LMFDB - Embedded newform 1470.3.f.d.391.14

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			391.14
Root			$-3.67087 + 6.35814i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.d.391.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} +10.2696 q^{11} +3.46410i q^{12} +7.02340i q^{13} +3.87298 q^{15} +4.00000 q^{16} -31.7481i q^{17} -4.24264 q^{18} -31.1086i q^{19} -4.47214i q^{20} +14.5234 q^{22} -23.6882 q^{23} +4.89898i q^{24} -5.00000 q^{25} +9.93258i q^{26} -5.19615i q^{27} +9.19673 q^{29} +5.47723 q^{30} -20.1508i q^{31} +5.65685 q^{32} +17.7874i q^{33} -44.8986i q^{34} -6.00000 q^{36} +48.1647 q^{37} -43.9942i q^{38} -12.1649 q^{39} -6.32456i q^{40} +65.1226i q^{41} -3.03497 q^{43} +20.5391 q^{44} +6.70820i q^{45} -33.5002 q^{46} -61.9464i q^{47} +6.92820i q^{48} -7.07107 q^{50} +54.9893 q^{51} +14.0468i q^{52} +1.38150 q^{53} -7.34847i q^{54} -22.9634i q^{55} +53.8817 q^{57} +13.0061 q^{58} +109.819i q^{59} +7.74597 q^{60} -39.6918i q^{61} -28.4976i q^{62} +8.00000 q^{64} +15.7048 q^{65} +25.1552i q^{66} +15.9190 q^{67} -63.4962i q^{68} -41.0292i q^{69} +53.3489 q^{71} -8.48528 q^{72} -72.3801i q^{73} +68.1151 q^{74} -8.66025i q^{75} -62.2172i q^{76} -17.2037 q^{78} +106.446 q^{79} -8.94427i q^{80} +9.00000 q^{81} +92.0973i q^{82} -49.4298i q^{83} -70.9909 q^{85} -4.29210 q^{86} +15.9292i q^{87} +29.0467 q^{88} -164.900i q^{89} +9.48683i q^{90} -47.3765 q^{92} +34.9023 q^{93} -87.6055i q^{94} -69.5610 q^{95} +9.79796i q^{96} -49.4799i q^{97} -30.8087 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	10.2696	0.933596	0.466798	−	0.884364i	$-0.345407\pi$
$11$	10.2696	0.933596	0.466798	+	0.884364i	$0.345407\pi$
$12$	3.46410i	0.288675i
$13$	7.02340i	0.540261i	0.962824	+	0.270131i	$0.0870669\pi$
$13$	7.02340i	0.540261i	−0.962824	+	0.270131i	$0.912933\pi$
$14$	0	0
$15$	3.87298	0.258199
$16$	4.00000	0.250000
$17$	− 31.7481i	− 1.86754i	−0.357879	−	0.933768i	$-0.616500\pi$
$17$	− 31.7481i	− 1.86754i	0.357879	−	0.933768i	$-0.383500\pi$
$18$	−4.24264	−0.235702
$19$	− 31.1086i	− 1.63730i	−0.574296	−	0.818648i	$-0.694724\pi$
$19$	− 31.1086i	− 1.63730i	0.574296	−	0.818648i	$-0.305276\pi$
$20$	− 4.47214i	− 0.223607i
$21$	0	0
$22$	14.5234	0.660152
$23$	−23.6882	−1.02992	−0.514962	−	0.857213i	$-0.672194\pi$
$23$	−23.6882	−1.02992	−0.514962	+	0.857213i	$0.672194\pi$
$24$	4.89898i	0.204124i
$25$	−5.00000	−0.200000
$26$	9.93258i	0.382022i
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	9.19673	0.317129	0.158564	−	0.987349i	$-0.449313\pi$
$29$	9.19673	0.317129	0.158564	+	0.987349i	$0.449313\pi$
$30$	5.47723	0.182574
$31$	− 20.1508i	− 0.650027i	−0.945709	−	0.325013i	$-0.894631\pi$
$31$	− 20.1508i	− 0.650027i	0.945709	−	0.325013i	$-0.105369\pi$
$32$	5.65685	0.176777
$33$	17.7874i	0.539012i
$34$	− 44.8986i	− 1.32055i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	48.1647	1.30175	0.650874	−	0.759186i	$-0.274403\pi$
$37$	48.1647	1.30175	0.650874	+	0.759186i	$0.274403\pi$
$38$	− 43.9942i	− 1.15774i
$39$	−12.1649	−0.311920
$40$	− 6.32456i	− 0.158114i
$41$	65.1226i	1.58836i	0.607685	+	0.794178i	$0.292098\pi$
$41$	65.1226i	1.58836i	−0.607685	+	0.794178i	$0.707902\pi$
$42$	0	0
$43$	−3.03497	−0.0705807	−0.0352904	−	0.999377i	$-0.511236\pi$
$43$	−3.03497	−0.0705807	−0.0352904	+	0.999377i	$0.511236\pi$
$44$	20.5391	0.466798
$45$	6.70820i	0.149071i
$46$	−33.5002	−0.728266
$47$	− 61.9464i	− 1.31801i	−0.752139	−	0.659005i	$-0.770978\pi$
$47$	− 61.9464i	− 1.31801i	0.752139	−	0.659005i	$-0.229022\pi$
$48$	6.92820i	0.144338i
$49$	0	0
$50$	−7.07107	−0.141421
$51$	54.9893	1.07822
$52$	14.0468i	0.270131i
$53$	1.38150	0.0260661	0.0130330	−	0.999915i	$-0.495851\pi$
$53$	1.38150	0.0260661	0.0130330	+	0.999915i	$0.495851\pi$
$54$	− 7.34847i	− 0.136083i
$55$	− 22.9634i	− 0.417517i
$56$	0	0
$57$	53.8817	0.945293
$58$	13.0061	0.224244
$59$	109.819i	1.86135i	0.365851	+	0.930673i	$0.380778\pi$
$59$	109.819i	1.86135i	−0.365851	+	0.930673i	$0.619222\pi$
$60$	7.74597	0.129099
$61$	− 39.6918i	− 0.650686i	−0.945596	−	0.325343i	$-0.894520\pi$
$61$	− 39.6918i	− 0.650686i	0.945596	−	0.325343i	$-0.105480\pi$
$62$	− 28.4976i	− 0.459638i
$63$	0	0
$64$	8.00000	0.125000
$65$	15.7048	0.241612
$66$	25.1552i	0.381139i
$67$	15.9190	0.237598	0.118799	−	0.992918i	$-0.462096\pi$
$67$	15.9190	0.237598	0.118799	+	0.992918i	$0.462096\pi$
$68$	− 63.4962i	− 0.933768i
$69$	− 41.0292i	− 0.594626i
$70$	0	0
$71$	53.3489	0.751393	0.375696	−	0.926743i	$-0.377404\pi$
$71$	53.3489	0.751393	0.375696	+	0.926743i	$0.377404\pi$
$72$	−8.48528	−0.117851
$73$	− 72.3801i	− 0.991509i	−0.868463	−	0.495754i	$-0.834892\pi$
$73$	− 72.3801i	− 0.991509i	0.868463	−	0.495754i	$-0.165108\pi$
$74$	68.1151	0.920475
$75$	− 8.66025i	− 0.115470i
$76$	− 62.2172i	− 0.818648i
$77$	0	0
$78$	−17.2037	−0.220561
$79$	106.446	1.34741	0.673707	−	0.738998i	$-0.264701\pi$
$79$	106.446	1.34741	0.673707	+	0.738998i	$0.264701\pi$
$80$	− 8.94427i	− 0.111803i
$81$	9.00000	0.111111
$82$	92.0973i	1.12314i
$83$	− 49.4298i	− 0.595540i	−0.954638	−	0.297770i	$-0.903757\pi$
$83$	− 49.4298i	− 0.595540i	0.954638	−	0.297770i	$-0.0962429\pi$
$84$	0	0
$85$	−70.9909	−0.835187
$86$	−4.29210	−0.0499081
$87$	15.9292i	0.183094i
$88$	29.0467	0.330076
$89$	− 164.900i	− 1.85281i	−0.376534	−	0.926403i	$-0.622884\pi$
$89$	− 164.900i	− 1.85281i	0.376534	−	0.926403i	$-0.377116\pi$
$90$	9.48683i	0.105409i
$91$	0	0
$92$	−47.3765	−0.514962
$93$	34.9023	0.375293
$94$	− 87.6055i	− 0.931973i
