LMFDB - Embedded newform 2366.2.d.g.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.d (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:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2366.337
Dual form			2366.2.d.g.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +4.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +4.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +4.00000 q^{10} +1.00000i q^{11} -1.00000 q^{12} +1.00000 q^{14} +4.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +2.00000i q^{18} +2.00000i q^{19} -4.00000i q^{20} +1.00000i q^{21} +1.00000 q^{22} +7.00000 q^{23} +1.00000i q^{24} -11.0000 q^{25} -5.00000 q^{27} -1.00000i q^{28} -8.00000 q^{29} +4.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} +1.00000i q^{33} +4.00000i q^{34} -4.00000 q^{35} +2.00000 q^{36} -7.00000i q^{37} +2.00000 q^{38} -4.00000 q^{40} -7.00000i q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000i q^{44} -8.00000i q^{45} -7.00000i q^{46} -3.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +11.0000i q^{50} -4.00000 q^{51} +5.00000i q^{54} -4.00000 q^{55} -1.00000 q^{56} +2.00000i q^{57} +8.00000i q^{58} +6.00000i q^{59} -4.00000i q^{60} -13.0000 q^{61} +3.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} +7.00000i q^{67} +4.00000 q^{68} +7.00000 q^{69} +4.00000i q^{70} +4.00000i q^{71} -2.00000i q^{72} -9.00000i q^{73} -7.00000 q^{74} -11.0000 q^{75} -2.00000i q^{76} -1.00000 q^{77} -13.0000 q^{79} +4.00000i q^{80} +1.00000 q^{81} -7.00000 q^{82} -16.0000i q^{83} -1.00000i q^{84} -16.0000i q^{85} -8.00000i q^{86} -8.00000 q^{87} -1.00000 q^{88} +6.00000i q^{89} -8.00000 q^{90} -7.00000 q^{92} +3.00000i q^{93} -3.00000 q^{94} -8.00000 q^{95} -1.00000i q^{96} +11.0000i q^{97} +1.00000i q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{22} + 14 q^{23} - 22 q^{25} - 10 q^{27} - 16 q^{29} + 8 q^{30} - 8 q^{35} + 4 q^{36} + 4 q^{38} - 8 q^{40} + 2 q^{42} + 16 q^{43} + 2 q^{48} - 2 q^{49} - 8 q^{51} - 8 q^{55} - 2 q^{56} - 26 q^{61} + 6 q^{62} - 2 q^{64} + 2 q^{66} + 8 q^{68} + 14 q^{69} - 14 q^{74} - 22 q^{75} - 2 q^{77} - 26 q^{79} + 2 q^{81} - 14 q^{82} - 16 q^{87} - 2 q^{88} - 16 q^{90} - 14 q^{92} - 6 q^{94} - 16 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 + 8 * q^10 - 2 * q^12 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^22 + 14 * q^23 - 22 * q^25 - 10 * q^27 - 16 * q^29 + 8 * q^30 - 8 * q^35 + 4 * q^36 + 4 * q^38 - 8 * q^40 + 2 * q^42 + 16 * q^43 + 2 * q^48 - 2 * q^49 - 8 * q^51 - 8 * q^55 - 2 * q^56 - 26 * q^61 + 6 * q^62 - 2 * q^64 + 2 * q^66 + 8 * q^68 + 14 * q^69 - 14 * q^74 - 22 * q^75 - 2 * q^77 - 26 * q^79 + 2 * q^81 - 14 * q^82 - 16 * q^87 - 2 * q^88 - 16 * q^90 - 14 * q^92 - 6 * q^94 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$.

$n$	$339$	$2199$
$\chi(n)$	$1$	$-1$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.00000	−0.500000
$5$	4.00000i	1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$	4.00000i	1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$	− 1.00000i	− 0.408248i
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	−2.00000	−0.666667
$10$	4.00000	1.26491
$11$	1.00000i	0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$	1.00000i	0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	4.00000i	1.03280i
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	2.00000i	0.471405i
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	− 4.00000i	− 0.894427i
$21$	1.00000i	0.218218i
$22$	1.00000	0.213201
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	1.00000i	0.204124i
$25$	−11.0000	−2.20000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	− 1.00000i	− 0.188982i
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	4.00000	0.730297
$31$	3.00000i	0.538816i	0.963026	+	0.269408i	$0.0868280\pi$
$31$	3.00000i	0.538816i	−0.963026	+	0.269408i	$0.913172\pi$
$32$	− 1.00000i	− 0.176777i
$33$	1.00000i	0.174078i
$34$	4.00000i	0.685994i
$35$	−4.00000	−0.676123
$36$	2.00000	0.333333
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−4.00000	−0.632456
$41$	− 7.00000i	− 1.09322i	−0.837389	−	0.546608i	$-0.815919\pi$
$41$	− 7.00000i	− 1.09322i	0.837389	−	0.546608i	$-0.184081\pi$
$42$	1.00000	0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	− 1.00000i	− 0.150756i
$45$	− 8.00000i	− 1.19257i
$46$	− 7.00000i	− 1.03209i
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	11.0000i	1.55563i
$51$	−4.00000	−0.560112
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	5.00000i	0.680414i
$55$	−4.00000	−0.539360
$56$	−1.00000	−0.133631
$57$	2.00000i	0.264906i
$58$	8.00000i	1.05045i
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	− 4.00000i	− 0.516398i
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	3.00000	0.381000
