LMFDB - Embedded newform 33.3.b.a.23.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,3,Mod(23,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("33.23");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	33.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:	$0.899184872389$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 3 $ x^2 - x + 3
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			23.1
Root			$0.500000 + 1.65831i$ of defining polynomial
Character	$\chi$	$=$	33.23
Dual form			33.3.b.a.23.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} +6.63325i q^{5} -9.94987i q^{6} -8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} +22.0000 q^{10} -3.31662i q^{11} -21.0000 q^{12} +4.00000 q^{13} +26.5330i q^{14} +19.8997i q^{15} +5.00000 q^{16} -13.2665i q^{17} -29.8496i q^{18} -6.00000 q^{19} -46.4327i q^{20} -24.0000 q^{21} -11.0000 q^{22} -6.63325i q^{23} +29.8496i q^{24} -19.0000 q^{25} -13.2665i q^{26} +27.0000 q^{27} +56.0000 q^{28} +39.7995i q^{29} +66.0000 q^{30} -26.0000 q^{31} +23.2164i q^{32} -9.94987i q^{33} -44.0000 q^{34} -53.0660i q^{35} -63.0000 q^{36} +30.0000 q^{37} +19.8997i q^{38} +12.0000 q^{39} -66.0000 q^{40} -13.2665i q^{41} +79.5990i q^{42} +42.0000 q^{43} +23.2164i q^{44} +59.6992i q^{45} -22.0000 q^{46} -86.2322i q^{47} +15.0000 q^{48} +15.0000 q^{49} +63.0159i q^{50} -39.7995i q^{51} -28.0000 q^{52} +59.6992i q^{53} -89.5489i q^{54} +22.0000 q^{55} -79.5990i q^{56} -18.0000 q^{57} +132.000 q^{58} -66.3325i q^{59} -139.298i q^{60} +12.0000 q^{61} +86.2322i q^{62} -72.0000 q^{63} +97.0000 q^{64} +26.5330i q^{65} -33.0000 q^{66} +2.00000 q^{67} +92.8655i q^{68} -19.8997i q^{69} -176.000 q^{70} +59.6992i q^{71} +89.5489i q^{72} -74.0000 q^{73} -99.4987i q^{74} -57.0000 q^{75} +42.0000 q^{76} +26.5330i q^{77} -39.7995i q^{78} -40.0000 q^{79} +33.1662i q^{80} +81.0000 q^{81} -44.0000 q^{82} +39.7995i q^{83} +168.000 q^{84} +88.0000 q^{85} -139.298i q^{86} +119.398i q^{87} +33.0000 q^{88} +119.398i q^{89} +198.000 q^{90} -32.0000 q^{91} +46.4327i q^{92} -78.0000 q^{93} -286.000 q^{94} -39.7995i q^{95} +69.6491i q^{96} +62.0000 q^{97} -49.7494i q^{98} -29.8496i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9} + 44 q^{10} - 42 q^{12} + 8 q^{13} + 10 q^{16} - 12 q^{19} - 48 q^{21} - 22 q^{22} - 38 q^{25} + 54 q^{27} + 112 q^{28} + 132 q^{30} - 52 q^{31} - 88 q^{34}+ \cdots + 124 q^{97}+O(q^{100}) $ 2 * q + 6 * q^3 - 14 * q^4 - 16 * q^7 + 18 * q^9 + 44 * q^10 - 42 * q^12 + 8 * q^13 + 10 * q^16 - 12 * q^19 - 48 * q^21 - 22 * q^22 - 38 * q^25 + 54 * q^27 + 112 * q^28 + 132 * q^30 - 52 * q^31 - 88 * q^34 - 126 * q^36 + 60 * q^37 + 24 * q^39 - 132 * q^40 + 84 * q^43 - 44 * q^46 + 30 * q^48 + 30 * q^49 - 56 * q^52 + 44 * q^55 - 36 * q^57 + 264 * q^58 + 24 * q^61 - 144 * q^63 + 194 * q^64 - 66 * q^66 + 4 * q^67 - 352 * q^70 - 148 * q^73 - 114 * q^75 + 84 * q^76 - 80 * q^79 + 162 * q^81 - 88 * q^82 + 336 * q^84 + 176 * q^85 + 66 * q^88 + 396 * q^90 - 64 * q^91 - 156 * q^93 - 572 * q^94 + 124 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.31662i	− 1.65831i	−0.559017	−	0.829156i	$-0.688821\pi$
$2$	− 3.31662i	− 1.65831i	0.559017	−	0.829156i	$-0.311179\pi$
$3$	3.00000	1.00000
$3$	3.00000	1.00000
$4$	−7.00000	−1.75000
$5$	6.63325i	1.32665i	0.748331	+	0.663325i	$0.230855\pi$
$5$	6.63325i	1.32665i	−0.748331	+	0.663325i	$0.769145\pi$
$6$	− 9.94987i	− 1.65831i
$7$	−8.00000	−1.14286	−0.571429	−	0.820652i	$-0.693611\pi$