$95$	−69.5610	−0.732221
$96$	9.79796i	0.102062i
$97$	− 49.4799i	− 0.510102i	−0.966928	−	0.255051i	$-0.917908\pi$
$97$	− 49.4799i	− 0.510102i	0.966928	−	0.255051i	$-0.0820922\pi$
$98$	0	0
$99$	−30.8087	−0.311199
$100$	−10.0000	−0.100000
$101$	− 134.873i	− 1.33537i	−0.744442	−	0.667687i	$-0.767285\pi$
$101$	− 134.873i	− 1.33537i	0.744442	−	0.667687i	$-0.232715\pi$
$102$	77.7667	0.762418
$103$	− 37.4021i	− 0.363127i	−0.983379	−	0.181563i	$-0.941884\pi$
$103$	− 37.4021i	− 0.363127i	0.983379	−	0.181563i	$-0.0581158\pi$
$104$	19.8652i	0.191011i
$105$	0	0
$106$	1.95374	0.0184315
$107$	−24.7960	−0.231738	−0.115869	−	0.993264i	$-0.536965\pi$
$107$	−24.7960	−0.231738	−0.115869	+	0.993264i	$0.536965\pi$
$108$	− 10.3923i	− 0.0962250i
$109$	−56.2897	−0.516419	−0.258209	−	0.966089i	$-0.583132\pi$
$109$	−56.2897	−0.516419	−0.258209	+	0.966089i	$0.583132\pi$
$110$	− 32.4752i	− 0.295229i
$111$	83.4237i	0.751565i
$112$	0	0
$113$	74.9910	0.663637	0.331818	−	0.943343i	$-0.392338\pi$
$113$	74.9910	0.663637	0.331818	+	0.943343i	$0.392338\pi$
$114$	76.2002	0.668423
$115$	52.9685i	0.460596i
$116$	18.3935	0.158564
$117$	− 21.0702i	− 0.180087i
$118$	155.308i	1.31617i
$119$	0	0
$120$	10.9545	0.0912871
$121$	−15.5361	−0.128398
$122$	− 56.1327i	− 0.460104i
$123$	−112.796	−0.917038
$124$	− 40.3017i	− 0.325013i
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	128.504	1.01184	0.505921	−	0.862580i	$-0.331153\pi$
$127$	128.504	1.01184	0.505921	+	0.862580i	$0.331153\pi$
$128$	11.3137	0.0883883
$129$	− 5.25673i	− 0.0407498i
$130$	22.2099	0.170846
$131$	75.4964i	0.576309i	0.957584	+	0.288154i	$0.0930416\pi$
$131$	75.4964i	0.576309i	−0.957584	+	0.288154i	$0.906958\pi$
$132$	35.5748i	0.269506i
$133$	0	0
$134$	22.5129	0.168007
$135$	−11.6190	−0.0860663
$136$	− 89.7972i	− 0.660274i
$137$	107.517	0.784792	0.392396	−	0.919796i	$-0.371646\pi$
$137$	107.517	0.784792	0.392396	+	0.919796i	$0.371646\pi$
$138$	− 58.0241i	− 0.420464i
$139$	272.004i	1.95686i	0.206576	+	0.978431i	$0.433768\pi$
$139$	272.004i	1.95686i	−0.206576	+	0.978431i	$0.566232\pi$
$140$	0	0
$141$	107.294	0.760953
$142$	75.4467	0.531315
$143$	72.1272i	0.504386i
$144$	−12.0000	−0.0833333
$145$	− 20.5645i	− 0.141824i
$146$	− 102.361i	− 0.701102i
$147$	0	0
$148$	96.3294	0.650874
$149$	−83.4269	−0.559912	−0.279956	−	0.960013i	$-0.590320\pi$
$149$	−83.4269	−0.559912	−0.279956	+	0.960013i	$0.590320\pi$
$150$	− 12.2474i	− 0.0816497i
$151$	126.795	0.839700	0.419850	−	0.907594i	$-0.362083\pi$
$151$	126.795	0.839700	0.419850	+	0.907594i	$0.362083\pi$
$152$	− 87.9884i	− 0.578871i
$153$	95.2443i	0.622512i
$154$	0	0
$155$	−45.0586	−0.290701
$156$	−24.3298	−0.155960
$157$	99.2368i	0.632082i	0.948746	+	0.316041i	$0.102354\pi$
$157$	99.2368i	0.632082i	−0.948746	+	0.316041i	$0.897646\pi$
$158$	150.537	0.952766
$159$	2.39283i	0.0150493i
$160$	− 12.6491i	− 0.0790569i
$161$	0	0
$162$	12.7279	0.0785674
$163$	278.980	1.71153	0.855765	−	0.517364i	$-0.173087\pi$
$163$	278.980	1.71153	0.855765	+	0.517364i	$0.173087\pi$
$164$	130.245i	0.794178i
$165$	39.7738	0.241054
$166$	− 69.9043i	− 0.421110i
$167$	29.9435i	0.179302i	0.995973	+	0.0896511i	$0.0285752\pi$
$167$	29.9435i	0.179302i	−0.995973	+	0.0896511i	$0.971425\pi$
$168$	0	0
$169$	119.672	0.708118
$170$	−100.396	−0.590567
$171$	93.3258i	0.545765i
$172$	−6.06994	−0.0352904
$173$	106.506i	0.615641i	0.951444	+	0.307821i	$0.0995997\pi$
$173$	106.506i	0.615641i	−0.951444	+	0.307821i	$0.900400\pi$
$174$	22.5273i	0.129467i
$175$	0	0
$176$	41.0782	0.233399
$177$	−190.213	−1.07465
$178$	− 233.203i	− 1.31013i
$179$	−239.973	−1.34063	−0.670315	−	0.742077i	$-0.733841\pi$
$179$	−239.973	−1.34063	−0.670315	+	0.742077i	$0.733841\pi$
$180$	13.4164i	0.0745356i
$181$	309.322i	1.70896i	0.519482	+	0.854482i	$0.326125\pi$
$181$	309.322i	1.70896i	−0.519482	+	0.854482i	$0.673875\pi$
$182$	0	0
$183$	68.7483	0.375674
$184$	−67.0004	−0.364133
$185$	− 107.699i	− 0.582159i
$186$	49.3592	0.265372
$187$	− 326.039i	− 1.74352i
$188$	− 123.893i	− 0.659005i
$189$	0	0
$190$	−98.3741	−0.517758
$191$	3.08960	0.0161759	0.00808796	−	0.999967i	$-0.497425\pi$
$191$	3.08960	0.0161759	0.00808796	+	0.999967i	$0.497425\pi$
$192$	13.8564i	0.0721688i
$193$	−238.697	−1.23677	−0.618387	−	0.785874i	$-0.712214\pi$
$193$	−238.697	−1.23677	−0.618387	+	0.785874i	$0.712214\pi$
$194$	− 69.9751i	− 0.360696i
$195$	27.2015i	0.139495i
$196$	0	0
$197$	−291.539	−1.47989	−0.739946	−	0.672666i	$-0.765149\pi$
$197$	−291.539	−1.47989	−0.739946	+	0.672666i	$0.765149\pi$
$198$	−43.5701	−0.220051
$199$	− 241.591i	− 1.21403i	−0.794692	−	0.607013i	$-0.792368\pi$
$199$	− 241.591i	− 1.21403i	0.794692	−	0.607013i	$-0.207632\pi$
$200$	−14.1421	−0.0707107
$201$	27.5726i	0.137177i
$202$	− 190.739i	− 0.944252i
$203$	0	0
$204$	109.979	0.539111
$205$	145.619	0.710335
$206$	− 52.8945i	− 0.256769i
$207$	71.0647	0.343308
$208$	28.0936i	0.135065i
$209$	− 319.472i	− 1.52857i
$210$	0	0
$211$	−263.018	−1.24653	−0.623266	−	0.782010i	$-0.714195\pi$
$211$	−263.018	−1.24653	−0.623266	+	0.782010i	$0.714195\pi$
$212$	2.76300	0.0130330
$213$	92.4030i	0.433817i
$214$	−35.0668	−0.163864
$215$	6.78640i	0.0315647i
$216$	− 14.6969i	− 0.0680414i
$217$	0	0
$218$	−79.6056	−0.365163
$219$	125.366	0.572448
$220$	− 45.9269i	− 0.208759i
$221$	222.980	1.00896
$222$	117.979i	0.531436i
$223$	112.658i	0.505193i	0.967572	+	0.252597i	$0.0812845\pi$
$223$	112.658i	0.505193i	−0.967572	+	0.252597i	$0.918715\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	106.053	0.469262
