LMFDB - Embedded newform 37.2.b.a.36.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,2,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	37.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.295446487479$
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:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.2.b.a.36.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{5} -2.00000i q^{6} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{5} -2.00000i q^{6} +3.00000 q^{7} -2.00000 q^{9} +4.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} -6.00000i q^{13} +6.00000i q^{14} +2.00000i q^{15} -4.00000 q^{16} +2.00000i q^{17} -4.00000i q^{18} +6.00000i q^{19} +4.00000i q^{20} -3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +1.00000 q^{25} +12.0000 q^{26} +5.00000 q^{27} -6.00000 q^{28} -4.00000i q^{29} -4.00000 q^{30} -8.00000i q^{32} +3.00000 q^{33} -4.00000 q^{34} -6.00000i q^{35} +4.00000 q^{36} +(-1.00000 + 6.00000i) q^{37} -12.0000 q^{38} +6.00000i q^{39} -3.00000 q^{41} -6.00000i q^{42} -6.00000i q^{43} +6.00000 q^{44} +4.00000i q^{45} -8.00000 q^{46} +3.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000i q^{50} -2.00000i q^{51} +12.0000i q^{52} +9.00000 q^{53} +10.0000i q^{54} +6.00000i q^{55} -6.00000i q^{57} +8.00000 q^{58} -4.00000i q^{59} -4.00000i q^{60} -6.00000 q^{63} +8.00000 q^{64} -12.0000 q^{65} +6.00000i q^{66} -12.0000 q^{67} -4.00000i q^{68} -4.00000i q^{69} +12.0000 q^{70} -3.00000 q^{71} +9.00000 q^{73} +(-12.0000 - 2.00000i) q^{74} -1.00000 q^{75} -12.0000i q^{76} -9.00000 q^{77} -12.0000 q^{78} +6.00000i q^{79} +8.00000i q^{80} +1.00000 q^{81} -6.00000i q^{82} +9.00000 q^{83} +6.00000 q^{84} +4.00000 q^{85} +12.0000 q^{86} +4.00000i q^{87} -14.0000i q^{89} -8.00000 q^{90} -18.0000i q^{91} -8.00000i q^{92} +6.00000i q^{94} +12.0000 q^{95} +8.00000i q^{96} +12.0000i q^{97} +4.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} + 6 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^4 + 6 * q^7 - 4 * q^9	$ 2 q - 2 q^{3} - 4 q^{4} + 6 q^{7} - 4 q^{9} + 8 q^{10} - 6 q^{11} + 4 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 24 q^{26} + 10 q^{27} - 12 q^{28} - 8 q^{30} + 6 q^{33} - 8 q^{34} + 8 q^{36} - 2 q^{37} - 24 q^{38} - 6 q^{41} + 12 q^{44} - 16 q^{46} + 6 q^{47} + 8 q^{48} + 4 q^{49} + 18 q^{53} + 16 q^{58} - 12 q^{63} + 16 q^{64} - 24 q^{65} - 24 q^{67} + 24 q^{70} - 6 q^{71} + 18 q^{73} - 24 q^{74} - 2 q^{75} - 18 q^{77} - 24 q^{78} + 2 q^{81} + 18 q^{83} + 12 q^{84} + 8 q^{85} + 24 q^{86} - 16 q^{90} + 24 q^{95} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 + 6 * q^7 - 4 * q^9 + 8 * q^10 - 6 * q^11 + 4 * q^12 - 8 * q^16 - 6 * q^21 + 2 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^28 - 8 * q^30 + 6 * q^33 - 8 * q^34 + 8 * q^36 - 2 * q^37 - 24 * q^38 - 6 * q^41 + 12 * q^44 - 16 * q^46 + 6 * q^47 + 8 * q^48 + 4 * q^49 + 18 * q^53 + 16 * q^58 - 12 * q^63 + 16 * q^64 - 24 * q^65 - 24 * q^67 + 24 * q^70 - 6 * q^71 + 18 * q^73 - 24 * q^74 - 2 * q^75 - 18 * q^77 - 24 * q^78 + 2 * q^81 + 18 * q^83 + 12 * q^84 + 8 * q^85 + 24 * q^86 - 16 * q^90 + 24 * q^95 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$			2.00000i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$			2.00000i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$	−2.00000			−1.00000
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		−	2.00000i		−	0.816497i
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$	4.00000			1.26491
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$	2.00000			0.577350
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$			6.00000i			1.60357i
$15$			2.00000i			0.516398i
$16$	−4.00000			−1.00000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		−	4.00000i		−	0.942809i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$			4.00000i			0.894427i
$21$	−3.00000			−0.654654
$22$		−	6.00000i		−	1.27920i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	12.0000			2.35339
$27$	5.00000			0.962250
$28$	−6.00000			−1.13389
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	−4.00000			−0.730297
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	8.00000i		−	1.41421i
$33$	3.00000			0.522233
$34$	−4.00000			−0.685994
$35$		−	6.00000i		−	1.01419i
$36$	4.00000			0.666667
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$38$	−12.0000			−1.94666
$39$			6.00000i			0.960769i
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		−	6.00000i		−	0.925820i
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	6.00000			0.904534
$45$			4.00000i			0.596285i
$46$	−8.00000			−1.17954
$47$	3.00000			0.437595			0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000			0.437595			0.218797	+	0.975770i	$0.429787\pi$
$48$	4.00000			0.577350
$49$	2.00000			0.285714
$50$			2.00000i			0.282843i
$51$		−	2.00000i		−	0.280056i
$52$			12.0000i			1.66410i
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$			10.0000i			1.36083i
$55$			6.00000i			0.809040i
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$	8.00000			1.05045
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		−	4.00000i		−	0.516398i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−6.00000			−0.755929
$64$	8.00000			1.00000
$65$	−12.0000			−1.48842
$66$			6.00000i			0.738549i
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		−	4.00000i		−	0.485071i
$69$		−	4.00000i		−	0.481543i
$70$	12.0000			1.43427
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		0			0
$73$	9.00000			1.05337			0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000			1.05337			0.526685	+	0.850060i	$0.323435\pi$
$74$	−12.0000	−	2.00000i	−1.39497	−	0.232495i
$75$	−1.00000			−0.115470
$76$		−	12.0000i		−	1.37649i
