LMFDB - Embedded newform 5070.2.b.r.1351.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not 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:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.3
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.r.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -3.56155i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -3.56155i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.12311i q^{11} -1.00000 q^{12} +3.56155 q^{14} +1.00000i q^{15} +1.00000 q^{16} -5.12311 q^{17} +1.00000i q^{18} -3.56155i q^{19} -1.00000i q^{20} -3.56155i q^{21} -4.12311 q^{22} +7.68466 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.56155i q^{28} +6.56155 q^{29} -1.00000 q^{30} +5.68466i q^{31} +1.00000i q^{32} +4.12311i q^{33} -5.12311i q^{34} +3.56155 q^{35} -1.00000 q^{36} -4.12311i q^{37} +3.56155 q^{38} +1.00000 q^{40} -4.24621i q^{41} +3.56155 q^{42} -4.56155 q^{43} -4.12311i q^{44} +1.00000i q^{45} +7.68466i q^{46} -7.00000i q^{47} +1.00000 q^{48} -5.68466 q^{49} -1.00000i q^{50} -5.12311 q^{51} +4.43845 q^{53} +1.00000i q^{54} -4.12311 q^{55} -3.56155 q^{56} -3.56155i q^{57} +6.56155i q^{58} -10.5616i q^{59} -1.00000i q^{60} +6.00000 q^{61} -5.68466 q^{62} -3.56155i q^{63} -1.00000 q^{64} -4.12311 q^{66} +14.2462i q^{67} +5.12311 q^{68} +7.68466 q^{69} +3.56155i q^{70} -4.87689i q^{71} -1.00000i q^{72} +15.3693i q^{73} +4.12311 q^{74} -1.00000 q^{75} +3.56155i q^{76} +14.6847 q^{77} +7.43845 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.24621 q^{82} +1.12311i q^{83} +3.56155i q^{84} -5.12311i q^{85} -4.56155i q^{86} +6.56155 q^{87} +4.12311 q^{88} -1.80776i q^{89} -1.00000 q^{90} -7.68466 q^{92} +5.68466i q^{93} +7.00000 q^{94} +3.56155 q^{95} +1.00000i q^{96} +1.12311i q^{97} -5.68466i q^{98} +4.12311i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 6 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{23} - 4 q^{25} + 4 q^{27} + 18 q^{29} - 4 q^{30} + 6 q^{35} - 4 q^{36} + 6 q^{38} + 4 q^{40} + 6 q^{42} - 10 q^{43} + 4 q^{48} + 2 q^{49} - 4 q^{51} + 26 q^{53} - 6 q^{56} + 24 q^{61} + 2 q^{62} - 4 q^{64} + 4 q^{68} + 6 q^{69} - 4 q^{75} + 34 q^{77} + 38 q^{79} + 4 q^{81} - 16 q^{82} + 18 q^{87} - 4 q^{90} - 6 q^{92} + 28 q^{94} + 6 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 6 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^23 - 4 * q^25 + 4 * q^27 + 18 * q^29 - 4 * q^30 + 6 * q^35 - 4 * q^36 + 6 * q^38 + 4 * q^40 + 6 * q^42 - 10 * q^43 + 4 * q^48 + 2 * q^49 - 4 * q^51 + 26 * q^53 - 6 * q^56 + 24 * q^61 + 2 * q^62 - 4 * q^64 + 4 * q^68 + 6 * q^69 - 4 * q^75 + 34 * q^77 + 38 * q^79 + 4 * q^81 - 16 * q^82 + 18 * q^87 - 4 * q^90 - 6 * q^92 + 28 * q^94 + 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.00000i	0.408248i
$7$	− 3.56155i	− 1.34614i	−0.739579	−	0.673070i	$-0.764975\pi$
$7$	− 3.56155i	− 1.34614i	0.739579	−	0.673070i	$-0.235025\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.12311i	1.24316i	0.783349	+	0.621582i	$0.213510\pi$
$11$	4.12311i	1.24316i	−0.783349	+	0.621582i	$0.786490\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.56155	0.951865
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	1.00000i	0.235702i
$19$	− 3.56155i	− 0.817076i	−0.912741	−	0.408538i	$-0.866039\pi$
$19$	− 3.56155i	− 0.817076i	0.912741	−	0.408538i	$-0.133961\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 3.56155i	− 0.777195i
$22$	−4.12311	−0.879049
$23$	7.68466	1.60236	0.801181	−	0.598422i	$-0.204205\pi$
$23$	7.68466	1.60236	0.801181	+	0.598422i	$0.204205\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	3.56155i	0.673070i
$29$	6.56155	1.21845	0.609225	−	0.792998i	$-0.291481\pi$
$29$	6.56155	1.21845	0.609225	+	0.792998i	$0.291481\pi$
$30$	−1.00000	−0.182574
$31$	5.68466i	1.02099i	0.859879	+	0.510497i	$0.170539\pi$
$31$	5.68466i	1.02099i	−0.859879	+	0.510497i	$0.829461\pi$
$32$	1.00000i	0.176777i
$33$	4.12311i	0.717741i
$34$	− 5.12311i	− 0.878605i
$35$	3.56155	0.602012
$36$	−1.00000	−0.166667
$37$	− 4.12311i	− 0.677834i	−0.940816	−	0.338917i	$-0.889939\pi$
$37$	− 4.12311i	− 0.677834i	0.940816	−	0.338917i	$-0.110061\pi$
$38$	3.56155	0.577760
$39$	0	0
$40$	1.00000	0.158114
$41$	− 4.24621i	− 0.663147i	−0.943429	−	0.331573i	$-0.892421\pi$
$41$	− 4.24621i	− 0.663147i	0.943429	−	0.331573i	$-0.107579\pi$
$42$	3.56155	0.549560
$43$	−4.56155	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$43$	−4.56155	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$44$	− 4.12311i	− 0.621582i
$45$	1.00000i	0.149071i
$46$	7.68466i	1.13304i
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	1.00000	0.144338
$49$	−5.68466	−0.812094
$50$	− 1.00000i	− 0.141421i
$51$	−5.12311	−0.717378
$52$	0	0
$53$	4.43845	0.609668	0.304834	−	0.952406i	$-0.401399\pi$
$53$	4.43845	0.609668	0.304834	+	0.952406i	$0.401399\pi$
$54$	1.00000i	0.136083i
$55$	−4.12311	−0.555959
$56$	−3.56155	−0.475933
$57$	− 3.56155i	− 0.471739i
$58$	6.56155i	0.861574i
$59$	− 10.5616i	− 1.37500i	−0.726186	−	0.687499i	$-0.758709\pi$
$59$	− 10.5616i	− 1.37500i	0.726186	−	0.687499i	$-0.241291\pi$
$60$	− 1.00000i	− 0.129099i
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−5.68466	−0.721952
$63$	− 3.56155i	− 0.448713i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.12311	−0.507519
$67$	14.2462i	1.74045i	0.492653	+	0.870226i	$0.336027\pi$
$67$	14.2462i	1.74045i	−0.492653	+	0.870226i	$0.663973\pi$
$68$	5.12311	0.621268
$69$	7.68466	0.925124
$70$	3.56155i	0.425687i
$71$	− 4.87689i	− 0.578781i	−0.957211	−	0.289390i	$-0.906547\pi$
$71$	− 4.87689i	− 0.578781i	0.957211	−	0.289390i	$-0.0934526\pi$
$72$	− 1.00000i	− 0.117851i
$73$	15.3693i	1.79884i	0.437083	+	0.899421i	$0.356012\pi$
$73$	15.3693i	1.79884i	−0.437083	+	0.899421i	$0.643988\pi$
