 23.7768i	− 0.389783i	0.980825	−	0.194892i	$-0.0624355\pi$
$62$	59.0937i	0.953125i
$63$	0	0
$64$	8.00000	0.125000
$65$	−12.0919	−0.186029
$66$	4.50419i	0.0682453i
$67$	−14.9268	−0.222788	−0.111394	−	0.993776i	$-0.535532\pi$
$67$	−14.9268	−0.222788	−0.111394	+	0.993776i	$0.535532\pi$
$68$	20.1170i	0.295839i
$69$	− 1.59572i	− 0.0231263i
$70$	0	0
$71$	−17.9620	−0.252987	−0.126493	−	0.991967i	$-0.540372\pi$
$71$	−17.9620	−0.252987	−0.126493	+	0.991967i	$0.540372\pi$
$72$	8.48528	0.117851
$73$	− 124.367i	− 1.70365i	−0.523824	−	0.851826i	$-0.675495\pi$
$73$	− 124.367i	− 1.70365i	0.523824	−	0.851826i	$-0.324505\pi$
$74$	−10.3010	−0.139202
$75$	− 8.66025i	− 0.115470i
$76$	18.3848i	0.241905i
$77$	0	0
$78$	−13.2460	−0.169820
$79$	−80.9365	−1.02451	−0.512257	−	0.858832i	$-0.671190\pi$
$79$	−80.9365	−1.02451	−0.512257	+	0.858832i	$0.671190\pi$
$80$	− 8.94427i	− 0.111803i
$81$	9.00000	0.111111
$82$	− 74.0875i	− 0.903506i
$83$	154.716i	1.86405i	0.362395	+	0.932025i	$0.381959\pi$
$83$	154.716i	1.86405i	−0.362395	+	0.932025i	$0.618041\pi$
$84$	0	0
$85$	22.4915	0.264606
$86$	11.4878	0.133579
$87$	21.7500i	0.249999i
$88$	5.20099	0.0591021
$89$	− 68.1223i	− 0.765419i	−0.923869	−	0.382709i	$-0.874991\pi$
$89$	− 68.1223i	− 0.765419i	0.923869	−	0.382709i	$-0.125009\pi$
$90$	− 9.48683i	− 0.105409i
$91$	0	0
$92$	−1.84258	−0.0200280
$93$	72.3747	0.778223
$94$	48.0291i	0.510948i
$95$	20.5548	0.216366
$96$	− 9.79796i	− 0.102062i
$97$	88.1736i	0.909006i	0.890745	+	0.454503i	$0.150183\pi$
$97$	88.1736i	0.909006i	−0.890745	+	0.454503i	$0.849817\pi$
$98$	0	0
$99$	5.51648	0.0557220
$100$	−10.0000	−0.100000
$101$	− 111.375i	− 1.10272i	−0.834266	−	0.551362i	$-0.814108\pi$
$101$	− 111.375i	− 1.10272i	0.834266	−	0.551362i	$-0.185892\pi$
$102$	24.6382	0.241551
$103$	46.9786i	0.456103i	0.973649	+	0.228051i	$0.0732354\pi$
$103$	46.9786i	0.456103i	−0.973649	+	0.228051i	$0.926765\pi$
$104$	15.2951i	0.147069i
$105$	0	0
$106$	147.190	1.38859
$107$	−185.682	−1.73534	−0.867672	−	0.497138i	$-0.834384\pi$
$107$	−185.682	−1.73534	−0.867672	+	0.497138i	$0.834384\pi$
$108$	− 10.3923i	− 0.0962250i
$109$	86.2897	0.791648	0.395824	−	0.918326i	$-0.370459\pi$
$109$	86.2897	0.791648	0.395824	+	0.918326i	$0.370459\pi$
$110$	− 5.81488i	− 0.0528626i
$111$	12.6161i	0.113658i
$112$	0	0
$113$	−85.8206	−0.759474	−0.379737	−	0.925094i	$-0.623986\pi$
$113$	−85.8206	−0.759474	−0.379737	+	0.925094i	$0.623986\pi$
$114$	22.5167	0.197514
$115$	2.06006i	0.0179136i
$116$	25.1147	0.216506
$117$	16.2229i	0.138658i
$118$	− 20.5624i	− 0.174257i
$119$	0	0
$120$	−10.9545	−0.0912871
$121$	−117.619	−0.972055
$122$	33.6254i	0.275618i
$123$	−90.7382	−0.737709
$124$	− 83.5711i	− 0.673961i
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−131.740	−1.03733	−0.518663	−	0.854979i	$-0.673570\pi$
$127$	−131.740	−1.03733	−0.518663	+	0.854979i	$0.673570\pi$
$128$	−11.3137	−0.0883883
$129$	− 14.0697i	− 0.109067i
$130$	17.1005	0.131542
$131$	118.768i	0.906625i	0.891352	+	0.453312i	$0.149758\pi$
$131$	118.768i	0.906625i	−0.891352	+	0.453312i	$0.850242\pi$
$132$	− 6.36988i	− 0.0482567i
$133$	0	0
$134$	21.1096	0.157535
$135$	−11.6190	−0.0860663
$136$	− 28.4498i	− 0.209190i
$137$	81.1530	0.592357	0.296179	−	0.955133i	$-0.404288\pi$
$137$	81.1530	0.592357	0.296179	+	0.955133i	$0.404288\pi$
$138$	2.25668i	0.0163528i
$139$	− 248.311i	− 1.78641i	−0.449646	−	0.893207i	$-0.648450\pi$
$139$	− 248.311i	− 1.78641i	0.449646	−	0.893207i	$-0.351550\pi$
$140$	0	0
$141$	58.8234	0.417187
$142$	25.4022	0.178888
$143$	9.94373i	0.0695366i
$144$	−12.0000	−0.0833333
$145$	− 28.0791i	− 0.193649i
$146$	175.881i	1.20466i
$147$	0	0
$148$	14.5678	0.0984308
$149$	−150.010	−1.00678	−0.503388	−	0.864060i	$-0.667913\pi$
$149$	−150.010	−1.00678	−0.503388	+	0.864060i	$0.667913\pi$
$150$	12.2474i	0.0816497i
$151$	183.030	1.21212	0.606059	−	0.795420i	$-0.292749\pi$
$151$	183.030	1.21212	0.606059	+	0.795420i	$0.292749\pi$
$152$	− 26.0000i	− 0.171053i
$153$	− 30.1755i	− 0.197226i
$154$	0	0
$155$	−93.4354	−0.602809
$156$	18.7326	0.120081
$157$	− 295.285i	− 1.88080i	−0.340075	−	0.940398i	$-0.610452\pi$
$157$	− 295.285i	− 1.88080i	0.340075	−	0.940398i	$-0.389548\pi$
$158$	114.462	0.724440
$159$	− 180.271i	− 1.13378i
$160$	12.6491i	0.0790569i
$161$	0	0
$162$	−12.7279	−0.0785674
$163$	−118.326	−0.725927	−0.362964	−	0.931803i	$-0.618235\pi$
$163$	−118.326	−0.725927	−0.362964	+	0.931803i	$0.618235\pi$
$164$	104.775i	0.638875i
$165$	−7.12175	−0.0431621
$166$	− 218.802i	− 1.31808i
$167$	− 205.186i	− 1.22866i	−0.789050	−	0.614329i	$-0.789427\pi$
$167$	− 205.186i	− 1.22866i	0.789050	−	0.614329i	$-0.210573\pi$
$168$	0	0
$169$	139.757	0.826967
$170$	−31.8078	−0.187105
$171$	− 27.5772i	− 0.161270i
$172$	−16.2462	−0.0944549
$173$	− 112.534i	− 0.650485i	−0.945631	−	0.325242i	$-0.894554\pi$
$173$	− 112.534i	− 0.650485i	0.945631	−	0.325242i	$-0.105446\pi$
$174$	− 30.7591i	− 0.176776i
$175$	0	0
$176$	−7.35531	−0.0417915
$177$	−25.1836	−0.142280
$178$	96.3395i	0.541233i
$179$	−44.4698	−0.248435	−0.124217	−	0.992255i	$-0.539642\pi$
