LMFDB - Embedded newform 1470.3.f.d.391.11

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			391.11
Root			$2.96377 + 5.13339i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.d.391.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -17.3852 q^{11} -3.46410i q^{12} +7.22559i q^{13} +3.87298 q^{15} +4.00000 q^{16} -2.65199i q^{17} -4.24264 q^{18} -2.54182i q^{19} +4.47214i q^{20} -24.5864 q^{22} +40.1015 q^{23} -4.89898i q^{24} -5.00000 q^{25} +10.2185i q^{26} +5.19615i q^{27} +47.0080 q^{29} +5.47723 q^{30} +40.3019i q^{31} +5.65685 q^{32} +30.1120i q^{33} -3.75049i q^{34} -6.00000 q^{36} +32.5027 q^{37} -3.59468i q^{38} +12.5151 q^{39} +6.32456i q^{40} +70.6679i q^{41} +37.3732 q^{43} -34.7704 q^{44} -6.70820i q^{45} +56.7120 q^{46} +33.4485i q^{47} -6.92820i q^{48} -7.07107 q^{50} -4.59339 q^{51} +14.4512i q^{52} -70.8884 q^{53} +7.34847i q^{54} -38.8745i q^{55} -4.40256 q^{57} +66.4793 q^{58} +100.559i q^{59} +7.74597 q^{60} +12.8889i q^{61} +56.9955i q^{62} +8.00000 q^{64} -16.1569 q^{65} +42.5848i q^{66} +94.0722 q^{67} -5.30399i q^{68} -69.4578i q^{69} -11.5793 q^{71} -8.48528 q^{72} -22.6690i q^{73} +45.9658 q^{74} +8.66025i q^{75} -5.08364i q^{76} +17.6990 q^{78} -24.1084 q^{79} +8.94427i q^{80} +9.00000 q^{81} +99.9396i q^{82} -111.664i q^{83} +5.93004 q^{85} +52.8537 q^{86} -81.4202i q^{87} -49.1727 q^{88} +127.907i q^{89} -9.48683i q^{90} +80.2029 q^{92} +69.8050 q^{93} +47.3033i q^{94} +5.68368 q^{95} -9.79796i q^{96} +7.48256i q^{97} +52.1556 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−17.3852	−1.58047	−0.790236	−	0.612803i	$-0.790042\pi$
$11$	−17.3852	−1.58047	−0.790236	+	0.612803i	$0.790042\pi$
$12$	− 3.46410i	− 0.288675i
$13$	7.22559i	0.555815i	0.960608	+	0.277907i	$0.0896408\pi$
$13$	7.22559i	0.555815i	−0.960608	+	0.277907i	$0.910359\pi$
$14$	0	0
$15$	3.87298	0.258199
$16$	4.00000	0.250000
$17$	− 2.65199i	− 0.156000i	−0.996953	−	0.0779998i	$-0.975147\pi$
$17$	− 2.65199i	− 0.156000i	0.996953	−	0.0779998i	$-0.0248534\pi$
$18$	−4.24264	−0.235702
$19$	− 2.54182i	− 0.133780i	−0.997760	−	0.0668900i	$-0.978692\pi$
$19$	− 2.54182i	− 0.133780i	0.997760	−	0.0668900i	$-0.0213077\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	−24.5864	−1.11756
$23$	40.1015	1.74354	0.871771	−	0.489914i	$-0.162972\pi$
$23$	40.1015	1.74354	0.871771	+	0.489914i	$0.162972\pi$
$24$	− 4.89898i	− 0.204124i
$25$	−5.00000	−0.200000
$26$	10.2185i	0.393020i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	47.0080	1.62096	0.810482	−	0.585763i	$-0.199205\pi$
$29$	47.0080	1.62096	0.810482	+	0.585763i	$0.199205\pi$
$30$	5.47723	0.182574
$31$	40.3019i	1.30006i	0.759907	+	0.650031i	$0.225244\pi$
$31$	40.3019i	1.30006i	−0.759907	+	0.650031i	$0.774756\pi$
$32$	5.65685	0.176777
$33$	30.1120i	0.912486i
$34$	− 3.75049i	− 0.110308i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	32.5027	0.878453	0.439226	−	0.898376i	$-0.355253\pi$
$37$	32.5027	0.878453	0.439226	+	0.898376i	$0.355253\pi$
$38$	− 3.59468i	− 0.0945968i
$39$	12.5151	0.320900
$40$	6.32456i	0.158114i
$41$	70.6679i	1.72361i	0.507241	+	0.861804i	$0.330665\pi$
$41$	70.6679i	1.72361i	−0.507241	+	0.861804i	$0.669335\pi$
$42$	0	0
$43$	37.3732	0.869144	0.434572	−	0.900637i	$-0.356900\pi$
$43$	37.3732	0.869144	0.434572	+	0.900637i	$0.356900\pi$
$44$	−34.7704	−0.790236
$45$	− 6.70820i	− 0.149071i
$46$	56.7120	1.23287
$47$	33.4485i	0.711669i	0.934549	+	0.355835i	$0.115803\pi$
$47$	33.4485i	0.711669i	−0.934549	+	0.355835i	$0.884197\pi$
$48$	− 6.92820i	− 0.144338i
$49$	0	0
$50$	−7.07107	−0.141421
$51$	−4.59339	−0.0900665
$52$	14.4512i	0.277907i
$53$	−70.8884	−1.33752	−0.668758	−	0.743480i	$-0.733174\pi$
$53$	−70.8884	−1.33752	−0.668758	+	0.743480i	$0.733174\pi$
$54$	7.34847i	0.136083i
$55$	− 38.8745i	− 0.706808i
$56$	0	0
$57$	−4.40256	−0.0772379
$58$	66.4793	1.14620
$59$	100.559i	1.70438i	0.523229	+	0.852192i	$0.324727\pi$
$59$	100.559i	1.70438i	−0.523229	+	0.852192i	$0.675273\pi$
$60$	7.74597	0.129099
$61$	12.8889i	0.211294i	0.994404	+	0.105647i	$0.0336913\pi$
$61$	12.8889i	0.211294i	−0.994404	+	0.105647i	$0.966309\pi$
$62$	56.9955i	0.919283i
$63$	0	0
$64$	8.00000	0.125000
$65$	−16.1569	−0.248568
$66$	42.5848i	0.645225i
$67$	94.0722	1.40406	0.702031	−	0.712146i	$-0.252277\pi$
$67$	94.0722	1.40406	0.702031	+	0.712146i	$0.252277\pi$
$68$	− 5.30399i	− 0.0779998i
$69$	− 69.4578i	− 1.00663i
$70$	0	0
$71$	−11.5793	−0.163088	−0.0815440	−	0.996670i	$-0.525985\pi$
$71$	−11.5793	−0.163088	−0.0815440	+	0.996670i	$0.525985\pi$
$72$	−8.48528	−0.117851
$73$	− 22.6690i	− 0.310535i	−0.987872	−	0.155267i	$-0.950376\pi$
$73$	− 22.6690i	− 0.310535i	0.987872	−	0.155267i	$-0.0496239\pi$
$74$	45.9658	0.621160
$75$	8.66025i	0.115470i
$76$	− 5.08364i	− 0.0668900i
$77$	0	0
$78$	17.6990	0.226910
$79$	−24.1084	−0.305170	−0.152585	−	0.988290i	$-0.548760\pi$
$79$	−24.1084	−0.305170	−0.152585	+	0.988290i	$0.548760\pi$
$80$	8.94427i	0.111803i
$81$	9.00000	0.111111
$82$	99.9396i	1.21878i
$83$	− 111.664i	− 1.34535i	−0.739937	−	0.672676i	$-0.765145\pi$
$83$	− 111.664i	− 1.34535i	0.739937	−	0.672676i	$-0.234855\pi$
$84$	0	0
$85$	5.93004	0.0697652
$86$	52.8537	0.614578
$87$	− 81.4202i	− 0.935865i
$88$	−49.1727	−0.558781
$89$	127.907i	1.43715i	0.695448	+	0.718577i	$0.255206\pi$
$89$	127.907i	1.43715i	−0.695448	+	0.718577i	$0.744794\pi$
$90$	− 9.48683i	− 0.105409i
$91$	0	0
$92$	80.2029	0.871771
$93$	69.8050	0.750591
$94$	47.3033i	0.503226i
$95$	5.68368	0.0598282