$74$	4.12311	0.479301
$75$	−1.00000	−0.115470
$76$	3.56155i	0.408538i
$77$	14.6847	1.67347
$78$	0	0
$79$	7.43845	0.836891	0.418445	−	0.908242i	$-0.362575\pi$
$79$	7.43845	0.836891	0.418445	+	0.908242i	$0.362575\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	4.24621	0.468916
$83$	1.12311i	0.123277i	0.998099	+	0.0616384i	$0.0196326\pi$
$83$	1.12311i	0.123277i	−0.998099	+	0.0616384i	$0.980367\pi$
$84$	3.56155i	0.388597i
$85$	− 5.12311i	− 0.555679i
$86$	− 4.56155i	− 0.491885i
$87$	6.56155	0.703472
$88$	4.12311	0.439525
$89$	− 1.80776i	− 0.191623i	−0.995400	−	0.0958113i	$-0.969455\pi$
$89$	− 1.80776i	− 0.191623i	0.995400	−	0.0958113i	$-0.0305445\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−7.68466	−0.801181
$93$	5.68466i	0.589472i
$94$	7.00000	0.721995
$95$	3.56155	0.365408
$96$	1.00000i	0.102062i
$97$	1.12311i	0.114034i	0.998373	+	0.0570170i	$0.0181589\pi$
$97$	1.12311i	0.114034i	−0.998373	+	0.0570170i	$0.981841\pi$
$98$	− 5.68466i	− 0.574237i
$99$	4.12311i	0.414388i
$100$	1.00000	0.100000
$101$	17.1231	1.70381	0.851906	−	0.523694i	$-0.175447\pi$
$101$	17.1231	1.70381	0.851906	+	0.523694i	$0.175447\pi$
$102$	− 5.12311i	− 0.507263i
$103$	−0.438447	−0.0432015	−0.0216007	−	0.999767i	$-0.506876\pi$
$103$	−0.438447	−0.0432015	−0.0216007	+	0.999767i	$0.506876\pi$
$104$	0	0
$105$	3.56155	0.347572
$106$	4.43845i	0.431100i
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−1.00000	−0.0962250
$109$	− 20.2462i	− 1.93924i	−0.244625	−	0.969618i	$-0.578665\pi$
$109$	− 20.2462i	− 1.93924i	0.244625	−	0.969618i	$-0.421335\pi$
$110$	− 4.12311i	− 0.393123i
$111$	− 4.12311i	− 0.391348i
$112$	− 3.56155i	− 0.336535i
$113$	−3.68466	−0.346624	−0.173312	−	0.984867i	$-0.555447\pi$
$113$	−3.68466	−0.346624	−0.173312	+	0.984867i	$0.555447\pi$
$114$	3.56155	0.333570
$115$	7.68466i	0.716598i
$116$	−6.56155	−0.609225
$117$	0	0
$118$	10.5616	0.972270
$119$	18.2462i	1.67263i
$120$	1.00000	0.0912871
$121$	−6.00000	−0.545455
$122$	6.00000i	0.543214i
$123$	− 4.24621i	− 0.382868i
$124$	− 5.68466i	− 0.510497i
$125$	− 1.00000i	− 0.0894427i
$126$	3.56155	0.317288
$127$	4.43845	0.393849	0.196924	−	0.980419i	$-0.436905\pi$
$127$	4.43845	0.393849	0.196924	+	0.980419i	$0.436905\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.56155	−0.401622
$130$	0	0
$131$	6.12311	0.534978	0.267489	−	0.963561i	$-0.413806\pi$
$131$	6.12311	0.534978	0.267489	+	0.963561i	$0.413806\pi$
$132$	− 4.12311i	− 0.358870i
$133$	−12.6847	−1.09990
$134$	−14.2462	−1.23069
$135$	1.00000i	0.0860663i
$136$	5.12311i	0.439303i
$137$	8.80776i	0.752498i	0.926519	+	0.376249i	$0.122786\pi$
$137$	8.80776i	0.752498i	−0.926519	+	0.376249i	$0.877214\pi$
$138$	7.68466i	0.654162i
$139$	8.43845	0.715740	0.357870	−	0.933771i	$-0.383503\pi$
$139$	8.43845	0.715740	0.357870	+	0.933771i	$0.383503\pi$
$140$	−3.56155	−0.301006
$141$	− 7.00000i	− 0.589506i
$142$	4.87689	0.409260
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.56155i	0.544907i
$146$	−15.3693	−1.27197
$147$	−5.68466	−0.468863
$148$	4.12311i	0.338917i
$149$	− 22.8078i	− 1.86848i	−0.356639	−	0.934242i	$-0.616077\pi$
$149$	− 22.8078i	− 1.86848i	0.356639	−	0.934242i	$-0.383923\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 9.36932i	− 0.762464i	−0.924479	−	0.381232i	$-0.875500\pi$
$151$	− 9.36932i	− 0.762464i	0.924479	−	0.381232i	$-0.124500\pi$
$152$	−3.56155	−0.288880
$153$	−5.12311	−0.414179
$154$	14.6847i	1.18332i
$155$	−5.68466	−0.456603
$156$	0	0
$157$	22.1231	1.76562	0.882808	−	0.469734i	$-0.155650\pi$
$157$	22.1231	1.76562	0.882808	+	0.469734i	$0.155650\pi$
$158$	7.43845i	0.591771i
$159$	4.43845	0.351992
$160$	−1.00000	−0.0790569
$161$	− 27.3693i	− 2.15700i
$162$	1.00000i	0.0785674i
$163$	18.5616i	1.45385i	0.686715	+	0.726927i	$0.259052\pi$
$163$	18.5616i	1.45385i	−0.686715	+	0.726927i	$0.740948\pi$
$164$	4.24621i	0.331573i
$165$	−4.12311	−0.320983
$166$	−1.12311	−0.0871699
$167$	20.3693i	1.57623i	0.615531	+	0.788113i	$0.288942\pi$
$167$	20.3693i	1.57623i	−0.615531	+	0.788113i	$0.711058\pi$
$168$	−3.56155	−0.274780
$169$	0	0
$170$	5.12311	0.392924
$171$	− 3.56155i	− 0.272359i
$172$	4.56155	0.347815
$173$	25.1771	1.91418	0.957089	−	0.289794i	$-0.0935868\pi$
$173$	25.1771	1.91418	0.957089	+	0.289794i	$0.0935868\pi$
$174$	6.56155i	0.497430i
$175$	3.56155i	0.269228i
$176$	4.12311i	0.310791i
$177$	− 10.5616i	− 0.793855i
$178$	1.80776	0.135498
$179$	0.315342	0.0235697	0.0117849	−	0.999931i	$-0.496249\pi$
$179$	0.315342	0.0235697	0.0117849	+	0.999931i	$0.496249\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	−11.1231	−0.826774	−0.413387	−	0.910555i	$-0.635654\pi$
$181$	−11.1231	−0.826774	−0.413387	+	0.910555i	$0.635654\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	− 7.68466i	− 0.566521i
$185$	4.12311	0.303137
$186$	−5.68466	−0.416819
$187$	− 21.1231i	− 1.54467i
$188$	7.00000i	0.510527i
$189$	− 3.56155i	− 0.259065i
$190$	3.56155i	0.258382i
$191$	19.1231	1.38370	0.691850	−	0.722042i	$-0.256796\pi$
$191$	19.1231	1.38370	0.691850	+	0.722042i	$0.256796\pi$
$192$	−1.00000	−0.0721688
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−1.12311	−0.0806343
$195$	0	0
$196$	5.68466	0.406047
$197$	− 7.80776i	− 0.556280i	−0.960541	−	0.278140i	$-0.910282\pi$
$197$	− 7.80776i	− 0.556280i	0.960541	−	0.278140i	$-0.0897179\pi$
$198$	−4.12311	−0.293016
$199$	−11.1231	−0.788496	−0.394248	−	0.919004i	$-0.628995\pi$
$199$	−11.1231	−0.788496	−0.394248	+	0.919004i	$0.628995\pi$
$200$	1.00000i	0.0707107i
$201$	14.2462i	1.00485i
$202$	17.1231i	1.20478i
$203$	− 23.3693i	− 1.64020i
$204$	5.12311	0.358689
$205$	4.24621	0.296568
$206$	− 0.438447i	− 0.0305481i
$207$	7.68466	0.534121
$208$	0	0
$209$	14.6847	1.01576
$210$	3.56155i	0.245770i
$211$	6.93087	0.477141	0.238570	−	0.971125i	$-0.423321\pi$