$179$	−44.4698	−0.248435	−0.124217	+	0.992255i	$0.539642\pi$
$180$	13.4164i	0.0745356i
$181$	9.24555i	0.0510804i	0.999674	+	0.0255402i	$0.00813058\pi$
$181$	9.24555i	0.0510804i	−0.999674	+	0.0255402i	$0.991869\pi$
$182$	0	0
$183$	41.1826	0.225041
$184$	2.60579	0.0141619
$185$	− 16.2873i	− 0.0880392i
$186$	−102.353	−0.550287
$187$	− 18.4959i	− 0.0989084i
$188$	− 67.9234i	− 0.361295i
$189$	0	0
$190$	−29.0689	−0.152994
$191$	−136.809	−0.716277	−0.358138	−	0.933669i	$-0.616588\pi$
$191$	−136.809	−0.716277	−0.358138	+	0.933669i	$0.616588\pi$
$192$	13.8564i	0.0721688i
$193$	−365.874	−1.89572	−0.947860	−	0.318687i	$-0.896758\pi$
$193$	−365.874	−1.89572	−0.947860	+	0.318687i	$0.896758\pi$
$194$	− 124.696i	− 0.642764i
$195$	− 20.9437i	− 0.107404i
$196$	0	0
$197$	−194.925	−0.989468	−0.494734	−	0.869045i	$-0.664734\pi$
$197$	−194.925	−0.989468	−0.494734	+	0.869045i	$0.664734\pi$
$198$	−7.80148	−0.0394014
$199$	− 242.835i	− 1.22028i	−0.792295	−	0.610138i	$-0.791114\pi$
$199$	− 242.835i	− 1.22028i	0.792295	−	0.610138i	$-0.208886\pi$
$200$	14.1421	0.0707107
$201$	− 25.8539i	− 0.128627i
$202$	157.508i	0.779744i
$203$	0	0
$204$	−34.8437	−0.170803
$205$	117.143	0.571427
$206$	− 66.4378i	− 0.322513i
$207$	2.76386	0.0133520
$208$	− 21.6306i	− 0.103993i
$209$	− 16.9032i	− 0.0808766i
$210$	0	0
$211$	185.930	0.881184	0.440592	−	0.897707i	$-0.354769\pi$
$211$	185.930	0.881184	0.440592	+	0.897707i	$0.354769\pi$
$212$	−208.159	−0.981880
$213$	− 31.1112i	− 0.146062i
$214$	262.594	1.22707
$215$	18.1638i	0.0844830i
$216$	14.6969i	0.0680414i
$217$	0	0
$218$	−122.032	−0.559780
$219$	215.409	0.983604
$220$	8.22348i	0.0373795i
$221$	54.3929	0.246122
$222$	− 17.8418i	− 0.0803684i
$223$	− 53.6348i	− 0.240515i	−0.992743	−	0.120257i	$-0.961628\pi$
$223$	− 53.6348i	− 0.240515i	0.992743	−	0.120257i	$-0.0383720\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	121.369	0.537030
$227$	169.039i	0.744665i	0.928099	+	0.372333i	$0.121442\pi$
$227$	169.039i	0.744665i	−0.928099	+	0.372333i	$0.878558\pi$
$228$	−31.8434	−0.139664
$229$	− 286.017i	− 1.24898i	−0.781031	−	0.624492i	$-0.785306\pi$
$229$	− 286.017i	− 1.24898i	0.781031	−	0.624492i	$-0.214694\pi$
$230$	− 2.91337i	− 0.0126668i
$231$	0	0
$232$	−35.5175	−0.153093
$233$	−181.839	−0.780424	−0.390212	−	0.920725i	$-0.627598\pi$
$233$	−181.839	−0.780424	−0.390212	+	0.920725i	$0.627598\pi$
$234$	− 22.9427i	− 0.0980458i
$235$	−75.9407	−0.323152
$236$	29.0796i	0.123218i
$237$	− 140.186i	− 0.591503i
$238$	0	0
$239$	−382.489	−1.60037	−0.800185	−	0.599753i	$-0.795266\pi$
$239$	−382.489	−1.60037	−0.800185	+	0.599753i	$0.795266\pi$
$240$	15.4919	0.0645497
$241$	− 208.220i	− 0.863982i	−0.901878	−	0.431991i	$-0.857811\pi$
$241$	− 208.220i	− 0.863982i	0.901878	−	0.431991i	$-0.142189\pi$
$242$	166.338	0.687347
$243$	15.5885i	0.0641500i
$244$	− 47.5536i	− 0.194892i
$245$	0	0
$246$	128.323	0.521639
$247$	49.7092	0.201252
$248$	118.187i	0.476562i
$249$	−267.976	−1.07621
$250$	− 15.8114i	− 0.0632456i
$251$	− 43.4959i	− 0.173291i	−0.996239	−	0.0866453i	$-0.972385\pi$
$251$	− 43.4959i	− 0.173291i	0.996239	−	0.0866453i	$-0.0276147\pi$
$252$	0	0
$253$	1.69409	0.00669600
$254$	186.309	0.733500
$255$	38.9565i	0.152770i
$256$	16.0000	0.0625000
$257$	− 4.25986i	− 0.0165753i	−0.999966	−	0.00828766i	$-0.997362\pi$
$257$	− 4.25986i	− 0.0165753i	0.999966	−	0.00828766i	$-0.00263807\pi$
$258$	19.8975i	0.0771221i
$259$	0	0
$260$	−24.1837	−0.0930144
$261$	−37.6720	−0.144337
$262$	− 167.963i	− 0.641080i
$263$	−418.374	−1.59078	−0.795388	−	0.606100i	$-0.792733\pi$
$263$	−418.374	−1.59078	−0.795388	+	0.606100i	$0.792733\pi$
$264$	9.00838i	0.0341226i
$265$	232.728i	0.878220i
$266$	0	0
$267$	117.991	0.441915
$268$	−29.8536	−0.111394
$269$	− 368.448i	− 1.36970i	−0.728686	−	0.684848i	$-0.759869\pi$
$269$	− 368.448i	− 1.36970i	0.728686	−	0.684848i	$-0.240131\pi$
$270$	16.4317	0.0608581
$271$	221.022i	0.815579i	0.913076	+	0.407790i	$0.133700\pi$
$271$	221.022i	0.815579i	−0.913076	+	0.407790i	$0.866300\pi$
$272$	40.2341i	0.147919i
$273$	0	0
$274$	−114.768	−0.418860
$275$	9.19414	0.0334332
$276$	− 3.19143i	− 0.0115632i
$277$	−396.854	−1.43269	−0.716343	−	0.697748i	$-0.754185\pi$
$277$	−396.854	−1.43269	−0.716343	+	0.697748i	$0.754185\pi$
$278$	351.165i	1.26319i
$279$	125.357i	0.449307i
$280$	0	0
$281$	−114.244	−0.406562	−0.203281	−	0.979120i	$-0.565161\pi$
$281$	−114.244	−0.406562	−0.203281	+	0.979120i	$0.565161\pi$
$282$	−83.1888	−0.294996
$283$	452.434i	1.59871i	0.600862	+	0.799353i	$0.294824\pi$
$283$	452.434i	1.59871i	−0.600862	+	0.799353i	$0.705176\pi$
$284$	−35.9241	−0.126493
$285$	35.6020i	0.124919i
$286$	− 14.0626i	− 0.0491698i
$287$	0	0
$288$	16.9706	0.0589256
$289$	187.826	0.649918
$290$	39.7098i	0.136930i
$291$	−152.721	−0.524815
$292$	− 248.733i	− 0.851826i
$293$	119.134i	0.406600i	0.979116	+	0.203300i	$0.0651667\pi$
$293$	119.134i	0.406600i	−0.979116	+	0.203300i	$0.934833\pi$
$294$	0	0
$295$	32.5119	0.110210
$296$	−20.6019	−0.0696011
$297$	9.55483i	0.0321711i
$298$	212.146	0.711898
$299$	4.98200i	0.0166622i
$300$	− 17.3205i	− 0.0577350i