$63$	− 2.00000i	− 0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	4.00000	0.485071
$69$	7.00000	0.842701
$70$	4.00000i	0.478091i
$71$	4.00000i	0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$71$	4.00000i	0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$72$	− 2.00000i	− 0.235702i
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	−7.00000	−0.813733
$75$	−11.0000	−1.27017
$76$	− 2.00000i	− 0.229416i
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−13.0000	−1.46261	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000	−1.46261	−0.731307	+	0.682048i	$0.761089\pi$
$80$	4.00000i	0.447214i
$81$	1.00000	0.111111
$82$	−7.00000	−0.773021
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	− 1.00000i	− 0.109109i
$85$	− 16.0000i	− 1.73544i
$86$	− 8.00000i	− 0.862662i
$87$	−8.00000	−0.857690
$88$	−1.00000	−0.106600
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	−8.00000	−0.843274
$91$	0	0
$92$	−7.00000	−0.729800
$93$	3.00000i	0.311086i
$94$	−3.00000	−0.309426
$95$	−8.00000	−0.820783
$96$	− 1.00000i	− 0.102062i
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	1.00000i	0.101015i
$99$	− 2.00000i	− 0.201008i
$100$	11.0000	1.10000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	4.00000i	0.396059i
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	5.00000	0.481125
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	4.00000i	0.381385i
$111$	− 7.00000i	− 0.664411i
$112$	1.00000i	0.0944911i
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	2.00000	0.187317
$115$	28.0000i	2.61101i
$116$	8.00000	0.742781
$117$	0	0
$118$	6.00000	0.552345
$119$	− 4.00000i	− 0.366679i
$120$	−4.00000	−0.365148
$121$	10.0000	0.909091
$122$	13.0000i	1.17696i
$123$	− 7.00000i	− 0.631169i
$124$	− 3.00000i	− 0.269408i
$125$	− 24.0000i	− 2.14663i
$126$	−2.00000	−0.178174
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000i	0.0883883i
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	−2.00000	−0.173422
$134$	7.00000	0.604708
$135$	− 20.0000i	− 1.72133i
$136$	− 4.00000i	− 0.342997i
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	− 7.00000i	− 0.595880i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.00000	0.338062
$141$	− 3.00000i	− 0.252646i
$142$	4.00000	0.335673
$143$	0	0
$144$	−2.00000	−0.166667
$145$	− 32.0000i	− 2.65746i
$146$	−9.00000	−0.744845
$147$	−1.00000	−0.0824786
$148$	7.00000i	0.575396i
$149$	15.0000i	1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$149$	15.0000i	1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$150$	11.0000i	0.898146i
$151$	4.00000i	0.325515i	0.986666	+	0.162758i	$0.0520389\pi$
$151$	4.00000i	0.325515i	−0.986666	+	0.162758i	$0.947961\pi$
$152$	−2.00000	−0.162221
$153$	8.00000	0.646762
$154$	1.00000i	0.0805823i
$155$	−12.0000	−0.963863
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	13.0000i	1.03422i
$159$	0	0
$160$	4.00000	0.316228
$161$	7.00000i	0.551677i
$162$	− 1.00000i	− 0.0785674i
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	7.00000i	0.546608i
$165$	−4.00000	−0.311400
$166$	−16.0000	−1.24184
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	−16.0000	−1.22714
$171$	− 4.00000i	− 0.305888i
$172$	−8.00000	−0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	8.00000i	0.606478i
$175$	− 11.0000i	− 0.831522i
$176$	1.00000i	0.0753778i
$177$	6.00000i	0.450988i
$178$	6.00000	0.449719
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	8.00000i	0.596285i
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	7.00000i	0.516047i
$185$	28.0000	2.05860
$186$	3.00000	0.219971
$187$	− 4.00000i	− 0.292509i
$188$	3.00000i	0.218797i
$189$	− 5.00000i	− 0.363696i
$190$	8.00000i	0.580381i
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	11.0000	0.789754
$195$	0	0
$196$	1.00000	0.0714286
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	−2.00000	−0.142134
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	− 11.0000i	− 0.777817i
$201$	7.00000i	0.493742i
$202$	9.00000i	0.633238i
$203$	− 8.00000i	− 0.561490i
$204$	4.00000	0.280056
$205$	28.0000	1.95560
$206$	10.0000i	0.696733i
$207$	−14.0000	−0.973067
$208$	0	0
$209$	−2.00000	−0.138343
$210$	4.00000i	0.276026i
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	0	0
$213$	4.00000i	0.274075i
$214$	− 12.0000i	− 0.820303i
$215$	32.0000i	2.18238i
$216$	− 5.00000i	− 0.340207i
$217$	−3.00000	−0.203653
$218$	14.0000	0.948200
$219$	− 9.00000i	− 0.608164i
$220$	4.00000	0.269680
$221$	0	0
$222$	−7.00000	−0.469809
$223$	9.00000i	0.602685i	0.953516	+	0.301342i	$0.0974347\pi$
$223$	9.00000i	0.602685i	−0.953516	+	0.301342i	$0.902565\pi$
$224$	1.00000	0.0668153
$225$	22.0000	1.46667
$226$	− 1.00000i	− 0.0665190i
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	− 2.00000i	− 0.132453i