$211$	6.93087	0.477141	0.238570	+	0.971125i	$0.423321\pi$
$212$	−4.43845	−0.304834
$213$	− 4.87689i	− 0.334159i
$214$	2.00000i	0.136717i
$215$	− 4.56155i	− 0.311095i
$216$	− 1.00000i	− 0.0680414i
$217$	20.2462	1.37440
$218$	20.2462	1.37125
$219$	15.3693i	1.03856i
$220$	4.12311	0.277980
$221$	0	0
$222$	4.12311	0.276725
$223$	− 26.3002i	− 1.76119i	−0.473869	−	0.880595i	$-0.657143\pi$
$223$	− 26.3002i	− 1.76119i	0.473869	−	0.880595i	$-0.342857\pi$
$224$	3.56155	0.237966
$225$	−1.00000	−0.0666667
$226$	− 3.68466i	− 0.245100i
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	3.56155i	0.235870i
$229$	7.75379i	0.512385i	0.966626	+	0.256192i	$0.0824681\pi$
$229$	7.75379i	0.512385i	−0.966626	+	0.256192i	$0.917532\pi$
$230$	−7.68466	−0.506711
$231$	14.6847	0.966180
$232$	− 6.56155i	− 0.430787i
$233$	17.6847	1.15856	0.579280	−	0.815128i	$-0.303334\pi$
$233$	17.6847	1.15856	0.579280	+	0.815128i	$0.303334\pi$
$234$	0	0
$235$	7.00000	0.456630
$236$	10.5616i	0.687499i
$237$	7.43845	0.483179
$238$	−18.2462	−1.18273
$239$	13.3693i	0.864789i	0.901684	+	0.432395i	$0.142331\pi$
$239$	13.3693i	0.864789i	−0.901684	+	0.432395i	$0.857669\pi$
$240$	1.00000i	0.0645497i
$241$	19.8769i	1.28038i	0.768215	+	0.640192i	$0.221145\pi$
$241$	19.8769i	1.28038i	−0.768215	+	0.640192i	$0.778855\pi$
$242$	− 6.00000i	− 0.385695i
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	− 5.68466i	− 0.363180i
$246$	4.24621	0.270729
$247$	0	0
$248$	5.68466	0.360976
$249$	1.12311i	0.0711739i
$250$	1.00000	0.0632456
$251$	−0.123106	−0.00777036	−0.00388518	−	0.999992i	$-0.501237\pi$
$251$	−0.123106	−0.00777036	−0.00388518	+	0.999992i	$0.501237\pi$
$252$	3.56155i	0.224357i
$253$	31.6847i	1.99200i
$254$	4.43845i	0.278493i
$255$	− 5.12311i	− 0.320821i
$256$	1.00000	0.0625000
$257$	1.43845	0.0897279	0.0448639	−	0.998993i	$-0.485715\pi$
$257$	1.43845	0.0897279	0.0448639	+	0.998993i	$0.485715\pi$
$258$	− 4.56155i	− 0.283990i
$259$	−14.6847	−0.912460
$260$	0	0
$261$	6.56155	0.406150
$262$	6.12311i	0.378287i
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	4.12311	0.253760
$265$	4.43845i	0.272652i
$266$	− 12.6847i	− 0.777746i
$267$	− 1.80776i	− 0.110633i
$268$	− 14.2462i	− 0.870226i
$269$	−3.36932	−0.205431	−0.102715	−	0.994711i	$-0.532753\pi$
$269$	−3.36932	−0.205431	−0.102715	+	0.994711i	$0.532753\pi$
$270$	−1.00000	−0.0608581
$271$	− 32.1771i	− 1.95462i	−0.211818	−	0.977309i	$-0.567938\pi$
$271$	− 32.1771i	− 1.95462i	0.211818	−	0.977309i	$-0.432062\pi$
$272$	−5.12311	−0.310634
$273$	0	0
$274$	−8.80776	−0.532096
$275$	− 4.12311i	− 0.248633i
$276$	−7.68466	−0.462562
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	8.43845i	0.506104i
$279$	5.68466i	0.340332i
$280$	− 3.56155i	− 0.212843i
$281$	0.246211i	0.0146877i	0.999973	+	0.00734387i	$0.00233765\pi$
$281$	0.246211i	0.0146877i	−0.999973	+	0.00734387i	$0.997662\pi$
$282$	7.00000	0.416844
$283$	−11.4384	−0.679945	−0.339973	−	0.940435i	$-0.610418\pi$
$283$	−11.4384	−0.679945	−0.339973	+	0.940435i	$0.610418\pi$
$284$	4.87689i	0.289390i
$285$	3.56155	0.210968
$286$	0	0
$287$	−15.1231	−0.892689
$288$	1.00000i	0.0589256i
$289$	9.24621	0.543895
$290$	−6.56155	−0.385308
$291$	1.12311i	0.0658376i
$292$	− 15.3693i	− 0.899421i
$293$	24.9309i	1.45648i	0.685324	+	0.728238i	$0.259661\pi$
$293$	24.9309i	1.45648i	−0.685324	+	0.728238i	$0.740339\pi$
$294$	− 5.68466i	− 0.331536i
$295$	10.5616	0.614917
$296$	−4.12311	−0.239651
$297$	4.12311i	0.239247i
$298$	22.8078	1.32122
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.2462i	0.936416i
$302$	9.36932	0.539144
$303$	17.1231	0.983697
$304$	− 3.56155i	− 0.204269i
$305$	6.00000i	0.343559i
$306$	− 5.12311i	− 0.292868i
$307$	15.6155i	0.891225i	0.895226	+	0.445613i	$0.147014\pi$
$307$	15.6155i	0.891225i	−0.895226	+	0.445613i	$0.852986\pi$
$308$	−14.6847	−0.836736
$309$	−0.438447	−0.0249424
$310$	− 5.68466i	− 0.322867i
$311$	18.7386	1.06257	0.531285	−	0.847193i	$-0.321709\pi$
$311$	18.7386	1.06257	0.531285	+	0.847193i	$0.321709\pi$
$312$	0	0
$313$	6.63068	0.374788	0.187394	−	0.982285i	$-0.439996\pi$
$313$	6.63068	0.374788	0.187394	+	0.982285i	$0.439996\pi$
$314$	22.1231i	1.24848i
$315$	3.56155	0.200671
$316$	−7.43845	−0.418445
$317$	− 4.19224i	− 0.235459i	−0.993046	−	0.117730i	$-0.962438\pi$
$317$	− 4.19224i	− 0.235459i	0.993046	−	0.117730i	$-0.0375616\pi$
$318$	4.43845i	0.248896i
$319$	27.0540i	1.51473i
$320$	− 1.00000i	− 0.0559017i
$321$	2.00000	0.111629
$322$	27.3693	1.52523
$323$	18.2462i	1.01525i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−18.5616	−1.02803
$327$	− 20.2462i	− 1.11962i
$328$	−4.24621	−0.234458
$329$	−24.9309	−1.37448
$330$	− 4.12311i	− 0.226969i
$331$	− 18.7386i	− 1.02997i	−0.857200	−	0.514984i	$-0.827798\pi$
$331$	− 18.7386i	− 1.02997i	0.857200	−	0.514984i	$-0.172202\pi$
$332$	− 1.12311i	− 0.0616384i
$333$	− 4.12311i	− 0.225945i
$334$	−20.3693	−1.11456
$335$	−14.2462	−0.778354
$336$	− 3.56155i	− 0.194299i
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	−3.68466	−0.200123
$340$	5.12311i	0.277839i
$341$	−23.4384	−1.26926
$342$	3.56155	0.192587
$343$	− 4.68466i	− 0.252948i
$344$	4.56155i	0.245942i
$345$	7.68466i	0.413728i
$346$	25.1771i	1.35353i
$347$	9.12311	0.489754	0.244877	−	0.969554i	$-0.421252\pi$
$347$	9.12311	0.489754	0.244877	+	0.969554i	$0.421252\pi$
$348$	−6.56155	−0.351736
$349$	24.4924i	1.31105i	0.755174	+	0.655525i	$0.227552\pi$
$349$	24.4924i	1.31105i	−0.755174	+	0.655525i	$0.772448\pi$
$350$	−3.56155	−0.190373
$351$	0	0
$352$	−4.12311	−0.219762
$353$	31.8617i	1.69583i	0.530133	+	0.847915i	$0.322142\pi$
$353$	31.8617i	1.69583i	−0.530133	+	0.847915i	$0.677858\pi$
$354$	10.5616	0.561340
$355$	4.87689	0.258839
$356$	1.80776i	0.0958113i
$357$	18.2462i	0.965692i
$358$	0.315342i	0.0166663i