$227$	− 49.0204i	− 0.215949i	−0.994154	−	0.107974i	$-0.965564\pi$
$227$	− 49.0204i	− 0.215949i	0.994154	−	0.107974i	$-0.0344365\pi$
$228$	107.763	0.472646
$229$	28.1141i	0.122769i	0.998114	+	0.0613846i	$0.0195516\pi$
$229$	28.1141i	0.122769i	−0.998114	+	0.0613846i	$0.980448\pi$
$230$	74.9088i	0.325690i
$231$	0	0
$232$	26.0123	0.112122
$233$	176.203	0.756235	0.378117	−	0.925758i	$-0.376572\pi$
$233$	176.203	0.756235	0.378117	+	0.925758i	$0.376572\pi$
$234$	− 29.7978i	− 0.127341i
$235$	−138.516	−0.589432
$236$	219.639i	0.930673i
$237$	184.369i	0.777930i
$238$	0	0
$239$	34.7150	0.145251	0.0726255	−	0.997359i	$-0.476862\pi$
$239$	34.7150	0.145251	0.0726255	+	0.997359i	$0.476862\pi$
$240$	15.4919	0.0645497
$241$	265.432i	1.10138i	0.834710	+	0.550690i	$0.185635\pi$
$241$	265.432i	1.10138i	−0.834710	+	0.550690i	$0.814365\pi$
$242$	−21.9714	−0.0907909
$243$	15.5885i	0.0641500i
$244$	− 79.3837i	− 0.325343i
$245$	0	0
$246$	−159.517	−0.648444
$247$	218.488	0.884567
$248$	− 56.9951i	− 0.229819i
$249$	85.6150	0.343835
$250$	15.8114i	0.0632456i
$251$	24.1723i	0.0963040i	0.998840	+	0.0481520i	$0.0153332\pi$
$251$	24.1723i	0.0963040i	−0.998840	+	0.0481520i	$0.984667\pi$
$252$	0	0
$253$	−243.268	−0.961533
$254$	181.732	0.715480
$255$	− 122.960i	− 0.482196i
$256$	16.0000	0.0625000
$257$	− 9.53747i	− 0.0371108i	−0.999828	−	0.0185554i	$-0.994093\pi$
$257$	− 9.53747i	− 0.0371108i	0.999828	−	0.0185554i	$-0.00590670\pi$
$258$	− 7.43413i	− 0.0288145i
$259$	0	0
$260$	31.4096	0.120806
$261$	−27.5902	−0.105710
$262$	106.768i	0.407512i
$263$	−131.863	−0.501381	−0.250691	−	0.968067i	$-0.580658\pi$
$263$	−131.863	−0.501381	−0.250691	+	0.968067i	$0.580658\pi$
$264$	50.3104i	0.190570i
$265$	− 3.08913i	− 0.0116571i
$266$	0	0
$267$	285.615	1.06972
$268$	31.8381	0.118799
$269$	37.8790i	0.140814i	0.997518	+	0.0704070i	$0.0224298\pi$
$269$	37.8790i	0.140814i	−0.997518	+	0.0704070i	$0.977570\pi$
$270$	−16.4317	−0.0608581
$271$	362.346i	1.33707i	0.743680	+	0.668535i	$0.233078\pi$
$271$	362.346i	1.33707i	−0.743680	+	0.668535i	$0.766922\pi$
$272$	− 126.992i	− 0.466884i
$273$	0	0
$274$	152.051	0.554932
$275$	−51.3478	−0.186719
$276$	− 82.0584i	− 0.297313i
$277$	−113.299	−0.409021	−0.204511	−	0.978864i	$-0.565560\pi$
$277$	−113.299	−0.409021	−0.204511	+	0.978864i	$0.565560\pi$
$278$	384.671i	1.38371i
$279$	60.4525i	0.216676i
$280$	0	0
$281$	−178.735	−0.636069	−0.318034	−	0.948079i	$-0.603023\pi$
$281$	−178.735	−0.636069	−0.318034	+	0.948079i	$0.603023\pi$
$282$	151.737	0.538075
$283$	− 43.1685i	− 0.152539i	−0.997087	−	0.0762695i	$-0.975699\pi$
$283$	− 43.1685i	− 0.152539i	0.997087	−	0.0762695i	$-0.0243009\pi$
$284$	106.698	0.375696
$285$	− 120.483i	− 0.422748i
$286$	102.003i	0.356655i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	−718.942	−2.48769
$290$	− 29.0826i	− 0.100285i
$291$	85.7016	0.294507
$292$	− 144.760i	− 0.495754i
$293$	15.4426i	0.0527050i	0.999653	+	0.0263525i	$0.00838923\pi$
$293$	15.4426i	0.0527050i	−0.999653	+	0.0263525i	$0.991611\pi$
$294$	0	0
$295$	245.564	0.832420
$296$	136.230	0.460237
$297$	− 53.3622i	− 0.179671i
$298$	−117.983	−0.395918
$299$	− 166.372i	− 0.556428i
$300$	− 17.3205i	− 0.0577350i
$301$	0	0
$302$	179.315	0.593757
$303$	233.606	0.770978
$304$	− 124.434i	− 0.409324i
$305$	−88.7536	−0.290996
$306$	134.696i	0.440182i
$307$	− 234.648i	− 0.764327i	−0.924095	−	0.382163i	$-0.875179\pi$
$307$	− 234.648i	− 0.764327i	0.924095	−	0.382163i	$-0.124821\pi$
$308$	0	0
$309$	64.7823	0.209651
$310$	−63.7225	−0.205556
$311$	398.778i	1.28224i	0.767439	+	0.641121i	$0.221531\pi$
$311$	398.778i	1.28224i	−0.767439	+	0.641121i	$0.778469\pi$
$312$	−34.4075	−0.110280
$313$	129.200i	0.412781i	0.978470	+	0.206390i	$0.0661717\pi$
$313$	129.200i	0.412781i	−0.978470	+	0.206390i	$0.933828\pi$
$314$	140.342i	0.446949i
$315$	0	0
$316$	212.892	0.673707
$317$	−402.565	−1.26992	−0.634961	−	0.772544i	$-0.718984\pi$
$317$	−402.565	−1.26992	−0.634961	+	0.772544i	$0.718984\pi$
$318$	3.38397i	0.0106414i
$319$	94.4464	0.296070
$320$	− 17.8885i	− 0.0559017i
$321$	− 42.9479i	− 0.133794i
$322$	0	0
$323$	−987.640	−3.05771
$324$	18.0000	0.0555556
$325$	− 35.1170i	− 0.108052i
$326$	394.537	1.21024
$327$	− 97.4966i	− 0.298155i
$328$	184.195i	0.561569i
$329$	0	0
$330$	56.2487	0.170451
$331$	−166.893	−0.504207	−0.252104	−	0.967700i	$-0.581122\pi$
$331$	−166.893	−0.504207	−0.252104	+	0.967700i	$0.581122\pi$
$332$	− 98.8597i	− 0.297770i
$333$	−144.494	−0.433916
$334$	42.3465i	0.126786i
$335$	− 35.5960i	− 0.106257i
$336$	0	0
$337$	−541.392	−1.60651	−0.803253	−	0.595638i	$-0.796899\pi$
$337$	−541.392	−1.60651	−0.803253	+	0.595638i	$0.796899\pi$
$338$	169.242	0.500715
$339$	129.888i	0.383151i
$340$	−141.982	−0.417594
$341$	− 206.940i	− 0.606863i
$342$	131.983i	0.385914i
$343$	0	0
$344$	−8.58420	−0.0249541
$345$	−91.7441	−0.265925
$346$	150.622i	0.435324i
$347$	428.684	1.23540	0.617700	−	0.786414i	$-0.288065\pi$
$347$	428.684	1.23540	0.617700	+	0.786414i	$0.288065\pi$
$348$	31.8584i	0.0915472i
$349$	74.6851i	0.213998i	0.994259	+	0.106999i	$0.0341241\pi$
$349$	74.6851i	0.213998i	−0.994259	+	0.106999i	$0.965876\pi$
$350$	0	0
$351$	36.4946	0.103973
$352$	58.0934	0.165038
$353$	− 365.913i	− 1.03658i	−0.855205	−	0.518290i	$-0.826569\pi$
$353$	− 365.913i	− 1.03658i	0.855205	−	0.518290i	$-0.173431\pi$
$354$	−269.002	−0.759892
$355$	− 119.292i	− 0.336033i
$356$	− 329.799i	− 0.926403i