$229$	20.0000i	1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$	20.0000i	1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$	28.0000	1.84627
$231$	−1.00000	−0.0657952
$232$	− 8.00000i	− 0.525226i
$233$	13.0000	0.851658	0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000	0.851658	0.425829	+	0.904804i	$0.359982\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	− 6.00000i	− 0.390567i
$237$	−13.0000	−0.844441
$238$	−4.00000	−0.259281
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	4.00000i	0.258199i
$241$	10.0000i	0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$	10.0000i	0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$	− 10.0000i	− 0.642824i
$243$	16.0000	1.02640
$244$	13.0000	0.832240
$245$	− 4.00000i	− 0.255551i
$246$	−7.00000	−0.446304
$247$	0	0
$248$	−3.00000	−0.190500
$249$	− 16.0000i	− 1.01396i
$250$	−24.0000	−1.51789
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	2.00000i	0.125988i
$253$	7.00000i	0.440086i
$254$	13.0000i	0.815693i
$255$	− 16.0000i	− 1.00196i
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	− 8.00000i	− 0.498058i
$259$	7.00000	0.434959
$260$	0	0
$261$	16.0000	0.990375
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	2.00000i	0.122628i
$267$	6.00000i	0.367194i
$268$	− 7.00000i	− 0.427593i
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	−20.0000	−1.21716
$271$	3.00000i	0.182237i	0.995840	+	0.0911185i	$0.0290442\pi$
$271$	3.00000i	0.182237i	−0.995840	+	0.0911185i	$0.970956\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	14.0000	0.845771
$275$	− 11.0000i	− 0.663325i
$276$	−7.00000	−0.421350
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	20.0000i	1.19952i
$279$	− 6.00000i	− 0.359211i
$280$	− 4.00000i	− 0.239046i
$281$	20.0000i	1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$	20.0000i	1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$	−3.00000	−0.178647
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	− 4.00000i	− 0.237356i
$285$	−8.00000	−0.473879
$286$	0	0
$287$	7.00000	0.413197
$288$	2.00000i	0.117851i
$289$	−1.00000	−0.0588235
$290$	−32.0000	−1.87910
$291$	11.0000i	0.644831i
$292$	9.00000i	0.526685i
$293$	− 14.0000i	− 0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$	− 14.0000i	− 0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	1.00000i	0.0583212i
$295$	−24.0000	−1.39733
$296$	7.00000	0.406867
$297$	− 5.00000i	− 0.290129i
$298$	15.0000	0.868927
$299$	0	0
$300$	11.0000	0.635085
$301$	8.00000i	0.461112i
$302$	4.00000	0.230174
$303$	−9.00000	−0.517036
$304$	2.00000i	0.114708i
$305$	− 52.0000i	− 2.97751i
$306$	− 8.00000i	− 0.457330i
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	1.00000	0.0569803
$309$	−10.0000	−0.568880
$310$	12.0000i	0.681554i
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	− 7.00000i	− 0.395033i
$315$	8.00000	0.450749
$316$	13.0000	0.731307
$317$	− 13.0000i	− 0.730153i	−0.930978	−	0.365076i	$-0.881043\pi$
$317$	− 13.0000i	− 0.730153i	0.930978	−	0.365076i	$-0.118957\pi$
$318$	0	0
$319$	− 8.00000i	− 0.447914i
$320$	− 4.00000i	− 0.223607i
$321$	12.0000	0.669775
$322$	7.00000	0.390095
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	14.0000i	0.774202i
$328$	7.00000	0.386510
$329$	3.00000	0.165395
$330$	4.00000i	0.220193i
$331$	15.0000i	0.824475i	0.911077	+	0.412237i	$0.135253\pi$
$331$	15.0000i	0.824475i	−0.911077	+	0.412237i	$0.864747\pi$
$332$	16.0000i	0.878114i
$333$	14.0000i	0.767195i
$334$	0	0
$335$	−28.0000	−1.52980
$336$	1.00000i	0.0545545i
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	1.00000	0.0543125
$340$	16.0000i	0.867722i
$341$	−3.00000	−0.162459
$342$	−4.00000	−0.216295
$343$	− 1.00000i	− 0.0539949i
$344$	8.00000i	0.431331i
$345$	28.0000i	1.50747i
$346$	14.0000i	0.752645i
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	8.00000	0.428845
$349$	− 2.00000i	− 0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$	− 2.00000i	− 0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$	−11.0000	−0.587975
$351$	0	0
$352$	1.00000	0.0533002
$353$	− 11.0000i	− 0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$	− 11.0000i	− 0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$	6.00000	0.318896
$355$	−16.0000	−0.849192
$356$	− 6.00000i	− 0.317999i
$357$	− 4.00000i	− 0.211702i
$358$	− 18.0000i	− 0.951330i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	8.00000	0.421637
$361$	15.0000	0.789474
$362$	13.0000i	0.683265i
$363$	10.0000	0.524864
$364$	0	0
$365$	36.0000	1.88433
$366$	13.0000i	0.679521i
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	7.00000	0.364900
$369$	14.0000i	0.728811i
$370$	− 28.0000i	− 1.45565i
$371$	0	0
$372$	− 3.00000i	− 0.155543i
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	−4.00000	−0.206835