$359$	− 4.87689i	− 0.257393i	−0.991684	−	0.128696i	$-0.958921\pi$
$359$	− 4.87689i	− 0.257393i	0.991684	−	0.128696i	$-0.0410792\pi$
$360$	1.00000	0.0527046
$361$	6.31534	0.332386
$362$	− 11.1231i	− 0.584617i
$363$	−6.00000	−0.314918
$364$	0	0
$365$	−15.3693	−0.804467
$366$	6.00000i	0.313625i
$367$	−20.4924	−1.06970	−0.534848	−	0.844948i	$-0.679631\pi$
$367$	−20.4924	−1.06970	−0.534848	+	0.844948i	$0.679631\pi$
$368$	7.68466	0.400591
$369$	− 4.24621i	− 0.221049i
$370$	4.12311i	0.214350i
$371$	− 15.8078i	− 0.820698i
$372$	− 5.68466i	− 0.294736i
$373$	−5.19224	−0.268844	−0.134422	−	0.990924i	$-0.542918\pi$
$373$	−5.19224	−0.268844	−0.134422	+	0.990924i	$0.542918\pi$
$374$	21.1231	1.09225
$375$	− 1.00000i	− 0.0516398i
$376$	−7.00000	−0.360997
$377$	0	0
$378$	3.56155	0.183187
$379$	− 5.31534i	− 0.273031i	−0.990638	−	0.136515i	$-0.956410\pi$
$379$	− 5.31534i	− 0.273031i	0.990638	−	0.136515i	$-0.0435903\pi$
$380$	−3.56155	−0.182704
$381$	4.43845	0.227389
$382$	19.1231i	0.978423i
$383$	20.8078i	1.06323i	0.846987	+	0.531614i	$0.178414\pi$
$383$	20.8078i	1.06323i	−0.846987	+	0.531614i	$0.821586\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	14.6847i	0.748399i
$386$	0	0
$387$	−4.56155	−0.231877
$388$	− 1.12311i	− 0.0570170i
$389$	−19.0540	−0.966075	−0.483037	−	0.875600i	$-0.660467\pi$
$389$	−19.0540	−0.966075	−0.483037	+	0.875600i	$0.660467\pi$
$390$	0	0
$391$	−39.3693	−1.99099
$392$	5.68466i	0.287119i
$393$	6.12311	0.308870
$394$	7.80776	0.393349
$395$	7.43845i	0.374269i
$396$	− 4.12311i	− 0.207194i
$397$	11.8769i	0.596084i	0.954553	+	0.298042i	$0.0963336\pi$
$397$	11.8769i	0.596084i	−0.954553	+	0.298042i	$0.903666\pi$
$398$	− 11.1231i	− 0.557551i
$399$	−12.6847	−0.635027
$400$	−1.00000	−0.0500000
$401$	− 12.6847i	− 0.633442i	−0.948519	−	0.316721i	$-0.897418\pi$
$401$	− 12.6847i	− 0.633442i	0.948519	−	0.316721i	$-0.102582\pi$
$402$	−14.2462	−0.710536
$403$	0	0
$404$	−17.1231	−0.851906
$405$	1.00000i	0.0496904i
$406$	23.3693	1.15980
$407$	17.0000	0.842659
$408$	5.12311i	0.253632i
$409$	1.80776i	0.0893882i	0.999001	+	0.0446941i	$0.0142313\pi$
$409$	1.80776i	0.0893882i	−0.999001	+	0.0446941i	$0.985769\pi$
$410$	4.24621i	0.209705i
$411$	8.80776i	0.434455i
$412$	0.438447	0.0216007
$413$	−37.6155	−1.85094
$414$	7.68466i	0.377680i
$415$	−1.12311	−0.0551311
$416$	0	0
$417$	8.43845	0.413233
$418$	14.6847i	0.718250i
$419$	−0.492423	−0.0240564	−0.0120282	−	0.999928i	$-0.503829\pi$
$419$	−0.492423	−0.0240564	−0.0120282	+	0.999928i	$0.503829\pi$
$420$	−3.56155	−0.173786
$421$	0.492423i	0.0239992i	0.999928	+	0.0119996i	$0.00381968\pi$
$421$	0.492423i	0.0239992i	−0.999928	+	0.0119996i	$0.996180\pi$
$422$	6.93087i	0.337389i
$423$	− 7.00000i	− 0.340352i
$424$	− 4.43845i	− 0.215550i
$425$	5.12311	0.248507
$426$	4.87689	0.236286
$427$	− 21.3693i	− 1.03413i
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	4.56155	0.219978
$431$	26.7386i	1.28795i	0.765045	+	0.643977i	$0.222717\pi$
$431$	26.7386i	1.28795i	−0.765045	+	0.643977i	$0.777283\pi$
$432$	1.00000	0.0481125
$433$	−30.7386	−1.47720	−0.738602	−	0.674141i	$-0.764514\pi$
$433$	−30.7386	−1.47720	−0.738602	+	0.674141i	$0.764514\pi$
$434$	20.2462i	0.971849i
$435$	6.56155i	0.314602i
$436$	20.2462i	0.969618i
$437$	− 27.3693i	− 1.30925i
$438$	−15.3693	−0.734374
$439$	−16.8769	−0.805490	−0.402745	−	0.915312i	$-0.631944\pi$
$439$	−16.8769	−0.805490	−0.402745	+	0.915312i	$0.631944\pi$
$440$	4.12311i	0.196561i
$441$	−5.68466	−0.270698
$442$	0	0
$443$	−4.87689	−0.231708	−0.115854	−	0.993266i	$-0.536961\pi$
$443$	−4.87689	−0.231708	−0.115854	+	0.993266i	$0.536961\pi$
$444$	4.12311i	0.195674i
$445$	1.80776	0.0856962
$446$	26.3002	1.24535
$447$	− 22.8078i	− 1.07877i
$448$	3.56155i	0.168268i
$449$	− 25.1771i	− 1.18818i	−0.804399	−	0.594090i	$-0.797512\pi$
$449$	− 25.1771i	− 1.18818i	0.804399	−	0.594090i	$-0.202488\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	17.5076	0.824400
$452$	3.68466	0.173312
$453$	− 9.36932i	− 0.440209i
$454$	20.0000	0.938647
$455$	0	0
$456$	−3.56155	−0.166785
$457$	3.75379i	0.175595i	0.996138	+	0.0877974i	$0.0279828\pi$
$457$	3.75379i	0.175595i	−0.996138	+	0.0877974i	$0.972017\pi$
$458$	−7.75379	−0.362311
$459$	−5.12311	−0.239126
$460$	− 7.68466i	− 0.358299i
$461$	7.05398i	0.328536i	0.986416	+	0.164268i	$0.0525262\pi$
$461$	7.05398i	0.328536i	−0.986416	+	0.164268i	$0.947474\pi$
$462$	14.6847i	0.683192i
$463$	− 33.6155i	− 1.56225i	−0.624377	−	0.781123i	$-0.714647\pi$
$463$	− 33.6155i	− 1.56225i	0.624377	−	0.781123i	$-0.285353\pi$
$464$	6.56155	0.304612
$465$	−5.68466	−0.263620
$466$	17.6847i	0.819226i
$467$	−39.8617	−1.84458	−0.922291	−	0.386497i	$-0.873685\pi$
$467$	−39.8617	−1.84458	−0.922291	+	0.386497i	$0.873685\pi$
$468$	0	0
$469$	50.7386	2.34289
$470$	7.00000i	0.322886i
$471$	22.1231	1.01938
$472$	−10.5616	−0.486135
$473$	− 18.8078i	− 0.864782i
$474$	7.43845i	0.341659i
$475$	3.56155i	0.163415i
$476$	− 18.2462i	− 0.836314i
$477$	4.43845	0.203223
$478$	−13.3693	−0.611498
$479$	21.7538i	0.993956i	0.867763	+	0.496978i	$0.165557\pi$
$479$	21.7538i	0.993956i	−0.867763	+	0.496978i	$0.834443\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−19.8769	−0.905368
$483$	− 27.3693i	− 1.24535i
$484$	6.00000	0.272727
$485$	−1.12311	−0.0509976
$486$	1.00000i	0.0453609i
$487$	− 24.0540i	− 1.08999i	−0.838439	−	0.544995i	$-0.816532\pi$
$487$	− 24.0540i	− 1.08999i	0.838439	−	0.544995i	$-0.183468\pi$
$488$	− 6.00000i	− 0.271607i
$489$	18.5616i	0.839382i
$490$	5.68466	0.256807
$491$	21.5616	0.973059	0.486530	−	0.873664i	$-0.338263\pi$
$491$	21.5616	0.973059	0.486530	+	0.873664i	$0.338263\pi$
$492$	4.24621i	0.191434i
$493$	−33.6155	−1.51397
$494$	0	0
$495$	−4.12311	−0.185320
$496$	5.68466i	0.255249i
$497$	−17.3693	−0.779120
$498$	−1.12311	−0.0503276