$301$	0	0
$302$	−258.843	−0.857097
$303$	192.907	0.636658
$304$	36.7695i	0.120952i
$305$	−53.1665	−0.174316
$306$	42.6747i	0.139460i
$307$	− 272.643i	− 0.888087i	−0.896005	−	0.444043i	$-0.853544\pi$
$307$	− 272.643i	− 0.888087i	0.896005	−	0.444043i	$-0.146456\pi$
$308$	0	0
$309$	−81.3693	−0.263331
$310$	132.138	0.426250
$311$	75.5724i	0.242998i	0.992592	+	0.121499i	$0.0387701\pi$
$311$	75.5724i	0.242998i	−0.992592	+	0.121499i	$0.961230\pi$
$312$	−26.4920	−0.0849101
$313$	63.6306i	0.203293i	0.994821	+	0.101646i	$0.0324110\pi$
$313$	63.6306i	0.203293i	−0.994821	+	0.101646i	$0.967589\pi$
$314$	417.596i	1.32992i
$315$	0	0
$316$	−161.873	−0.512257
$317$	15.4521	0.0487448	0.0243724	−	0.999703i	$-0.492241\pi$
$317$	15.4521	0.0487448	0.0243724	+	0.999703i	$0.492241\pi$
$318$	254.941i	0.801702i
$319$	−23.0908	−0.0723849
$320$	− 17.8885i	− 0.0559017i
$321$	− 321.610i	− 1.00190i
$322$	0	0
$323$	−92.4617	−0.286259
$324$	18.0000	0.0555556
$325$	27.0382i	0.0831946i
$326$	167.338	0.513308
$327$	149.458i	0.457058i
$328$	− 148.175i	− 0.451753i
$329$	0	0
$330$	10.0717	0.0305202
$331$	−373.079	−1.12713	−0.563564	−	0.826073i	$-0.690570\pi$
$331$	−373.079	−1.12713	−0.563564	+	0.826073i	$0.690570\pi$
$332$	309.432i	0.932025i
$333$	−21.8516	−0.0656206
$334$	290.177i	0.868793i
$335$	33.3773i	0.0996337i
$336$	0	0
$337$	642.919	1.90777	0.953885	−	0.300171i	$-0.0970437\pi$
$337$	642.919	1.90777	0.953885	+	0.300171i	$0.0970437\pi$
$338$	−197.647	−0.584754
$339$	− 148.646i	− 0.438483i
$340$	44.9830	0.132303
$341$	76.8364i	0.225327i
$342$	39.0000i	0.114035i
$343$	0	0
$344$	22.9757	0.0667897
$345$	−3.56813	−0.0103424
$346$	159.147i	0.459962i
$347$	144.009	0.415010	0.207505	−	0.978234i	$-0.433466\pi$
$347$	144.009	0.415010	0.207505	+	0.978234i	$0.433466\pi$
$348$	43.4999i	0.125000i
$349$	− 566.507i	− 1.62323i	−0.584194	−	0.811614i	$-0.698589\pi$
$349$	− 566.507i	− 1.62323i	0.584194	−	0.811614i	$-0.301411\pi$
$350$	0	0
$351$	−28.0990	−0.0800540
$352$	10.4020	0.0295511
$353$	510.163i	1.44522i	0.691255	+	0.722611i	$0.257058\pi$
$353$	510.163i	1.44522i	−0.691255	+	0.722611i	$0.742942\pi$
$354$	35.6150	0.100607
$355$	40.1644i	0.113139i
$356$	− 136.245i	− 0.382709i
$357$	0	0
$358$	62.8898	0.175670
$359$	667.863	1.86034	0.930171	−	0.367126i	$-0.119658\pi$
$359$	667.863	1.86034	0.930171	+	0.367126i	$0.119658\pi$
$360$	− 18.9737i	− 0.0527046i
$361$	276.500	0.765928
$362$	− 13.0752i	− 0.0361193i
$363$	− 203.722i	− 0.561217i
$364$	0	0
$365$	−278.092	−0.761897
$366$	−58.2410	−0.159128
$367$	− 34.8969i	− 0.0950870i	−0.998869	−	0.0475435i	$-0.984861\pi$
$367$	− 34.8969i	− 0.0950870i	0.998869	−	0.0475435i	$-0.0151393\pi$
$368$	−3.68515	−0.0100140
$369$	− 157.163i	− 0.425917i
$370$	23.0337i	0.0622531i
$371$	0	0
$372$	144.749	0.389111
$373$	−408.024	−1.09390	−0.546950	−	0.837166i	$-0.684211\pi$
$373$	−408.024	−1.09390	−0.546950	+	0.837166i	$0.684211\pi$
$374$	26.1571i	0.0699388i
$375$	−19.3649	−0.0516398
$376$	96.0582i	0.255474i
$377$	− 67.9057i	− 0.180121i
$378$	0	0
$379$	−10.4706	−0.0276268	−0.0138134	−	0.999905i	$-0.504397\pi$
$379$	−10.4706	−0.0276268	−0.0138134	+	0.999905i	$0.504397\pi$
$380$	41.1096	0.108183
$381$	− 228.181i	− 0.598900i
$382$	193.477	0.506484
$383$	230.561i	0.601988i	0.953626	+	0.300994i	$0.0973184\pi$
$383$	230.561i	0.601988i	−0.953626	+	0.300994i	$0.902682\pi$
$384$	− 19.5959i	− 0.0510310i
$385$	0	0
$386$	517.424	1.34048
$387$	24.3694	0.0629699
$388$	176.347i	0.454503i
$389$	676.214	1.73834	0.869170	−	0.494514i	$-0.164654\pi$
$389$	676.214	1.73834	0.869170	+	0.494514i	$0.164654\pi$
$390$	29.6189i	0.0759459i
$391$	− 9.26678i	− 0.0237002i
$392$	0	0
$393$	−205.712	−0.523440
$394$	275.666	0.699659
$395$	180.980i	0.458176i
$396$	11.0330	0.0278610
$397$	163.553i	0.411972i	0.978555	+	0.205986i	$0.0660401\pi$
$397$	163.553i	0.411972i	−0.978555	+	0.205986i	$0.933960\pi$
$398$	343.421i	0.862866i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	−299.247	−0.746253	−0.373127	−	0.927780i	$-0.621714\pi$
$401$	−299.247	−0.746253	−0.373127	+	0.927780i	$0.621714\pi$
$402$	36.5630i	0.0909527i
$403$	−225.962	−0.560699
$404$	− 222.750i	− 0.551362i
$405$	− 20.1246i	− 0.0496904i
$406$	0	0
$407$	−13.3938	−0.0329086
$408$	49.2765	0.120776
$409$	465.649i	1.13851i	0.822162	+	0.569253i	$0.192768\pi$
$409$	465.649i	1.13851i	−0.822162	+	0.569253i	$0.807232\pi$
$410$	−165.665	−0.404060
$411$	140.561i	0.341998i
$412$	93.9572i	0.228051i
$413$	0	0
$414$	−3.90869	−0.00944129
$415$	345.956	0.833628
$416$	30.5903i	0.0735343i
$417$	430.088	1.03139
$418$	23.9047i	0.0571884i
$419$	575.882i	1.37442i	0.726458	+	0.687211i	$0.241165\pi$
$419$	575.882i	1.37442i	−0.726458	+	0.687211i	$0.758835\pi$
$420$	0	0
$421$	571.149	1.35665	0.678324	−	0.734763i	$-0.262706\pi$
$421$	571.149	1.35665	0.678324	+	0.734763i	$0.262706\pi$
$422$	−262.945	−0.623091
$423$	101.885i	0.240863i
$424$	294.381	0.694294
$425$	− 50.2926i	− 0.118335i
$426$	43.9978i	0.103281i
$427$	0	0
$428$	−371.363	−0.867672
$429$	−17.2230	−0.0401470
$430$	− 25.6876i	− 0.0597385i