$375$	− 24.0000i	− 1.23935i
$376$	3.00000	0.154713
$377$	0	0
$378$	−5.00000	−0.257172
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	8.00000	0.410391
$381$	−13.0000	−0.666010
$382$	− 24.0000i	− 1.22795i
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	1.00000i	0.0510310i
$385$	− 4.00000i	− 0.203859i
$386$	4.00000	0.203595
$387$	−16.0000	−0.813326
$388$	− 11.0000i	− 0.558440i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	3.00000	0.151138
$395$	− 52.0000i	− 2.61640i
$396$	2.00000i	0.100504i
$397$	− 8.00000i	− 0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$	− 8.00000i	− 0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$	16.0000i	0.802008i
$399$	−2.00000	−0.100125
$400$	−11.0000	−0.550000
$401$	− 8.00000i	− 0.399501i	−0.979847	−	0.199750i	$-0.935987\pi$
$401$	− 8.00000i	− 0.399501i	0.979847	−	0.199750i	$-0.0640132\pi$
$402$	7.00000	0.349128
$403$	0	0
$404$	9.00000	0.447767
$405$	4.00000i	0.198762i
$406$	−8.00000	−0.397033
$407$	7.00000	0.346977
$408$	− 4.00000i	− 0.198030i
$409$	− 10.0000i	− 0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$	− 10.0000i	− 0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$	− 28.0000i	− 1.38282i
$411$	14.0000i	0.690569i
$412$	10.0000	0.492665
$413$	−6.00000	−0.295241
$414$	14.0000i	0.688062i
$415$	64.0000	3.14164
$416$	0	0
$417$	−20.0000	−0.979404
$418$	2.00000i	0.0978232i
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	4.00000	0.195180
$421$	− 11.0000i	− 0.536107i	−0.963404	−	0.268054i	$-0.913620\pi$
$421$	− 11.0000i	− 0.536107i	0.963404	−	0.268054i	$-0.0863804\pi$
$422$	− 22.0000i	− 1.07094i
$423$	6.00000i	0.291730i
$424$	0	0
$425$	44.0000	2.13431
$426$	4.00000	0.193801
$427$	− 13.0000i	− 0.629114i
$428$	−12.0000	−0.580042
$429$	0	0
$430$	32.0000	1.54318
$431$	2.00000i	0.0963366i	0.998839	+	0.0481683i	$0.0153384\pi$
$431$	2.00000i	0.0963366i	−0.998839	+	0.0481683i	$0.984662\pi$
$432$	−5.00000	−0.240563
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	3.00000i	0.144005i
$435$	− 32.0000i	− 1.53428i
$436$	− 14.0000i	− 0.670478i
$437$	14.0000i	0.669711i
$438$	−9.00000	−0.430037
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	− 4.00000i	− 0.190693i
$441$	2.00000	0.0952381
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	7.00000i	0.332205i
$445$	−24.0000	−1.13771
$446$	9.00000	0.426162
$447$	15.0000i	0.709476i
$448$	− 1.00000i	− 0.0472456i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	− 22.0000i	− 1.03709i
$451$	7.00000	0.329617
$452$	−1.00000	−0.0470360
$453$	4.00000i	0.187936i
$454$	28.0000	1.31411
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	20.0000	0.934539
$459$	20.0000	0.933520
$460$	− 28.0000i	− 1.30551i
$461$	16.0000i	0.745194i	0.927993	+	0.372597i	$0.121533\pi$
$461$	16.0000i	0.745194i	−0.927993	+	0.372597i	$0.878467\pi$
$462$	1.00000i	0.0465242i
$463$	12.0000i	0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$	12.0000i	0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$	−8.00000	−0.371391
$465$	−12.0000	−0.556487
$466$	− 13.0000i	− 0.602213i
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	− 12.0000i	− 0.553519i
$471$	7.00000	0.322543
$472$	−6.00000	−0.276172
$473$	8.00000i	0.367840i
$474$	13.0000i	0.597110i
$475$	− 22.0000i	− 1.00943i
$476$	4.00000i	0.183340i
$477$	0	0
$478$	−6.00000	−0.274434
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	4.00000	0.182574
$481$	0	0
$482$	10.0000	0.455488
$483$	7.00000i	0.318511i
$484$	−10.0000	−0.454545
$485$	−44.0000	−1.99794
$486$	− 16.0000i	− 0.725775i
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	− 13.0000i	− 0.588482i
$489$	− 12.0000i	− 0.542659i
$490$	−4.00000	−0.180702
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	7.00000i	0.315584i
$493$	32.0000	1.44121
$494$	0	0
$495$	8.00000	0.359573
$496$	3.00000i	0.134704i
$497$	−4.00000	−0.179425
$498$	−16.0000	−0.716977
$499$	37.0000i	1.65635i	0.560471	+	0.828174i	$0.310620\pi$
$499$	37.0000i	1.65635i	−0.560471	+	0.828174i	$0.689380\pi$
$500$	24.0000i	1.07331i
$501$	0	0
$502$	17.0000i	0.758747i
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	2.00000	0.0890871
$505$	− 36.0000i	− 1.60198i
$506$	7.00000	0.311188
$507$	0	0
$508$	13.0000	0.576782
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	−16.0000	−0.708492
$511$	9.00000	0.398137
$512$	− 1.00000i	− 0.0441942i
$513$	− 10.0000i	− 0.441511i
$514$	− 12.0000i	− 0.529297i
$515$	− 40.0000i	− 1.76261i
$516$	−8.00000	−0.352180
$517$	3.00000	0.131940
$518$	− 7.00000i	− 0.307562i
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	− 16.0000i	− 0.700301i
$523$	−15.0000	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$523$	−15.0000	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$524$	0	0
$525$	− 11.0000i	− 0.480079i
$526$	− 16.0000i	− 0.697633i