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	1.00000i	0.0447214i
$501$	20.3693i	0.910034i
$502$	− 0.123106i	− 0.00549447i
$503$	−5.94602	−0.265120	−0.132560	−	0.991175i	$-0.542320\pi$
$503$	−5.94602	−0.265120	−0.132560	+	0.991175i	$0.542320\pi$
$504$	−3.56155	−0.158644
$505$	17.1231i	0.761968i
$506$	−31.6847	−1.40855
$507$	0	0
$508$	−4.43845	−0.196924
$509$	− 29.5464i	− 1.30962i	−0.755793	−	0.654811i	$-0.772748\pi$
$509$	− 29.5464i	− 1.30962i	0.755793	−	0.654811i	$-0.227252\pi$
$510$	5.12311	0.226855
$511$	54.7386	2.42149
$512$	1.00000i	0.0441942i
$513$	− 3.56155i	− 0.157246i
$514$	1.43845i	0.0634472i
$515$	− 0.438447i	− 0.0193203i
$516$	4.56155	0.200811
$517$	28.8617	1.26934
$518$	− 14.6847i	− 0.645207i
$519$	25.1771	1.10515
$520$	0	0
$521$	−18.6847	−0.818590	−0.409295	−	0.912402i	$-0.634225\pi$
$521$	−18.6847	−0.818590	−0.409295	+	0.912402i	$0.634225\pi$
$522$	6.56155i	0.287191i
$523$	−9.19224	−0.401948	−0.200974	−	0.979597i	$-0.564411\pi$
$523$	−9.19224	−0.401948	−0.200974	+	0.979597i	$0.564411\pi$
$524$	−6.12311	−0.267489
$525$	3.56155i	0.155439i
$526$	− 13.0000i	− 0.566827i
$527$	− 29.1231i	− 1.26862i
$528$	4.12311i	0.179435i
$529$	36.0540	1.56756
$530$	−4.43845	−0.192794
$531$	− 10.5616i	− 0.458332i
$532$	12.6847	0.549950
$533$	0	0
$534$	1.80776	0.0782296
$535$	2.00000i	0.0864675i
$536$	14.2462	0.615343
$537$	0.315342	0.0136080
$538$	− 3.36932i	− 0.145262i
$539$	− 23.4384i	− 1.00957i
$540$	− 1.00000i	− 0.0430331i
$541$	2.63068i	0.113102i	0.998400	+	0.0565510i	$0.0180103\pi$
$541$	2.63068i	0.113102i	−0.998400	+	0.0565510i	$0.981990\pi$
$542$	32.1771	1.38212
$543$	−11.1231	−0.477338
$544$	− 5.12311i	− 0.219651i
$545$	20.2462	0.867252
$546$	0	0
$547$	35.6155	1.52281	0.761405	−	0.648276i	$-0.224510\pi$
$547$	35.6155	1.52281	0.761405	+	0.648276i	$0.224510\pi$
$548$	− 8.80776i	− 0.376249i
$549$	6.00000	0.256074
$550$	4.12311	0.175810
$551$	− 23.3693i	− 0.995566i
$552$	− 7.68466i	− 0.327081i
$553$	− 26.4924i	− 1.12657i
$554$	1.00000i	0.0424859i
$555$	4.12311	0.175016
$556$	−8.43845	−0.357870
$557$	4.43845i	0.188063i	0.995569	+	0.0940315i	$0.0299754\pi$
$557$	4.43845i	0.188063i	−0.995569	+	0.0940315i	$0.970025\pi$
$558$	−5.68466	−0.240651
$559$	0	0
$560$	3.56155	0.150503
$561$	− 21.1231i	− 0.891818i
$562$	−0.246211	−0.0103858
$563$	−32.9848	−1.39015	−0.695073	−	0.718939i	$-0.744628\pi$
$563$	−32.9848	−1.39015	−0.695073	+	0.718939i	$0.744628\pi$
$564$	7.00000i	0.294753i
$565$	− 3.68466i	− 0.155015i
$566$	− 11.4384i	− 0.480794i
$567$	− 3.56155i	− 0.149571i
$568$	−4.87689	−0.204630
$569$	19.1771	0.803945	0.401973	−	0.915652i	$-0.368325\pi$
$569$	19.1771	0.803945	0.401973	+	0.915652i	$0.368325\pi$
$570$	3.56155i	0.149177i
$571$	−11.3153	−0.473532	−0.236766	−	0.971567i	$-0.576088\pi$
$571$	−11.3153	−0.473532	−0.236766	+	0.971567i	$0.576088\pi$
$572$	0	0
$573$	19.1231	0.798879
$574$	− 15.1231i	− 0.631226i
$575$	−7.68466	−0.320472
$576$	−1.00000	−0.0416667
$577$	8.73863i	0.363794i	0.983318	+	0.181897i	$0.0582238\pi$
$577$	8.73863i	0.363794i	−0.983318	+	0.181897i	$0.941776\pi$
$578$	9.24621i	0.384592i
$579$	0	0
$580$	− 6.56155i	− 0.272454i
$581$	4.00000	0.165948
$582$	−1.12311	−0.0465542
$583$	18.3002i	0.757916i
$584$	15.3693	0.635987
$585$	0	0
$586$	−24.9309	−1.02988
$587$	− 23.7538i	− 0.980424i	−0.871603	−	0.490212i	$-0.836919\pi$
$587$	− 23.7538i	− 0.980424i	0.871603	−	0.490212i	$-0.163081\pi$
$588$	5.68466	0.234431
$589$	20.2462	0.834231
$590$	10.5616i	0.434812i
$591$	− 7.80776i	− 0.321168i
$592$	− 4.12311i	− 0.169459i
$593$	12.1771i	0.500053i	0.968239	+	0.250026i	$0.0804393\pi$
$593$	12.1771i	0.500053i	−0.968239	+	0.250026i	$0.919561\pi$
$594$	−4.12311	−0.169173
$595$	−18.2462	−0.748022
$596$	22.8078i	0.934242i
$597$	−11.1231	−0.455238
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	1.00000i	0.0408248i
$601$	−35.9848	−1.46785	−0.733926	−	0.679229i	$-0.762314\pi$
$601$	−35.9848	−1.46785	−0.733926	+	0.679229i	$0.762314\pi$
$602$	−16.2462	−0.662146
$603$	14.2462i	0.580151i
$604$	9.36932i	0.381232i
$605$	− 6.00000i	− 0.243935i
$606$	17.1231i	0.695579i
$607$	−29.4233	−1.19425	−0.597127	−	0.802146i	$-0.703691\pi$
$607$	−29.4233	−1.19425	−0.597127	+	0.802146i	$0.703691\pi$
$608$	3.56155	0.144440
$609$	− 23.3693i	− 0.946973i
$610$	−6.00000	−0.242933
$611$	0	0
$612$	5.12311	0.207089
$613$	28.1231i	1.13588i	0.823069	+	0.567941i	$0.192260\pi$
$613$	28.1231i	1.13588i	−0.823069	+	0.567941i	$0.807740\pi$
$614$	−15.6155	−0.630191
$615$	4.24621	0.171224
$616$	− 14.6847i	− 0.591662i
$617$	− 41.6847i	− 1.67816i	−0.544007	−	0.839081i	$-0.683094\pi$
$617$	− 41.6847i	− 1.67816i	0.544007	−	0.839081i	$-0.316906\pi$
$618$	− 0.438447i	− 0.0176369i
$619$	16.4384i	0.660717i	0.943856	+	0.330358i	$0.107170\pi$
$619$	16.4384i	0.660717i	−0.943856	+	0.330358i	$0.892830\pi$
$620$	5.68466	0.228301
$621$	7.68466	0.308375
$622$	18.7386i	0.751351i
$623$	−6.43845	−0.257951
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.63068i	0.265015i
$627$	14.6847	0.586449
$628$	−22.1231	−0.882808
$629$	21.1231i	0.842233i
$630$	3.56155i	0.141896i
$631$	30.2462i	1.20408i	0.798465	+	0.602041i	$0.205646\pi$
$631$	30.2462i	1.20408i	−0.798465	+	0.602041i	$0.794354\pi$
$632$	− 7.43845i	− 0.295886i
$633$	6.93087	0.275477
$634$	4.19224	0.166495
$635$	4.43845i	0.176134i
$636$	−4.43845	−0.175996
$637$	0	0
$638$	−27.0540	−1.07108
$639$	− 4.87689i	− 0.192927i
$640$	1.00000	0.0395285
$641$	15.5616	0.614644	0.307322	−	0.951606i	$-0.400567\pi$
$641$	15.5616	0.614644	0.307322	+	0.951606i	$0.400567\pi$
$642$	2.00000i	0.0789337i
$643$	38.2462i	1.50828i	0.656712	+	0.754142i	$0.271947\pi$
$643$	38.2462i	1.50828i	−0.656712	+	0.754142i	$0.728053\pi$
$644$	27.3693i	1.07850i
$645$	− 4.56155i	− 0.179611i
$646$	−18.2462	−0.717888
$647$	2.05398	0.0807501	0.0403751	−	0.999185i	$-0.487145\pi$