$96$	− 9.79796i	− 0.102062i
$97$	7.48256i	0.0771398i	0.999256	+	0.0385699i	$0.0122802\pi$
$97$	7.48256i	0.0771398i	−0.999256	+	0.0385699i	$0.987720\pi$
$98$	0	0
$99$	52.1556	0.526824
$100$	−10.0000	−0.100000
$101$	− 94.5963i	− 0.936597i	−0.883570	−	0.468299i	$-0.844867\pi$
$101$	− 94.5963i	− 0.936597i	0.883570	−	0.468299i	$-0.155133\pi$
$102$	−6.49603	−0.0636866
$103$	59.8217i	0.580794i	0.956906	+	0.290397i	$0.0937873\pi$
$103$	59.8217i	0.580794i	−0.956906	+	0.290397i	$0.906213\pi$
$104$	20.4371i	0.196510i
$105$	0	0
$106$	−100.251	−0.945767
$107$	6.46161	0.0603889	0.0301945	−	0.999544i	$-0.490387\pi$
$107$	6.46161	0.0603889	0.0301945	+	0.999544i	$0.490387\pi$
$108$	10.3923i	0.0962250i
$109$	−163.840	−1.50312	−0.751560	−	0.659664i	$-0.770698\pi$
$109$	−163.840	−1.50312	−0.751560	+	0.659664i	$0.770698\pi$
$110$	− 54.9768i	− 0.499789i
$111$	− 56.2964i	− 0.507175i
$112$	0	0
$113$	−105.434	−0.933040	−0.466520	−	0.884511i	$-0.654492\pi$
$113$	−105.434	−0.933040	−0.466520	+	0.884511i	$0.654492\pi$
$114$	−6.22616	−0.0546155
$115$	89.6696i	0.779735i
$116$	94.0160	0.810482
$117$	− 21.6768i	− 0.185272i
$118$	142.211i	1.20518i
$119$	0	0
$120$	10.9545	0.0912871
$121$	181.245	1.49789
$122$	18.2277i	0.149407i
$123$	122.400	0.995126
$124$	80.6039i	0.650031i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	53.7033	0.422860	0.211430	−	0.977393i	$-0.432188\pi$
$127$	53.7033	0.422860	0.211430	+	0.977393i	$0.432188\pi$
$128$	11.3137	0.0883883
$129$	− 64.7323i	− 0.501801i
$130$	−22.8493	−0.175764
$131$	58.4843i	0.446445i	0.974768	+	0.223222i	$0.0716576\pi$
$131$	58.4843i	0.446445i	−0.974768	+	0.223222i	$0.928342\pi$
$132$	60.2240i	0.456243i
$133$	0	0
$134$	133.038	0.992822
$135$	−11.6190	−0.0860663
$136$	− 7.50097i	− 0.0551542i
$137$	−12.1170	−0.0884450	−0.0442225	−	0.999022i	$-0.514081\pi$
$137$	−12.1170	−0.0884450	−0.0442225	+	0.999022i	$0.514081\pi$
$138$	− 98.2281i	− 0.711798i
$139$	− 45.2562i	− 0.325584i	−0.986660	−	0.162792i	$-0.947950\pi$
$139$	− 45.2562i	− 0.325584i	0.986660	−	0.162792i	$-0.0520500\pi$
$140$	0	0
$141$	57.9344	0.410882
$142$	−16.3755	−0.115321
$143$	− 125.618i	− 0.878449i
$144$	−12.0000	−0.0833333
$145$	105.113i	0.724918i
$146$	− 32.0589i	− 0.219581i
$147$	0	0
$148$	65.0055	0.439226
$149$	−50.8948	−0.341576	−0.170788	−	0.985308i	$-0.554631\pi$
$149$	−50.8948	−0.341576	−0.170788	+	0.985308i	$0.554631\pi$
$150$	12.2474i	0.0816497i
$151$	−157.695	−1.04434	−0.522169	−	0.852842i	$-0.674877\pi$
$151$	−157.695	−1.04434	−0.522169	+	0.852842i	$0.674877\pi$
$152$	− 7.18935i	− 0.0472984i
$153$	7.95598i	0.0519999i
$154$	0	0
$155$	−90.1179	−0.581406
$156$	25.0302	0.160450
$157$	167.846i	1.06908i	0.845142	+	0.534542i	$0.179516\pi$
$157$	167.846i	1.06908i	−0.845142	+	0.534542i	$0.820484\pi$
$158$	−34.0944	−0.215787
$159$	122.782i	0.772216i
$160$	12.6491i	0.0790569i
$161$	0	0
$162$	12.7279	0.0785674
$163$	276.570	1.69675	0.848374	−	0.529397i	$-0.177582\pi$
$163$	276.570	1.69675	0.848374	+	0.529397i	$0.177582\pi$
$164$	141.336i	0.861804i
$165$	−67.3325	−0.408076
$166$	− 157.917i	− 0.951308i
$167$	48.4258i	0.289975i	0.989433	+	0.144987i	$0.0463142\pi$
$167$	48.4258i	0.289975i	−0.989433	+	0.144987i	$0.953686\pi$
$168$	0	0
$169$	116.791	0.691070
$170$	8.38634	0.0493314
$171$	7.62546i	0.0445933i
$172$	74.7464	0.434572
$173$	112.969i	0.652998i	0.945198	+	0.326499i	$0.105869\pi$
$173$	112.969i	0.652998i	−0.945198	+	0.326499i	$0.894131\pi$
$174$	− 115.146i	− 0.661756i
$175$	0	0
$176$	−69.5407	−0.395118
$177$	174.173	0.984026
$178$	180.887i	1.01622i
$179$	331.416	1.85149	0.925744	−	0.378151i	$-0.123440\pi$
$179$	331.416	1.85149	0.925744	+	0.378151i	$0.123440\pi$
$180$	− 13.4164i	− 0.0745356i
$181$	− 213.328i	− 1.17861i	−0.807912	−	0.589303i	$-0.799402\pi$
$181$	− 213.328i	− 1.17861i	0.807912	−	0.589303i	$-0.200598\pi$
$182$	0	0
$183$	22.3243	0.121991
$184$	113.424	0.616435
$185$	72.6783i	0.392856i
$186$	98.7192	0.530748
$187$	46.1054i	0.246553i
$188$	66.8969i	0.355835i
$189$	0	0
$190$	8.03794	0.0423050
$191$	−66.9033	−0.350279	−0.175140	−	0.984544i	$-0.556038\pi$
$191$	−66.9033	−0.350279	−0.175140	+	0.984544i	$0.556038\pi$
$192$	− 13.8564i	− 0.0721688i
$193$	−112.604	−0.583440	−0.291720	−	0.956504i	$-0.594228\pi$
$193$	−112.604	−0.583440	−0.291720	+	0.956504i	$0.594228\pi$
$194$	10.5819i	0.0545460i
$195$	27.9846i	0.143511i
$196$	0	0
$197$	−61.0211	−0.309752	−0.154876	−	0.987934i	$-0.549498\pi$
$197$	−61.0211	−0.309752	−0.154876	+	0.987934i	$0.549498\pi$
$198$	73.7591	0.372521
$199$	− 191.165i	− 0.960629i	−0.877096	−	0.480314i	$-0.840523\pi$
$199$	− 191.165i	− 0.960629i	0.877096	−	0.480314i	$-0.159477\pi$
$200$	−14.1421	−0.0707107
$201$	− 162.938i	− 0.810636i
$202$	− 133.779i	− 0.662274i
$203$	0	0
$204$	−9.18678	−0.0450332
$205$	−158.018	−0.770821
$206$	84.6007i	0.410683i
$207$	−120.304	−0.581180
$208$	28.9024i	0.138954i
$209$	44.1900i	0.211435i
$210$	0	0
$211$	280.115	1.32756	0.663781	−	0.747927i	$-0.268951\pi$
$211$	280.115	1.32756	0.663781	+	0.747927i	$0.268951\pi$
$212$	−141.777	−0.668758
$213$	20.0559i	0.0941589i
$214$	9.13810	0.0427014
$215$	83.5690i	0.388693i
$216$	14.6969i	0.0680414i
$217$	0	0
$218$	−231.705	−1.06287
$219$	−39.2639	−0.179287
$220$	− 77.7489i	− 0.353404i
$221$	19.1622	0.0867069
$222$	− 79.6151i	− 0.358627i
$223$	− 272.759i	− 1.22313i	−0.791192	−	0.611567i	$-0.790539\pi$
$223$	− 272.759i	− 1.22313i	0.791192	−	0.611567i	$-0.209461\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−149.106	−0.659759