$77$	−9.00000			−1.02565
$78$	−12.0000			−1.35873
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$			8.00000i			0.894427i
$81$	1.00000			0.111111
$82$		−	6.00000i		−	0.662589i
$83$	9.00000			0.987878			0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000			0.987878			0.493939	+	0.869496i	$0.335557\pi$
$84$	6.00000			0.654654
$85$	4.00000			0.433861
$86$	12.0000			1.29399
$87$			4.00000i			0.428845i
$88$		0			0
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	−8.00000			−0.843274
$91$		−	18.0000i		−	1.88691i
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$			6.00000i			0.618853i
$95$	12.0000			1.23117
$96$			8.00000i			0.816497i
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$			4.00000i			0.404061i
$99$	6.00000			0.603023
$100$	−2.00000			−0.200000
$101$	−3.00000			−0.298511			−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000			−0.298511			−0.149256	+	0.988799i	$0.547688\pi$
$102$	4.00000			0.396059
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$			6.00000i			0.585540i
$106$			18.0000i			1.74831i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$	−10.0000			−0.962250
$109$			6.00000i			0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$			6.00000i			0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	−12.0000			−1.14416
$111$	1.00000	−	6.00000i	0.0949158	−	0.569495i
$112$	−12.0000			−1.13389
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	12.0000			1.12390
$115$	8.00000			0.746004
$116$			8.00000i			0.742781i
$117$			12.0000i			1.10940i
$118$	8.00000			0.736460
$119$			6.00000i			0.550019i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	3.00000			0.270501
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		−	12.0000i		−	1.06904i
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		−	24.0000i		−	2.10494i
$131$		−	10.0000i		−	0.873704i	−0.899533	−	0.436852i	$-0.856093\pi$
$131$		−	10.0000i		−	0.873704i	0.899533	−	0.436852i	$-0.143907\pi$
$132$	−6.00000			−0.522233
$133$			18.0000i			1.56080i
$134$		−	24.0000i		−	2.07328i
$135$		−	10.0000i		−	0.860663i
$136$		0			0
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$	8.00000			0.681005
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$			12.0000i			1.01419i
$141$	−3.00000			−0.252646
$142$		−	6.00000i		−	0.503509i
$143$			18.0000i			1.50524i
$144$	8.00000			0.666667
$145$	−8.00000			−0.664364
$146$			18.0000i			1.48969i
$147$	−2.00000			−0.164957
$148$	2.00000	−	12.0000i	0.164399	−	0.986394i
$149$	15.0000			1.22885			0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000			1.22885			0.614424	+	0.788976i	$0.289388\pi$
$150$		−	2.00000i		−	0.163299i
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		−	18.0000i		−	1.45048i
$155$		0			0
$156$		−	12.0000i		−	0.960769i
$157$	3.00000			0.239426			0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000			0.239426			0.119713	+	0.992809i	$0.461803\pi$
$158$	−12.0000			−0.954669
$159$	−9.00000			−0.713746
$160$	−16.0000			−1.26491
$161$			12.0000i			0.945732i
$162$			2.00000i			0.157135i
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	6.00000			0.468521
$165$		−	6.00000i		−	0.467099i
$166$			18.0000i			1.39707i
$167$			2.00000i			0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$			2.00000i			0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$			8.00000i			0.613572i
$171$		−	12.0000i		−	0.917663i
$172$			12.0000i			0.914991i
$173$	−21.0000			−1.59660			−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000			−1.59660			−0.798300	+	0.602260i	$0.794267\pi$
$174$	−8.00000			−0.606478
$175$	3.00000			0.226779
$176$	12.0000			0.904534
$177$			4.00000i			0.300658i
$178$	28.0000			2.09869
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$		−	8.00000i		−	0.596285i
$181$	−3.00000			−0.222988			−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000			−0.222988			−0.111494	+	0.993765i	$0.535564\pi$
$182$	36.0000			2.66850
$183$		0			0
$184$		0			0
$185$	12.0000	+	2.00000i	0.882258	+	0.147043i
$186$		0			0
$187$		−	6.00000i		−	0.438763i
$188$	−6.00000			−0.437595
$189$	15.0000			1.09109
$190$			24.0000i			1.74114i
$191$			20.0000i			1.44715i	0.690246	+	0.723575i	$0.257502\pi$
$191$			20.0000i			1.44715i	−0.690246	+	0.723575i	$0.742498\pi$
$192$	−8.00000			−0.577350
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−24.0000			−1.72310
$195$	12.0000			0.859338
$196$	−4.00000			−0.285714
$197$	3.00000			0.213741			0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$			12.0000i			0.852803i
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$	12.0000			0.846415
$202$		−	6.00000i		−	0.422159i
$203$		−	12.0000i		−	0.842235i
$204$			4.00000i			0.280056i
$205$			6.00000i			0.419058i
$206$	12.0000			0.836080
$207$		−	8.00000i		−	0.556038i
$208$			24.0000i			1.66410i
$209$		−	18.0000i		−	1.24509i
$210$	−12.0000			−0.828079
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$	−18.0000			−1.23625
$213$	3.00000			0.205557
$214$		−	24.0000i		−	1.64061i
$215$	−12.0000			−0.818393
$216$		0			0
$217$		0			0
$218$	−12.0000			−0.812743
$219$	−9.00000			−0.608164
$220$		−	12.0000i		−	0.809040i
$221$	12.0000			0.807207
$222$	12.0000	+	2.00000i	0.805387	+	0.134231i
$223$	19.0000			1.27233			0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000			1.27233			0.636167	+	0.771551i	$0.280519\pi$
$224$		−	24.0000i		−	1.60357i
$225$	−2.00000			−0.133333
$226$	−8.00000			−0.532152
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$			12.0000i			0.794719i
$229$	−5.00000			−0.330409			−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000			−0.330409			−0.165205	+	0.986259i	$0.552828\pi$
$230$			16.0000i			1.05501i
$231$	9.00000			0.592157