$647$	2.05398	0.0807501	0.0403751	+	0.999185i	$0.487145\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	43.5464	1.70935
$650$	0	0
$651$	20.2462	0.793512
$652$	− 18.5616i	− 0.726927i
$653$	−40.4384	−1.58248	−0.791239	−	0.611507i	$-0.790564\pi$
$653$	−40.4384	−1.58248	−0.791239	+	0.611507i	$0.790564\pi$
$654$	20.2462	0.791690
$655$	6.12311i	0.239250i
$656$	− 4.24621i	− 0.165787i
$657$	15.3693i	0.599614i
$658$	− 24.9309i	− 0.971906i
$659$	31.0540	1.20969	0.604846	−	0.796343i	$-0.293235\pi$
$659$	31.0540	1.20969	0.604846	+	0.796343i	$0.293235\pi$
$660$	4.12311	0.160492
$661$	11.6155i	0.451792i	0.974151	+	0.225896i	$0.0725309\pi$
$661$	11.6155i	0.451792i	−0.974151	+	0.225896i	$0.927469\pi$
$662$	18.7386	0.728298
$663$	0	0
$664$	1.12311	0.0435850
$665$	− 12.6847i	− 0.491890i
$666$	4.12311	0.159767
$667$	50.4233	1.95240
$668$	− 20.3693i	− 0.788113i
$669$	− 26.3002i	− 1.01682i
$670$	− 14.2462i	− 0.550379i
$671$	24.7386i	0.955024i
$672$	3.56155	0.137390
$673$	−42.2462	−1.62847	−0.814236	−	0.580534i	$-0.802844\pi$
$673$	−42.2462	−1.62847	−0.814236	+	0.580534i	$0.802844\pi$
$674$	6.00000i	0.231111i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−28.8769	−1.10983	−0.554915	−	0.831907i	$-0.687249\pi$
$677$	−28.8769	−1.10983	−0.554915	+	0.831907i	$0.687249\pi$
$678$	− 3.68466i	− 0.141508i
$679$	4.00000	0.153506
$680$	−5.12311	−0.196462
$681$	− 20.0000i	− 0.766402i
$682$	− 23.4384i	− 0.897505i
$683$	− 8.87689i	− 0.339665i	−0.985473	−	0.169832i	$-0.945677\pi$
$683$	− 8.87689i	− 0.339665i	0.985473	−	0.169832i	$-0.0543227\pi$
$684$	3.56155i	0.136179i
$685$	−8.80776	−0.336527
$686$	4.68466	0.178861
$687$	7.75379i	0.295825i
$688$	−4.56155	−0.173908
$689$	0	0
$690$	−7.68466	−0.292550
$691$	− 16.4384i	− 0.625348i	−0.949860	−	0.312674i	$-0.898775\pi$
$691$	− 16.4384i	− 0.625348i	0.949860	−	0.312674i	$-0.101225\pi$
$692$	−25.1771	−0.957089
$693$	14.6847	0.557824
$694$	9.12311i	0.346308i
$695$	8.43845i	0.320089i
$696$	− 6.56155i	− 0.248715i
$697$	21.7538i	0.823984i
$698$	−24.4924	−0.927052
$699$	17.6847	0.668895
$700$	− 3.56155i	− 0.134614i
$701$	−17.3002	−0.653419	−0.326710	−	0.945125i	$-0.605940\pi$
$701$	−17.3002	−0.653419	−0.326710	+	0.945125i	$0.605940\pi$
$702$	0	0
$703$	−14.6847	−0.553842
$704$	− 4.12311i	− 0.155395i
$705$	7.00000	0.263635
$706$	−31.8617	−1.19913
$707$	− 60.9848i	− 2.29357i
$708$	10.5616i	0.396927i
$709$	− 17.7538i	− 0.666758i	−0.942793	−	0.333379i	$-0.891811\pi$
$709$	− 17.7538i	− 0.666758i	0.942793	−	0.333379i	$-0.108189\pi$
$710$	4.87689i	0.183027i
$711$	7.43845	0.278964
$712$	−1.80776	−0.0677488
$713$	43.6847i	1.63600i
$714$	−18.2462	−0.682847
$715$	0	0
$716$	−0.315342	−0.0117849
$717$	13.3693i	0.499286i
$718$	4.87689	0.182004
$719$	20.9848	0.782603	0.391301	−	0.920263i	$-0.372025\pi$
$719$	20.9848	0.782603	0.391301	+	0.920263i	$0.372025\pi$
$720$	1.00000i	0.0372678i
$721$	1.56155i	0.0581553i
$722$	6.31534i	0.235033i
$723$	19.8769i	0.739230i
$724$	11.1231	0.413387
$725$	−6.56155	−0.243690
$726$	− 6.00000i	− 0.222681i
$727$	23.4233	0.868722	0.434361	−	0.900739i	$-0.356974\pi$
$727$	23.4233	0.868722	0.434361	+	0.900739i	$0.356974\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 15.3693i	− 0.568844i
$731$	23.3693	0.864345
$732$	−6.00000	−0.221766
$733$	48.9309i	1.80730i	0.428269	+	0.903651i	$0.359124\pi$
$733$	48.9309i	1.80730i	−0.428269	+	0.903651i	$0.640876\pi$
$734$	− 20.4924i	− 0.756389i
$735$	− 5.68466i	− 0.209682i
$736$	7.68466i	0.283260i
$737$	−58.7386	−2.16367
$738$	4.24621	0.156305
$739$	− 42.5464i	− 1.56509i	−0.622591	−	0.782547i	$-0.713920\pi$
$739$	− 42.5464i	− 1.56509i	0.622591	−	0.782547i	$-0.286080\pi$
$740$	−4.12311	−0.151568
$741$	0	0
$742$	15.8078	0.580321
$743$	− 32.1771i	− 1.18046i	−0.807234	−	0.590231i	$-0.799037\pi$
$743$	− 32.1771i	− 1.18046i	0.807234	−	0.590231i	$-0.200963\pi$
$744$	5.68466	0.208410
$745$	22.8078	0.835612
$746$	− 5.19224i	− 0.190101i
$747$	1.12311i	0.0410923i
$748$	21.1231i	0.772337i
$749$	− 7.12311i	− 0.260273i
$750$	1.00000	0.0365148
$751$	3.19224	0.116486	0.0582432	−	0.998302i	$-0.481450\pi$
$751$	3.19224	0.116486	0.0582432	+	0.998302i	$0.481450\pi$
$752$	− 7.00000i	− 0.255264i
$753$	−0.123106	−0.00448622
$754$	0	0
$755$	9.36932	0.340984
$756$	3.56155i	0.129532i
$757$	33.4233	1.21479	0.607395	−	0.794400i	$-0.292215\pi$
$757$	33.4233	1.21479	0.607395	+	0.794400i	$0.292215\pi$
$758$	5.31534	0.193062
$759$	31.6847i	1.15008i
$760$	− 3.56155i	− 0.129191i
$761$	− 10.9309i	− 0.396244i	−0.980177	−	0.198122i	$-0.936516\pi$
$761$	− 10.9309i	− 0.396244i	0.980177	−	0.198122i	$-0.0634842\pi$
$762$	4.43845i	0.160788i
$763$	−72.1080	−2.61048
$764$	−19.1231	−0.691850
$765$	− 5.12311i	− 0.185226i
$766$	−20.8078	−0.751815
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 45.6847i	− 1.64743i	−0.567003	−	0.823715i	$-0.691897\pi$
$769$	− 45.6847i	− 1.64743i	0.567003	−	0.823715i	$-0.308103\pi$
$770$	−14.6847	−0.529198
$771$	1.43845	0.0518044
$772$	0	0
$773$	− 21.1771i	− 0.761687i	−0.924640	−	0.380843i	$-0.875634\pi$
$773$	− 21.1771i	− 0.761687i	0.924640	−	0.380843i	$-0.124366\pi$
$774$	− 4.56155i	− 0.163962i
$775$	− 5.68466i	− 0.204199i
$776$	1.12311	0.0403171
$777$	−14.6847	−0.526809
$778$	− 19.0540i	− 0.683118i
$779$	−15.1231	−0.541841
$780$	0	0
$781$	20.1080	0.719519
$782$	− 39.3693i	− 1.40784i
$783$	6.56155	0.234491
$784$	−5.68466	−0.203024
$785$	22.1231i	0.789607i
$786$	6.12311i	0.218404i
$787$	− 43.6847i	− 1.55719i	−0.627527	−	0.778595i	$-0.715933\pi$
$787$	− 43.6847i	− 1.55719i	0.627527	−	0.778595i	$-0.284067\pi$
$788$	7.80776i	0.278140i
$789$	−13.0000	−0.462812
$790$	−7.43845	−0.264648
$791$	13.1231i	0.466604i
$792$	4.12311	0.146508
$793$	0	0
$794$	−11.8769	−0.421495
$795$	4.43845i	0.157415i
$796$	11.1231	0.394248