$357$	0	0
$358$	−339.373	−0.947968
$359$	−497.586	−1.38603	−0.693017	−	0.720921i	$-0.743719\pi$
$359$	−497.586	−1.38603	−0.693017	+	0.720921i	$0.743719\pi$
$360$	18.9737i	0.0527046i
$361$	−606.746	−1.68074
$362$	437.448i	1.20842i
$363$	− 26.9094i	− 0.0741305i
$364$	0	0
$365$	−161.847	−0.443416
$366$	97.2247	0.265641
$367$	− 139.372i	− 0.379761i	−0.981807	−	0.189880i	$-0.939190\pi$
$367$	− 139.372i	− 0.379761i	0.981807	−	0.189880i	$-0.0608100\pi$
$368$	−94.7529	−0.257481
$369$	− 195.368i	− 0.529452i
$370$	− 152.310i	− 0.411649i
$371$	0	0
$372$	69.8045	0.187647
$373$	347.520	0.931688	0.465844	−	0.884867i	$-0.345751\pi$
$373$	347.520	0.931688	0.465844	+	0.884867i	$0.345751\pi$
$374$	− 461.089i	− 1.23286i
$375$	−19.3649	−0.0516398
$376$	− 175.211i	− 0.465987i
$377$	64.5923i	0.171332i
$378$	0	0
$379$	307.387	0.811048	0.405524	−	0.914084i	$-0.367089\pi$
$379$	307.387	0.811048	0.405524	+	0.914084i	$0.367089\pi$
$380$	−139.122	−0.366110
$381$	222.575i	0.584187i
$382$	4.36936	0.0114381
$383$	− 508.728i	− 1.32827i	−0.747611	−	0.664136i	$-0.768799\pi$
$383$	− 508.728i	− 1.32827i	0.747611	−	0.664136i	$-0.231201\pi$
$384$	19.5959i	0.0510310i
$385$	0	0
$386$	−337.569	−0.874531
$387$	9.10492	0.0235269
$388$	− 98.9597i	− 0.255051i
$389$	−170.968	−0.439506	−0.219753	−	0.975555i	$-0.570525\pi$
$389$	−170.968	−0.439506	−0.219753	+	0.975555i	$0.570525\pi$
$390$	38.4687i	0.0986378i
$391$	752.057i	1.92342i
$392$	0	0
$393$	−130.764	−0.332732
$394$	−412.298	−1.04644
$395$	− 238.020i	− 0.602582i
$396$	−61.6174	−0.155599
$397$	636.498i	1.60327i	0.597815	+	0.801634i	$0.296036\pi$
$397$	636.498i	1.60327i	−0.597815	+	0.801634i	$0.703964\pi$
$398$	− 341.661i	− 0.858446i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	−592.220	−1.47686	−0.738429	−	0.674331i	$-0.764432\pi$
$401$	−592.220	−1.47686	−0.738429	+	0.674331i	$0.764432\pi$
$402$	38.9935i	0.0969988i
$403$	141.527	0.351184
$404$	− 269.745i	− 0.667687i
$405$	− 20.1246i	− 0.0496904i
$406$	0	0
$407$	494.630	1.21531
$408$	155.533	0.381209
$409$	283.730i	0.693717i	0.937918	+	0.346858i	$0.112752\pi$
$409$	283.730i	0.693717i	−0.937918	+	0.346858i	$0.887248\pi$
$410$	205.936	0.502282
$411$	186.224i	0.453100i
$412$	− 74.8041i	− 0.181563i
$413$	0	0
$414$	100.501	0.242755
$415$	−110.528	−0.266334
$416$	39.7303i	0.0955056i
$417$	−471.124	−1.12979
$418$	− 451.801i	− 1.08086i
$419$	− 482.511i	− 1.15158i	−0.817599	−	0.575789i	$-0.804695\pi$
$419$	− 482.511i	− 1.15158i	0.817599	−	0.575789i	$-0.195305\pi$
$420$	0	0
$421$	762.080	1.81017	0.905083	−	0.425234i	$-0.139808\pi$
$421$	762.080	1.81017	0.905083	+	0.425234i	$0.139808\pi$
$422$	−371.964	−0.881431
$423$	185.839i	0.439336i
$424$	3.90748	0.00921575
$425$	158.741i	0.373507i
$426$	130.678i	0.306755i
$427$	0	0
$428$	−49.5920	−0.115869
$429$	−124.928	−0.291207
$430$	9.59742i	0.0223196i
$431$	256.017	0.594006	0.297003	−	0.954877i	$-0.404013\pi$
$431$	256.017	0.594006	0.297003	+	0.954877i	$0.404013\pi$
$432$	− 20.7846i	− 0.0481125i
$433$	− 646.579i	− 1.49325i	−0.665243	−	0.746627i	$-0.731672\pi$
$433$	− 646.579i	− 1.49325i	0.665243	−	0.746627i	$-0.268328\pi$
$434$	0	0
$435$	35.6188	0.0818823
$436$	−112.579	−0.258209
$437$	736.908i	1.68629i
$438$	177.294	0.404782
$439$	335.270i	0.763712i	0.924222	+	0.381856i	$0.124715\pi$
$439$	335.270i	0.763712i	−0.924222	+	0.381856i	$0.875285\pi$
$440$	− 64.9504i	− 0.147615i
$441$	0	0
$442$	315.341	0.713441
$443$	−281.054	−0.634433	−0.317216	−	0.948353i	$-0.602748\pi$
$443$	−281.054	−0.634433	−0.317216	+	0.948353i	$0.602748\pi$
$444$	166.847i	0.375782i
$445$	−368.727	−0.828600
$446$	159.323i	0.357225i
$447$	− 144.500i	− 0.323265i
$448$	0	0
$449$	47.2320	0.105194	0.0525969	−	0.998616i	$-0.483250\pi$
$449$	47.2320	0.105194	0.0525969	+	0.998616i	$0.483250\pi$
$450$	21.2132	0.0471405
$451$	668.781i	1.48288i
$452$	149.982	0.331818
$453$	219.615i	0.484801i
$454$	− 69.3253i	− 0.152699i
$455$	0	0
$456$	152.400	0.334212
$457$	588.793	1.28839	0.644193	−	0.764863i	$-0.277193\pi$
$457$	588.793	1.28839	0.644193	+	0.764863i	$0.277193\pi$
$458$	39.7594i	0.0868109i
$459$	−164.968	−0.359407
$460$	105.937i	0.230298i
$461$	− 60.5606i	− 0.131368i	−0.997840	−	0.0656839i	$-0.979077\pi$
$461$	− 60.5606i	− 0.131368i	0.997840	−	0.0656839i	$-0.0209229\pi$
$462$	0	0
$463$	88.7592	0.191704	0.0958522	−	0.995396i	$-0.469442\pi$
$463$	88.7592	0.191704	0.0958522	+	0.995396i	$0.469442\pi$
$464$	36.7869	0.0792822
$465$	− 78.0438i	− 0.167836i
$466$	249.188	0.534739
$467$	302.224i	0.647160i	0.946201	+	0.323580i	$0.104886\pi$
$467$	302.224i	0.647160i	−0.946201	+	0.323580i	$0.895114\pi$
$468$	− 42.1404i	− 0.0900436i
$469$	0	0
$470$	−195.892	−0.416791
$471$	−171.883	−0.364933
$472$	310.616i	0.658085i
$473$	−31.1678	−0.0658939
$474$	260.738i	0.550080i
$475$	155.543i	0.327459i
$476$	0	0
$477$	−4.14451	−0.00868869
$478$	49.0944	0.102708
$479$	− 35.1303i	− 0.0733409i	−0.999327	−	0.0366704i	$-0.988325\pi$
$479$	− 35.1303i	− 0.0733409i	0.999327	−	0.0366704i	$-0.0116752\pi$
$480$	21.9089	0.0456435
$481$	338.280i	0.703284i
$482$	375.378i	0.778793i
$483$	0	0
$484$	−31.0723	−0.0641989
$485$	−110.640	−0.228124
$486$	22.0454i	0.0453609i
$487$	64.6333	0.132717	0.0663586	−	0.997796i	$-0.478862\pi$
$487$	64.6333	0.132717	0.0663586	+	0.997796i	$0.478862\pi$
$488$	− 112.265i	− 0.230052i
$489$	483.207i	0.988153i
$490$	0	0
$491$	241.365	0.491578	0.245789	−	0.969323i	$-0.420953\pi$
$491$	241.365	0.491578	0.245789	+	0.969323i	$0.420953\pi$