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	−24.0000			−1.56893
$235$		−	6.00000i		−	0.391397i
$236$			8.00000i			0.520756i
$237$		−	6.00000i		−	0.389742i
$238$	−12.0000			−0.777844
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		−	8.00000i		−	0.516398i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	4.00000i		−	0.257130i
$243$	−16.0000			−1.02640
$244$		0			0
$245$		−	4.00000i		−	0.255551i
$246$			6.00000i			0.382546i
$247$	36.0000			2.29063
$248$		0			0
$249$	−9.00000			−0.570352
$250$	24.0000			1.51789
$251$			10.0000i			0.631194i	0.948893	+	0.315597i	$0.102205\pi$
$251$			10.0000i			0.631194i	−0.948893	+	0.315597i	$0.897795\pi$
$252$	12.0000			0.755929
$253$		−	12.0000i		−	0.754434i
$254$		−	14.0000i		−	0.878438i
$255$	−4.00000			−0.250490
$256$	16.0000			1.00000
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	−12.0000			−0.747087
$259$	−3.00000	+	18.0000i	−0.186411	+	1.11847i
$260$	24.0000			1.48842
$261$			8.00000i			0.495188i
$262$	20.0000			1.23560
$263$	9.00000			0.554964			0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000			0.554964			0.277482	+	0.960731i	$0.410500\pi$
$264$		0			0
$265$		−	18.0000i		−	1.10573i
$266$	−36.0000			−2.20730
$267$			14.0000i			0.856786i
$268$	24.0000			1.46603
$269$	−30.0000			−1.82913			−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000			−1.82913			−0.914566	+	0.404436i	$0.867468\pi$
$270$	20.0000			1.21716
$271$	−3.00000			−0.182237			−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000			−0.182237			−0.0911185	+	0.995840i	$0.529044\pi$
$272$		−	8.00000i		−	0.485071i
$273$			18.0000i			1.08941i
$274$			36.0000i			2.17484i
$275$	−3.00000			−0.180907
$276$			8.00000i			0.481543i
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.0000i		−	1.19310i	−0.802576	−	0.596550i	$-0.796538\pi$
$281$		−	20.0000i		−	1.19310i	0.802576	−	0.596550i	$-0.203462\pi$
$282$		−	6.00000i		−	0.357295i
$283$			24.0000i			1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$			24.0000i			1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	6.00000			0.356034
$285$	−12.0000			−0.710819
$286$	−36.0000			−2.12872
$287$	−9.00000			−0.531253
$288$			16.0000i			0.942809i
$289$	13.0000			0.764706
$290$		−	16.0000i		−	0.939552i
$291$		−	12.0000i		−	0.703452i
$292$	−18.0000			−1.05337
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		−	4.00000i		−	0.233285i
$295$	−8.00000			−0.465778
$296$		0			0
$297$	−15.0000			−0.870388
$298$			30.0000i			1.73785i
$299$	24.0000			1.38796
$300$	2.00000			0.115470
$301$		−	18.0000i		−	1.03750i
$302$		−	16.0000i		−	0.920697i
$303$	3.00000			0.172345
$304$		−	24.0000i		−	1.37649i
$305$		0			0
$306$	8.00000			0.457330
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$	18.0000			1.02565
$309$			6.00000i			0.341328i
$310$		0			0
$311$			10.0000i			0.567048i	0.958965	+	0.283524i	$0.0915036\pi$
$311$			10.0000i			0.567048i	−0.958965	+	0.283524i	$0.908496\pi$
$312$		0			0
$313$		−	6.00000i		−	0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$		−	6.00000i		−	0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$			6.00000i			0.338600i
$315$			12.0000i			0.676123i
$316$		−	12.0000i		−	0.675053i
$317$	18.0000			1.01098			0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000			1.01098			0.505490	+	0.862832i	$0.331312\pi$
$318$		−	18.0000i		−	1.00939i
$319$			12.0000i			0.671871i
$320$		−	16.0000i		−	0.894427i
$321$	12.0000			0.669775
$322$	−24.0000			−1.33747
$323$	−12.0000			−0.667698
$324$	−2.00000			−0.111111
$325$		−	6.00000i		−	0.332820i
$326$	12.0000			0.664619
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	9.00000			0.496186
$330$	12.0000			0.660578
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−18.0000			−0.987878
$333$	2.00000	−	12.0000i	0.109599	−	0.657596i
$334$	−4.00000			−0.218870
$335$			24.0000i			1.31126i
$336$	12.0000			0.654654
$337$	13.0000			0.708155			0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000			0.708155			0.354078	+	0.935216i	$0.384795\pi$
$338$		−	46.0000i		−	2.50207i
$339$		−	4.00000i		−	0.217250i
$340$	−8.00000			−0.433861
$341$		0			0
$342$	24.0000			1.29777
$343$	−15.0000			−0.809924
$344$		0			0
$345$	−8.00000			−0.430706
$346$		−	42.0000i		−	2.25793i
$347$			32.0000i			1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$			32.0000i			1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$		−	8.00000i		−	0.428845i
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$			6.00000i			0.320713i
$351$		−	30.0000i		−	1.60128i
$352$			24.0000i			1.27920i
$353$			14.0000i			0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$			14.0000i			0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	−8.00000			−0.425195
$355$			6.00000i			0.318447i
$356$			28.0000i			1.48400i
$357$		−	6.00000i		−	0.317554i
$358$	−32.0000			−1.69125
$359$	−15.0000			−0.791670			−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000			−0.791670			−0.395835	+	0.918322i	$0.629545\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		−	6.00000i		−	0.315353i
$363$	2.00000			0.104973
$364$			36.0000i			1.88691i
$365$		−	18.0000i		−	0.942163i
$366$		0			0
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$		−	16.0000i		−	0.834058i
$369$	6.00000			0.312348
$370$	−4.00000	+	24.0000i	−0.207950	+	1.24770i
$371$	27.0000			1.40177
$372$		0			0
$373$	−21.0000			−1.08734			−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000			−1.08734			−0.543669	+	0.839299i	$0.682965\pi$
$374$	12.0000			0.620505
$375$			12.0000i			0.619677i
$376$		0			0
$377$	−24.0000			−1.23606
$378$			30.0000i			1.54303i
$379$	15.0000			0.770498			0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000			0.770498			0.385249	+	0.922813i	$0.374116\pi$