$527$	− 12.0000i	− 0.522728i
$528$	1.00000i	0.0435194i
$529$	26.0000	1.13043
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	2.00000	0.0867110
$533$	0	0
$534$	6.00000	0.259645
$535$	48.0000i	2.07522i
$536$	−7.00000	−0.302354
$537$	18.0000	0.776757
$538$	− 3.00000i	− 0.129339i
$539$	− 1.00000i	− 0.0430730i
$540$	20.0000i	0.860663i
$541$	38.0000i	1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$	38.0000i	1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	3.00000	0.128861
$543$	−13.0000	−0.557883
$544$	4.00000i	0.171499i
$545$	−56.0000	−2.39878
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	− 14.0000i	− 0.598050i
$549$	26.0000	1.10965
$550$	−11.0000	−0.469042
$551$	− 16.0000i	− 0.681623i
$552$	7.00000i	0.297940i
$553$	− 13.0000i	− 0.552816i
$554$	− 14.0000i	− 0.594803i
$555$	28.0000	1.18853
$556$	20.0000	0.848189
$557$	− 31.0000i	− 1.31351i	−0.754103	−	0.656756i	$-0.771928\pi$
$557$	− 31.0000i	− 1.31351i	0.754103	−	0.656756i	$-0.228072\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	−4.00000	−0.169031
$561$	− 4.00000i	− 0.168880i
$562$	20.0000	0.843649
$563$	31.0000	1.30649	0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000	1.30649	0.653247	+	0.757145i	$0.273406\pi$
$564$	3.00000i	0.126323i
$565$	4.00000i	0.168281i
$566$	17.0000i	0.714563i
$567$	1.00000i	0.0419961i
$568$	−4.00000	−0.167836
$569$	−29.0000	−1.21574	−0.607872	−	0.794035i	$-0.707976\pi$
$569$	−29.0000	−1.21574	−0.607872	+	0.794035i	$0.707976\pi$
$570$	8.00000i	0.335083i
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	− 7.00000i	− 0.292174i
$575$	−77.0000	−3.21112
$576$	2.00000	0.0833333
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	1.00000i	0.0415945i
$579$	4.00000i	0.166234i
$580$	32.0000i	1.32873i
$581$	16.0000	0.663792
$582$	11.0000	0.455965
$583$	0	0
$584$	9.00000	0.372423
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	1.00000	0.0412393
$589$	−6.00000	−0.247226
$590$	24.0000i	0.988064i
$591$	3.00000i	0.123404i
$592$	− 7.00000i	− 0.287698i
$593$	10.0000i	0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$	10.0000i	0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$	−5.00000	−0.205152
$595$	16.0000	0.655936
$596$	− 15.0000i	− 0.614424i
$597$	−16.0000	−0.654836
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	− 11.0000i	− 0.449073i
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	8.00000	0.326056
$603$	− 14.0000i	− 0.570124i
$604$	− 4.00000i	− 0.162758i
$605$	40.0000i	1.62623i
$606$	9.00000i	0.365600i
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	2.00000	0.0811107
$609$	− 8.00000i	− 0.324176i
$610$	−52.0000	−2.10542
$611$	0	0
$612$	−8.00000	−0.323381
$613$	17.0000i	0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$	17.0000i	0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$	12.0000	0.484281
$615$	28.0000	1.12907
$616$	− 1.00000i	− 0.0402911i
$617$	− 38.0000i	− 1.52982i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	− 38.0000i	− 1.52982i	0.644136	−	0.764911i	$-0.277217\pi$
$618$	10.0000i	0.402259i
$619$	− 14.0000i	− 0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$	− 14.0000i	− 0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$	12.0000	0.481932
$621$	−35.0000	−1.40450
$622$	− 30.0000i	− 1.20289i
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	− 6.00000i	− 0.239808i
$627$	−2.00000	−0.0798723
$628$	−7.00000	−0.279330
$629$	28.0000i	1.11643i
$630$	− 8.00000i	− 0.318728i
$631$	− 22.0000i	− 0.875806i	−0.899022	−	0.437903i	$-0.855721\pi$
$631$	− 22.0000i	− 0.875806i	0.899022	−	0.437903i	$-0.144279\pi$
$632$	− 13.0000i	− 0.517112i
$633$	22.0000	0.874421
$634$	−13.0000	−0.516296
$635$	− 52.0000i	− 2.06356i
$636$	0	0
$637$	0	0
$638$	−8.00000	−0.316723
$639$	− 8.00000i	− 0.316475i
$640$	−4.00000	−0.158114
$641$	25.0000	0.987441	0.493720	−	0.869621i	$-0.335637\pi$
$641$	25.0000	0.987441	0.493720	+	0.869621i	$0.335637\pi$
$642$	− 12.0000i	− 0.473602i
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	− 7.00000i	− 0.275839i
$645$	32.0000i	1.26000i
$646$	−8.00000	−0.314756
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	1.00000i	0.0392837i
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−3.00000	−0.117579
$652$	12.0000i	0.469956i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	− 7.00000i	− 0.273304i
$657$	18.0000i	0.702247i
$658$	− 3.00000i	− 0.116952i
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	4.00000	0.155700
$661$	8.00000i	0.311164i	0.987823	+	0.155582i	$0.0497253\pi$
$661$	8.00000i	0.311164i	−0.987823	+	0.155582i	$0.950275\pi$
$662$	15.0000	0.582992
$663$	0	0
$664$	16.0000	0.620920
$665$	− 8.00000i	− 0.310227i
$666$	14.0000	0.542489
$667$	−56.0000	−2.16833
$668$	0	0
$669$	9.00000i	0.347960i
$670$	28.0000i	1.08173i