$227$	16.0724i	0.0708035i	0.999373	+	0.0354018i	$0.0112711\pi$
$227$	16.0724i	0.0708035i	−0.999373	+	0.0354018i	$0.988729\pi$
$228$	−8.80512	−0.0386190
$229$	− 148.103i	− 0.646740i	−0.946273	−	0.323370i	$-0.895184\pi$
$229$	− 148.103i	− 0.646740i	0.946273	−	0.323370i	$-0.104816\pi$
$230$	126.812i	0.551356i
$231$	0	0
$232$	132.959	0.573098
$233$	−299.118	−1.28377	−0.641885	−	0.766801i	$-0.721847\pi$
$233$	−299.118	−1.28377	−0.641885	+	0.766801i	$0.721847\pi$
$234$	− 30.6556i	− 0.131007i
$235$	−74.7930	−0.318268
$236$	201.117i	0.852192i
$237$	41.7570i	0.176190i
$238$	0	0
$239$	−114.253	−0.478046	−0.239023	−	0.971014i	$-0.576827\pi$
$239$	−114.253	−0.478046	−0.239023	+	0.971014i	$0.576827\pi$
$240$	15.4919	0.0645497
$241$	− 136.442i	− 0.566149i	−0.959098	−	0.283074i	$-0.908646\pi$
$241$	− 136.442i	− 0.566149i	0.959098	−	0.283074i	$-0.0913544\pi$
$242$	256.319	1.05917
$243$	− 15.5885i	− 0.0641500i
$244$	25.7778i	0.105647i
$245$	0	0
$246$	173.100	0.703660
$247$	18.3662	0.0743569
$248$	113.991i	0.459642i
$249$	−193.408	−0.776740
$250$	− 15.8114i	− 0.0632456i
$251$	− 457.024i	− 1.82081i	−0.413717	−	0.910406i	$-0.635770\pi$
$251$	− 457.024i	− 1.82081i	0.413717	−	0.910406i	$-0.364230\pi$
$252$	0	0
$253$	−697.171	−2.75562
$254$	75.9479	0.299007
$255$	− 10.2711i	− 0.0402789i
$256$	16.0000	0.0625000
$257$	− 49.9981i	− 0.194545i	−0.995258	−	0.0972726i	$-0.968988\pi$
$257$	− 49.9981i	− 0.194545i	0.995258	−	0.0972726i	$-0.0310119\pi$
$258$	− 91.5453i	− 0.354827i
$259$	0	0
$260$	−32.3138	−0.124284
$261$	−141.024	−0.540322
$262$	82.7092i	0.315684i
$263$	182.278	0.693071	0.346535	−	0.938037i	$-0.387358\pi$
$263$	182.278	0.693071	0.346535	+	0.938037i	$0.387358\pi$
$264$	85.1697i	0.322612i
$265$	− 158.511i	− 0.598156i
$266$	0	0
$267$	221.541	0.829741
$268$	188.144	0.702031
$269$	− 37.6217i	− 0.139858i	−0.997552	−	0.0699289i	$-0.977723\pi$
$269$	− 37.6217i	− 0.139858i	0.997552	−	0.0699289i	$-0.0222772\pi$
$270$	−16.4317	−0.0608581
$271$	− 202.737i	− 0.748108i	−0.927407	−	0.374054i	$-0.877968\pi$
$271$	− 202.737i	− 0.748108i	0.927407	−	0.374054i	$-0.122032\pi$
$272$	− 10.6080i	− 0.0389999i
$273$	0	0
$274$	−17.1360	−0.0625401
$275$	86.9259	0.316094
$276$	− 138.916i	− 0.503317i
$277$	256.242	0.925060	0.462530	−	0.886604i	$-0.346942\pi$
$277$	256.242	0.925060	0.462530	+	0.886604i	$0.346942\pi$
$278$	− 64.0020i	− 0.230223i
$279$	− 120.906i	− 0.433354i
$280$	0	0
$281$	−141.462	−0.503423	−0.251711	−	0.967802i	$-0.580993\pi$
$281$	−141.462	−0.503423	−0.251711	+	0.967802i	$0.580993\pi$
$282$	81.9317	0.290538
$283$	543.368i	1.92003i	0.279951	+	0.960014i	$0.409682\pi$
$283$	543.368i	1.92003i	−0.279951	+	0.960014i	$0.590318\pi$
$284$	−23.1585	−0.0815440
$285$	− 9.84443i	− 0.0345419i
$286$	− 177.651i	− 0.621157i
$287$	0	0
$288$	−16.9706	−0.0589256
$289$	281.967	0.975664
$290$	148.652i	0.512594i
$291$	12.9602	0.0445367
$292$	− 45.3381i	− 0.155267i
$293$	375.289i	1.28085i	0.768021	+	0.640424i	$0.221241\pi$
$293$	375.289i	1.28085i	−0.768021	+	0.640424i	$0.778759\pi$
$294$	0	0
$295$	−224.856	−0.762224
$296$	91.9316	0.310580
$297$	− 90.3361i	− 0.304162i
$298$	−71.9762	−0.241531
$299$	289.757i	0.969086i
$300$	17.3205i	0.0577350i
$301$	0	0
$302$	−223.015	−0.738459
$303$	−163.846	−0.540745
$304$	− 10.1673i	− 0.0334450i
$305$	−28.8205	−0.0944935
$306$	11.2515i	0.0367695i
$307$	− 41.3436i	− 0.134670i	−0.997730	−	0.0673349i	$-0.978550\pi$
$307$	− 41.3436i	− 0.134670i	0.997730	−	0.0673349i	$-0.0214496\pi$
$308$	0	0
$309$	103.614	0.335321
$310$	−127.446	−0.411116
$311$	− 146.559i	− 0.471250i	−0.971844	−	0.235625i	$-0.924286\pi$
$311$	− 146.559i	− 0.471250i	0.971844	−	0.235625i	$-0.0757137\pi$
$312$	35.3980	0.113455
$313$	− 224.871i	− 0.718437i	−0.933254	−	0.359218i	$-0.883043\pi$
$313$	− 224.871i	− 0.718437i	0.933254	−	0.359218i	$-0.116957\pi$
$314$	237.370i	0.755957i
$315$	0	0
$316$	−48.2168	−0.152585
$317$	−226.227	−0.713649	−0.356825	−	0.934171i	$-0.616141\pi$
$317$	−226.227	−0.713649	−0.356825	+	0.934171i	$0.616141\pi$
$318$	173.640i	0.546039i
$319$	−817.242	−2.56189
$320$	17.8885i	0.0559017i
$321$	− 11.1918i	− 0.0348656i
$322$	0	0
$323$	−6.74089	−0.0208696
$324$	18.0000	0.0555556
$325$	− 36.1280i	− 0.111163i
$326$	391.129	1.19978
$327$	283.779i	0.867827i
$328$	199.879i	0.609388i
$329$	0	0
$330$	−95.2226	−0.288553
$331$	139.816	0.422406	0.211203	−	0.977442i	$-0.432262\pi$
$331$	139.816	0.422406	0.211203	+	0.977442i	$0.432262\pi$
$332$	− 223.329i	− 0.672676i
$333$	−97.5082	−0.292818
$334$	68.4844i	0.205043i
$335$	210.352i	0.627916i
$336$	0	0
$337$	30.1128	0.0893556	0.0446778	−	0.999001i	$-0.485774\pi$
$337$	30.1128	0.0893556	0.0446778	+	0.999001i	$0.485774\pi$
$338$	165.167	0.488660
$339$	182.616i	0.538691i
$340$	11.8601	0.0348826
$341$	− 700.657i	− 2.05471i
$342$	10.7840i	0.0315323i
$343$	0	0
$344$	105.707	0.307289
$345$	155.312	0.450180
$346$	159.762i	0.461739i
$347$	−415.907	−1.19858	−0.599290	−	0.800532i	$-0.704550\pi$
$347$	−415.907	−1.19858	−0.599290	+	0.800532i	$0.704550\pi$
$348$	− 162.840i	− 0.467932i
$349$	594.950i	1.70473i	0.522950	+	0.852363i	$0.324831\pi$
$349$	594.950i	1.70473i	−0.522950	+	0.852363i	$0.675169\pi$
$350$	0	0
$351$	−37.5453	−0.106967
$352$	−98.3455	−0.279390
$353$	364.318i	1.03206i	0.856570	+	0.516031i	$0.172591\pi$
$353$	364.318i	1.03206i	−0.856570	+	0.516031i	$0.827409\pi$
$354$	246.317	0.695812
$355$	− 25.8920i	− 0.0729352i
$356$	255.813i	0.718577i
$357$	0	0
$358$	468.693	1.30920