$492$	−225.591	−0.458519
$493$	− 291.979i	− 0.592249i
$494$	308.989	0.625484
$495$	68.8903i	0.139172i
$496$	− 80.6033i	− 0.162507i
$497$	0	0
$498$	121.078	0.243128
$499$	−191.625	−0.384017	−0.192009	−	0.981393i	$-0.561500\pi$
$499$	−191.625	−0.384017	−0.192009	+	0.981393i	$0.561500\pi$
$500$	22.3607i	0.0447214i
$501$	−51.8636	−0.103520
$502$	34.1848i	0.0680972i
$503$	− 919.711i	− 1.82845i	−0.405205	−	0.914226i	$-0.632800\pi$
$503$	− 919.711i	− 1.82845i	0.405205	−	0.914226i	$-0.367200\pi$
$504$	0	0
$505$	−301.585	−0.597197
$506$	−344.033	−0.679906
$507$	207.278i	0.408832i
$508$	257.008	0.505921
$509$	289.802i	0.569356i	0.958623	+	0.284678i	$0.0918866\pi$
$509$	289.802i	0.569356i	−0.958623	+	0.284678i	$0.908113\pi$
$510$	− 173.892i	− 0.340964i
$511$	0	0
$512$	22.6274	0.0441942
$513$	−161.645	−0.315098
$514$	− 13.4880i	− 0.0262413i
$515$	−83.6336	−0.162395
$516$	− 10.5135i	− 0.0203749i
$517$	− 636.163i	− 1.23049i
$518$	0	0
$519$	−184.474	−0.355441
$520$	44.4199	0.0854228
$521$	754.223i	1.44765i	0.689986	+	0.723823i	$0.257617\pi$
$521$	754.223i	1.44765i	−0.689986	+	0.723823i	$0.742383\pi$
$522$	−39.0184	−0.0747479
$523$	75.7277i	0.144795i	0.997376	+	0.0723974i	$0.0230650\pi$
$523$	75.7277i	0.144795i	−0.997376	+	0.0723974i	$0.976935\pi$
$524$	150.993i	0.288154i
$525$	0	0
$526$	−186.483	−0.354530
$527$	−639.751	−1.21395
$528$	71.1496i	0.134753i
$529$	32.1324	0.0607418
$530$	− 4.36869i	− 0.00824282i
$531$	− 329.458i	− 0.620449i
$532$	0	0
$533$	−457.382	−0.858128
$534$	403.920	0.756405
$535$	55.4455i	0.103636i
$536$	45.0258	0.0840034
$537$	− 415.645i	− 0.774013i
$538$	53.5689i	0.0995705i
$539$	0	0
$540$	−23.2379	−0.0430331
$541$	986.353	1.82320	0.911602	−	0.411074i	$-0.134846\pi$
$541$	986.353	1.82320	0.911602	+	0.411074i	$0.134846\pi$
$542$	512.435i	0.945452i
$543$	−535.762	−0.986670
$544$	− 179.594i	− 0.330137i
$545$	125.868i	0.230950i
$546$	0	0
$547$	346.700	0.633820	0.316910	−	0.948456i	$-0.397355\pi$
$547$	346.700	0.633820	0.316910	+	0.948456i	$0.397355\pi$
$548$	215.033	0.392396
$549$	119.075i	0.216895i
$550$	−72.6168	−0.132030
$551$	− 286.098i	− 0.519233i
$552$	− 116.048i	− 0.210232i
$553$	0	0
$554$	−160.229	−0.289222
$555$	186.541	0.336110
$556$	544.007i	0.978431i
$557$	−153.449	−0.275492	−0.137746	−	0.990468i	$-0.543986\pi$
$557$	−153.449	−0.275492	−0.137746	+	0.990468i	$0.543986\pi$
$558$	85.4927i	0.153213i
$559$	− 21.3158i	− 0.0381320i
$560$	0	0
$561$	564.716	1.00662
$562$	−252.770	−0.449768
$563$	− 174.139i	− 0.309305i	−0.987969	−	0.154653i	$-0.950574\pi$
$563$	− 174.139i	− 0.309305i	0.987969	−	0.154653i	$-0.0494259\pi$
$564$	214.589	0.380476
$565$	− 167.685i	− 0.296787i
$566$	− 61.0495i	− 0.107861i
$567$	0	0
$568$	150.893	0.265658
$569$	−219.182	−0.385205	−0.192603	−	0.981277i	$-0.561693\pi$
$569$	−219.182	−0.385205	−0.192603	+	0.981277i	$0.561693\pi$
$570$	− 170.389i	− 0.298928i
$571$	957.829	1.67746	0.838729	−	0.544549i	$-0.183299\pi$
$571$	957.829	1.67746	0.838729	+	0.544549i	$0.183299\pi$
$572$	144.254i	0.252193i
$573$	5.35135i	0.00933917i
$574$	0	0
$575$	118.441	0.205985
$576$	−24.0000	−0.0416667
$577$	1037.89i	1.79877i	0.437160	+	0.899384i	$0.355984\pi$
$577$	1037.89i	1.79877i	−0.437160	+	0.899384i	$0.644016\pi$
$578$	−1016.74	−1.75906
$579$	− 413.436i	− 0.714052i
$580$	− 41.1290i	− 0.0709121i
$581$	0	0
$582$	121.200	0.208248
$583$	14.1874	0.0243352
$584$	− 204.722i	− 0.350551i
$585$	−47.1144	−0.0805374
$586$	21.8391i	0.0372681i
$587$	819.162i	1.39551i	0.716339	+	0.697753i	$0.245817\pi$
$587$	819.162i	1.39551i	−0.716339	+	0.697753i	$0.754183\pi$
$588$	0	0
$589$	−626.864	−1.06429
$590$	347.280	0.588609
$591$	− 504.960i	− 0.854416i
$592$	192.659	0.325437
$593$	533.379i	0.899458i	0.893165	+	0.449729i	$0.148479\pi$
$593$	533.379i	0.899458i	−0.893165	+	0.449729i	$0.851521\pi$
$594$	− 75.4655i	− 0.127046i
$595$	0	0
$596$	−166.854	−0.279956
$597$	418.448	0.700918
$598$	− 235.285i	− 0.393454i
$599$	−342.903	−0.572460	−0.286230	−	0.958161i	$-0.592402\pi$
$599$	−342.903	−0.572460	−0.286230	+	0.958161i	$0.592402\pi$
$600$	− 24.4949i	− 0.0408248i
$601$	− 418.941i	− 0.697073i	−0.937295	−	0.348536i	$-0.886679\pi$
$601$	− 418.941i	− 0.697073i	0.937295	−	0.348536i	$-0.113321\pi$
$602$	0	0
$603$	−47.7571	−0.0791992
$604$	253.589	0.419850
$605$	34.7398i	0.0574212i
$606$	330.369	0.545164
$607$	141.575i	0.233237i	0.993177	+	0.116619i	$0.0372055\pi$
$607$	141.575i	0.233237i	−0.993177	+	0.116619i	$0.962794\pi$
$608$	− 175.977i	− 0.289436i
$609$	0	0
$610$	−125.517	−0.205765
$611$	435.074	0.712069
$612$	190.489i	0.311256i
$613$	−297.040	−0.484567	−0.242284	−	0.970205i	$-0.577896\pi$
$613$	−297.040	−0.484567	−0.242284	+	0.970205i	$0.577896\pi$
$614$	− 331.843i	− 0.540461i
$615$	252.219i	0.410112i
$616$	0	0
$617$	−674.329	−1.09292	−0.546458	−	0.837486i	$-0.684024\pi$
$617$	−674.329	−1.09292	−0.546458	+	0.837486i	$0.684024\pi$
$618$	91.6160	0.148246
$619$	961.930i	1.55401i	0.629497	+	0.777003i	$0.283261\pi$
$619$	961.930i	1.55401i	−0.629497	+	0.777003i	$0.716739\pi$
$620$	−90.1172	−0.145350
$621$	123.088i	0.198209i
$622$	563.957i	0.906683i
$623$	0	0
$624$	−48.6595	−0.0779800
$625$	25.0000	0.0400000
$626$	182.717i	0.291880i
$627$	553.341	0.882522
$628$	198.474i	0.316041i
$629$	− 1529.14i	− 2.43106i
$630$	0	0
$631$	1185.17	1.87824	0.939122	−	0.343584i	$-0.111641\pi$
$631$	1185.17	1.87824	0.939122	+	0.343584i	$0.111641\pi$
$632$	301.074	0.476383
$633$	− 455.561i	− 0.719686i