$431$	−27.1394	−0.0629684	−0.0314842	−	0.999504i	$-0.510023\pi$
$431$	−27.1394	−0.0629684	−0.0314842	+	0.999504i	$0.510023\pi$
$432$	− 20.7846i	− 0.0481125i
$433$	− 363.408i	− 0.839279i	−0.907691	−	0.419640i	$-0.862156\pi$
$433$	− 363.408i	− 0.839279i	0.907691	−	0.419640i	$-0.137844\pi$
$434$	0	0
$435$	48.6344	0.111803
$436$	172.579	0.395824
$437$	− 8.46883i	− 0.0193795i
$438$	−304.635	−0.695513
$439$	465.410i	1.06016i	0.847948	+	0.530079i	$0.177838\pi$
$439$	465.410i	1.06016i	−0.847948	+	0.530079i	$0.822162\pi$
$440$	− 11.6298i	− 0.0264313i
$441$	0	0
$442$	−76.9232	−0.174034
$443$	487.936	1.10144	0.550718	−	0.834692i	$-0.314354\pi$
$443$	487.936	1.10144	0.550718	+	0.834692i	$0.314354\pi$
$444$	25.2321i	0.0568291i
$445$	−152.326	−0.342306
$446$	75.8510i	0.170070i
$447$	− 259.824i	− 0.581263i
$448$	0	0
$449$	−285.837	−0.636609	−0.318304	−	0.947989i	$-0.603113\pi$
$449$	−285.837	−0.636609	−0.318304	+	0.947989i	$0.603113\pi$
$450$	−21.2132	−0.0471405
$451$	− 96.3320i	− 0.213596i
$452$	−171.641	−0.379737
$453$	317.017i	0.699817i
$454$	− 239.057i	− 0.526558i
$455$	0	0
$456$	45.0333	0.0987572
$457$	423.247	0.926142	0.463071	−	0.886321i	$-0.346747\pi$
$457$	423.247	0.926142	0.463071	+	0.886321i	$0.346747\pi$
$458$	404.490i	0.883165i
$459$	52.2656	0.113868
$460$	4.12012i	0.00895679i
$461$	471.748i	1.02331i	0.859190	+	0.511657i	$0.170968\pi$
$461$	471.748i	1.02331i	−0.859190	+	0.511657i	$0.829032\pi$
$462$	0	0
$463$	−194.019	−0.419046	−0.209523	−	0.977804i	$-0.567191\pi$
$463$	−194.019	−0.419046	−0.209523	+	0.977804i	$0.567191\pi$
$464$	50.2294	0.108253
$465$	− 161.835i	− 0.348032i
$466$	257.159	0.551843
$467$	11.3271i	0.0242550i	0.999926	+	0.0121275i	$0.00386040\pi$
$467$	11.3271i	0.0242550i	−0.999926	+	0.0121275i	$0.996140\pi$
$468$	32.4459i	0.0693288i
$469$	0	0
$470$	107.396	0.228503
$471$	511.449	1.08588
$472$	− 41.1247i	− 0.0871286i
$473$	14.9370	0.0315793
$474$	198.253i	0.418256i
$475$	− 45.9619i	− 0.0967619i
$476$	0	0
$477$	312.238	0.654587
$478$	540.921	1.13163
$479$	− 632.697i	− 1.32087i	−0.750883	−	0.660435i	$-0.770372\pi$
$479$	− 632.697i	− 1.32087i	0.750883	−	0.660435i	$-0.229628\pi$
$480$	−21.9089	−0.0456435
$481$	− 39.3887i	− 0.0818891i
$482$	294.467i	0.610927i
$483$	0	0
$484$	−235.237	−0.486028
$485$	197.162	0.406520
$486$	− 22.0454i	− 0.0453609i
$487$	−63.0204	−0.129405	−0.0647027	−	0.997905i	$-0.520610\pi$
$487$	−63.0204	−0.129405	−0.0647027	+	0.997905i	$0.520610\pi$
$488$	67.2509i	0.137809i
$489$	− 204.947i	− 0.419114i
$490$	0	0
$491$	261.092	0.531756	0.265878	−	0.964007i	$-0.414338\pi$
$491$	261.092	0.531756	0.265878	+	0.964007i	$0.414338\pi$
$492$	−181.476	−0.368855
$493$	126.308i	0.256203i
$494$	−70.2994	−0.142306
$495$	− 12.3352i	− 0.0249197i
$496$	− 167.142i	− 0.336980i
$497$	0	0
$498$	378.976	0.760995
$499$	438.963	0.879685	0.439843	−	0.898075i	$-0.355034\pi$
$499$	438.963	0.879685	0.439843	+	0.898075i	$0.355034\pi$
$500$	22.3607i	0.0447214i
$501$	355.393	0.709367
$502$	61.5126i	0.122535i
$503$	702.372i	1.39637i	0.715919	+	0.698183i	$0.246008\pi$
$503$	702.372i	1.39637i	−0.715919	+	0.698183i	$0.753992\pi$
$504$	0	0
$505$	−249.042	−0.493153
$506$	−2.39580	−0.00473479
$507$	242.067i	0.477449i
$508$	−263.481	−0.518663
$509$	119.152i	0.234089i	0.993127	+	0.117045i	$0.0373421\pi$
$509$	119.152i	0.234089i	−0.993127	+	0.117045i	$0.962658\pi$
$510$	− 55.0928i	− 0.108025i
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	47.7650	0.0931092
$514$	6.02435i	0.0117205i
$515$	105.047	0.203975
$516$	− 28.1393i	− 0.0545336i
$517$	62.4497i	0.120792i
$518$	0	0
$519$	194.914	0.375558
$520$	34.2010	0.0657711
$521$	− 451.976i	− 0.867516i	−0.901029	−	0.433758i	$-0.857187\pi$
$521$	− 451.976i	− 0.867516i	0.901029	−	0.433758i	$-0.142813\pi$
$522$	53.2763	0.102062
$523$	894.901i	1.71109i	0.517726	+	0.855546i	$0.326779\pi$
$523$	894.901i	1.71109i	−0.517726	+	0.855546i	$0.673221\pi$
$524$	237.536i	0.453312i
$525$	0	0
$526$	591.670	1.12485
$527$	420.301	0.797535
$528$	− 12.7398i	− 0.0241283i
$529$	−528.151	−0.998396
$530$	− 329.128i	− 0.620996i
$531$	− 43.6193i	− 0.0821457i
$532$	0	0
$533$	283.294	0.531509
$534$	−166.865	−0.312481
$535$	415.197i	0.776069i
$536$	42.2193	0.0787674
$537$	− 77.0240i	− 0.143434i
$538$	521.065i	0.968522i
$539$	0	0
$540$	−23.2379	−0.0430331
$541$	−316.525	−0.585074	−0.292537	−	0.956254i	$-0.594499\pi$
$541$	−316.525	−0.585074	−0.292537	+	0.956254i	$0.594499\pi$
$542$	− 312.572i	− 0.576701i
$543$	−16.0138	−0.0294913
$544$	− 56.8996i	− 0.104595i
$545$	− 192.950i	− 0.354036i
$546$	0	0
$547$	796.193	1.45556	0.727782	−	0.685809i	$-0.240551\pi$
$547$	796.193	1.45556	0.727782	+	0.685809i	$0.240551\pi$
$548$	162.306	0.296179
$549$	71.3303i	0.129928i
$550$	−13.0025	−0.0236409
$551$	115.432i	0.209495i
$552$	4.51337i	0.00817639i
$553$	0	0
$554$	561.236	1.01306
$555$	28.2104	0.0508295
$556$	− 496.623i	− 0.893207i
$557$	311.952	0.560058	0.280029	−	0.959992i	$-0.409656\pi$
$557$	311.952	0.560058	0.280029	+	0.959992i	$0.409656\pi$
$558$	− 177.281i	− 0.317708i
$559$	43.9270i	0.0785813i
$560$	0	0
$561$	32.0358	0.0571048
$562$	161.565	0.287483