$359$	−612.762	−1.70686	−0.853430	−	0.521208i	$-0.825482\pi$
$359$	−612.762	−1.70686	−0.853430	+	0.521208i	$0.825482\pi$
$360$	− 18.9737i	− 0.0527046i
$361$	354.539	0.982103
$362$	− 301.691i	− 0.833401i
$363$	− 313.925i	− 0.864807i
$364$	0	0
$365$	50.6895	0.138875
$366$	31.5713	0.0862603
$367$	− 493.019i	− 1.34337i	−0.740835	−	0.671687i	$-0.765570\pi$
$367$	− 493.019i	− 1.34337i	0.740835	−	0.671687i	$-0.234430\pi$
$368$	160.406	0.435885
$369$	− 212.004i	− 0.574536i
$370$	102.783i	0.277791i
$371$	0	0
$372$	139.610	0.375296
$373$	165.403	0.443439	0.221719	−	0.975111i	$-0.428833\pi$
$373$	165.403	0.443439	0.221719	+	0.975111i	$0.428833\pi$
$374$	65.2029i	0.174339i
$375$	−19.3649	−0.0516398
$376$	94.6065i	0.251613i
$377$	339.660i	0.900956i
$378$	0	0
$379$	−179.349	−0.473215	−0.236608	−	0.971605i	$-0.576036\pi$
$379$	−179.349	−0.473215	−0.236608	+	0.971605i	$0.576036\pi$
$380$	11.3674	0.0299141
$381$	− 93.0168i	− 0.244139i
$382$	−94.6156	−0.247685
$383$	− 269.170i	− 0.702793i	−0.936227	−	0.351396i	$-0.885707\pi$
$383$	− 269.170i	− 0.702793i	0.936227	−	0.351396i	$-0.114293\pi$
$384$	− 19.5959i	− 0.0510310i
$385$	0	0
$386$	−159.246	−0.412554
$387$	−112.120	−0.289715
$388$	14.9651i	0.0385699i
$389$	−473.748	−1.21786	−0.608930	−	0.793224i	$-0.708401\pi$
$389$	−473.748	−1.21786	−0.608930	+	0.793224i	$0.708401\pi$
$390$	39.5762i	0.101477i
$391$	− 106.349i	− 0.271992i
$392$	0	0
$393$	101.298	0.257755
$394$	−86.2969	−0.219028
$395$	− 53.9080i	− 0.136476i
$396$	104.311	0.263412
$397$	513.934i	1.29454i	0.762260	+	0.647271i	$0.224090\pi$
$397$	513.934i	1.29454i	−0.762260	+	0.647271i	$0.775910\pi$
$398$	− 270.348i	− 0.679267i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	−406.734	−1.01430	−0.507150	−	0.861858i	$-0.669301\pi$
$401$	−406.734	−1.01430	−0.507150	+	0.861858i	$0.669301\pi$
$402$	− 230.429i	− 0.573206i
$403$	−291.205	−0.722594
$404$	− 189.193i	− 0.468299i
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	−565.066	−1.38837
$408$	−12.9921	−0.0318433
$409$	− 487.484i	− 1.19189i	−0.803025	−	0.595946i	$-0.796777\pi$
$409$	− 487.484i	− 1.19189i	0.803025	−	0.595946i	$-0.203223\pi$
$410$	−223.472	−0.545053
$411$	20.9872i	0.0510638i
$412$	119.643i	0.290397i
$413$	0	0
$414$	−170.136	−0.410957
$415$	249.689	0.601660
$416$	40.8741i	0.0982551i
$417$	−78.3861	−0.187976
$418$	62.4941i	0.149507i
$419$	552.257i	1.31804i	0.752127	+	0.659018i	$0.229028\pi$
$419$	552.257i	1.31804i	−0.752127	+	0.659018i	$0.770972\pi$
$420$	0	0
$421$	−74.6870	−0.177404	−0.0887019	−	0.996058i	$-0.528272\pi$
$421$	−74.6870	−0.177404	−0.0887019	+	0.996058i	$0.528272\pi$
$422$	396.143	0.938728
$423$	− 100.345i	− 0.237223i
$424$	−200.503	−0.472884
$425$	13.2600i	0.0311999i
$426$	28.3633i	0.0665804i
$427$	0	0
$428$	12.9232	0.0301945
$429$	−217.577	−0.507173
$430$	118.184i	0.274848i
$431$	484.677	1.12454	0.562271	−	0.826953i	$-0.309928\pi$
$431$	484.677	1.12454	0.562271	+	0.826953i	$0.309928\pi$
$432$	20.7846i	0.0481125i
$433$	− 458.196i	− 1.05819i	−0.848563	−	0.529094i	$-0.822532\pi$
$433$	− 458.196i	− 1.05819i	0.848563	−	0.529094i	$-0.177468\pi$
$434$	0	0
$435$	182.061	0.418531
$436$	−327.680	−0.751560
$437$	− 101.931i	− 0.233251i
$438$	−55.5276	−0.126775
$439$	− 139.912i	− 0.318706i	−0.987222	−	0.159353i	$-0.949059\pi$
$439$	− 139.912i	− 0.318706i	0.987222	−	0.159353i	$-0.0509407\pi$
$440$	− 109.954i	− 0.249894i
$441$	0	0
$442$	27.0995	0.0613110
$443$	−16.0923	−0.0363258	−0.0181629	−	0.999835i	$-0.505782\pi$
$443$	−16.0923	−0.0363258	−0.0181629	+	0.999835i	$0.505782\pi$
$444$	− 112.593i	− 0.253587i
$445$	−286.008	−0.642715
$446$	− 385.740i	− 0.864887i
$447$	88.1524i	0.197209i
$448$	0	0
$449$	−329.314	−0.733439	−0.366720	−	0.930332i	$-0.619519\pi$
$449$	−329.314	−0.733439	−0.366720	+	0.930332i	$0.619519\pi$
$450$	21.2132	0.0471405
$451$	− 1228.58i	− 2.72411i
$452$	−210.867	−0.466520
$453$	273.136i	0.602949i
$454$	22.7298i	0.0500657i
$455$	0	0
$456$	−12.4523	−0.0273077
$457$	−890.917	−1.94949	−0.974745	−	0.223320i	$-0.928311\pi$
$457$	−890.917	−1.94949	−0.974745	+	0.223320i	$0.928311\pi$
$458$	− 209.450i	− 0.457314i
$459$	13.7802	0.0300222
$460$	179.339i	0.389868i
$461$	534.019i	1.15839i	0.815188	+	0.579196i	$0.196633\pi$
$461$	534.019i	1.15839i	−0.815188	+	0.579196i	$0.803367\pi$
$462$	0	0
$463$	158.679	0.342719	0.171359	−	0.985209i	$-0.445184\pi$
$463$	158.679	0.342719	0.171359	+	0.985209i	$0.445184\pi$
$464$	188.032	0.405241
$465$	156.089i	0.335675i
$466$	−423.017	−0.907762
$467$	112.199i	0.240255i	0.992758	+	0.120127i	$0.0383303\pi$
$467$	112.199i	0.240255i	−0.992758	+	0.120127i	$0.961670\pi$
$468$	− 43.3535i	− 0.0926358i
$469$	0	0
$470$	−105.773	−0.225050
$471$	290.718	0.617236
$472$	284.423i	0.602591i
$473$	−649.740	−1.37366
$474$	59.0533i	0.124585i
$475$	12.7091i	0.0267560i
$476$	0	0
$477$	212.665	0.445839
$478$	−161.578	−0.338030
$479$	− 691.283i	− 1.44318i	−0.692321	−	0.721590i	$-0.743412\pi$
$479$	− 691.283i	− 1.44318i	0.692321	−	0.721590i	$-0.256588\pi$
$480$	21.9089	0.0456435
$481$	234.852i	0.488257i
$482$	− 192.958i	− 0.400328i
$483$	0	0
$484$	362.489	0.748945
$485$	−16.7315	−0.0344979
$486$	− 22.0454i	− 0.0453609i
$487$	720.820	1.48012	0.740062	−	0.672539i	$-0.234796\pi$
$487$	720.820	1.48012	0.740062	+	0.672539i	$0.234796\pi$
$488$	36.4554i	0.0747036i
$489$	− 479.033i	− 0.979618i
$490$	0	0
$491$	589.995	1.20162	0.600809	−	0.799392i	$-0.294845\pi$
$491$	589.995	1.20162	0.600809	+	0.799392i	$0.294845\pi$