$634$	−569.314	−0.897971
$635$	− 287.343i	− 0.452509i
$636$	4.78566i	0.00752463i
$637$	0	0
$638$	133.567	0.209353
$639$	−160.047	−0.250464
$640$	− 25.2982i	− 0.0395285i
$641$	407.098	0.635098	0.317549	−	0.948242i	$-0.397140\pi$
$641$	407.098	0.635098	0.317549	+	0.948242i	$0.397140\pi$
$642$	− 60.7375i	− 0.0946067i
$643$	1077.83i	1.67626i	0.545474	+	0.838128i	$0.316350\pi$
$643$	1077.83i	1.67626i	−0.545474	+	0.838128i	$0.683650\pi$
$644$	0	0
$645$	−11.7544	−0.0182239
$646$	−1396.73	−2.16213
$647$	− 1176.70i	− 1.81870i	−0.416032	−	0.909350i	$-0.636580\pi$
$647$	− 1176.70i	− 1.81870i	0.416032	−	0.909350i	$-0.363420\pi$
$648$	25.4558	0.0392837
$649$	1127.80i	1.73775i
$650$	− 49.6629i	− 0.0764045i
$651$	0	0
$652$	557.959	0.855765
$653$	−879.126	−1.34629	−0.673144	−	0.739512i	$-0.735056\pi$
$653$	−879.126	−1.34629	−0.673144	+	0.739512i	$0.735056\pi$
$654$	− 137.881i	− 0.210827i
$655$	168.815	0.257733
$656$	260.490i	0.397089i
$657$	217.140i	0.330503i
$658$	0	0
$659$	65.1550	0.0988696	0.0494348	−	0.998777i	$-0.484258\pi$
$659$	65.1550	0.0988696	0.0494348	+	0.998777i	$0.484258\pi$
$660$	79.5477	0.120527
$661$	− 25.4468i	− 0.0384974i	−0.999815	−	0.0192487i	$-0.993873\pi$
$661$	− 25.4468i	− 0.0384974i	0.999815	−	0.0192487i	$-0.00612743\pi$
$662$	−236.022	−0.356528
$663$	386.212i	0.582522i
$664$	− 139.809i	− 0.210555i
$665$	0	0
$666$	−204.345	−0.306825
$667$	−217.854	−0.326618
$668$	59.8869i	0.0896511i
$669$	−195.129	−0.291673
$670$	− 50.3404i	− 0.0751349i
$671$	− 407.618i	− 0.607478i
$672$	0	0
$673$	23.1893	0.0344566	0.0172283	−	0.999852i	$-0.494516\pi$
$673$	23.1893	0.0344566	0.0172283	+	0.999852i	$0.494516\pi$
$674$	−765.645	−1.13597
$675$	25.9808i	0.0384900i
$676$	239.344	0.354059
$677$	− 142.132i	− 0.209943i	−0.994475	−	0.104972i	$-0.966525\pi$
$677$	− 142.132i	− 0.209943i	0.994475	−	0.104972i	$-0.0334752\pi$
$678$	183.690i	0.270929i
$679$	0	0
$680$	−200.793	−0.295283
$681$	84.9058	0.124678
$682$	− 292.658i	− 0.429117i
$683$	656.704	0.961499	0.480749	−	0.876858i	$-0.340365\pi$
$683$	656.704	0.961499	0.480749	+	0.876858i	$0.340365\pi$
$684$	186.652i	0.272883i
$685$	− 240.414i	− 0.350970i
$686$	0	0
$687$	−48.6951	−0.0708808
$688$	−12.1399	−0.0176452
$689$	9.70284i	0.0140825i
$690$	−129.746	−0.188037
$691$	245.201i	0.354849i	0.984134	+	0.177425i	$0.0567766\pi$
$691$	245.201i	0.354849i	−0.984134	+	0.177425i	$0.943223\pi$
$692$	213.012i	0.307821i
$693$	0	0
$694$	606.250	0.873560
$695$	608.219	0.875135
$696$	45.0546i	0.0647336i
$697$	2067.52	2.96631
$698$	105.621i	0.151319i
$699$	305.192i	0.436612i
$700$	0	0
$701$	379.419	0.541254	0.270627	−	0.962684i	$-0.412769\pi$
$701$	379.419	0.541254	0.270627	+	0.962684i	$0.412769\pi$
$702$	51.6112	0.0735203
$703$	− 1498.34i	− 2.13135i
$704$	82.1565	0.116700
$705$	− 239.918i	− 0.340309i
$706$	− 517.479i	− 0.732973i
$707$	0	0
$708$	−380.426	−0.537324
$709$	−884.109	−1.24698	−0.623490	−	0.781831i	$-0.714286\pi$
$709$	−884.109	−1.24698	−0.623490	+	0.781831i	$0.714286\pi$
$710$	− 168.704i	− 0.237611i
$711$	−319.337	−0.449138
$712$	− 466.407i	− 0.655066i
$713$	477.337i	0.669477i
$714$	0	0
$715$	161.281	0.225568
$716$	−479.945	−0.670315
$717$	60.1282i	0.0838608i
$718$	−703.693	−0.980074
$719$	− 953.588i	− 1.32627i	−0.748500	−	0.663135i	$-0.769226\pi$
$719$	− 953.588i	− 1.32627i	0.748500	−	0.663135i	$-0.230774\pi$
$720$	26.8328i	0.0372678i
$721$	0	0
$722$	−858.068	−1.18846
$723$	−459.743	−0.635882
$724$	618.645i	0.854482i
$725$	−45.9837	−0.0634257
$726$	− 38.0556i	− 0.0524182i
$727$	1110.82i	1.52795i	0.645248	+	0.763974i	$0.276754\pi$
$727$	1110.82i	1.52795i	−0.645248	+	0.763974i	$0.723246\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−228.886	−0.313543
$731$	96.3546i	0.131812i
$732$	137.497	0.187837
$733$	40.7141i	0.0555445i	0.999614	+	0.0277722i	$0.00884132\pi$
$733$	40.7141i	0.0555445i	−0.999614	+	0.0277722i	$0.991159\pi$
$734$	− 197.102i	− 0.268531i
$735$	0	0
$736$	−134.001	−0.182066
$737$	163.481	0.221820
$738$	− 276.292i	− 0.374379i
$739$	845.471	1.14407	0.572037	−	0.820228i	$-0.306153\pi$
$739$	845.471	1.14407	0.572037	+	0.820228i	$0.306153\pi$
$740$	− 215.399i	− 0.291080i
$741$	378.433i	0.510705i
$742$	0	0
$743$	355.319	0.478222	0.239111	−	0.970992i	$-0.423144\pi$
$743$	355.319	0.478222	0.239111	+	0.970992i	$0.423144\pi$
$744$	98.7185	0.132686
$745$	186.548i	0.250400i
$746$	491.467	0.658803
$747$	148.289i	0.198513i
$748$	− 652.078i	− 0.871762i
$749$	0	0
$750$	−27.3861	−0.0365148
$751$	217.537	0.289663	0.144831	−	0.989456i	$-0.453736\pi$
$751$	217.537	0.289663	0.144831	+	0.989456i	$0.453736\pi$
$752$	− 247.786i	− 0.329502i
$753$	−41.8676	−0.0556011
$754$	91.3473i	0.121150i
$755$	− 283.521i	− 0.375525i
$756$	0	0
$757$	1178.25	1.55647	0.778233	−	0.627975i	$-0.216116\pi$
$757$	1178.25	1.55647	0.778233	+	0.627975i	$0.216116\pi$
$758$	434.711	0.573497
$759$	− 421.352i	− 0.555141i
$760$	−196.748	−0.258879
$761$	821.727i	1.07980i	0.841730	+	0.539899i	$0.181538\pi$
$761$	821.727i	1.07980i	−0.841730	+	0.539899i	$0.818462\pi$
$762$	314.769i	0.413082i
$763$	0	0
$764$	6.17920	0.00808796
$765$	212.973	0.278396
$766$	− 719.451i	− 0.939231i
$767$	−771.306	−1.00561
$768$	27.7128i	0.0360844i
$769$	− 230.888i	− 0.300244i	−0.988667	−	0.150122i	$-0.952033\pi$
$769$	− 230.888i	− 0.300244i	0.988667	−	0.150122i	$-0.0479667\pi$
$770$	0	0
$771$	16.5194	0.0214259
$772$	−477.395	−0.618387
$773$	− 674.468i	− 0.872533i	−0.899817	−	0.436267i	$-0.856300\pi$