$671$	− 13.0000i	− 0.501859i
$672$	1.00000	0.0385758
$673$	33.0000	1.27206	0.636028	−	0.771666i	$-0.280576\pi$
$673$	33.0000	1.27206	0.636028	+	0.771666i	$0.280576\pi$
$674$	9.00000i	0.346667i
$675$	55.0000	2.11695
$676$	0	0
$677$	13.0000	0.499631	0.249815	−	0.968294i	$-0.419630\pi$
$677$	13.0000	0.499631	0.249815	+	0.968294i	$0.419630\pi$
$678$	− 1.00000i	− 0.0384048i
$679$	−11.0000	−0.422141
$680$	16.0000	0.613572
$681$	28.0000i	1.07296i
$682$	3.00000i	0.114876i
$683$	− 31.0000i	− 1.18618i	−0.805135	−	0.593091i	$-0.797907\pi$
$683$	− 31.0000i	− 1.18618i	0.805135	−	0.593091i	$-0.202093\pi$
$684$	4.00000i	0.152944i
$685$	−56.0000	−2.13965
$686$	−1.00000	−0.0381802
$687$	20.0000i	0.763048i
$688$	8.00000	0.304997
$689$	0	0
$690$	28.0000	1.06594
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	14.0000	0.532200
$693$	2.00000	0.0759737
$694$	32.0000i	1.21470i
$695$	− 80.0000i	− 3.03457i
$696$	− 8.00000i	− 0.303239i
$697$	28.0000i	1.06058i
$698$	−2.00000	−0.0757011
$699$	13.0000	0.491705
$700$	11.0000i	0.415761i
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	− 1.00000i	− 0.0376889i
$705$	12.0000	0.451946
$706$	−11.0000	−0.413990
$707$	− 9.00000i	− 0.338480i
$708$	− 6.00000i	− 0.225494i
$709$	51.0000i	1.91535i	0.287860	+	0.957673i	$0.407056\pi$
$709$	51.0000i	1.91535i	−0.287860	+	0.957673i	$0.592944\pi$
$710$	16.0000i	0.600469i
$711$	26.0000	0.975076
$712$	−6.00000	−0.224860
$713$	21.0000i	0.786456i
$714$	−4.00000	−0.149696
$715$	0	0
$716$	−18.0000	−0.672692
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	22.0000	0.820462	0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000	0.820462	0.410231	+	0.911982i	$0.365448\pi$
$720$	− 8.00000i	− 0.298142i
$721$	− 10.0000i	− 0.372419i
$722$	− 15.0000i	− 0.558242i
$723$	10.0000i	0.371904i
$724$	13.0000	0.483141
$725$	88.0000	3.26824
$726$	− 10.0000i	− 0.371135i
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	− 36.0000i	− 1.33242i
$731$	−32.0000	−1.18356
$732$	13.0000	0.480494
$733$	− 12.0000i	− 0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$	− 12.0000i	− 0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$	− 12.0000i	− 0.442928i
$735$	− 4.00000i	− 0.147542i
$736$	− 7.00000i	− 0.258023i
$737$	−7.00000	−0.257848
$738$	14.0000	0.515347
$739$	8.00000i	0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$	8.00000i	0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$	−28.0000	−1.02930
$741$	0	0
$742$	0	0
$743$	− 44.0000i	− 1.61420i	−0.590412	−	0.807102i	$-0.701035\pi$
$743$	− 44.0000i	− 1.61420i	0.590412	−	0.807102i	$-0.298965\pi$
$744$	−3.00000	−0.109985
$745$	−60.0000	−2.19823
$746$	32.0000i	1.17160i
$747$	32.0000i	1.17082i
$748$	4.00000i	0.146254i
$749$	12.0000i	0.438470i
$750$	−24.0000	−0.876356
$751$	−37.0000	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$751$	−37.0000	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$752$	− 3.00000i	− 0.109399i
$753$	−17.0000	−0.619514
$754$	0	0
$755$	−16.0000	−0.582300
$756$	5.00000i	0.181848i
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−8.00000	−0.290573
$759$	7.00000i	0.254084i
$760$	− 8.00000i	− 0.290191i
$761$	− 1.00000i	− 0.0362500i	−0.999836	−	0.0181250i	$-0.994230\pi$
$761$	− 1.00000i	− 0.0362500i	0.999836	−	0.0181250i	$-0.00576968\pi$
$762$	13.0000i	0.470940i
$763$	−14.0000	−0.506834
$764$	−24.0000	−0.868290
$765$	32.0000i	1.15696i
$766$	21.0000	0.758761
$767$	0	0
$768$	1.00000	0.0360844
$769$	25.0000i	0.901523i	0.892644	+	0.450762i	$0.148848\pi$
$769$	25.0000i	0.901523i	−0.892644	+	0.450762i	$0.851152\pi$
$770$	−4.00000	−0.144150
$771$	12.0000	0.432169
$772$	− 4.00000i	− 0.143963i
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	16.0000i	0.575108i
$775$	− 33.0000i	− 1.18539i
$776$	−11.0000	−0.394877
$777$	7.00000	0.251124
$778$	8.00000i	0.286814i
$779$	14.0000	0.501602
$780$	0	0
$781$	−4.00000	−0.143131
$782$	28.0000i	1.00128i
$783$	40.0000	1.42948
$784$	−1.00000	−0.0357143
$785$	28.0000i	0.999363i
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	− 3.00000i	− 0.106871i
$789$	16.0000	0.569615
$790$	−52.0000	−1.85008
$791$	1.00000i	0.0355559i
$792$	2.00000	0.0710669
$793$	0	0
$794$	−8.00000	−0.283909
$795$	0	0
$796$	16.0000	0.567105
$797$	−21.0000	−0.743858	−0.371929	−	0.928261i	$-0.621304\pi$
$797$	−21.0000	−0.743858	−0.371929	+	0.928261i	$0.621304\pi$
$798$	2.00000i	0.0707992i
$799$	12.0000i	0.424529i
$800$	11.0000i	0.388909i
$801$	− 12.0000i	− 0.423999i
$802$	−8.00000	−0.282490
$803$	9.00000	0.317603
$804$	− 7.00000i	− 0.246871i
$805$	−28.0000	−0.986870
$806$	0	0
$807$	3.00000	0.105605
$808$	− 9.00000i	− 0.316619i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	4.00000	0.140546
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	8.00000i	0.280745i