$563$	617.014i	1.09594i	0.836498	+	0.547970i	$0.184599\pi$
$563$	617.014i	1.09594i	−0.836498	+	0.547970i	$0.815401\pi$
$564$	117.647	0.208594
$565$	191.901i	0.339647i
$566$	− 639.838i	− 1.13046i
$567$	0	0
$568$	50.8043	0.0894442
$569$	−782.990	−1.37608	−0.688041	−	0.725672i	$-0.741529\pi$
$569$	−782.990	−1.37608	−0.688041	+	0.725672i	$0.741529\pi$
$570$	− 50.3488i	− 0.0883312i
$571$	2.13199	0.00373379	0.00186689	−	0.999998i	$-0.499406\pi$
$571$	2.13199	0.00373379	0.00186689	+	0.999998i	$0.499406\pi$
$572$	19.8875i	0.0347683i
$573$	− 236.960i	− 0.413543i
$574$	0	0
$575$	4.60644	0.00801120
$576$	−24.0000	−0.0416667
$577$	− 956.157i	− 1.65712i	−0.559902	−	0.828559i	$-0.689161\pi$
$577$	− 956.157i	− 1.65712i	0.559902	−	0.828559i	$-0.310839\pi$
$578$	−265.626	−0.459561
$579$	− 633.712i	− 1.09449i
$580$	− 56.1581i	− 0.0968244i
$581$	0	0
$582$	215.980	0.371100
$583$	191.384	0.328274
$584$	351.762i	0.602332i
$585$	36.2756	0.0620096
$586$	− 168.481i	− 0.287510i
$587$	− 628.961i	− 1.07148i	−0.844382	−	0.535742i	$-0.820032\pi$
$587$	− 628.961i	− 1.07148i	0.844382	−	0.535742i	$-0.179968\pi$
$588$	0	0
$589$	384.109	0.652138
$590$	−45.9788	−0.0779302
$591$	− 337.620i	− 0.571269i
$592$	29.1355	0.0492154
$593$	482.308i	0.813336i	0.913576	+	0.406668i	$0.133309\pi$
$593$	482.308i	0.813336i	−0.913576	+	0.406668i	$0.866691\pi$
$594$	− 13.5126i	− 0.0227484i
$595$	0	0
$596$	−300.019	−0.503388
$597$	420.603	0.704527
$598$	− 7.04561i	− 0.0117820i
$599$	362.465	0.605117	0.302559	−	0.953131i	$-0.402159\pi$
$599$	362.465	0.605117	0.302559	+	0.953131i	$0.402159\pi$
$600$	24.4949i	0.0408248i
$601$	545.450i	0.907570i	0.891111	+	0.453785i	$0.149927\pi$
$601$	545.450i	0.907570i	−0.891111	+	0.453785i	$0.850073\pi$
$602$	0	0
$603$	44.7803	0.0742626
$604$	366.060	0.606059
$605$	263.003i	0.434716i
$606$	−272.812	−0.450185
$607$	640.197i	1.05469i	0.849651	+	0.527345i	$0.176812\pi$
$607$	640.197i	1.05469i	−0.849651	+	0.527345i	$0.823188\pi$
$608$	− 52.0000i	− 0.0855263i
$609$	0	0
$610$	75.1888	0.123260
$611$	−183.653	−0.300578
$612$	− 60.3511i	− 0.0986129i
$613$	−396.544	−0.646890	−0.323445	−	0.946247i	$-0.604841\pi$
$613$	−396.544	−0.646890	−0.323445	+	0.946247i	$0.604841\pi$
$614$	385.575i	0.627972i
$615$	202.897i	0.329914i
$616$	0	0
$617$	−421.502	−0.683148	−0.341574	−	0.939855i	$-0.610960\pi$
$617$	−421.502	−0.683148	−0.341574	+	0.939855i	$0.610960\pi$
$618$	115.074	0.186203
$619$	726.609i	1.17384i	0.809644	+	0.586921i	$0.199660\pi$
$619$	726.609i	1.17384i	−0.809644	+	0.586921i	$0.800340\pi$
$620$	−186.871	−0.301404
$621$	4.78715i	0.00770878i
$622$	− 106.876i	− 0.171826i
$623$	0	0
$624$	37.4653	0.0600405
$625$	25.0000	0.0400000
$626$	− 89.9873i	− 0.143750i
$627$	29.2772	0.0466941
$628$	− 590.570i	− 0.940398i
$629$	73.2650i	0.116479i
$630$	0	0
$631$	−537.550	−0.851901	−0.425950	−	0.904746i	$-0.640060\pi$
$631$	−537.550	−0.851901	−0.425950	+	0.904746i	$0.640060\pi$
$632$	228.923	0.362220
$633$	322.040i	0.508752i
$634$	−21.8526	−0.0344678
$635$	294.580i	0.463906i
$636$	− 360.541i	− 0.566889i
$637$	0	0
$638$	32.6553	0.0511838
$639$	53.8861	0.0843288
$640$	25.2982i	0.0395285i
$641$	977.848	1.52550	0.762752	−	0.646691i	$-0.223848\pi$
$641$	977.848	1.52550	0.762752	+	0.646691i	$0.223848\pi$
$642$	454.826i	0.708451i
$643$	276.520i	0.430047i	0.976609	+	0.215023i	$0.0689828\pi$
$643$	276.520i	0.430047i	−0.976609	+	0.215023i	$0.931017\pi$
$644$	0	0
$645$	−31.4607	−0.0487763
$646$	130.761	0.202416
$647$	− 242.725i	− 0.375154i	−0.982250	−	0.187577i	$-0.939937\pi$
$647$	− 242.725i	− 0.375154i	0.982250	−	0.187577i	$-0.0600634\pi$
$648$	−25.4558	−0.0392837
$649$	− 26.7361i	− 0.0411959i
$650$	− 38.2378i	− 0.0588275i
$651$	0	0
$652$	−236.652	−0.362964
$653$	−53.2218	−0.0815035	−0.0407518	−	0.999169i	$-0.512975\pi$
$653$	−53.2218	−0.0815035	−0.0407518	+	0.999169i	$0.512975\pi$
$654$	− 211.366i	− 0.323189i
$655$	265.573	0.405455
$656$	209.551i	0.319437i
$657$	373.100i	0.567884i
$658$	0	0
$659$	640.732	0.972279	0.486139	−	0.873881i	$-0.338405\pi$
$659$	640.732	0.972279	0.486139	+	0.873881i	$0.338405\pi$
$660$	−14.2435	−0.0215811
$661$	12.0476i	0.0182263i	0.999958	+	0.00911317i	$0.00290085\pi$
$661$	12.0476i	0.0182263i	−0.999958	+	0.00911317i	$0.997099\pi$
$662$	527.614	0.796999
$663$	94.2113i	0.142098i
$664$	− 437.603i	− 0.659041i
$665$	0	0
$666$	30.9029	0.0464007
$667$	−11.5689	−0.0173447
$668$	− 410.372i	− 0.614329i
$669$	92.8982	0.138861
$670$	− 47.2026i	− 0.0704517i
$671$	43.7214i	0.0651586i
$672$	0	0
$673$	−952.008	−1.41457	−0.707286	−	0.706927i	$-0.750081\pi$
$673$	−952.008	−1.41457	−0.707286	+	0.706927i	$0.750081\pi$
$674$	−909.225	−1.34900
$675$	25.9808i	0.0384900i
$676$	279.515	0.413483
$677$	− 21.1270i	− 0.0312068i	−0.999878	−	0.0156034i	$-0.995033\pi$
$677$	− 21.1270i	− 0.0312068i	0.999878	−	0.0156034i	$-0.00496692\pi$
$678$	210.217i	0.310054i
$679$	0	0
$680$	−63.6156	−0.0935524
$681$	−292.784	−0.429933
$682$	− 108.663i	− 0.159330i
$683$	−392.897	−0.575251	−0.287626	−	0.957743i	$-0.592866\pi$
$683$	−392.897	−0.575251	−0.287626	+	0.957743i	$0.592866\pi$