$773$	− 674.468i	− 0.872533i	0.899817	−	0.436267i	$-0.143700\pi$
$774$	12.8763	0.0166360
$775$	100.754i	0.130005i
$776$	− 139.950i	− 0.180348i
$777$	0	0
$778$	−241.785	−0.310778
$779$	2025.87	2.60061
$780$	54.4030i	0.0697474i
$781$	547.870	0.701498
$782$	1063.57i	1.36006i
$783$	− 47.7876i	− 0.0610314i
$784$	0	0
$785$	221.900	0.282676
$786$	−184.928	−0.235277
$787$	− 622.890i	− 0.791473i	−0.918364	−	0.395737i	$-0.870489\pi$
$787$	− 622.890i	− 0.791473i	0.918364	−	0.395737i	$-0.129511\pi$
$788$	−583.077	−0.739946
$789$	− 228.394i	− 0.289472i
$790$	− 336.611i	− 0.426090i
$791$	0	0
$792$	−87.1401	−0.110025
$793$	278.772	0.351540
$794$	900.144i	1.13368i
$795$	5.35053	0.00673023
$796$	− 483.182i	− 0.607013i
$797$	− 1322.28i	− 1.65907i	−0.558452	−	0.829537i	$-0.688604\pi$
$797$	− 1322.28i	− 1.65907i	0.558452	−	0.829537i	$-0.311396\pi$
$798$	0	0
$799$	−1966.68	−2.46143
$800$	−28.2843	−0.0353553
$801$	494.699i	0.617602i
$802$	−837.526	−1.04430
$803$	− 743.312i	− 0.925669i
$804$	55.1451i	0.0685885i
$805$	0	0
$806$	200.150	0.248325
$807$	−65.6083	−0.0812990
$808$	− 381.478i	− 0.472126i
$809$	1042.21	1.28827	0.644135	−	0.764912i	$-0.277218\pi$
$809$	1042.21	1.28827	0.644135	+	0.764912i	$0.277218\pi$
$810$	− 28.4605i	− 0.0351364i
$811$	782.292i	0.964602i	0.876006	+	0.482301i	$0.160199\pi$
$811$	782.292i	0.964602i	−0.876006	+	0.482301i	$0.839801\pi$
$812$	0	0
$813$	−627.602	−0.771958
$814$	699.513	0.859352
$815$	− 623.817i	− 0.765420i
$816$	219.957	0.269556
$817$	94.4138i	0.115562i
$818$	401.255i	0.490532i
$819$	0	0
$820$	291.237	0.355167
$821$	−1011.61	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$821$	−1011.61	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$822$	263.361i	0.320390i
$823$	−1114.41	−1.35408	−0.677040	−	0.735946i	$-0.736738\pi$
$823$	−1114.41	−1.35408	−0.677040	+	0.735946i	$0.736738\pi$
$824$	− 105.789i	− 0.128385i
$825$	− 88.9370i	− 0.107802i
$826$	0	0
$827$	−1267.52	−1.53267	−0.766336	−	0.642440i	$-0.777922\pi$
$827$	−1267.52	−1.53267	−0.766336	+	0.642440i	$0.777922\pi$
$828$	142.129	0.171654
$829$	208.464i	0.251464i	0.992064	+	0.125732i	$0.0401280\pi$
$829$	208.464i	0.251464i	−0.992064	+	0.125732i	$0.959872\pi$
$830$	−156.311	−0.188326
$831$	− 196.239i	− 0.236149i
$832$	56.1872i	0.0675327i
$833$	0	0
$834$	−666.270	−0.798885
$835$	66.9556	0.0801864
$836$	− 638.944i	− 0.764287i
$837$	−104.707	−0.125098
$838$	− 682.373i	− 0.814288i
$839$	389.239i	0.463932i	0.972724	+	0.231966i	$0.0745159\pi$
$839$	389.239i	0.463932i	−0.972724	+	0.231966i	$0.925484\pi$
$840$	0	0
$841$	−756.420	−0.899429
$842$	1077.74	1.27998
$843$	− 309.579i	− 0.367234i
$844$	−526.037	−0.623266
$845$	− 267.594i	− 0.316680i
$846$	262.816i	0.310658i
$847$	0	0
$848$	5.52601	0.00651652
$849$	74.7701	0.0880684
$850$	224.493i	0.264109i
$851$	−1140.94	−1.34070
$852$	184.806i	0.216908i
$853$	1239.21i	1.45277i	0.687287	+	0.726386i	$0.258801\pi$
$853$	1239.21i	1.45277i	−0.687287	+	0.726386i	$0.741199\pi$
$854$	0	0
$855$	208.683	0.244074
$856$	−70.1336	−0.0819318
$857$	− 182.267i	− 0.212680i	−0.994330	−	0.106340i	$-0.966087\pi$
$857$	− 182.267i	− 0.212680i	0.994330	−	0.106340i	$-0.0339132\pi$
$858$	−176.675	−0.205915
$859$	− 423.766i	− 0.493325i	−0.969101	−	0.246662i	$-0.920666\pi$
$859$	− 423.766i	− 0.493325i	0.969101	−	0.246662i	$-0.0793339\pi$
$860$	13.5728i	0.0157823i
$861$	0	0
$862$	362.062	0.420026
$863$	−17.2630	−0.0200035	−0.0100018	−	0.999950i	$-0.503184\pi$
$863$	−17.2630	−0.0200035	−0.0100018	+	0.999950i	$0.503184\pi$
$864$	− 29.3939i	− 0.0340207i
$865$	238.155	0.275323
$866$	− 914.401i	− 1.05589i
$867$	− 1245.24i	− 1.43627i
$868$	0	0
$869$	1093.15	1.25794
$870$	50.3726	0.0578995
$871$	111.806i	0.128365i
$872$	−159.211	−0.182582
$873$	148.440i	0.170034i
$874$	1042.15i	1.19239i
$875$	0	0
$876$	250.732	0.286224
$877$	−367.336	−0.418855	−0.209428	−	0.977824i	$-0.567160\pi$
$877$	−367.336	−0.418855	−0.209428	+	0.977824i	$0.567160\pi$
$878$	474.143i	0.540026i
$879$	−26.7473	−0.0304292
$880$	− 91.8537i	− 0.104379i
$881$	376.890i	0.427798i	0.976856	+	0.213899i	$0.0686164\pi$
$881$	376.890i	0.427798i	−0.976856	+	0.213899i	$0.931384\pi$
$882$	0	0
$883$	−1101.06	−1.24695	−0.623476	−	0.781842i	$-0.714280\pi$
$883$	−1101.06	−1.24695	−0.623476	+	0.781842i	$0.714280\pi$
$884$	445.959	0.504479
$885$	425.329i	0.480598i
$886$	−397.470	−0.448612
$887$	− 121.375i	− 0.136837i	−0.997657	−	0.0684186i	$-0.978205\pi$
$887$	− 121.375i	− 0.136837i	0.997657	−	0.0684186i	$-0.0217953\pi$
$888$	235.958i	0.265718i
$889$	0	0
$890$	−521.459	−0.585909
$891$	92.4260	0.103733
$892$	225.316i	0.252597i
$893$	−1927.07	−2.15797
$894$	− 204.353i	− 0.228583i
$895$	536.595i	0.599548i
$896$	0	0
$897$	288.165	0.321254
$898$	66.7961	0.0743832
$899$	− 185.322i	− 0.206142i
$900$	30.0000	0.0333333
$901$	− 43.8601i	− 0.0486793i
$902$	945.799i	1.04856i
$903$	0	0
$904$	212.106	0.234631
$905$	691.666	0.764272
$906$	310.582i	0.342806i
$907$	78.1107	0.0861199	0.0430599	−	0.999072i	$-0.486289\pi$
$907$	78.1107	0.0861199	0.0430599	+	0.999072i	$0.486289\pi$
$908$	− 98.0408i	− 0.107974i
$909$	404.618i	0.445124i
$910$	0	0
$911$	863.281	0.947619	0.473809	−	0.880627i	$-0.342879\pi$
$911$	863.281	0.947619	0.473809	+	0.880627i	$0.342879\pi$
$912$	215.527	0.236323
$913$	− 507.623i	− 0.555994i
$914$	832.679	0.911027
$915$	− 153.726i	− 0.168006i
$916$	56.2283i	0.0613846i
$917$	0	0
$918$	−233.300	−0.254139
$919$	1357.85	1.47753	0.738766	−	0.673962i	$-0.235409\pi$