$492$	244.801	0.497563
$493$	− 124.665i	− 0.252870i
$494$	25.9737	0.0525783
$495$	116.623i	0.235603i
$496$	161.208i	0.325016i
$497$	0	0
$498$	−273.521	−0.549238
$499$	−875.690	−1.75489	−0.877445	−	0.479677i	$-0.840754\pi$
$499$	−875.690	−1.75489	−0.877445	+	0.479677i	$0.840754\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	83.8760	0.167417
$502$	− 646.329i	− 1.28751i
$503$	817.809i	1.62586i	0.582360	+	0.812931i	$0.302129\pi$
$503$	817.809i	1.62586i	−0.582360	+	0.812931i	$0.697871\pi$
$504$	0	0
$505$	211.524	0.418859
$506$	−985.949	−1.94852
$507$	− 202.288i	− 0.398989i
$508$	107.407	0.211430
$509$	759.313i	1.49177i	0.666072	+	0.745887i	$0.267974\pi$
$509$	759.313i	1.49177i	−0.666072	+	0.745887i	$0.732026\pi$
$510$	− 14.5256i	− 0.0284815i
$511$	0	0
$512$	22.6274	0.0441942
$513$	13.2077	0.0257460
$514$	− 70.7080i	− 0.137564i
$515$	−133.765	−0.259739
$516$	− 129.465i	− 0.250900i
$517$	− 581.508i	− 1.12477i
$518$	0	0
$519$	195.667	0.377009
$520$	−45.6987	−0.0878820
$521$	− 177.444i	− 0.340583i	−0.985394	−	0.170292i	$-0.945529\pi$
$521$	− 177.444i	− 0.340583i	0.985394	−	0.170292i	$-0.0544710\pi$
$522$	−199.438	−0.382065
$523$	− 105.219i	− 0.201183i	−0.994928	−	0.100592i	$-0.967926\pi$
$523$	− 105.219i	− 0.201183i	0.994928	−	0.100592i	$-0.0320735\pi$
$524$	116.969i	0.223222i
$525$	0	0
$526$	257.780	0.490075
$527$	106.881	0.202809
$528$	120.448i	0.228121i
$529$	1079.13	2.03994
$530$	− 224.169i	− 0.422960i
$531$	− 301.676i	− 0.568128i
$532$	0	0
$533$	−510.618	−0.958007
$534$	313.306	0.586716
$535$	14.4486i	0.0270067i
$536$	266.076	0.496411
$537$	− 574.030i	− 1.06896i
$538$	− 53.2051i	− 0.0988943i
$539$	0	0
$540$	−23.2379	−0.0430331
$541$	−276.363	−0.510837	−0.255419	−	0.966831i	$-0.582213\pi$
$541$	−276.363	−0.510837	−0.255419	+	0.966831i	$0.582213\pi$
$542$	− 286.714i	− 0.528992i
$543$	−369.495	−0.680469
$544$	− 15.0019i	− 0.0275771i
$545$	− 366.358i	− 0.672216i
$546$	0	0
$547$	918.409	1.67899	0.839496	−	0.543366i	$-0.182850\pi$
$547$	918.409	1.67899	0.839496	+	0.543366i	$0.182850\pi$
$548$	−24.2339	−0.0442225
$549$	− 38.6668i	− 0.0704313i
$550$	122.932	0.223512
$551$	− 119.486i	− 0.216853i
$552$	− 196.456i	− 0.355899i
$553$	0	0
$554$	362.380	0.654116
$555$	125.883	0.226815
$556$	− 90.5124i	− 0.162792i
$557$	735.010	1.31959	0.659794	−	0.751447i	$-0.270644\pi$
$557$	735.010	1.31959	0.659794	+	0.751447i	$0.270644\pi$
$558$	− 170.987i	− 0.306428i
$559$	270.044i	0.483083i
$560$	0	0
$561$	79.8569	0.142347
$562$	−200.057	−0.355974
$563$	826.566i	1.46815i	0.679070	+	0.734073i	$0.262383\pi$
$563$	826.566i	1.46815i	−0.679070	+	0.734073i	$0.737617\pi$
$564$	115.869	0.205441
$565$	− 235.757i	− 0.417268i
$566$	768.438i	1.35767i
$567$	0	0
$568$	−32.7511	−0.0576603
$569$	−433.092	−0.761146	−0.380573	−	0.924751i	$-0.624273\pi$
$569$	−433.092	−0.761146	−0.380573	+	0.924751i	$0.624273\pi$
$570$	− 13.9221i	− 0.0244248i
$571$	480.072	0.840756	0.420378	−	0.907349i	$-0.361897\pi$
$571$	480.072	0.840756	0.420378	+	0.907349i	$0.361897\pi$
$572$	− 251.236i	− 0.439225i
$573$	115.880i	0.202234i
$574$	0	0
$575$	−200.507	−0.348708
$576$	−24.0000	−0.0416667
$577$	− 87.1390i	− 0.151021i	−0.997145	−	0.0755104i	$-0.975941\pi$
$577$	− 87.1390i	− 0.151021i	0.997145	−	0.0755104i	$-0.0240586\pi$
$578$	398.761	0.689899
$579$	195.036i	0.336849i
$580$	210.226i	0.362459i
$581$	0	0
$582$	18.3284	0.0314922
$583$	1232.41	2.11391
$584$	− 64.1177i	− 0.109791i
$585$	48.4707	0.0828560
$586$	530.738i	0.905697i
$587$	− 104.332i	− 0.177737i	−0.996043	−	0.0888687i	$-0.971675\pi$
$587$	− 104.332i	− 0.177737i	0.996043	−	0.0888687i	$-0.0283252\pi$
$588$	0	0
$589$	102.440	0.173922
$590$	−317.994	−0.538973
$591$	105.692i	0.178835i
$592$	130.011	0.219613
$593$	− 291.471i	− 0.491519i	−0.969331	−	0.245760i	$-0.920963\pi$
$593$	− 291.471i	− 0.491519i	0.969331	−	0.245760i	$-0.0790374\pi$
$594$	− 127.754i	− 0.215075i
$595$	0	0
$596$	−101.790	−0.170788
$597$	−331.108	−0.554619
$598$	409.778i	0.685247i
$599$	597.678	0.997793	0.498896	−	0.866662i	$-0.333739\pi$
$599$	597.678	0.997793	0.498896	+	0.866662i	$0.333739\pi$
$600$	24.4949i	0.0408248i
$601$	− 447.444i	− 0.744499i	−0.928133	−	0.372250i	$-0.878587\pi$
$601$	− 447.444i	− 0.744499i	0.928133	−	0.372250i	$-0.121413\pi$
$602$	0	0
$603$	−282.217	−0.468021
$604$	−315.390	−0.522169
$605$	405.275i	0.669877i
$606$	−231.713	−0.382364
$607$	− 616.309i	− 1.01534i	−0.861553	−	0.507668i	$-0.830508\pi$
$607$	− 616.309i	− 1.01534i	0.861553	−	0.507668i	$-0.169492\pi$
$608$	− 14.3787i	− 0.0236492i
$609$	0	0
$610$	−40.7584	−0.0668170
$611$	−241.685	−0.395556
$612$	15.9120i	0.0259999i
$613$	−127.077	−0.207303	−0.103651	−	0.994614i	$-0.533053\pi$
$613$	−127.077	−0.207303	−0.103651	+	0.994614i	$0.533053\pi$
$614$	− 58.4687i	− 0.0952259i
$615$	273.696i	0.445034i
$616$	0	0
$617$	68.5630	0.111123	0.0555616	−	0.998455i	$-0.482305\pi$
$617$	68.5630	0.111123	0.0555616	+	0.998455i	$0.482305\pi$
$618$	146.533	0.237108
$619$	962.477i	1.55489i	0.628951	+	0.777445i	$0.283485\pi$
$619$	962.477i	1.55489i	−0.628951	+	0.777445i	$0.716515\pi$
$620$	−180.236	−0.290703
$621$	208.373i	0.335545i
$622$	− 207.265i	− 0.333224i
$623$	0	0
$624$	50.0604	0.0802249
$625$	25.0000	0.0400000
$626$	− 318.015i	− 0.508011i
$627$	76.5394	0.122072
$628$	335.693i	0.534542i
$629$	− 86.1971i	− 0.137038i
$630$	0	0
$631$	412.586	0.653860	0.326930	−	0.945048i	$-0.393986\pi$
$631$	412.586	0.653860	0.326930	+	0.945048i	$0.393986\pi$
$632$	−68.1888	−0.107894