$380$	−24.0000			−1.23117
$381$	7.00000			0.358621
$382$	−40.0000			−2.04658
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$			18.0000i			0.917365i
$386$	12.0000			0.610784
$387$			12.0000i			0.609994i
$388$		−	24.0000i		−	1.21842i
$389$			16.0000i			0.811232i	0.914044	+	0.405616i	$0.132943\pi$
$389$			16.0000i			0.811232i	−0.914044	+	0.405616i	$0.867057\pi$
$390$			24.0000i			1.21529i
$391$	−8.00000			−0.404577
$392$		0			0
$393$			10.0000i			0.504433i
$394$			6.00000i			0.302276i
$395$	12.0000			0.603786
$396$	−12.0000			−0.603023
$397$	33.0000			1.65622			0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000			1.65622			0.828111	+	0.560564i	$0.189416\pi$
$398$	48.0000			2.40602
$399$		−	18.0000i		−	0.901127i
$400$	−4.00000			−0.200000
$401$		−	10.0000i		−	0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$		−	10.0000i		−	0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$			24.0000i			1.19701i
$403$		0			0
$404$	6.00000			0.298511
$405$		−	2.00000i		−	0.0993808i
$406$	24.0000			1.19110
$407$	3.00000	−	18.0000i	0.148704	−	0.892227i
$408$		0			0
$409$		−	24.0000i		−	1.18672i	−0.804936	−	0.593362i	$-0.797800\pi$
$409$		−	24.0000i		−	1.18672i	0.804936	−	0.593362i	$-0.202200\pi$
$410$	−12.0000			−0.592638
$411$	−18.0000			−0.887875
$412$			12.0000i			0.591198i
$413$		−	12.0000i		−	0.590481i
$414$	16.0000			0.786357
$415$		−	18.0000i		−	0.883585i
$416$	−48.0000			−2.35339
$417$		0			0
$418$	36.0000			1.76082
$419$	15.0000			0.732798			0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000			0.732798			0.366399	+	0.930458i	$0.380591\pi$
$420$		−	12.0000i		−	0.585540i
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$		−	26.0000i		−	1.26566i
$423$	−6.00000			−0.291730
$424$		0			0
$425$			2.00000i			0.0970143i
$426$			6.00000i			0.290701i
$427$		0			0
$428$	24.0000			1.16008
$429$		−	18.0000i		−	0.869048i
$430$		−	24.0000i		−	1.15738i
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$	−20.0000			−0.962250
$433$	9.00000			0.432512			0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000			0.432512			0.216256	+	0.976337i	$0.430615\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		−	12.0000i		−	0.574696i
$437$	−24.0000			−1.14808
$438$		−	18.0000i		−	0.860073i
$439$		−	24.0000i		−	1.14546i	−0.819745	−	0.572729i	$-0.805885\pi$
$439$		−	24.0000i		−	1.14546i	0.819745	−	0.572729i	$-0.194115\pi$
$440$		0			0
$441$	−4.00000			−0.190476
$442$			24.0000i			1.14156i
$443$	−21.0000			−0.997740			−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000			−0.997740			−0.498870	+	0.866677i	$0.666252\pi$
$444$	−2.00000	+	12.0000i	−0.0949158	+	0.569495i
$445$	−28.0000			−1.32733
$446$			38.0000i			1.79935i
$447$	−15.0000			−0.709476
$448$	24.0000			1.13389
$449$			26.0000i			1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$			26.0000i			1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$		−	4.00000i		−	0.188562i
$451$	9.00000			0.423793
$452$		−	8.00000i		−	0.376288i
$453$	8.00000			0.375873
$454$	16.0000			0.750917
$455$	−36.0000			−1.68771
$456$		0			0
$457$		−	18.0000i		−	0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$		−	18.0000i		−	0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$		−	10.0000i		−	0.467269i
$459$			10.0000i			0.466760i
$460$	−16.0000			−0.746004
$461$			10.0000i			0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$			10.0000i			0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$			18.0000i			0.837436i
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$			16.0000i			0.742781i
$465$		0			0
$466$		−	12.0000i		−	0.555889i
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		−	24.0000i		−	1.10940i
$469$	−36.0000			−1.66233
$470$	12.0000			0.553519
$471$	−3.00000			−0.138233
$472$		0			0
$473$			18.0000i			0.827641i
$474$	12.0000			0.551178
$475$			6.00000i			0.275299i
$476$		−	12.0000i		−	0.550019i
$477$	−18.0000			−0.824163
$478$	−32.0000			−1.46365
$479$		−	4.00000i		−	0.182765i	−0.995816	−	0.0913823i	$-0.970871\pi$
$479$		−	4.00000i		−	0.182765i	0.995816	−	0.0913823i	$-0.0291285\pi$
$480$	16.0000			0.730297
$481$	36.0000	+	6.00000i	1.64146	+	0.273576i
$482$		0			0
$483$		−	12.0000i		−	0.546019i
$484$	4.00000			0.181818
$485$	24.0000			1.08978
$486$		−	32.0000i		−	1.45155i
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$	8.00000			0.361403
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000			−0.270501
$493$	8.00000			0.360302
$494$			72.0000i			3.23943i
$495$		−	12.0000i		−	0.539360i
$496$		0			0
$497$	−9.00000			−0.403705
$498$		−	18.0000i		−	0.806599i
$499$			36.0000i			1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$			36.0000i			1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$			24.0000i			1.07331i
$501$		−	2.00000i		−	0.0893534i
$502$	−20.0000			−0.892644
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$			6.00000i			0.266996i
$506$	24.0000			1.06693
$507$	23.0000			1.02147
$508$	14.0000			0.621150
$509$	15.0000			0.664863			0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000			0.664863			0.332432	+	0.943127i	$0.392131\pi$
$510$		−	8.00000i		−	0.354246i
$511$	27.0000			1.19441
$512$			32.0000i			1.41421i
$513$			30.0000i			1.32453i
$514$	16.0000			0.705730
$515$	−12.0000			−0.528783
$516$		−	12.0000i		−	0.528271i
$517$	−9.00000			−0.395820
$518$	−36.0000	−	6.00000i	−1.58175	−	0.263625i
$519$	21.0000			0.921798
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$	−16.0000			−0.700301
$523$		−	6.00000i		−	0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$		−	6.00000i		−	0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$			20.0000i			0.873704i
$525$	−3.00000			−0.130931
$526$			18.0000i			0.784837i
$527$		0			0