$919$	1357.85	1.47753	0.738766	+	0.673962i	$0.235409\pi$
$920$	149.818i	0.162845i
$921$	406.423	0.441284
$922$	− 85.6456i	− 0.0928911i
$923$	374.691i	0.405949i
$924$	0	0
$925$	−240.823	−0.260350
$926$	125.524	0.135556
$927$	112.206i	0.121042i
$928$	52.0246	0.0560610
$929$	924.823i	0.995504i	0.867320	+	0.497752i	$0.165841\pi$
$929$	924.823i	0.995504i	−0.867320	+	0.497752i	$0.834159\pi$
$930$	− 110.371i	− 0.118678i
$931$	0	0
$932$	352.405	0.378117
$933$	−690.703	−0.740303
$934$	427.409i	0.457611i
$935$	−729.046	−0.779728
$936$	− 59.5955i	− 0.0636704i
$937$	270.668i	0.288867i	0.989515	+	0.144433i	$0.0461359\pi$
$937$	270.668i	0.288867i	−0.989515	+	0.144433i	$0.953864\pi$
$938$	0	0
$939$	−223.782	−0.238319
$940$	−277.033	−0.294716
$941$	456.308i	0.484918i	0.970162	+	0.242459i	$0.0779540\pi$
$941$	456.308i	0.484918i	−0.970162	+	0.242459i	$0.922046\pi$
$942$	−243.080	−0.258046
$943$	− 1542.64i	− 1.63589i
$944$	439.278i	0.465337i
$945$	0	0
$946$	−44.0780	−0.0465940
$947$	−993.573	−1.04918	−0.524590	−	0.851355i	$-0.675781\pi$
$947$	−993.573	−1.04918	−0.524590	+	0.851355i	$0.675781\pi$
$948$	368.739i	0.388965i
$949$	508.354	0.535674
$950$	219.971i	0.231549i
$951$	− 697.264i	− 0.733190i
$952$	0	0
$953$	−206.385	−0.216563	−0.108281	−	0.994120i	$-0.534535\pi$
$953$	−206.385	−0.216563	−0.108281	+	0.994120i	$0.534535\pi$
$954$	−5.86122	−0.00614383
$955$	− 6.90856i	− 0.00723409i
$956$	69.4300	0.0726255
$957$	163.586i	0.170936i
$958$	− 49.6817i	− 0.0518598i
$959$	0	0
$960$	30.9839	0.0322749
$961$	554.944	0.577465
$962$	478.400i	0.497297i
$963$	74.3879	0.0772460
$964$	530.865i	0.550690i
$965$	533.744i	0.553102i
$966$	0	0
$967$	169.282	0.175058	0.0875292	−	0.996162i	$-0.472103\pi$
$967$	169.282	0.175058	0.0875292	+	0.996162i	$0.472103\pi$
$968$	−43.9428	−0.0453955
$969$	− 1710.64i	− 1.76537i
$970$	−156.469	−0.161308
$971$	− 141.902i	− 0.146140i	−0.997327	−	0.0730702i	$-0.976720\pi$
$971$	− 141.902i	− 0.146140i	0.997327	−	0.0730702i	$-0.0232797\pi$
$972$	31.1769i	0.0320750i
$973$	0	0
$974$	91.4053	0.0938453
$975$	60.8244	0.0623840
$976$	− 158.767i	− 0.162671i
$977$	−1407.80	−1.44094	−0.720472	−	0.693485i	$-0.756075\pi$
$977$	−1407.80	−1.44094	−0.720472	+	0.693485i	$0.756075\pi$
$978$	683.357i	0.698730i
$979$	− 1693.45i	− 1.72977i
$980$	0	0
$981$	168.869	0.172140
$982$	341.341	0.347598
$983$	− 897.780i	− 0.913306i	−0.889645	−	0.456653i	$-0.849048\pi$
$983$	− 897.780i	− 0.913306i	0.889645	−	0.456653i	$-0.150952\pi$
$984$	−319.034	−0.324222
$985$	651.900i	0.661828i
$986$	− 412.920i	− 0.418783i
$987$	0	0
$988$	436.976	0.442284
$989$	71.8931	0.0726927
$990$	97.4256i	0.0984097i
$991$	1524.28	1.53812	0.769061	−	0.639175i	$-0.220724\pi$
$991$	1524.28	1.53812	0.769061	+	0.639175i	$0.220724\pi$
$992$	− 113.990i	− 0.114910i
$993$	− 289.066i	− 0.291104i
$994$	0	0
$995$	−540.214	−0.542929
$996$	171.230	0.171918
$997$	− 889.869i	− 0.892547i	−0.894897	−	0.446273i	$-0.852751\pi$
$997$	− 889.869i	− 0.892547i	0.894897	−	0.446273i	$-0.147249\pi$
$998$	−270.998	−0.271541
$999$	− 250.271i	− 0.250522i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.d.391.14		16
7.2	even	3		210.3.o.b.31.1	✓	16
7.3	odd	6		210.3.o.b.61.1	yes	16
7.6	odd	2	inner	1470.3.f.d.391.12		16
21.2	odd	6		630.3.v.c.451.7		16
21.17	even	6		630.3.v.c.271.7		16
35.2	odd	12		1050.3.q.e.199.14		32
35.3	even	12		1050.3.q.e.649.13		32
35.9	even	6		1050.3.p.i.451.7		16
35.17	even	12		1050.3.q.e.649.3		32
35.23	odd	12		1050.3.q.e.199.3		32
35.24	odd	6		1050.3.p.i.901.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.b.31.1	✓	16	7.2	even	3
210.3.o.b.61.1	yes	16	7.3	odd	6
630.3.v.c.271.7		16	21.17	even	6
630.3.v.c.451.7		16	21.2	odd	6
1050.3.p.i.451.7		16	35.9	even	6
1050.3.p.i.901.7		16	35.24	odd	6
1050.3.q.e.199.3		32	35.23	odd	12
1050.3.q.e.199.14		32	35.2	odd	12
1050.3.q.e.649.3		32	35.17	even	12
1050.3.q.e.649.13		32	35.3	even	12
1470.3.f.d.391.12		16	7.6	odd	2	inner
1470.3.f.d.391.14		16	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} +10.2696 q^{11} +3.46410i q^{12} +7.02340i q^{13} +3.87298 q^{15} +4.00000 q^{16} -31.7481i q^{17} -4.24264 q^{18} -31.1086i q^{19} -4.47214i q^{20} +14.5234 q^{22} -23.6882 q^{23} +4.89898i q^{24} -5.00000 q^{25} +9.93258i q^{26} -5.19615i q^{27} +9.19673 q^{29} +5.47723 q^{30} -20.1508i q^{31} +5.65685 q^{32} +17.7874i q^{33} -44.8986i q^{34} -6.00000 q^{36} +48.1647 q^{37} -43.9942i q^{38} -12.1649 q^{39} -6.32456i q^{40} +65.1226i q^{41} -3.03497 q^{43} +20.5391 q^{44} +6.70820i q^{45} -33.5002 q^{46} -61.9464i q^{47} +6.92820i q^{48} -7.07107 q^{50} +54.9893 q^{51} +14.0468i q^{52} +1.38150 q^{53} -7.34847i q^{54} -22.9634i q^{55} +53.8817 q^{57} +13.0061 q^{58} +109.819i q^{59} +7.74597 q^{60} -39.6918i q^{61} -28.4976i q^{62} +8.00000 q^{64} +15.7048 q^{65} +25.1552i q^{66} +15.9190 q^{67} -63.4962i q^{68} -41.0292i q^{69} +53.3489 q^{71} -8.48528 q^{72} -72.3801i q^{73} +68.1151 q^{74} -8.66025i q^{75} -62.2172i q^{76} -17.2037 q^{78} +106.446 q^{79} -8.94427i q^{80} +9.00000 q^{81} +92.0973i q^{82} -49.4298i q^{83} -70.9909 q^{85} -4.29210 q^{86} +15.9292i q^{87} +29.0467 q^{88} -164.900i q^{89} +9.48683i q^{90} -47.3765 q^{92} +34.9023 q^{93} -87.6055i q^{94} -69.5610 q^{95} +9.79796i q^{96} -49.4799i q^{97} -30.8087 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 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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