$633$	− 485.174i	− 0.766468i
$634$	−319.933	−0.504626
$635$	120.084i	0.189109i
$636$	245.565i	0.386108i
$637$	0	0
$638$	−1155.76	−1.81153
$639$	34.7378	0.0543627
$640$	25.2982i	0.0395285i
$641$	−142.675	−0.222582	−0.111291	−	0.993788i	$-0.535498\pi$
$641$	−142.675	−0.222582	−0.111291	+	0.993788i	$0.535498\pi$
$642$	− 15.8277i	− 0.0246537i
$643$	239.942i	0.373160i	0.982440	+	0.186580i	$0.0597403\pi$
$643$	239.942i	0.373160i	−0.982440	+	0.186580i	$0.940260\pi$
$644$	0	0
$645$	144.746	0.224412
$646$	−9.53306	−0.0147571
$647$	− 148.745i	− 0.229899i	−0.993371	−	0.114950i	$-0.963329\pi$
$647$	− 148.745i	− 0.229899i	0.993371	−	0.114950i	$-0.0366706\pi$
$648$	25.4558	0.0392837
$649$	− 1748.23i	− 2.69373i
$650$	− 51.0926i	− 0.0786041i
$651$	0	0
$652$	553.140	0.848374
$653$	288.022	0.441074	0.220537	−	0.975379i	$-0.429219\pi$
$653$	288.022	0.441074	0.220537	+	0.975379i	$0.429219\pi$
$654$	401.325i	0.613646i
$655$	−130.775	−0.199656
$656$	282.672i	0.430902i
$657$	68.0071i	0.103512i
$658$	0	0
$659$	−175.647	−0.266536	−0.133268	−	0.991080i	$-0.542547\pi$
$659$	−175.647	−0.266536	−0.133268	+	0.991080i	$0.542547\pi$
$660$	−134.665	−0.204038
$661$	− 710.158i	− 1.07437i	−0.843465	−	0.537185i	$-0.819488\pi$
$661$	− 710.158i	− 1.07437i	0.843465	−	0.537185i	$-0.180512\pi$
$662$	197.730	0.298686
$663$	− 33.1900i	− 0.0500603i
$664$	− 315.834i	− 0.475654i
$665$	0	0
$666$	−137.897	−0.207053
$667$	1885.09	2.82622
$668$	96.8516i	0.144987i
$669$	−472.433	−0.706177
$670$	297.482i	0.444004i
$671$	− 224.076i	− 0.333944i
$672$	0	0
$673$	−1173.04	−1.74301	−0.871504	−	0.490389i	$-0.836855\pi$
$673$	−1173.04	−1.74301	−0.871504	+	0.490389i	$0.836855\pi$
$674$	42.5860	0.0631840
$675$	− 25.9808i	− 0.0384900i
$676$	233.582	0.345535
$677$	− 677.461i	− 1.00068i	−0.865829	−	0.500341i	$-0.833208\pi$
$677$	− 677.461i	− 1.00068i	0.865829	−	0.500341i	$-0.166792\pi$
$678$	258.258i	0.380912i
$679$	0	0
$680$	16.7727	0.0246657
$681$	27.8382	0.0408784
$682$	− 990.878i	− 1.45290i
$683$	−831.027	−1.21673	−0.608366	−	0.793657i	$-0.708175\pi$
$683$	−831.027	−1.21673	−0.608366	+	0.793657i	$0.708175\pi$
$684$	15.2509i	0.0222967i
$685$	− 27.0944i	− 0.0395538i
$686$	0	0
$687$	−256.523	−0.373395
$688$	149.493	0.217286
$689$	− 512.210i	− 0.743411i
$690$	219.645	0.318326
$691$	− 625.196i	− 0.904770i	−0.891823	−	0.452385i	$-0.850573\pi$
$691$	− 625.196i	− 0.904770i	0.891823	−	0.452385i	$-0.149427\pi$
$692$	225.937i	0.326499i
$693$	0	0
$694$	−588.181	−0.847524
$695$	101.196	0.145606
$696$	− 230.291i	− 0.330878i
$697$	187.411	0.268882
$698$	841.386i	1.20542i
$699$	518.088i	0.741185i
$700$	0	0
$701$	1030.02	1.46936	0.734678	−	0.678416i	$-0.237333\pi$
$701$	1030.02	1.46936	0.734678	+	0.678416i	$0.237333\pi$
$702$	−53.0970	−0.0756368
$703$	− 82.6161i	− 0.117519i
$704$	−139.081	−0.197559
$705$	129.545i	0.183752i
$706$	515.224i	0.729779i
$707$	0	0
$708$	348.345	0.492013
$709$	−658.251	−0.928422	−0.464211	−	0.885725i	$-0.653662\pi$
$709$	−658.251	−0.928422	−0.464211	+	0.885725i	$0.653662\pi$
$710$	− 36.6168i	− 0.0515730i
$711$	72.3252	0.101723
$712$	361.775i	0.508111i
$713$	1616.17i	2.26671i
$714$	0	0
$715$	280.891	0.392854
$716$	662.833	0.925744
$717$	197.892i	0.276000i
$718$	−866.577	−1.20693
$719$	1347.52i	1.87416i	0.349117	+	0.937079i	$0.386481\pi$
$719$	1347.52i	1.87416i	−0.349117	+	0.937079i	$0.613519\pi$
$720$	− 26.8328i	− 0.0372678i
$721$	0	0
$722$	501.394	0.694452
$723$	−236.324	−0.326866
$724$	− 426.656i	− 0.589303i
$725$	−235.040	−0.324193
$726$	− 443.957i	− 0.611511i
$727$	1126.18i	1.54907i	0.632530	+	0.774536i	$0.282017\pi$
$727$	1126.18i	1.54907i	−0.632530	+	0.774536i	$0.717983\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	71.6858	0.0981998
$731$	− 99.1135i	− 0.135586i
$732$	44.6485	0.0609953
$733$	− 606.674i	− 0.827658i	−0.910355	−	0.413829i	$-0.864191\pi$
$733$	− 606.674i	− 0.827658i	0.910355	−	0.413829i	$-0.135809\pi$
$734$	− 697.233i	− 0.949909i
$735$	0	0
$736$	226.848	0.308218
$737$	−1635.46	−2.21908
$738$	− 299.819i	− 0.406258i
$739$	922.169	1.24786	0.623930	−	0.781480i	$-0.285535\pi$
$739$	922.169	1.24786	0.623930	+	0.781480i	$0.285535\pi$
$740$	145.357i	0.196428i
$741$	− 31.8111i	− 0.0429300i
$742$	0	0
$743$	13.3994	0.0180342	0.00901712	−	0.999959i	$-0.497130\pi$
$743$	13.3994	0.0180342	0.00901712	+	0.999959i	$0.497130\pi$
$744$	197.438	0.265374
$745$	− 113.804i	− 0.152757i
$746$	233.915	0.313559
$747$	334.993i	0.448451i
$748$	92.2108i	0.123277i
$749$	0	0
$750$	−27.3861	−0.0365148
$751$	1076.14	1.43295	0.716473	−	0.697615i	$-0.245755\pi$
$751$	1076.14	1.43295	0.716473	+	0.697615i	$0.245755\pi$
$752$	133.794i	0.177917i
$753$	−791.588	−1.05125
$754$	480.352i	0.637072i
$755$	− 352.617i	− 0.467043i
$756$	0	0
$757$	−254.117	−0.335690	−0.167845	−	0.985813i	$-0.553681\pi$
$757$	−254.117	−0.335690	−0.167845	+	0.985813i	$0.553681\pi$
$758$	−253.637	−0.334614
$759$	1207.54i	1.59096i
$760$	16.0759	0.0211525
$761$	791.079i	1.03953i	0.854310	+	0.519763i	$0.173980\pi$
$761$	791.079i	1.03953i	−0.854310	+	0.519763i	$0.826020\pi$
$762$	− 131.546i	− 0.172632i
$763$	0	0
$764$	−133.807	−0.175140
$765$	−17.7901	−0.0232551
$766$	− 380.663i	− 0.496949i
$767$	−726.596	−0.947322
$768$	− 27.7128i	− 0.0360844i
$769$	− 464.403i	− 0.603905i	−0.953323	−	0.301952i	$-0.902362\pi$
$769$	− 464.403i	− 0.603905i	0.953323	−	0.301952i	$-0.0976384\pi$
$770$	0	0
$771$	−86.5993	−0.112321
$772$	−225.208	−0.291720
$773$	780.125i	1.00922i	0.863348	+	0.504609i	$0.168363\pi$