$528$	−12.0000			−0.522233
$529$	7.00000			0.304348
$530$	36.0000			1.56374
$531$			8.00000i			0.347170i
$532$		−	36.0000i		−	1.56080i
$533$			18.0000i			0.779667i
$534$	−28.0000			−1.21168
$535$			24.0000i			1.03761i
$536$		0			0
$537$		−	16.0000i		−	0.690451i
$538$		−	60.0000i		−	2.58678i
$539$	−6.00000			−0.258438
$540$			20.0000i			0.860663i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		−	6.00000i		−	0.257722i
$543$	3.00000			0.128742
$544$	16.0000			0.685994
$545$	12.0000			0.514024
$546$	−36.0000			−1.54066
$547$			12.0000i			0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$			12.0000i			0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	−36.0000			−1.53784
$549$		0			0
$550$		−	6.00000i		−	0.255841i
$551$	24.0000			1.02243
$552$		0			0
$553$			18.0000i			0.765438i
$554$	−24.0000			−1.01966
$555$	−12.0000	−	2.00000i	−0.509372	−	0.0848953i
$556$		0			0
$557$		−	28.0000i		−	1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$		−	28.0000i		−	1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$	−36.0000			−1.52264
$560$			24.0000i			1.01419i
$561$			6.00000i			0.253320i
$562$	40.0000			1.68730
$563$		−	26.0000i		−	1.09577i	−0.836554	−	0.547885i	$-0.815433\pi$
$563$		−	26.0000i		−	1.09577i	0.836554	−	0.547885i	$-0.184567\pi$
$564$	6.00000			0.252646
$565$	8.00000			0.336563
$566$	−48.0000			−2.01759
$567$	3.00000			0.125988
$568$		0			0
$569$		−	44.0000i		−	1.84458i	−0.386503	−	0.922288i	$-0.626317\pi$
$569$		−	44.0000i		−	1.84458i	0.386503	−	0.922288i	$-0.373683\pi$
$570$		−	24.0000i		−	1.00525i
$571$	−33.0000			−1.38101			−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000			−1.38101			−0.690504	+	0.723329i	$0.742611\pi$
$572$		−	36.0000i		−	1.50524i
$573$		−	20.0000i		−	0.835512i
$574$		−	18.0000i		−	0.751305i
$575$			4.00000i			0.166812i
$576$	−16.0000			−0.666667
$577$			42.0000i			1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$			42.0000i			1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$			26.0000i			1.08146i
$579$			6.00000i			0.249351i
$580$	16.0000			0.664364
$581$	27.0000			1.12015
$582$	24.0000			0.994832
$583$	−27.0000			−1.11823
$584$		0			0
$585$	24.0000			0.992278
$586$		−	12.0000i		−	0.495715i
$587$			22.0000i			0.908037i	0.890992	+	0.454019i	$0.150010\pi$
$587$			22.0000i			0.908037i	−0.890992	+	0.454019i	$0.849990\pi$
$588$	4.00000			0.164957
$589$		0			0
$590$		−	16.0000i		−	0.658710i
$591$	−3.00000			−0.123404
$592$	4.00000	−	24.0000i	0.164399	−	0.986394i
$593$	−21.0000			−0.862367			−0.431183	−	0.902264i	$-0.641904\pi$
$593$	−21.0000			−0.862367			−0.431183	+	0.902264i	$0.641904\pi$
$594$		−	30.0000i		−	1.23091i
$595$	12.0000			0.491952
$596$	−30.0000			−1.22885
$597$			24.0000i			0.982255i
$598$			48.0000i			1.96287i
$599$	45.0000			1.83865			0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000			1.83865			0.919325	+	0.393499i	$0.128735\pi$
$600$		0			0
$601$	−18.0000			−0.734235			−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000			−0.734235			−0.367118	+	0.930175i	$0.619655\pi$
$602$	36.0000			1.46725
$603$	24.0000			0.977356
$604$	16.0000			0.651031
$605$			4.00000i			0.162623i
$606$			6.00000i			0.243733i
$607$		−	18.0000i		−	0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$		−	18.0000i		−	0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	48.0000			1.94666
$609$			12.0000i			0.486265i
$610$		0			0
$611$		−	18.0000i		−	0.728202i
$612$			8.00000i			0.323381i
$613$	29.0000			1.17130			0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000			1.17130			0.585649	+	0.810564i	$0.300840\pi$
$614$		−	14.0000i		−	0.564994i
$615$		−	6.00000i		−	0.241943i
$616$		0			0
$617$	3.00000			0.120775			0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000			0.120775			0.0603877	+	0.998175i	$0.480766\pi$
$618$	−12.0000			−0.482711
$619$	−35.0000			−1.40677			−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000			−1.40677			−0.703384	+	0.710810i	$0.748329\pi$
$620$		0			0
$621$			20.0000i			0.802572i
$622$	−20.0000			−0.801927
$623$		−	42.0000i		−	1.68269i
$624$		−	24.0000i		−	0.960769i
$625$	−19.0000			−0.760000
$626$	12.0000			0.479616
$627$			18.0000i			0.718851i
$628$	−6.00000			−0.239426
$629$	−12.0000	−	2.00000i	−0.478471	−	0.0797452i
$630$	−24.0000			−0.956183
$631$		−	30.0000i		−	1.19428i	−0.802137	−	0.597141i	$-0.796303\pi$
$631$		−	30.0000i		−	1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$632$		0			0
$633$	13.0000			0.516704
$634$			36.0000i			1.42974i
$635$			14.0000i			0.555573i
$636$	18.0000			0.713746
$637$		−	12.0000i		−	0.475457i
$638$	−24.0000			−0.950169
$639$	6.00000			0.237356
$640$		0			0
$641$	−33.0000			−1.30342			−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000			−1.30342			−0.651711	+	0.758468i	$0.725948\pi$
$642$			24.0000i			0.947204i
$643$			24.0000i			0.946468i	0.880937	+	0.473234i	$0.156913\pi$
$643$			24.0000i			0.946468i	−0.880937	+	0.473234i	$0.843087\pi$
$644$		−	24.0000i		−	0.945732i
$645$	12.0000			0.472500
$646$		−	24.0000i		−	0.944267i
$647$			2.00000i			0.0786281i	0.999227	+	0.0393141i	$0.0125173\pi$
$647$			2.00000i			0.0786281i	−0.999227	+	0.0393141i	$0.987483\pi$
$648$		0			0
$649$			12.0000i			0.471041i
$650$	12.0000			0.470679
$651$		0			0
$652$			12.0000i			0.469956i
$653$			44.0000i			1.72185i	0.508729	+	0.860927i	$0.330115\pi$
$653$			44.0000i			1.72185i	−0.508729	+	0.860927i	$0.669885\pi$
$654$	12.0000			0.469237
$655$	−20.0000			−0.781465
$656$	12.0000			0.468521
$657$	−18.0000			−0.702247
$658$			18.0000i			0.701713i
$659$	−45.0000			−1.75295			−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000			−1.75295			−0.876476	+	0.481446i	$0.840112\pi$
$660$			12.0000i			0.467099i