$684$	− 55.1543i	− 0.0806350i
$685$	− 181.464i	− 0.264910i
$686$	0	0
$687$	495.397	0.721101
$688$	−32.4925	−0.0472274
$689$	562.824i	0.816871i
$690$	5.04610	0.00731319
$691$	− 266.683i	− 0.385938i	−0.981205	−	0.192969i	$-0.938188\pi$
$691$	− 266.683i	− 0.385938i	0.981205	−	0.192969i	$-0.0618117\pi$
$692$	− 225.068i	− 0.325242i
$693$	0	0
$694$	−203.659	−0.293457
$695$	−555.241	−0.798908
$696$	− 61.5182i	− 0.0883882i
$697$	−526.943	−0.756016
$698$	801.161i	1.14780i
$699$	− 314.954i	− 0.450578i
$700$	0	0
$701$	207.973	0.296680	0.148340	−	0.988936i	$-0.452607\pi$
$701$	207.973	0.296680	0.148340	+	0.988936i	$0.452607\pi$
$702$	39.7379	0.0566067
$703$	66.9562i	0.0952436i
$704$	−14.7106	−0.0208958
$705$	− 131.533i	− 0.186572i
$706$	− 721.480i	− 1.02193i
$707$	0	0
$708$	−50.3673	−0.0711402
$709$	−895.427	−1.26294	−0.631472	−	0.775399i	$-0.717549\pi$
$709$	−895.427	−1.26294	−0.631472	+	0.775399i	$0.717549\pi$
$710$	− 56.8010i	− 0.0800014i
$711$	242.810	0.341504
$712$	192.679i	0.270616i
$713$	38.4965i	0.0539923i
$714$	0	0
$715$	22.2349	0.0310977
$716$	−88.9396	−0.124217
$717$	− 662.490i	− 0.923974i
$718$	−944.501	−1.31546
$719$	378.429i	0.526327i	0.964751	+	0.263163i	$0.0847659\pi$
$719$	378.429i	0.526327i	−0.964751	+	0.263163i	$0.915234\pi$
$720$	26.8328i	0.0372678i
$721$	0	0
$722$	−391.030	−0.541593
$723$	360.647	0.498820
$724$	18.4911i	0.0255402i
$725$	−62.7867	−0.0866024
$726$	288.106i	0.396840i
$727$	− 456.052i	− 0.627307i	−0.949538	−	0.313653i	$-0.898447\pi$
$727$	− 456.052i	− 0.627307i	0.949538	−	0.313653i	$-0.101553\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	393.282	0.538742
$731$	− 81.7065i	− 0.111774i
$732$	82.3652	0.112521
$733$	− 760.823i	− 1.03796i	−0.854787	−	0.518979i	$-0.826312\pi$
$733$	− 760.823i	− 1.03796i	0.854787	−	0.518979i	$-0.173688\pi$
$734$	49.3517i	0.0672367i
$735$	0	0
$736$	5.21159	0.00708096
$737$	27.4478	0.0372426
$738$	222.262i	0.301169i
$739$	−28.1666	−0.0381145	−0.0190573	−	0.999818i	$-0.506066\pi$
$739$	−28.1666	−0.0381145	−0.0190573	+	0.999818i	$0.506066\pi$
$740$	− 32.5745i	− 0.0440196i
$741$	86.0988i	0.116193i
$742$	0	0
$743$	−1077.90	−1.45073	−0.725367	−	0.688362i	$-0.758330\pi$
$743$	−1077.90	−1.45073	−0.725367	+	0.688362i	$0.758330\pi$
$744$	−204.707	−0.275143
$745$	335.432i	0.450244i
$746$	577.034	0.773504
$747$	− 464.148i	− 0.621350i
$748$	− 36.9917i	− 0.0494542i
$749$	0	0
$750$	27.3861	0.0365148
$751$	180.117	0.239836	0.119918	−	0.992784i	$-0.461737\pi$
$751$	180.117	0.239836	0.119918	+	0.992784i	$0.461737\pi$
$752$	− 135.847i	− 0.180647i
$753$	75.3372	0.100049
$754$	96.0331i	0.127365i
$755$	− 409.267i	− 0.542076i
$756$	0	0
$757$	219.675	0.290192	0.145096	−	0.989418i	$-0.453651\pi$
$757$	219.675	0.290192	0.145096	+	0.989418i	$0.453651\pi$
$758$	14.8076	0.0195351
$759$	2.93425i	0.00386594i
$760$	−58.1377	−0.0764970
$761$	− 1286.30i	− 1.69028i	−0.534548	−	0.845138i	$-0.679518\pi$
$761$	− 1286.30i	− 1.69028i	0.534548	−	0.845138i	$-0.320482\pi$
$762$	322.697i	0.423486i
$763$	0	0
$764$	−273.618	−0.358138
$765$	−67.4746	−0.0882020
$766$	− 326.063i	− 0.425670i
$767$	78.6260	0.102511
$768$	27.7128i	0.0360844i
$769$	365.543i	0.475348i	0.971345	+	0.237674i	$0.0763850\pi$
$769$	365.543i	0.475348i	−0.971345	+	0.237674i	$0.923615\pi$
$770$	0	0
$771$	7.37829	0.00956977
$772$	−731.748	−0.947860
$773$	308.517i	0.399116i	0.979886	+	0.199558i	$0.0639507\pi$
$773$	308.517i	0.399116i	−0.979886	+	0.199558i	$0.936049\pi$
$774$	−34.4635	−0.0445265
$775$	208.928i	0.269584i
$776$	− 249.393i	− 0.321382i
$777$	0	0
$778$	−956.311	−1.22919
$779$	−481.568	−0.618188
$780$	− 41.8875i	− 0.0537019i
$781$	33.0291	0.0422908
$782$	13.1052i	0.0167586i
$783$	− 65.2499i	− 0.0833332i
$784$	0	0
$785$	−660.277	−0.841118
$786$	290.921	0.370128
$787$	1423.57i	1.80886i	0.426620	+	0.904431i	$0.359704\pi$
$787$	1423.57i	1.80886i	−0.426620	+	0.904431i	$0.640296\pi$
$788$	−389.850	−0.494734
$789$	− 724.645i	− 0.918435i
$790$	− 255.944i	− 0.323980i
$791$	0	0
$792$	−15.6030	−0.0197007
$793$	−128.576	−0.162139
$794$	− 231.299i	− 0.291308i
$795$	−403.097	−0.507041
$796$	− 485.670i	− 0.610138i
$797$	475.713i	0.596880i	0.954428	+	0.298440i	$0.0964663\pi$
$797$	475.713i	0.596880i	−0.954428	+	0.298440i	$0.903534\pi$
$798$	0	0
$799$	341.604	0.427540
$800$	28.2843	0.0353553
$801$	204.367i	0.255140i
$802$	423.200	0.527681
$803$	228.689i	0.284793i
$804$	− 51.7079i	− 0.0643133i
$805$	0	0
$806$	319.558	0.396474
$807$	638.171	0.790795
$808$	315.017i	0.389872i
$809$	727.421	0.899160	0.449580	−	0.893240i	$-0.351574\pi$
$809$	727.421	0.899160	0.449580	+	0.893240i	$0.351574\pi$
$810$	28.4605i	0.0351364i
$811$	673.804i	0.830831i	0.909632	+	0.415415i	$0.136364\pi$
$811$	673.804i	0.830831i	−0.909632	+	0.415415i	$0.863636\pi$
$812$	0	0
$813$	−382.821	−0.470875
$814$	18.9417	0.0232699
$815$	264.585i	0.324644i
$816$	−69.6874	−0.0854013
$817$	− 74.6708i	− 0.0913964i
$818$	− 658.527i	− 0.805046i
$819$	0	0
$820$	234.285	0.285714
$821$	−890.930	−1.08518	−0.542588	−	0.839999i	$-0.682556\pi$