$773$	780.125i	1.00922i	−0.863348	+	0.504609i	$0.831637\pi$
$774$	−158.561	−0.204859
$775$	− 201.510i	− 0.260013i
$776$	21.1639i	0.0272730i
$777$	0	0
$778$	−669.980	−0.861157
$779$	179.625	0.230584
$780$	55.9692i	0.0717554i
$781$	201.307	0.257756
$782$	− 150.400i	− 0.192327i
$783$	244.261i	0.311955i
$784$	0	0
$785$	−375.316	−0.478109
$786$	143.257	0.182260
$787$	1152.80i	1.46481i	0.680870	+	0.732404i	$0.261602\pi$
$787$	1152.80i	1.46481i	−0.680870	+	0.732404i	$0.738398\pi$
$788$	−122.042	−0.154876
$789$	− 315.714i	− 0.400145i
$790$	− 76.2374i	− 0.0965031i
$791$	0	0
$792$	147.518	0.186260
$793$	−93.1301	−0.117440
$794$	726.812i	0.915380i
$795$	−274.550	−0.345345
$796$	− 382.330i	− 0.480314i
$797$	− 1475.63i	− 1.85148i	−0.378156	−	0.925742i	$-0.623442\pi$
$797$	− 1475.63i	− 1.85148i	0.378156	−	0.925742i	$-0.376558\pi$
$798$	0	0
$799$	88.7051	0.111020
$800$	−28.2843	−0.0353553
$801$	− 383.720i	− 0.479051i
$802$	−575.209	−0.717218
$803$	394.106i	0.490791i
$804$	− 325.876i	− 0.405318i
$805$	0	0
$806$	−411.827	−0.510951
$807$	−65.1627	−0.0807469
$808$	− 267.559i	− 0.331137i
$809$	−13.7610	−0.0170099	−0.00850495	−	0.999964i	$-0.502707\pi$
$809$	−13.7610	−0.0170099	−0.00850495	+	0.999964i	$0.502707\pi$
$810$	28.4605i	0.0351364i
$811$	− 1274.65i	− 1.57170i	−0.618416	−	0.785851i	$-0.712225\pi$
$811$	− 1274.65i	− 1.57170i	0.618416	−	0.785851i	$-0.287775\pi$
$812$	0	0
$813$	−351.151	−0.431920
$814$	−799.124	−0.981725
$815$	618.429i	0.758809i
$816$	−18.3736	−0.0225166
$817$	− 94.9960i	− 0.116274i
$818$	− 689.406i	− 0.842795i
$819$	0	0
$820$	−316.037	−0.385410
$821$	584.498	0.711934	0.355967	−	0.934499i	$-0.384152\pi$
$821$	584.498	0.711934	0.355967	+	0.934499i	$0.384152\pi$
$822$	29.6804i	0.0361075i
$823$	−80.9671	−0.0983804	−0.0491902	−	0.998789i	$-0.515664\pi$
$823$	−80.9671	−0.0983804	−0.0491902	+	0.998789i	$0.515664\pi$
$824$	169.201i	0.205342i
$825$	− 150.560i	− 0.182497i
$826$	0	0
$827$	−1628.46	−1.96911	−0.984556	−	0.175072i	$-0.943984\pi$
$827$	−1628.46	−1.96911	−0.984556	+	0.175072i	$0.943984\pi$
$828$	−240.609	−0.290590
$829$	− 879.861i	− 1.06135i	−0.847575	−	0.530676i	$-0.821938\pi$
$829$	− 879.861i	− 1.06135i	0.847575	−	0.530676i	$-0.178062\pi$
$830$	353.113	0.425438
$831$	− 443.824i	− 0.534084i
$832$	57.8047i	0.0694768i
$833$	0	0
$834$	−110.855	−0.132919
$835$	−108.283	−0.129681
$836$	88.3800i	0.105718i
$837$	−209.415	−0.250197
$838$	781.009i	0.931992i
$839$	− 647.389i	− 0.771619i	−0.922578	−	0.385810i	$-0.873922\pi$
$839$	− 647.389i	− 0.771619i	0.922578	−	0.385810i	$-0.126078\pi$
$840$	0	0
$841$	1368.75	1.62753
$842$	−105.623	−0.125443
$843$	245.019i	0.290651i
$844$	560.231	0.663781
$845$	261.152i	0.309056i
$846$	− 141.910i	− 0.167742i
$847$	0	0
$848$	−283.554	−0.334379
$849$	941.141	1.10853
$850$	18.7524i	0.0220617i
$851$	1303.41	1.53162
$852$	40.1117i	0.0470795i
$853$	569.518i	0.667665i	0.942632	+	0.333833i	$0.108342\pi$
$853$	569.518i	0.667665i	−0.942632	+	0.333833i	$0.891658\pi$
$854$	0	0
$855$	−17.0511	−0.0199427
$856$	18.2762	0.0213507
$857$	638.597i	0.745154i	0.928001	+	0.372577i	$0.121526\pi$
$857$	638.597i	0.745154i	−0.928001	+	0.372577i	$0.878474\pi$
$858$	−307.701	−0.358625
$859$	− 1267.12i	− 1.47512i	−0.675284	−	0.737558i	$-0.735979\pi$
$859$	− 1267.12i	− 1.47512i	0.675284	−	0.737558i	$-0.264021\pi$
$860$	167.138i	0.194347i
$861$	0	0
$862$	685.437	0.795171
$863$	−206.034	−0.238741	−0.119371	−	0.992850i	$-0.538088\pi$
$863$	−206.034	−0.238741	−0.119371	+	0.992850i	$0.538088\pi$
$864$	29.3939i	0.0340207i
$865$	−252.606	−0.292030
$866$	− 647.987i	− 0.748252i
$867$	− 488.381i	− 0.563300i
$868$	0	0
$869$	419.129	0.482312
$870$	257.473	0.295946
$871$	679.727i	0.780399i
$872$	−463.410	−0.531433
$873$	− 22.4477i	− 0.0257133i
$874$	− 144.152i	− 0.164933i
$875$	0	0
$876$	−78.5279	−0.0896437
$877$	824.544	0.940187	0.470093	−	0.882617i	$-0.344220\pi$
$877$	824.544	0.940187	0.470093	+	0.882617i	$0.344220\pi$
$878$	− 197.865i	− 0.225359i
$879$	650.019	0.739498
$880$	− 155.498i	− 0.176702i
$881$	416.337i	0.472574i	0.971683	+	0.236287i	$0.0759305\pi$
$881$	416.337i	0.472574i	−0.971683	+	0.236287i	$0.924070\pi$
$882$	0	0
$883$	650.529	0.736726	0.368363	−	0.929682i	$-0.379918\pi$
$883$	650.529	0.736726	0.368363	+	0.929682i	$0.379918\pi$
$884$	38.3245	0.0433535
$885$	389.462i	0.440070i
$886$	−22.7580	−0.0256862
$887$	346.396i	0.390525i	0.980751	+	0.195262i	$0.0625559\pi$
$887$	346.396i	0.390525i	−0.980751	+	0.195262i	$0.937444\pi$
$888$	− 159.230i	− 0.179313i
$889$	0	0
$890$	−404.476	−0.454468
$891$	−156.467	−0.175608
$892$	− 545.518i	− 0.611567i
$893$	85.0200	0.0952071
$894$	124.666i	0.139448i
$895$	741.069i	0.828011i
$896$	0	0
$897$	501.873	0.559502
$898$	−465.721	−0.518620
$899$	1894.51i	2.10736i
$900$	30.0000	0.0333333
$901$	187.996i	0.208652i
$902$	− 1737.47i	− 1.92624i
$903$	0	0
$904$	−298.211	−0.329879
$905$	477.015	0.527089
$906$	386.273i	0.426350i
$907$	420.182	0.463265	0.231633	−	0.972803i	$-0.425593\pi$
$907$	420.182	0.463265	0.231633	+	0.972803i	$0.425593\pi$
$908$	32.1448i	0.0354018i
$909$	283.789i	0.312199i
$910$	0	0
$911$	717.087	0.787142	0.393571	−	0.919294i	$-0.371239\pi$
$911$	717.087	0.787142	0.393571	+	0.919294i	$0.371239\pi$
$912$	−17.6102	−0.0193095
$913$	1941.30i	2.12629i
$914$	−1259.95	−1.37850
$915$	49.9186i	0.0545558i
$916$	− 296.207i	− 0.323370i
$917$	0	0
$918$	19.4881	0.0212289
$919$	−119.027	−0.129518	−0.0647592	−	0.997901i	$-0.520628\pi$