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$	−12.0000			−0.466041
$664$		0			0
$665$	36.0000			1.39602
$666$	24.0000	+	4.00000i	0.929981	+	0.154997i
$667$	16.0000			0.619522
$668$		−	4.00000i		−	0.154765i
$669$	−19.0000			−0.734582
$670$	−48.0000			−1.85440
$671$		0			0
$672$			24.0000i			0.925820i
$673$	−21.0000			−0.809491			−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000			−0.809491			−0.404745	+	0.914429i	$0.632640\pi$
$674$			26.0000i			1.00148i
$675$	5.00000			0.192450
$676$	46.0000			1.76923
$677$	3.00000			0.115299			0.0576497	−	0.998337i	$-0.481639\pi$
$677$	3.00000			0.115299			0.0576497	+	0.998337i	$0.481639\pi$
$678$	8.00000			0.307238
$679$			36.0000i			1.38155i
$680$		0			0
$681$			8.00000i			0.306561i
$682$		0			0
$683$		−	46.0000i		−	1.76014i	−0.474843	−	0.880071i	$-0.657495\pi$
$683$		−	46.0000i		−	1.76014i	0.474843	−	0.880071i	$-0.342505\pi$
$684$			24.0000i			0.917663i
$685$		−	36.0000i		−	1.37549i
$686$		−	30.0000i		−	1.14541i
$687$	5.00000			0.190762
$688$			24.0000i			0.914991i
$689$		−	54.0000i		−	2.05724i
$690$		−	16.0000i		−	0.609110i
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$	42.0000			1.59660
$693$	18.0000			0.683763
$694$	−64.0000			−2.42941
$695$		0			0
$696$		0			0
$697$		−	6.00000i		−	0.227266i
$698$			60.0000i			2.27103i
$699$	6.00000			0.226941
$700$	−6.00000			−0.226779
$701$			20.0000i			0.755390i	0.925930	+	0.377695i	$0.123283\pi$
$701$			20.0000i			0.755390i	−0.925930	+	0.377695i	$0.876717\pi$
$702$	60.0000			2.26455
$703$	−36.0000	−	6.00000i	−1.35777	−	0.226294i
$704$	−24.0000			−0.904534
$705$			6.00000i			0.225973i
$706$	−28.0000			−1.05379
$707$	−9.00000			−0.338480
$708$		−	8.00000i		−	0.300658i
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$	−12.0000			−0.450352
$711$		−	12.0000i		−	0.450035i
$712$		0			0
$713$		0			0
$714$	12.0000			0.449089
$715$	36.0000			1.34632
$716$		−	32.0000i		−	1.19590i
$717$		−	16.0000i		−	0.597531i
$718$		−	30.0000i		−	1.11959i
$719$	15.0000			0.559406			0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000			0.559406			0.279703	+	0.960087i	$0.409764\pi$
$720$		−	16.0000i		−	0.596285i
$721$		−	18.0000i		−	0.670355i
$722$		−	34.0000i		−	1.26535i
$723$		0			0
$724$	6.00000			0.222988
$725$		−	4.00000i		−	0.148556i
$726$			4.00000i			0.148454i
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$	36.0000			1.33242
$731$	12.0000			0.443836
$732$		0			0
$733$	−31.0000			−1.14501			−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000			−1.14501			−0.572506	+	0.819901i	$0.694029\pi$
$734$			16.0000i			0.590571i
$735$			4.00000i			0.147542i
$736$	32.0000			1.17954
$737$	36.0000			1.32608
$738$			12.0000i			0.441726i
$739$	25.0000			0.919640			0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000			0.919640			0.459820	+	0.888012i	$0.347914\pi$
$740$	−24.0000	−	4.00000i	−0.882258	−	0.147043i
$741$	−36.0000			−1.32249
$742$			54.0000i			1.98240i
$743$	39.0000			1.43077			0.715386	−	0.698730i	$-0.246251\pi$
$743$	39.0000			1.43077			0.715386	+	0.698730i	$0.246251\pi$
$744$		0			0
$745$		−	30.0000i		−	1.09911i
$746$		−	42.0000i		−	1.53773i
$747$	−18.0000			−0.658586
$748$			12.0000i			0.438763i
$749$	−36.0000			−1.31541
$750$	−24.0000			−0.876356
$751$	27.0000			0.985244			0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000			0.985244			0.492622	+	0.870243i	$0.336039\pi$
$752$	−12.0000			−0.437595
$753$		−	10.0000i		−	0.364420i
$754$		−	48.0000i		−	1.74806i
$755$			16.0000i			0.582300i
$756$	−30.0000			−1.09109
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$			30.0000i			1.08965i
$759$			12.0000i			0.435572i
$760$		0			0
$761$	−33.0000			−1.19625			−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000			−1.19625			−0.598125	+	0.801403i	$0.704087\pi$
$762$			14.0000i			0.507166i
$763$			18.0000i			0.651644i
$764$		−	40.0000i		−	1.44715i
$765$	−8.00000			−0.289241
$766$	32.0000			1.15621
$767$	−24.0000			−0.866590
$768$	−16.0000			−0.577350
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$	−36.0000			−1.29735
$771$			8.00000i			0.288113i
$772$			12.0000i			0.431889i
$773$	39.0000			1.40273			0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000			1.40273			0.701366	+	0.712801i	$0.252574\pi$
$774$	−24.0000			−0.862662
$775$		0			0
$776$		0			0
$777$	3.00000	−	18.0000i	0.107624	−	0.645746i
$778$	−32.0000			−1.14726
$779$		−	18.0000i		−	0.644917i
$780$	−24.0000			−0.859338
$781$	9.00000			0.322045
$782$		−	16.0000i		−	0.572159i
$783$		−	20.0000i		−	0.714742i
$784$	−8.00000			−0.285714
$785$		−	6.00000i		−	0.214149i
$786$	−20.0000			−0.713376
$787$	23.0000			0.819861			0.409931	−	0.912117i	$-0.365553\pi$
$787$	23.0000			0.819861			0.409931	+	0.912117i	$0.365553\pi$
$788$	−6.00000			−0.213741
$789$	−9.00000			−0.320408
$790$			24.0000i			0.853882i
$791$			12.0000i			0.426671i
$792$		0			0
$793$		0			0
$794$			66.0000i			2.34225i
$795$			18.0000i			0.638394i
$796$			48.0000i			1.70131i
$797$		−	8.00000i		−	0.283375i	−0.989911	−	0.141687i	$-0.954747\pi$
$797$		−	8.00000i		−	0.283375i	0.989911	−	0.141687i	$-0.0452527\pi$
$798$	36.0000			1.27439
$799$			6.00000i			0.212265i
$800$		−	8.00000i		−	0.282843i
$801$			28.0000i			0.989331i
$802$	20.0000			0.706225
$803$	−27.0000			−0.952809
$804$	−24.0000			−0.846415
$805$	24.0000			0.845889
$806$		0			0
$807$	30.0000			1.05605
$808$		0			0
$809$		−	44.0000i		−	1.54696i	−0.633822	−	0.773479i	$-0.718515\pi$
$809$		−	44.0000i		−	1.54696i	0.633822	−	0.773479i	$-0.281485\pi$
$810$	4.00000			0.140546
$811$	−33.0000			−1.15879			−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000			−1.15879			−0.579393	+	0.815048i	$0.696710\pi$
$812$			24.0000i			0.842235i
$813$	3.00000			0.105215