$797$	−4.87689	−0.172748	−0.0863742	−	0.996263i	$-0.527528\pi$
$797$	−4.87689	−0.172748	−0.0863742	+	0.996263i	$0.527528\pi$
$798$	− 12.6847i	− 0.449032i
$799$	35.8617i	1.26870i
$800$	− 1.00000i	− 0.0353553i
$801$	− 1.80776i	− 0.0638742i
$802$	12.6847	0.447911
$803$	−63.3693	−2.23625
$804$	− 14.2462i	− 0.502425i
$805$	27.3693	0.964642
$806$	0	0
$807$	−3.36932	−0.118606
$808$	− 17.1231i	− 0.602389i
$809$	18.4924	0.650159	0.325079	−	0.945687i	$-0.394609\pi$
$809$	18.4924	0.650159	0.325079	+	0.945687i	$0.394609\pi$
$810$	−1.00000	−0.0351364
$811$	− 22.5464i	− 0.791711i	−0.918313	−	0.395856i	$-0.870448\pi$
$811$	− 22.5464i	− 0.791711i	0.918313	−	0.395856i	$-0.129552\pi$
$812$	23.3693i	0.820102i
$813$	− 32.1771i	− 1.12850i
$814$	17.0000i	0.595850i
$815$	−18.5616	−0.650183
$816$	−5.12311	−0.179345
$817$	16.2462i	0.568383i
$818$	−1.80776	−0.0632070
$819$	0	0
$820$	−4.24621	−0.148284
$821$	− 12.5616i	− 0.438401i	−0.975680	−	0.219201i	$-0.929655\pi$
$821$	− 12.5616i	− 0.438401i	0.975680	−	0.219201i	$-0.0703449\pi$
$822$	−8.80776	−0.307206
$823$	32.0540	1.11733	0.558666	−	0.829393i	$-0.311313\pi$
$823$	32.0540	1.11733	0.558666	+	0.829393i	$0.311313\pi$
$824$	0.438447i	0.0152740i
$825$	− 4.12311i	− 0.143548i
$826$	− 37.6155i	− 1.30881i
$827$	11.7538i	0.408719i	0.978896	+	0.204360i	$0.0655112\pi$
$827$	11.7538i	0.408719i	−0.978896	+	0.204360i	$0.934489\pi$
$828$	−7.68466	−0.267060
$829$	−53.6155	−1.86214	−0.931072	−	0.364835i	$-0.881125\pi$
$829$	−53.6155	−1.86214	−0.931072	+	0.364835i	$0.881125\pi$
$830$	− 1.12311i	− 0.0389836i
$831$	1.00000	0.0346896
$832$	0	0
$833$	29.1231	1.00906
$834$	8.43845i	0.292200i
$835$	−20.3693	−0.704909
$836$	−14.6847	−0.507880
$837$	5.68466i	0.196491i
$838$	− 0.492423i	− 0.0170105i
$839$	12.7386i	0.439786i	0.975524	+	0.219893i	$0.0705709\pi$
$839$	12.7386i	0.439786i	−0.975524	+	0.219893i	$0.929429\pi$
$840$	− 3.56155i	− 0.122885i
$841$	14.0540	0.484620
$842$	−0.492423	−0.0169700
$843$	0.246211i	0.00847997i
$844$	−6.93087	−0.238570
$845$	0	0
$846$	7.00000	0.240665
$847$	21.3693i	0.734258i
$848$	4.43845	0.152417
$849$	−11.4384	−0.392566
$850$	5.12311i	0.175721i
$851$	− 31.6847i	− 1.08614i
$852$	4.87689i	0.167080i
$853$	51.3002i	1.75648i	0.478216	+	0.878242i	$0.341284\pi$
$853$	51.3002i	1.75648i	−0.478216	+	0.878242i	$0.658716\pi$
$854$	21.3693	0.731243
$855$	3.56155	0.121803
$856$	− 2.00000i	− 0.0683586i
$857$	26.8078	0.915736	0.457868	−	0.889020i	$-0.348613\pi$
$857$	26.8078	0.915736	0.457868	+	0.889020i	$0.348613\pi$
$858$	0	0
$859$	−46.7926	−1.59654	−0.798272	−	0.602298i	$-0.794252\pi$
$859$	−46.7926	−1.59654	−0.798272	+	0.602298i	$0.794252\pi$
$860$	4.56155i	0.155548i
$861$	−15.1231	−0.515394
$862$	−26.7386	−0.910721
$863$	6.56155i	0.223358i	0.993744	+	0.111679i	$0.0356228\pi$
$863$	6.56155i	0.223358i	−0.993744	+	0.111679i	$0.964377\pi$
$864$	1.00000i	0.0340207i
$865$	25.1771i	0.856046i
$866$	− 30.7386i	− 1.04454i
$867$	9.24621	0.314018
$868$	−20.2462	−0.687201
$869$	30.6695i	1.04039i
$870$	−6.56155	−0.222457
$871$	0	0
$872$	−20.2462	−0.685623
$873$	1.12311i	0.0380114i
$874$	27.3693	0.925781
$875$	−3.56155	−0.120402
$876$	− 15.3693i	− 0.519281i
$877$	12.5616i	0.424173i	0.977251	+	0.212087i	$0.0680259\pi$
$877$	12.5616i	0.424173i	−0.977251	+	0.212087i	$0.931974\pi$
$878$	− 16.8769i	− 0.569568i
$879$	24.9309i	0.840897i
$880$	−4.12311	−0.138990
$881$	20.4384	0.688589	0.344294	−	0.938862i	$-0.388118\pi$
$881$	20.4384	0.688589	0.344294	+	0.938862i	$0.388118\pi$
$882$	− 5.68466i	− 0.191412i
$883$	−26.1771	−0.880929	−0.440464	−	0.897770i	$-0.645186\pi$
$883$	−26.1771	−0.880929	−0.440464	+	0.897770i	$0.645186\pi$
$884$	0	0
$885$	10.5616	0.355023
$886$	− 4.87689i	− 0.163842i
$887$	41.1080	1.38027	0.690135	−	0.723681i	$-0.257551\pi$
$887$	41.1080	1.38027	0.690135	+	0.723681i	$0.257551\pi$
$888$	−4.12311	−0.138362
$889$	− 15.8078i	− 0.530175i
$890$	1.80776i	0.0605964i
$891$	4.12311i	0.138129i
$892$	26.3002i	0.880595i
$893$	−24.9309	−0.834280
$894$	22.8078	0.762806
$895$	0.315342i	0.0105407i
$896$	−3.56155	−0.118983
$897$	0	0
$898$	25.1771	0.840170
$899$	37.3002i	1.24403i
$900$	1.00000	0.0333333
$901$	−22.7386	−0.757534
$902$	17.5076i	0.582939i
$903$	16.2462i	0.540640i
$904$	3.68466i	0.122550i
$905$	− 11.1231i	− 0.369745i
$906$	9.36932	0.311275
$907$	−40.9157	−1.35858	−0.679292	−	0.733868i	$-0.737713\pi$
$907$	−40.9157	−1.35858	−0.679292	+	0.733868i	$0.737713\pi$
$908$	20.0000i	0.663723i
$909$	17.1231	0.567938
$910$	0	0
$911$	−43.3693	−1.43689	−0.718445	−	0.695584i	$-0.755146\pi$
$911$	−43.3693	−1.43689	−0.718445	+	0.695584i	$0.755146\pi$
$912$	− 3.56155i	− 0.117935i
$913$	−4.63068	−0.153253
$914$	−3.75379	−0.124164
$915$	6.00000i	0.198354i
$916$	− 7.75379i	− 0.256192i
$917$	− 21.8078i	− 0.720156i
$918$	− 5.12311i	− 0.169088i
$919$	41.3693	1.36465	0.682324	−	0.731050i	$-0.260969\pi$
$919$	41.3693	1.36465	0.682324	+	0.731050i	$0.260969\pi$
$920$	7.68466	0.253356
$921$	15.6155i	0.514549i
$922$	−7.05398	−0.232310
$923$	0	0
$924$	−14.6847	−0.483090
$925$	4.12311i	0.135567i
$926$	33.6155	1.10467
$927$	−0.438447	−0.0144005
$928$	6.56155i	0.215394i
$929$	− 16.8769i	− 0.553713i	−0.960911	−	0.276856i	$-0.910707\pi$
$929$	− 16.8769i	− 0.553713i	0.960911	−	0.276856i	$-0.0892927\pi$
$930$	− 5.68466i	− 0.186407i
$931$	20.2462i	0.663543i
$932$	−17.6847	−0.579280
$933$	18.7386	0.613475
$934$	− 39.8617i	− 1.30432i
$935$	21.1231	0.690799
$936$	0	0
$937$	−29.2311	−0.954937	−0.477468	−	0.878649i	$-0.658446\pi$
$937$	−29.2311	−0.954937	−0.477468	+	0.878649i	$0.658446\pi$
$938$	50.7386i	1.65668i
$939$	6.63068	0.216384
$940$	−7.00000	−0.228315
$941$	− 0.630683i	− 0.0205597i	−0.999947	−	0.0102798i	$-0.996728\pi$
$941$	− 0.630683i	− 0.0205597i	0.999947	−	0.0102798i	$-0.00327223\pi$