$813$	3.00000i	0.105215i
$814$	− 7.00000i	− 0.245350i
$815$	48.0000	1.68137
$816$	−4.00000	−0.140028
$817$	16.0000i	0.559769i
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−28.0000	−0.977802
$821$	46.0000i	1.60541i	0.596376	+	0.802706i	$0.296607\pi$
$821$	46.0000i	1.60541i	−0.596376	+	0.802706i	$0.703393\pi$
$822$	14.0000	0.488306
$823$	35.0000	1.22002	0.610012	−	0.792392i	$-0.291165\pi$
$823$	35.0000	1.22002	0.610012	+	0.792392i	$0.291165\pi$
$824$	− 10.0000i	− 0.348367i
$825$	− 11.0000i	− 0.382971i
$826$	6.00000i	0.208767i
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	14.0000	0.486534
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	− 64.0000i	− 2.22147i
$831$	14.0000	0.485655
$832$	0	0
$833$	4.00000	0.138592
$834$	20.0000i	0.692543i
$835$	0	0
$836$	2.00000	0.0691714
$837$	− 15.0000i	− 0.518476i
$838$	19.0000i	0.656344i
$839$	− 21.0000i	− 0.725001i	−0.931984	−	0.362500i	$-0.881923\pi$
$839$	− 21.0000i	− 0.725001i	0.931984	−	0.362500i	$-0.118077\pi$
$840$	− 4.00000i	− 0.138013i
$841$	35.0000	1.20690
$842$	−11.0000	−0.379085
$843$	20.0000i	0.688837i
$844$	−22.0000	−0.757271
$845$	0	0
$846$	6.00000	0.206284
$847$	10.0000i	0.343604i
$848$	0	0
$849$	−17.0000	−0.583438
$850$	− 44.0000i	− 1.50919i
$851$	− 49.0000i	− 1.67970i
$852$	− 4.00000i	− 0.137038i
$853$	44.0000i	1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$	44.0000i	1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$	−13.0000	−0.444851
$855$	16.0000	0.547188
$856$	12.0000i	0.410152i
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	− 32.0000i	− 1.09119i
$861$	7.00000	0.238559
$862$	2.00000	0.0681203
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	5.00000i	0.170103i
$865$	− 56.0000i	− 1.90406i
$866$	− 18.0000i	− 0.611665i
$867$	−1.00000	−0.0339618
$868$	3.00000	0.101827
$869$	− 13.0000i	− 0.440995i
$870$	−32.0000	−1.08490
$871$	0	0
$872$	−14.0000	−0.474100
$873$	− 22.0000i	− 0.744587i
$874$	14.0000	0.473557
$875$	24.0000	0.811348
$876$	9.00000i	0.304082i
$877$	29.0000i	0.979260i	0.871930	+	0.489630i	$0.162868\pi$
$877$	29.0000i	0.979260i	−0.871930	+	0.489630i	$0.837132\pi$
$878$	− 2.00000i	− 0.0674967i
$879$	− 14.0000i	− 0.472208i
$880$	−4.00000	−0.134840
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	14.0000	0.470074	0.235037	−	0.971986i	$-0.424479\pi$
$887$	14.0000	0.470074	0.235037	+	0.971986i	$0.424479\pi$
$888$	7.00000	0.234905
$889$	− 13.0000i	− 0.436006i
$890$	24.0000i	0.804482i
$891$	1.00000i	0.0335013i
$892$	− 9.00000i	− 0.301342i
$893$	6.00000	0.200782
$894$	15.0000	0.501675
$895$	72.0000i	2.40669i
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	− 24.0000i	− 0.800445i
$900$	−22.0000	−0.733333
$901$	0	0
$902$	− 7.00000i	− 0.233075i
$903$	8.00000i	0.266223i
$904$	1.00000i	0.0332595i
$905$	− 52.0000i	− 1.72854i
$906$	4.00000	0.132891
$907$	−54.0000	−1.79304	−0.896520	−	0.443003i	$-0.853913\pi$
$907$	−54.0000	−1.79304	−0.896520	+	0.443003i	$0.853913\pi$
$908$	− 28.0000i	− 0.929213i
$909$	18.0000	0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	2.00000i	0.0662266i
$913$	16.0000	0.529523
$914$	18.0000	0.595387
$915$	− 52.0000i	− 1.71907i
$916$	− 20.0000i	− 0.660819i
$917$	0	0
$918$	− 20.0000i	− 0.660098i
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	−28.0000	−0.923133
$921$	12.0000i	0.395413i
$922$	16.0000	0.526932
$923$	0	0
$924$	1.00000	0.0328976
$925$	77.0000i	2.53174i
$926$	12.0000	0.394344
$927$	20.0000	0.656886
$928$	8.00000i	0.262613i
$929$	− 27.0000i	− 0.885841i	−0.896561	−	0.442921i	$-0.853942\pi$
$929$	− 27.0000i	− 0.885841i	0.896561	−	0.442921i	$-0.146058\pi$
$930$	12.0000i	0.393496i
$931$	− 2.00000i	− 0.0655474i
$932$	−13.0000	−0.425829
$933$	30.0000	0.982156
$934$	− 20.0000i	− 0.654420i
$935$	16.0000	0.523256
$936$	0	0
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	7.00000i	0.228558i
$939$	6.00000	0.195803
$940$	−12.0000	−0.391397
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	− 7.00000i	− 0.228072i
$943$	− 49.0000i	− 1.59566i
$944$	6.00000i	0.195283i
$945$	20.0000	0.650600
$946$	8.00000	0.260102
$947$	8.00000i	0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$	8.00000i	0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$	13.0000	0.422220
$949$	0	0
$950$	−22.0000	−0.713774
$951$	− 13.0000i	− 0.421554i
$952$	4.00000	0.129641
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	96.0000i	3.10649i
$956$	6.00000i	0.194054i
$957$	− 8.00000i	− 0.258603i
$958$	0	0
$959$	−14.0000	−0.452084
$960$	− 4.00000i	− 0.129099i
$961$	22.0000	0.709677
$962$	0	0
$963$	−24.0000	−0.773389
$964$	− 10.0000i	− 0.322078i
$965$	−16.0000	−0.515058
$966$	7.00000	0.225221
$967$	18.0000i	0.578841i	0.957202	+	0.289420i	$0.0934626\pi$
$967$	18.0000i	0.578841i	−0.957202	+	0.289420i	$0.906537\pi$