$814$	36.0000	+	6.00000i	1.26180	+	0.210300i
$815$	−12.0000			−0.420342
$816$			8.00000i			0.280056i
$817$	36.0000			1.25948
$818$	48.0000			1.67828
$819$			36.0000i			1.25794i
$820$		−	12.0000i		−	0.419058i
$821$	−3.00000			−0.104701			−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000			−0.104701			−0.0523504	+	0.998629i	$0.516671\pi$
$822$		−	36.0000i		−	1.25564i
$823$	4.00000			0.139431			0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000			0.139431			0.0697156	+	0.997567i	$0.477791\pi$
$824$		0			0
$825$	3.00000			0.104447
$826$	24.0000			0.835067
$827$		−	8.00000i		−	0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$		−	8.00000i		−	0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$			16.0000i			0.556038i
$829$		−	24.0000i		−	0.833554i	−0.909009	−	0.416777i	$-0.863160\pi$
$829$		−	24.0000i		−	0.833554i	0.909009	−	0.416777i	$-0.136840\pi$
$830$	36.0000			1.24958
$831$		−	12.0000i		−	0.416275i
$832$		−	48.0000i		−	1.66410i
$833$			4.00000i			0.138592i
$834$		0			0
$835$	4.00000			0.138426
$836$			36.0000i			1.24509i
$837$		0			0
$838$			30.0000i			1.03633i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$	−60.0000			−2.06774
$843$			20.0000i			0.688837i
$844$	26.0000			0.894957
$845$			46.0000i			1.58245i
$846$		−	12.0000i		−	0.412568i
$847$	−6.00000			−0.206162
$848$	−36.0000			−1.23625
$849$		−	24.0000i		−	0.823678i
$850$	−4.00000			−0.137199
$851$	−24.0000	−	4.00000i	−0.822709	−	0.137118i
$852$	−6.00000			−0.205557
$853$			24.0000i			0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$			24.0000i			0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$		0			0
$855$	−24.0000			−0.820783
$856$		0			0
$857$			22.0000i			0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$			22.0000i			0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	36.0000			1.22902
$859$			6.00000i			0.204717i	0.994748	+	0.102359i	$0.0326389\pi$
$859$			6.00000i			0.204717i	−0.994748	+	0.102359i	$0.967361\pi$
$860$	24.0000			0.818393
$861$	9.00000			0.306719
$862$	−20.0000			−0.681203
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		−	40.0000i		−	1.36083i
$865$			42.0000i			1.42804i
$866$			18.0000i			0.611665i
$867$	−13.0000			−0.441503
$868$		0			0
$869$		−	18.0000i		−	0.610608i
$870$			16.0000i			0.542451i
$871$			72.0000i			2.43963i
$872$		0			0
$873$		−	24.0000i		−	0.812277i
$874$		−	48.0000i		−	1.62362i
$875$		−	36.0000i		−	1.21702i
$876$	18.0000			0.608164
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$	48.0000			1.61992
$879$	6.00000			0.202375
$880$		−	24.0000i		−	0.809040i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		−	8.00000i		−	0.269374i
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	−24.0000			−0.807207
$885$	8.00000			0.268917
$886$		−	42.0000i		−	1.41102i
$887$	−57.0000			−1.91387			−0.956936	−	0.290298i	$-0.906246\pi$
$887$	−57.0000			−1.91387			−0.956936	+	0.290298i	$0.906246\pi$
$888$		0			0
$889$	−21.0000			−0.704317
$890$		−	56.0000i		−	1.87712i
$891$	−3.00000			−0.100504
$892$	−38.0000			−1.27233
$893$			18.0000i			0.602347i
$894$		−	30.0000i		−	1.00335i
$895$	32.0000			1.06964
$896$		0			0
$897$	−24.0000			−0.801337
$898$	−52.0000			−1.73526
$899$		0			0
$900$	4.00000			0.133333
$901$			18.0000i			0.599667i
$902$			18.0000i			0.599334i
$903$			18.0000i			0.599002i
$904$		0			0
$905$			6.00000i			0.199447i
$906$			16.0000i			0.531564i
$907$			12.0000i			0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$			12.0000i			0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$			16.0000i			0.530979i
$909$	6.00000			0.199007
$910$		−	72.0000i		−	2.38678i
$911$			50.0000i			1.65657i	0.560304	+	0.828287i	$0.310684\pi$
$911$			50.0000i			1.65657i	−0.560304	+	0.828287i	$0.689316\pi$
$912$			24.0000i			0.794719i
$913$	−27.0000			−0.893570
$914$	36.0000			1.19077
$915$		0			0
$916$	10.0000			0.330409
$917$		−	30.0000i		−	0.990687i
$918$	−20.0000			−0.660098
$919$			6.00000i			0.197922i	0.995091	+	0.0989609i	$0.0315519\pi$
$919$			6.00000i			0.197922i	−0.995091	+	0.0989609i	$0.968448\pi$
$920$		0			0
$921$	7.00000			0.230658
$922$	−20.0000			−0.658665
$923$			18.0000i			0.592477i
$924$	−18.0000			−0.592157
$925$	−1.00000	+	6.00000i	−0.0328798	+	0.197279i
$926$	72.0000			2.36607
$927$			12.0000i			0.394132i
$928$	−32.0000			−1.05045
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$			12.0000i			0.393284i
$932$	12.0000			0.393073
$933$		−	10.0000i		−	0.327385i
$934$	56.0000			1.83238
$935$	−12.0000			−0.392442
$936$		0			0
$937$	−47.0000			−1.53542			−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000			−1.53542			−0.767712	+	0.640796i	$0.778605\pi$
$938$		−	72.0000i		−	2.35088i
$939$			6.00000i			0.195803i
$940$			12.0000i			0.391397i
$941$	−18.0000			−0.586783			−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000			−0.586783			−0.293392	+	0.955992i	$0.594784\pi$
$942$		−	6.00000i		−	0.195491i
$943$		−	12.0000i		−	0.390774i
$944$			16.0000i			0.520756i
$945$		−	30.0000i		−	0.975900i
$946$	−36.0000			−1.17046
$947$		−	38.0000i		−	1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$947$		−	38.0000i		−	1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$948$			12.0000i			0.389742i
$949$		−	54.0000i		−	1.75291i
$950$	−12.0000			−0.389331
$951$	−18.0000			−0.583690
$952$		0			0
$953$	39.0000			1.26333			0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000			1.26333			0.631667	+	0.775240i	$0.282371\pi$
$954$		−	36.0000i		−	1.16554i
$955$	40.0000			1.29437
$956$		−	32.0000i		−	1.03495i
$957$		−	12.0000i		−	0.387905i
$958$	8.00000			0.258468
$959$	54.0000			1.74375
$960$			16.0000i			0.516398i
$961$	31.0000			1.00000
$962$	−12.0000	+	72.0000i	−0.386896	+	2.32137i