$942$	22.1231i	0.720810i
$943$	− 32.6307i	− 1.06260i
$944$	− 10.5616i	− 0.343749i
$945$	3.56155	0.115857
$946$	18.8078	0.611493
$947$	− 13.8617i	− 0.450446i	−0.974307	−	0.225223i	$-0.927689\pi$
$947$	− 13.8617i	− 0.450446i	0.974307	−	0.225223i	$-0.0723111\pi$
$948$	−7.43845	−0.241590
$949$	0	0
$950$	−3.56155	−0.115552
$951$	− 4.19224i	− 0.135943i
$952$	18.2462	0.591363
$953$	−5.82292	−0.188623	−0.0943114	−	0.995543i	$-0.530065\pi$
$953$	−5.82292	−0.188623	−0.0943114	+	0.995543i	$0.530065\pi$
$954$	4.43845i	0.143700i
$955$	19.1231i	0.618809i
$956$	− 13.3693i	− 0.432395i
$957$	27.0540i	0.874531i
$958$	−21.7538	−0.702833
$959$	31.3693	1.01297
$960$	− 1.00000i	− 0.0322749i
$961$	−1.31534	−0.0424304
$962$	0	0
$963$	2.00000	0.0644491
$964$	− 19.8769i	− 0.640192i
$965$	0	0
$966$	27.3693	0.880593
$967$	3.31534i	0.106614i	0.998578	+	0.0533071i	$0.0169762\pi$
$967$	3.31534i	0.106614i	−0.998578	+	0.0533071i	$0.983024\pi$
$968$	6.00000i	0.192847i
$969$	18.2462i	0.586153i
$970$	− 1.12311i	− 0.0360607i
$971$	−16.6847	−0.535436	−0.267718	−	0.963497i	$-0.586270\pi$
$971$	−16.6847	−0.535436	−0.267718	+	0.963497i	$0.586270\pi$
$972$	−1.00000	−0.0320750
$973$	− 30.0540i	− 0.963486i
$974$	24.0540	0.770739
$975$	0	0
$976$	6.00000	0.192055
$977$	5.30019i	0.169568i	0.996399	+	0.0847840i	$0.0270200\pi$
$977$	5.30019i	0.169568i	−0.996399	+	0.0847840i	$0.972980\pi$
$978$	−18.5616	−0.593533
$979$	7.45360	0.238218
$980$	5.68466i	0.181590i
$981$	− 20.2462i	− 0.646412i
$982$	21.5616i	0.688057i
$983$	− 23.8769i	− 0.761555i	−0.924667	−	0.380777i	$-0.875656\pi$
$983$	− 23.8769i	− 0.761555i	0.924667	−	0.380777i	$-0.124344\pi$
$984$	−4.24621	−0.135364
$985$	7.80776	0.248776
$986$	− 33.6155i	− 1.07054i
$987$	−24.9309	−0.793558
$988$	0	0
$989$	−35.0540	−1.11465
$990$	− 4.12311i	− 0.131041i
$991$	−12.4233	−0.394639	−0.197319	−	0.980339i	$-0.563224\pi$
$991$	−12.4233	−0.394639	−0.197319	+	0.980339i	$0.563224\pi$
$992$	−5.68466	−0.180488
$993$	− 18.7386i	− 0.594653i
$994$	− 17.3693i	− 0.550921i
$995$	− 11.1231i	− 0.352626i
$996$	− 1.12311i	− 0.0355870i
$997$	32.5464	1.03075	0.515377	−	0.856963i	$-0.327652\pi$
$997$	32.5464	1.03075	0.515377	+	0.856963i	$0.327652\pi$
$998$	−4.00000	−0.126618
$999$	− 4.12311i	− 0.130449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.r.1351.3		4
13.2	odd	12		390.2.i.g.61.1	✓	4
13.5	odd	4		5070.2.a.bi.1.2		2
13.6	odd	12		390.2.i.g.211.1	yes	4
13.8	odd	4		5070.2.a.bb.1.1		2
13.12	even	2	inner	5070.2.b.r.1351.2		4
39.2	even	12		1170.2.i.o.451.1		4
39.32	even	12		1170.2.i.o.991.1		4
65.2	even	12		1950.2.z.n.1699.2		8
65.19	odd	12		1950.2.i.bi.601.2		4
65.28	even	12		1950.2.z.n.1699.3		8
65.32	even	12		1950.2.z.n.1849.3		8
65.54	odd	12		1950.2.i.bi.451.2		4
65.58	even	12		1950.2.z.n.1849.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.i.g.61.1	✓	4	13.2	odd	12
390.2.i.g.211.1	yes	4	13.6	odd	12
1170.2.i.o.451.1		4	39.2	even	12
1170.2.i.o.991.1		4	39.32	even	12
1950.2.i.bi.451.2		4	65.54	odd	12
1950.2.i.bi.601.2		4	65.19	odd	12
1950.2.z.n.1699.2		8	65.2	even	12
1950.2.z.n.1699.3		8	65.28	even	12
1950.2.z.n.1849.2		8	65.58	even	12
1950.2.z.n.1849.3		8	65.32	even	12
5070.2.a.bb.1.1		2	13.8	odd	4
5070.2.a.bi.1.2		2	13.5	odd	4
5070.2.b.r.1351.2		4	13.12	even	2	inner
5070.2.b.r.1351.3		4	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -3.56155i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -3.56155i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.12311i q^{11} -1.00000 q^{12} +3.56155 q^{14} +1.00000i q^{15} +1.00000 q^{16} -5.12311 q^{17} +1.00000i q^{18} -3.56155i q^{19} -1.00000i q^{20} -3.56155i q^{21} -4.12311 q^{22} +7.68466 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.56155i q^{28} +6.56155 q^{29} -1.00000 q^{30} +5.68466i q^{31} +1.00000i q^{32} +4.12311i q^{33} -5.12311i q^{34} +3.56155 q^{35} -1.00000 q^{36} -4.12311i q^{37} +3.56155 q^{38} +1.00000 q^{40} -4.24621i q^{41} +3.56155 q^{42} -4.56155 q^{43} -4.12311i q^{44} +1.00000i q^{45} +7.68466i q^{46} -7.00000i q^{47} +1.00000 q^{48} -5.68466 q^{49} -1.00000i q^{50} -5.12311 q^{51} +4.43845 q^{53} +1.00000i q^{54} -4.12311 q^{55} -3.56155 q^{56} -3.56155i q^{57} +6.56155i q^{58} -10.5616i q^{59} -1.00000i q^{60} +6.00000 q^{61} -5.68466 q^{62} -3.56155i q^{63} -1.00000 q^{64} -4.12311 q^{66} +14.2462i q^{67} +5.12311 q^{68} +7.68466 q^{69} +3.56155i q^{70} -4.87689i q^{71} -1.00000i q^{72} +15.3693i q^{73} +4.12311 q^{74} -1.00000 q^{75} +3.56155i q^{76} +14.6847 q^{77} +7.43845 q^{79} +1.00000i q^{80} +1.00000 q^{81} +4.24621 q^{82} +1.12311i q^{83} +3.56155i q^{84} -5.12311i q^{85} -4.56155i q^{86} +6.56155 q^{87} +4.12311 q^{88} -1.80776i q^{89} -1.00000 q^{90} -7.68466 q^{92} +5.68466i q^{93} +7.00000 q^{94} +3.56155 q^{95} +1.00000i q^{96} +1.12311i q^{97} -5.68466i q^{98} +4.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 6 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{23} - 4 q^{25} + 4 q^{27} + 18 q^{29} - 4 q^{30} + 6 q^{35} - 4 q^{36} + 6 q^{38} + 4 q^{40} + 6 q^{42} - 10 q^{43} + 4 q^{48} + 2 q^{49} - 4 q^{51} + 26 q^{53} - 6 q^{56} + 24 q^{61} + 2 q^{62} - 4 q^{64} + 4 q^{68} + 6 q^{69} - 4 q^{75} + 34 q^{77} + 38 q^{79} + 4 q^{81} - 16 q^{82} + 18 q^{87} - 4 q^{90} - 6 q^{92} + 28 q^{94} + 6 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 6 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^23 - 4 * q^25 + 4 * q^27 + 18 * q^29 - 4 * q^30 + 6 * q^35 - 4 * q^36 + 6 * q^38 + 4 * q^40 + 6 * q^42 - 10 * q^43 + 4 * q^48 + 2 * q^49 - 4 * q^51 + 26 * q^53 - 6 * q^56 + 24 * q^61 + 2 * q^62 - 4 * q^64 + 4 * q^68 + 6 * q^69 - 4 * q^75 + 34 * q^77 + 38 * q^79 + 4 * q^81 - 16 * q^82 + 18 * q^87 - 4 * q^90 - 6 * q^92 + 28 * q^94 + 6 * q^95

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