$821$	−890.930	−1.08518	−0.542588	+	0.839999i	$0.682556\pi$
$822$	− 198.783i	− 0.241829i
$823$	−991.746	−1.20504	−0.602519	−	0.798105i	$-0.705836\pi$
$823$	−991.746	−1.20504	−0.602519	+	0.798105i	$0.705836\pi$
$824$	− 132.876i	− 0.161257i
$825$	15.9247i	0.0193027i
$826$	0	0
$827$	1264.47	1.52899	0.764493	−	0.644632i	$-0.222989\pi$
$827$	1264.47	1.52899	0.764493	+	0.644632i	$0.222989\pi$
$828$	5.52773	0.00667600
$829$	− 90.4298i	− 0.109083i	−0.998512	−	0.0545415i	$-0.982630\pi$
$829$	− 90.4298i	− 0.109083i	0.998512	−	0.0545415i	$-0.0173697\pi$
$830$	−489.255	−0.589464
$831$	− 687.371i	− 0.827162i
$832$	− 43.2612i	− 0.0519966i
$833$	0	0
$834$	−608.236	−0.729300
$835$	−458.810	−0.549473
$836$	− 33.8064i	− 0.0404383i
$837$	−217.124	−0.259408
$838$	− 814.421i	− 0.971863i
$839$	− 148.508i	− 0.177006i	−0.996076	−	0.0885032i	$-0.971792\pi$
$839$	− 148.508i	− 0.177006i	0.996076	−	0.0885032i	$-0.0282083\pi$
$840$	0	0
$841$	−683.313	−0.812501
$842$	−807.727	−0.959296
$843$	− 197.876i	− 0.234729i
$844$	371.860	0.440592
$845$	− 312.507i	− 0.369831i
$846$	− 144.087i	− 0.170316i
$847$	0	0
$848$	−416.317	−0.490940
$849$	−783.638	−0.923013
$850$	71.1244i	0.0836758i
$851$	−6.71055	−0.00788549
$852$	− 62.2223i	− 0.0730309i
$853$	642.996i	0.753805i	0.926253	+	0.376902i	$0.123011\pi$
$853$	642.996i	0.753805i	−0.926253	+	0.376902i	$0.876989\pi$
$854$	0	0
$855$	−61.6644	−0.0721221
$856$	525.187	0.613537
$857$	− 126.646i	− 0.147779i	−0.997266	−	0.0738894i	$-0.976459\pi$
$857$	− 126.646i	− 0.147779i	0.997266	−	0.0738894i	$-0.0235412\pi$
$858$	24.3571	0.0283882
$859$	503.168i	0.585760i	0.956149	+	0.292880i	$0.0946136\pi$
$859$	503.168i	0.585760i	−0.956149	+	0.292880i	$0.905386\pi$
$860$	36.3277i	0.0422415i
$861$	0	0
$862$	38.3809	0.0445254
$863$	1267.78	1.46904	0.734518	−	0.678589i	$-0.237408\pi$
$863$	1267.78	1.46904	0.734518	+	0.678589i	$0.237408\pi$
$864$	29.3939i	0.0340207i
$865$	−251.633	−0.290906
$866$	513.936i	0.593460i
$867$	325.325i	0.375230i
$868$	0	0
$869$	148.828	0.171264
$870$	−68.7794	−0.0790568
$871$	80.7187i	0.0926736i
$872$	−244.064	−0.279890
$873$	− 264.521i	− 0.303002i
$874$	11.9767i	0.0137034i
$875$	0	0
$876$	430.819	0.491802
$877$	−329.701	−0.375942	−0.187971	−	0.982175i	$-0.560191\pi$
$877$	−329.701	−0.375942	−0.187971	+	0.982175i	$0.560191\pi$
$878$	− 658.189i	− 0.749645i
$879$	−206.346	−0.234751
$880$	16.4470i	0.0186897i
$881$	583.618i	0.662449i	0.943552	+	0.331224i	$0.107462\pi$
$881$	583.618i	0.662449i	−0.943552	+	0.331224i	$0.892538\pi$
$882$	0	0
$883$	−1172.36	−1.32770	−0.663849	−	0.747867i	$-0.731078\pi$
$883$	−1172.36	−1.32770	−0.663849	+	0.747867i	$0.731078\pi$
$884$	108.786	0.123061
$885$	56.3123i	0.0636298i
$886$	−690.045	−0.778832
$887$	− 1471.83i	− 1.65933i	−0.558258	−	0.829667i	$-0.688530\pi$
$887$	− 1471.83i	− 1.65933i	0.558258	−	0.829667i	$-0.311470\pi$
$888$	− 35.6836i	− 0.0401842i
$889$	0	0
$890$	215.422	0.242047
$891$	−16.5494	−0.0185740
$892$	− 107.270i	− 0.120257i
$893$	312.189	0.349596
$894$	367.447i	0.411015i
$895$	99.4375i	0.111103i
$896$	0	0
$897$	−8.62907	−0.00961993
$898$	404.235	0.450151
$899$	− 524.716i	− 0.583666i
$900$	30.0000	0.0333333
$901$	− 1046.88i	− 1.16191i
$902$	136.234i	0.151035i
$903$	0	0
$904$	242.737	0.268515
$905$	20.6737	0.0228438
$906$	− 448.330i	− 0.494845i
$907$	997.198	1.09945	0.549723	−	0.835347i	$-0.314733\pi$
$907$	997.198	1.09945	0.549723	+	0.835347i	$0.314733\pi$
$908$	338.078i	0.372333i
$909$	334.126i	0.367575i
$910$	0	0
$911$	−758.955	−0.833101	−0.416550	−	0.909113i	$-0.636761\pi$
$911$	−758.955	−0.833101	−0.416550	+	0.909113i	$0.636761\pi$
$912$	−63.6867	−0.0698319
$913$	− 284.496i	− 0.311606i
$914$	−598.562	−0.654882
$915$	− 92.0871i	− 0.100642i
$916$	− 572.035i	− 0.624492i
$917$	0	0
$918$	−73.9147	−0.0805171
$919$	1407.56	1.53162	0.765812	−	0.643064i	$-0.222337\pi$
$919$	1407.56	1.53162	0.765812	+	0.643064i	$0.222337\pi$
$920$	− 5.82673i	− 0.00633341i
$921$	472.231	0.512737
$922$	− 667.152i	− 0.723592i
$923$	97.1324i	0.105236i
$924$	0	0
$925$	−36.4194	−0.0393723
$926$	274.384	0.296311
$927$	− 140.936i	− 0.152034i
$928$	−71.0351	−0.0765464
$929$	− 363.240i	− 0.391001i	−0.980704	−	0.195500i	$-0.937367\pi$
$929$	− 363.240i	− 0.391001i	0.980704	−	0.195500i	$-0.0626331\pi$
$930$	228.869i	0.246096i
$931$	0	0
$932$	−363.677	−0.390212
$933$	−130.895	−0.140295
$934$	− 16.0189i	− 0.0171509i
$935$	−41.3580	−0.0442332
$936$	− 45.8854i	− 0.0490229i
$937$	401.784i	0.428798i	0.976746	+	0.214399i	$0.0687793\pi$
$937$	401.784i	0.428798i	−0.976746	+	0.214399i	$0.931221\pi$
$938$	0	0
$939$	−110.211	−0.117371
$940$	−151.881	−0.161576
$941$	− 844.405i	− 0.897348i	−0.893695	−	0.448674i	$-0.851896\pi$
$941$	− 844.405i	− 0.897348i	0.893695	−	0.448674i	$-0.148104\pi$
$942$	−723.298	−0.767832
$943$	− 48.2642i	− 0.0511815i
$944$	58.1591i	0.0616092i
$945$	0	0
$946$	−21.1241	−0.0223299
$947$	288.167	0.304295	0.152147	−	0.988358i	$-0.451381\pi$
$947$	288.167	0.304295	0.152147	+	0.988358i	$0.451381\pi$
$948$	− 280.372i	− 0.295751i
$949$	−672.531	−0.708673
$950$	65.0000i	0.0684210i
$951$	26.7638i	0.0281428i