$963$	24.0000			0.773389
$964$		0			0
$965$	−12.0000			−0.386294
$966$	24.0000			0.772187
$967$		−	48.0000i		−	1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$		−	48.0000i		−	1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$		0			0
$969$	12.0000			0.385496
$970$			48.0000i			1.54119i
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$	32.0000			1.02640
$973$		0			0
$974$	36.0000			1.15351
$975$			6.00000i			0.192154i
$976$		0			0
$977$			32.0000i			1.02377i	0.859054	+	0.511885i	$0.171053\pi$
$977$			32.0000i			1.02377i	−0.859054	+	0.511885i	$0.828947\pi$
$978$	−12.0000			−0.383718
$979$			42.0000i			1.34233i
$980$			8.00000i			0.255551i
$981$		−	12.0000i		−	0.383131i
$982$			24.0000i			0.765871i
$983$	39.0000			1.24391			0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000			1.24391			0.621953	+	0.783054i	$0.286339\pi$
$984$		0			0
$985$		−	6.00000i		−	0.191176i
$986$			16.0000i			0.509544i
$987$	−9.00000			−0.286473
$988$	−72.0000			−2.29063
$989$	24.0000			0.763156
$990$	24.0000			0.762770
$991$			30.0000i			0.952981i	0.879180	+	0.476491i	$0.158091\pi$
$991$			30.0000i			0.952981i	−0.879180	+	0.476491i	$0.841909\pi$
$992$		0			0
$993$		0			0
$994$		−	18.0000i		−	0.570925i
$995$	−48.0000			−1.52170
$996$	18.0000			0.570352
$997$		−	48.0000i		−	1.52018i	−0.649821	−	0.760088i	$-0.725156\pi$
$997$		−	48.0000i		−	1.52018i	0.649821	−	0.760088i	$-0.274844\pi$
$998$	−72.0000			−2.27912
$999$	−5.00000	+	30.0000i	−0.158193	+	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.2.b.a.36.2	yes	2
3.2	odd	2		333.2.c.a.73.1		2
4.3	odd	2		592.2.g.b.369.1		2
5.2	odd	4		925.2.d.a.924.2		2
5.3	odd	4		925.2.d.d.924.1		2
5.4	even	2		925.2.c.b.776.1		2
8.3	odd	2		2368.2.g.b.961.2		2
8.5	even	2		2368.2.g.f.961.2		2
12.11	even	2		5328.2.h.c.2737.2		2
37.6	odd	4		1369.2.a.f.1.1		1
37.31	odd	4		1369.2.a.a.1.1		1
37.36	even	2	inner	37.2.b.a.36.1	✓	2
111.110	odd	2		333.2.c.a.73.2		2
148.147	odd	2		592.2.g.b.369.2		2
185.73	odd	4		925.2.d.a.924.1		2
185.147	odd	4		925.2.d.d.924.2		2
185.184	even	2		925.2.c.b.776.2		2
296.147	odd	2		2368.2.g.b.961.1		2
296.221	even	2		2368.2.g.f.961.1		2
444.443	even	2		5328.2.h.c.2737.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.2.b.a.36.1	✓	2	37.36	even	2	inner
37.2.b.a.36.2	yes	2	1.1	even	1	trivial
333.2.c.a.73.1		2	3.2	odd	2
333.2.c.a.73.2		2	111.110	odd	2
592.2.g.b.369.1		2	4.3	odd	2
592.2.g.b.369.2		2	148.147	odd	2
925.2.c.b.776.1		2	5.4	even	2
925.2.c.b.776.2		2	185.184	even	2
925.2.d.a.924.1		2	185.73	odd	4
925.2.d.a.924.2		2	5.2	odd	4
925.2.d.d.924.1		2	5.3	odd	4
925.2.d.d.924.2		2	185.147	odd	4
1369.2.a.a.1.1		1	37.31	odd	4
1369.2.a.f.1.1		1	37.6	odd	4
2368.2.g.b.961.1		2	296.147	odd	2
2368.2.g.b.961.2		2	8.3	odd	2
2368.2.g.f.961.1		2	296.221	even	2
2368.2.g.f.961.2		2	8.5	even	2
5328.2.h.c.2737.1		2	444.443	even	2
5328.2.h.c.2737.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{5} -2.00000i q^{6} +3.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} -1.00000 q^{3} -2.00000 q^{4} -2.00000i q^{5} -2.00000i q^{6} +3.00000 q^{7} -2.00000 q^{9} +4.00000 q^{10} -3.00000 q^{11} +2.00000 q^{12} -6.00000i q^{13} +6.00000i q^{14} +2.00000i q^{15} -4.00000 q^{16} +2.00000i q^{17} -4.00000i q^{18} +6.00000i q^{19} +4.00000i q^{20} -3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +1.00000 q^{25} +12.0000 q^{26} +5.00000 q^{27} -6.00000 q^{28} -4.00000i q^{29} -4.00000 q^{30} -8.00000i q^{32} +3.00000 q^{33} -4.00000 q^{34} -6.00000i q^{35} +4.00000 q^{36} +(-1.00000 + 6.00000i) q^{37} -12.0000 q^{38} +6.00000i q^{39} -3.00000 q^{41} -6.00000i q^{42} -6.00000i q^{43} +6.00000 q^{44} +4.00000i q^{45} -8.00000 q^{46} +3.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} +2.00000i q^{50} -2.00000i q^{51} +12.0000i q^{52} +9.00000 q^{53} +10.0000i q^{54} +6.00000i q^{55} -6.00000i q^{57} +8.00000 q^{58} -4.00000i q^{59} -4.00000i q^{60} -6.00000 q^{63} +8.00000 q^{64} -12.0000 q^{65} +6.00000i q^{66} -12.0000 q^{67} -4.00000i q^{68} -4.00000i q^{69} +12.0000 q^{70} -3.00000 q^{71} +9.00000 q^{73} +(-12.0000 - 2.00000i) q^{74} -1.00000 q^{75} -12.0000i q^{76} -9.00000 q^{77} -12.0000 q^{78} +6.00000i q^{79} +8.00000i q^{80} +1.00000 q^{81} -6.00000i q^{82} +9.00000 q^{83} +6.00000 q^{84} +4.00000 q^{85} +12.0000 q^{86} +4.00000i q^{87} -14.0000i q^{89} -8.00000 q^{90} -18.0000i q^{91} -8.00000i q^{92} +6.00000i q^{94} +12.0000 q^{95} +8.00000i q^{96} +12.0000i q^{97} +4.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^4 + 6 * q^7 - 4 * q^9	\( 2 q - 2 q^{3} - 4 q^{4} + 6 q^{7} - 4 q^{9} + 8 q^{10} - 6 q^{11} + 4 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 24 q^{26} + 10 q^{27} - 12 q^{28} - 8 q^{30} + 6 q^{33} - 8 q^{34} + 8 q^{36} - 2 q^{37} - 24 q^{38} - 6 q^{41} + 12 q^{44} - 16 q^{46} + 6 q^{47} + 8 q^{48} + 4 q^{49} + 18 q^{53} + 16 q^{58} - 12 q^{63} + 16 q^{64} - 24 q^{65} - 24 q^{67} + 24 q^{70} - 6 q^{71} + 18 q^{73} - 24 q^{74} - 2 q^{75} - 18 q^{77} - 24 q^{78} + 2 q^{81} + 18 q^{83} + 12 q^{84} + 8 q^{85} + 24 q^{86} - 16 q^{90} + 24 q^{95} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 + 6 * q^7 - 4 * q^9 + 8 * q^10 - 6 * q^11 + 4 * q^12 - 8 * q^16 - 6 * q^21 + 2 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^28 - 8 * q^30 + 6 * q^33 - 8 * q^34 + 8 * q^36 - 2 * q^37 - 24 * q^38 - 6 * q^41 + 12 * q^44 - 16 * q^46 + 6 * q^47 + 8 * q^48 + 4 * q^49 + 18 * q^53 + 16 * q^58 - 12 * q^63 + 16 * q^64 - 24 * q^65 - 24 * q^67 + 24 * q^70 - 6 * q^71 + 18 * q^73 - 24 * q^74 - 2 * q^75 - 18 * q^77 - 24 * q^78 + 2 * q^81 + 18 * q^83 + 12 * q^84 + 8 * q^85 + 24 * q^86 - 16 * q^90 + 24 * q^95 + 12 * q^99

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