$968$	10.0000i	0.321412i
$969$	− 8.00000i	− 0.256997i
$970$	44.0000i	1.41275i
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	−16.0000	−0.513200
$973$	− 20.0000i	− 0.641171i
$974$	26.0000	0.833094
$975$	0	0
$976$	−13.0000	−0.416120
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	−12.0000	−0.383718
$979$	−6.00000	−0.191761
$980$	4.00000i	0.127775i
$981$	− 28.0000i	− 0.893971i
$982$	40.0000i	1.27645i
$983$	− 32.0000i	− 1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$	− 32.0000i	− 1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$	7.00000	0.223152
$985$	−12.0000	−0.382352
$986$	− 32.0000i	− 1.01909i
$987$	3.00000	0.0954911
$988$	0	0
$989$	56.0000	1.78070
$990$	− 8.00000i	− 0.254257i
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	3.00000	0.0952501
$993$	15.0000i	0.476011i
$994$	4.00000i	0.126872i
$995$	− 64.0000i	− 2.02894i
$996$	16.0000i	0.506979i
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	37.0000	1.17121
$999$	35.0000i	1.10735i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.d.g.337.1		2
13.5	odd	4		182.2.a.a.1.1	✓	1
13.8	odd	4		2366.2.a.m.1.1		1
13.12	even	2	inner	2366.2.d.g.337.2		2
39.5	even	4		1638.2.a.k.1.1		1
52.31	even	4		1456.2.a.e.1.1		1
65.44	odd	4		4550.2.a.t.1.1		1
91.5	even	12		1274.2.f.t.1145.1		2
91.18	odd	12		1274.2.f.n.79.1		2
91.31	even	12		1274.2.f.t.79.1		2
91.44	odd	12		1274.2.f.n.1145.1		2
91.83	even	4		1274.2.a.b.1.1		1
104.5	odd	4		5824.2.a.g.1.1		1
104.83	even	4		5824.2.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.a.a.1.1	✓	1	13.5	odd	4
1274.2.a.b.1.1		1	91.83	even	4
1274.2.f.n.79.1		2	91.18	odd	12
1274.2.f.n.1145.1		2	91.44	odd	12
1274.2.f.t.79.1		2	91.31	even	12
1274.2.f.t.1145.1		2	91.5	even	12
1456.2.a.e.1.1		1	52.31	even	4
1638.2.a.k.1.1		1	39.5	even	4
2366.2.a.m.1.1		1	13.8	odd	4
2366.2.d.g.337.1		2	1.1	even	1	trivial
2366.2.d.g.337.2		2	13.12	even	2	inner
4550.2.a.t.1.1		1	65.44	odd	4
5824.2.a.g.1.1		1	104.5	odd	4
5824.2.a.w.1.1		1	104.83	even	4

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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +4.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +4.00000i q^{5} -1.00000i q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +4.00000 q^{10} +1.00000i q^{11} -1.00000 q^{12} +1.00000 q^{14} +4.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +2.00000i q^{18} +2.00000i q^{19} -4.00000i q^{20} +1.00000i q^{21} +1.00000 q^{22} +7.00000 q^{23} +1.00000i q^{24} -11.0000 q^{25} -5.00000 q^{27} -1.00000i q^{28} -8.00000 q^{29} +4.00000 q^{30} +3.00000i q^{31} -1.00000i q^{32} +1.00000i q^{33} +4.00000i q^{34} -4.00000 q^{35} +2.00000 q^{36} -7.00000i q^{37} +2.00000 q^{38} -4.00000 q^{40} -7.00000i q^{41} +1.00000 q^{42} +8.00000 q^{43} -1.00000i q^{44} -8.00000i q^{45} -7.00000i q^{46} -3.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +11.0000i q^{50} -4.00000 q^{51} +5.00000i q^{54} -4.00000 q^{55} -1.00000 q^{56} +2.00000i q^{57} +8.00000i q^{58} +6.00000i q^{59} -4.00000i q^{60} -13.0000 q^{61} +3.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +1.00000 q^{66} +7.00000i q^{67} +4.00000 q^{68} +7.00000 q^{69} +4.00000i q^{70} +4.00000i q^{71} -2.00000i q^{72} -9.00000i q^{73} -7.00000 q^{74} -11.0000 q^{75} -2.00000i q^{76} -1.00000 q^{77} -13.0000 q^{79} +4.00000i q^{80} +1.00000 q^{81} -7.00000 q^{82} -16.0000i q^{83} -1.00000i q^{84} -16.0000i q^{85} -8.00000i q^{86} -8.00000 q^{87} -1.00000 q^{88} +6.00000i q^{89} -8.00000 q^{90} -7.00000 q^{92} +3.00000i q^{93} -3.00000 q^{94} -8.00000 q^{95} -1.00000i q^{96} +11.0000i q^{97} +1.00000i q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} + 8 q^{10} - 2 q^{12} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{22} + 14 q^{23} - 22 q^{25} - 10 q^{27} - 16 q^{29} + 8 q^{30} - 8 q^{35} + 4 q^{36} + 4 q^{38} - 8 q^{40} + 2 q^{42} + 16 q^{43} + 2 q^{48} - 2 q^{49} - 8 q^{51} - 8 q^{55} - 2 q^{56} - 26 q^{61} + 6 q^{62} - 2 q^{64} + 2 q^{66} + 8 q^{68} + 14 q^{69} - 14 q^{74} - 22 q^{75} - 2 q^{77} - 26 q^{79} + 2 q^{81} - 14 q^{82} - 16 q^{87} - 2 q^{88} - 16 q^{90} - 14 q^{92} - 6 q^{94} - 16 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 + 8 * q^10 - 2 * q^12 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^22 + 14 * q^23 - 22 * q^25 - 10 * q^27 - 16 * q^29 + 8 * q^30 - 8 * q^35 + 4 * q^36 + 4 * q^38 - 8 * q^40 + 2 * q^42 + 16 * q^43 + 2 * q^48 - 2 * q^49 - 8 * q^51 - 8 * q^55 - 2 * q^56 - 26 * q^61 + 6 * q^62 - 2 * q^64 + 2 * q^66 + 8 * q^68 + 14 * q^69 - 14 * q^74 - 22 * q^75 - 2 * q^77 - 26 * q^79 + 2 * q^81 - 14 * q^82 - 16 * q^87 - 2 * q^88 - 16 * q^90 - 14 * q^92 - 6 * q^94 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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