$952$	0	0
$953$	293.080	0.307534	0.153767	−	0.988107i	$-0.450859\pi$
$953$	293.080	0.307534	0.153767	+	0.988107i	$0.450859\pi$
$954$	−441.571	−0.462863
$955$	305.914i	0.320329i
$956$	−764.977	−0.800185
$957$	− 39.9944i	− 0.0417914i
$958$	894.769i	0.933997i
$959$	0	0
$960$	30.9839	0.0322749
$961$	−785.034	−0.816893
$962$	55.7040i	0.0579044i
$963$	557.045	0.578448
$964$	− 416.439i	− 0.431991i
$965$	818.119i	0.847792i
$966$	0	0
$967$	68.4003	0.0707345	0.0353673	−	0.999374i	$-0.488740\pi$
$967$	68.4003	0.0707345	0.0353673	+	0.999374i	$0.488740\pi$
$968$	332.676	0.343674
$969$	− 160.148i	− 0.165272i
$970$	−278.829	−0.287453
$971$	− 292.387i	− 0.301119i	−0.988601	−	0.150560i	$-0.951892\pi$
$971$	− 292.387i	− 0.301119i	0.988601	−	0.150560i	$-0.0481076\pi$
$972$	31.1769i	0.0320750i
$973$	0	0
$974$	89.1243	0.0915034
$975$	−46.8316	−0.0480324
$976$	− 95.1071i	− 0.0974458i
$977$	−1077.39	−1.10276	−0.551379	−	0.834255i	$-0.685898\pi$
$977$	−1077.39	−1.10276	−0.551379	+	0.834255i	$0.685898\pi$
$978$	289.839i	0.296358i
$979$	125.265i	0.127952i
$980$	0	0
$981$	−258.869	−0.263883
$982$	−369.240	−0.376008
$983$	1564.56i	1.59162i	0.605545	+	0.795811i	$0.292955\pi$
$983$	1564.56i	1.59162i	−0.605545	+	0.795811i	$0.707045\pi$
$984$	256.646	0.260820
$985$	435.866i	0.442503i
$986$	− 178.627i	− 0.181163i
$987$	0	0
$988$	99.4184	0.100626
$989$	7.48373	0.00756697
$990$	17.4446i	0.0176209i
$991$	1453.93	1.46714	0.733570	−	0.679614i	$-0.237853\pi$
$991$	1453.93	1.46714	0.733570	+	0.679614i	$0.237853\pi$
$992$	236.375i	0.238281i
$993$	− 646.192i	− 0.650747i
$994$	0	0
$995$	−542.996	−0.545724
$996$	−535.952	−0.538105
$997$	− 809.472i	− 0.811908i	−0.913894	−	0.405954i	$-0.866939\pi$
$997$	− 809.472i	− 0.811908i	0.913894	−	0.405954i	$-0.133061\pi$
$998$	−620.787	−0.622031
$999$	− 37.8482i	− 0.0378860i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.a.391.3		8
7.4	even	3		210.3.o.a.61.4	yes	8
7.5	odd	6		210.3.o.a.31.4	✓	8
7.6	odd	2	inner	1470.3.f.a.391.2		8
21.5	even	6		630.3.v.b.451.1		8
21.11	odd	6		630.3.v.b.271.1		8
35.4	even	6		1050.3.p.b.901.1		8
35.12	even	12		1050.3.q.c.199.3		16
35.18	odd	12		1050.3.q.c.649.3		16
35.19	odd	6		1050.3.p.b.451.1		8
35.32	odd	12		1050.3.q.c.649.6		16
35.33	even	12		1050.3.q.c.199.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.a.31.4	✓	8	7.5	odd	6
210.3.o.a.61.4	yes	8	7.4	even	3
630.3.v.b.271.1		8	21.11	odd	6
630.3.v.b.451.1		8	21.5	even	6
1050.3.p.b.451.1		8	35.19	odd	6
1050.3.p.b.901.1		8	35.4	even	6
1050.3.q.c.199.3		16	35.12	even	12
1050.3.q.c.199.6		16	35.33	even	12
1050.3.q.c.649.3		16	35.18	odd	12
1050.3.q.c.649.6		16	35.32	odd	12
1470.3.f.a.391.2		8	7.6	odd	2	inner
1470.3.f.a.391.3		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -1.83883 q^{11} +3.46410i q^{12} -5.40765i q^{13} +3.87298 q^{15} +4.00000 q^{16} +10.0585i q^{17} +4.24264 q^{18} +9.19238i q^{19} -4.47214i q^{20} +2.60049 q^{22} -0.921288 q^{23} -4.89898i q^{24} -5.00000 q^{25} +7.64757i q^{26} -5.19615i q^{27} +12.5573 q^{29} -5.47723 q^{30} -41.7856i q^{31} -5.65685 q^{32} -3.18494i q^{33} -14.2249i q^{34} -6.00000 q^{36} +7.28388 q^{37} -13.0000i q^{38} +9.36632 q^{39} +6.32456i q^{40} +52.3877i q^{41} -8.12312 q^{43} -3.67765 q^{44} +6.70820i q^{45} +1.30290 q^{46} -33.9617i q^{47} +6.92820i q^{48} +7.07107 q^{50} -17.4219 q^{51} -10.8153i q^{52} -104.079 q^{53} +7.34847i q^{54} +4.11174i q^{55} -15.9217 q^{57} -17.7588 q^{58} +14.5398i q^{59} +7.74597 q^{60} -23.7768i q^{61} +59.0937i q^{62} +8.00000 q^{64} -12.0919 q^{65} +4.50419i q^{66} -14.9268 q^{67} +20.1170i q^{68} -1.59572i q^{69} -17.9620 q^{71} +8.48528 q^{72} -124.367i q^{73} -10.3010 q^{74} -8.66025i q^{75} +18.3848i q^{76} -13.2460 q^{78} -80.9365 q^{79} -8.94427i q^{80} +9.00000 q^{81} -74.0875i q^{82} +154.716i q^{83} +22.4915 q^{85} +11.4878 q^{86} +21.7500i q^{87} +5.20099 q^{88} -68.1223i q^{89} -9.48683i q^{90} -1.84258 q^{92} +72.3747 q^{93} +48.0291i q^{94} +20.5548 q^{95} -9.79796i q^{96} +88.1736i q^{97} +5.51648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 24 q^{9}+O(q^{10}) \) 8 * q + 16 * q^4 - 24 * q^9	\( 8 q + 16 q^{4} - 24 q^{9} + 8 q^{11} + 32 q^{16} - 48 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{29} - 48 q^{36} + 192 q^{37} - 48 q^{39} - 112 q^{43} + 16 q^{44} - 16 q^{46} - 168 q^{51} - 64 q^{53} + 216 q^{57} - 208 q^{58} + 64 q^{64} - 40 q^{65} + 240 q^{67} + 8 q^{71} + 32 q^{74} - 192 q^{78} - 24 q^{79} + 72 q^{81} + 120 q^{85} + 80 q^{86} - 96 q^{88} - 48 q^{92} + 264 q^{93} + 80 q^{95} - 24 q^{99}+O(q^{100}) \) 8 * q + 16 * q^4 - 24 * q^9 + 8 * q^11 + 32 * q^16 - 48 * q^22 - 24 * q^23 - 40 * q^25 + 72 * q^29 - 48 * q^36 + 192 * q^37 - 48 * q^39 - 112 * q^43 + 16 * q^44 - 16 * q^46 - 168 * q^51 - 64 * q^53 + 216 * q^57 - 208 * q^58 + 64 * q^64 - 40 * q^65 + 240 * q^67 + 8 * q^71 + 32 * q^74 - 192 * q^78 - 24 * q^79 + 72 * q^81 + 120 * q^85 + 80 * q^86 - 96 * q^88 - 48 * q^92 + 264 * q^93 + 80 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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