$919$	−119.027	−0.129518	−0.0647592	+	0.997901i	$0.520628\pi$
$920$	253.624i	0.275678i
$921$	−71.6093	−0.0777516
$922$	755.216i	0.819107i
$923$	− 83.6669i	− 0.0906467i
$924$	0	0
$925$	−162.514	−0.175691
$926$	224.406	0.242339
$927$	− 179.465i	− 0.193598i
$928$	265.917	0.286549
$929$	− 378.798i	− 0.407749i	−0.978997	−	0.203874i	$-0.934647\pi$
$929$	− 378.798i	− 0.407749i	0.978997	−	0.203874i	$-0.0653534\pi$
$930$	220.743i	0.237358i
$931$	0	0
$932$	−598.237	−0.641885
$933$	−253.847	−0.272076
$934$	158.673i	0.169886i
$935$	−103.095	−0.110262
$936$	− 61.3112i	− 0.0655034i
$937$	365.585i	0.390165i	0.980787	+	0.195083i	$0.0624975\pi$
$937$	365.585i	0.390165i	−0.980787	+	0.195083i	$0.937503\pi$
$938$	0	0
$939$	−389.487	−0.414790
$940$	−149.586	−0.159134
$941$	− 1377.81i	− 1.46419i	−0.681201	−	0.732097i	$-0.738542\pi$
$941$	− 1377.81i	− 1.46419i	0.681201	−	0.732097i	$-0.261458\pi$
$942$	411.138	0.436452
$943$	2833.89i	3.00518i
$944$	402.235i	0.426096i
$945$	0	0
$946$	−918.871	−0.971323
$947$	474.833	0.501408	0.250704	−	0.968064i	$-0.419338\pi$
$947$	474.833	0.501408	0.250704	+	0.968064i	$0.419338\pi$
$948$	83.5139i	0.0880949i
$949$	163.797	0.172600
$950$	17.9734i	0.0189194i
$951$	391.836i	0.412025i
$952$	0	0
$953$	−172.839	−0.181363	−0.0906813	−	0.995880i	$-0.528904\pi$
$953$	−172.839	−0.181363	−0.0906813	+	0.995880i	$0.528904\pi$
$954$	300.754	0.315256
$955$	− 149.600i	− 0.156650i
$956$	−228.506	−0.239023
$957$	1415.51i	1.47911i
$958$	− 977.622i	− 1.02048i
$959$	0	0
$960$	30.9839	0.0322749
$961$	−663.246	−0.690163
$962$	332.130i	0.345250i
$963$	−19.3848	−0.0201296
$964$	− 272.884i	− 0.283074i
$965$	− 251.790i	− 0.260922i
$966$	0	0
$967$	−1209.88	−1.25117	−0.625583	−	0.780158i	$-0.715139\pi$
$967$	−1209.88	−1.25117	−0.625583	+	0.780158i	$0.715139\pi$
$968$	512.637	0.529584
$969$	11.6756i	0.0120491i
$970$	−23.6619	−0.0243937
$971$	780.566i	0.803879i	0.915666	+	0.401939i	$0.131664\pi$
$971$	780.566i	0.803879i	−0.915666	+	0.401939i	$0.868336\pi$
$972$	− 31.1769i	− 0.0320750i
$973$	0	0
$974$	1019.39	1.04661
$975$	−62.5755	−0.0641800
$976$	51.5557i	0.0528235i
$977$	1589.86	1.62729	0.813644	−	0.581363i	$-0.197480\pi$
$977$	1589.86	1.62729	0.813644	+	0.581363i	$0.197480\pi$
$978$	− 677.455i	− 0.692695i
$979$	− 2223.68i	− 2.27138i
$980$	0	0
$981$	491.520	0.501040
$982$	834.378	0.849673
$983$	701.569i	0.713702i	0.934161	+	0.356851i	$0.116150\pi$
$983$	701.569i	0.713702i	−0.934161	+	0.356851i	$0.883850\pi$
$984$	346.201	0.351830
$985$	− 136.447i	− 0.138525i
$986$	− 176.303i	− 0.178806i
$987$	0	0
$988$	36.7323	0.0371785
$989$	1498.72	1.51539
$990$	164.930i	0.166596i
$991$	406.718	0.410412	0.205206	−	0.978719i	$-0.434214\pi$
$991$	406.718	0.410412	0.205206	+	0.978719i	$0.434214\pi$
$992$	227.982i	0.229821i
$993$	− 242.169i	− 0.243876i
$994$	0	0
$995$	427.458	0.429606
$996$	−386.816	−0.388370
$997$	− 756.759i	− 0.759036i	−0.925184	−	0.379518i	$-0.876090\pi$
$997$	− 756.759i	− 0.759036i	0.925184	−	0.379518i	$-0.123910\pi$
$998$	−1238.41	−1.24090
$999$	168.889i	0.169058i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.d.391.11		16
7.4	even	3		210.3.o.b.61.2	yes	16
7.5	odd	6		210.3.o.b.31.2	✓	16
7.6	odd	2	inner	1470.3.f.d.391.13		16
21.5	even	6		630.3.v.c.451.8		16
21.11	odd	6		630.3.v.c.271.8		16
35.4	even	6		1050.3.p.i.901.5		16
35.12	even	12		1050.3.q.e.199.15		32
35.18	odd	12		1050.3.q.e.649.15		32
35.19	odd	6		1050.3.p.i.451.5		16
35.32	odd	12		1050.3.q.e.649.2		32
35.33	even	12		1050.3.q.e.199.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.b.31.2	✓	16	7.5	odd	6
210.3.o.b.61.2	yes	16	7.4	even	3
630.3.v.c.271.8		16	21.11	odd	6
630.3.v.c.451.8		16	21.5	even	6
1050.3.p.i.451.5		16	35.19	odd	6
1050.3.p.i.901.5		16	35.4	even	6
1050.3.q.e.199.2		32	35.33	even	12
1050.3.q.e.199.15		32	35.12	even	12
1050.3.q.e.649.2		32	35.32	odd	12
1050.3.q.e.649.15		32	35.18	odd	12
1470.3.f.d.391.11		16	1.1	even	1	trivial
1470.3.f.d.391.13		16	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949i q^{6} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -17.3852 q^{11} -3.46410i q^{12} +7.22559i q^{13} +3.87298 q^{15} +4.00000 q^{16} -2.65199i q^{17} -4.24264 q^{18} -2.54182i q^{19} +4.47214i q^{20} -24.5864 q^{22} +40.1015 q^{23} -4.89898i q^{24} -5.00000 q^{25} +10.2185i q^{26} +5.19615i q^{27} +47.0080 q^{29} +5.47723 q^{30} +40.3019i q^{31} +5.65685 q^{32} +30.1120i q^{33} -3.75049i q^{34} -6.00000 q^{36} +32.5027 q^{37} -3.59468i q^{38} +12.5151 q^{39} +6.32456i q^{40} +70.6679i q^{41} +37.3732 q^{43} -34.7704 q^{44} -6.70820i q^{45} +56.7120 q^{46} +33.4485i q^{47} -6.92820i q^{48} -7.07107 q^{50} -4.59339 q^{51} +14.4512i q^{52} -70.8884 q^{53} +7.34847i q^{54} -38.8745i q^{55} -4.40256 q^{57} +66.4793 q^{58} +100.559i q^{59} +7.74597 q^{60} +12.8889i q^{61} +56.9955i q^{62} +8.00000 q^{64} -16.1569 q^{65} +42.5848i q^{66} +94.0722 q^{67} -5.30399i q^{68} -69.4578i q^{69} -11.5793 q^{71} -8.48528 q^{72} -22.6690i q^{73} +45.9658 q^{74} +8.66025i q^{75} -5.08364i q^{76} +17.6990 q^{78} -24.1084 q^{79} +8.94427i q^{80} +9.00000 q^{81} +99.9396i q^{82} -111.664i q^{83} +5.93004 q^{85} +52.8537 q^{86} -81.4202i q^{87} -49.1727 q^{88} +127.907i q^{89} -9.48683i q^{90} +80.2029 q^{92} +69.8050 q^{93} +47.3033i q^{94} +5.68368 q^{95} -9.79796i q^{96} +7.48256i q^{97} +52.1556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 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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