$7$	−8.00000	−1.14286	−0.571429	+	0.820652i	$0.693611\pi$
$8$	9.94987i	1.24373i
$9$	9.00000	1.00000
$10$	22.0000	2.20000
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	−21.0000	−1.75000
$13$	4.00000	0.307692	0.153846	−	0.988095i	$-0.450834\pi$
$13$	4.00000	0.307692	0.153846	+	0.988095i	$0.450834\pi$
$14$	26.5330i	1.89521i
$15$	19.8997i	1.32665i
$16$	5.00000	0.312500
$17$	− 13.2665i	− 0.780382i	−0.920734	−	0.390191i	$-0.872409\pi$
$17$	− 13.2665i	− 0.780382i	0.920734	−	0.390191i	$-0.127591\pi$
$18$	− 29.8496i	− 1.65831i
$19$	−6.00000	−0.315789	−0.157895	−	0.987456i	$-0.550471\pi$
$19$	−6.00000	−0.315789	−0.157895	+	0.987456i	$0.550471\pi$
$20$	− 46.4327i	− 2.32164i
$21$	−24.0000	−1.14286
$22$	−11.0000	−0.500000
$23$	− 6.63325i	− 0.288402i	−0.989548	−	0.144201i	$-0.953939\pi$
$23$	− 6.63325i	− 0.288402i	0.989548	−	0.144201i	$-0.0460612\pi$
$24$	29.8496i	1.24373i
$25$	−19.0000	−0.760000
$26$	− 13.2665i	− 0.510250i
$27$	27.0000	1.00000
$28$	56.0000	2.00000
$29$	39.7995i	1.37240i	0.727415	+	0.686198i	$0.240722\pi$
$29$	39.7995i	1.37240i	−0.727415	+	0.686198i	$0.759278\pi$
$30$	66.0000	2.20000
$31$	−26.0000	−0.838710	−0.419355	−	0.907822i	$-0.637744\pi$
$31$	−26.0000	−0.838710	−0.419355	+	0.907822i	$0.637744\pi$
$32$	23.2164i	0.725512i
$33$	− 9.94987i	− 0.301511i
$34$	−44.0000	−1.29412
$35$	− 53.0660i	− 1.51617i
$36$	−63.0000	−1.75000
$37$	30.0000	0.810811	0.405405	−	0.914137i	$-0.367130\pi$
$37$	30.0000	0.810811	0.405405	+	0.914137i	$0.367130\pi$
$38$	19.8997i	0.523678i
$39$	12.0000	0.307692
$40$	−66.0000	−1.65000
$41$	− 13.2665i	− 0.323573i	−0.986826	−	0.161787i	$-0.948274\pi$
$41$	− 13.2665i	− 0.323573i	0.986826	−	0.161787i	$-0.0517256\pi$
$42$	79.5990i	1.89521i
$43$	42.0000	0.976744	0.488372	−	0.872635i	$-0.337591\pi$
$43$	42.0000	0.976744	0.488372	+	0.872635i	$0.337591\pi$
$44$	23.2164i	0.527645i
$45$	59.6992i	1.32665i
$46$	−22.0000	−0.478261
$47$	− 86.2322i	− 1.83473i	−0.398049	−	0.917364i	$-0.630312\pi$
$47$	− 86.2322i	− 1.83473i	0.398049	−	0.917364i	$-0.369688\pi$
$48$	15.0000	0.312500
$49$	15.0000	0.306122
$50$	63.0159i	1.26032i
$51$	− 39.7995i	− 0.780382i
$52$	−28.0000	−0.538462
$53$	59.6992i	1.12640i	0.826320	+	0.563200i	$0.190430\pi$
$53$	59.6992i	1.12640i	−0.826320	+	0.563200i	$0.809570\pi$
$54$	− 89.5489i	− 1.65831i
$55$	22.0000	0.400000
$56$	− 79.5990i	− 1.42141i
$57$	−18.0000	−0.315789
$58$	132.000	2.27586
$59$	− 66.3325i	− 1.12428i	−0.827042	−	0.562140i	$-0.809978\pi$
$59$	− 66.3325i	− 1.12428i	0.827042	−	0.562140i	$-0.190022\pi$
$60$	− 139.298i	− 2.32164i
$61$	12.0000	0.196721	0.0983607	−	0.995151i	$-0.468640\pi$
$61$	12.0000	0.196721	0.0983607	+	0.995151i	$0.468640\pi$
$62$	86.2322i	1.39084i
$63$	−72.0000	−1.14286
$64$	97.0000	1.51562
$65$	26.5330i	0.408200i
$66$	−33.0000	−0.500000
$67$	2.00000	0.0298507	0.0149254	−	0.999889i	$-0.495249\pi$
$67$	2.00000	0.0298507	0.0149254	+	0.999889i	$0.495249\pi$
$68$	92.8655i	1.36567i
$69$	− 19.8997i	− 0.288402i
$70$	−176.000	−2.51429
$71$	59.6992i	0.840834i	0.907331	+	0.420417i	$0.138116\pi$
$71$	59.6992i	0.840834i	−0.907331	+	0.420417i	$0.861884\pi$
$72$	89.5489i	1.24373i
$73$	−74.0000	−1.01370	−0.506849	−	0.862035i	$-0.669190\pi$
$73$	−74.0000	−1.01370	−0.506849	+	0.862035i	$0.669190\pi$
$74$	− 99.4987i	− 1.34458i
$75$	−57.0000	−0.760000
$76$	42.0000	0.552632
$77$	26.5330i	0.344584i
$78$	− 39.7995i	− 0.510250i
$79$	−40.0000	−0.506329	−0.253165	−	0.967423i	$-0.581471\pi$
$79$	−40.0000	−0.506329	−0.253165	+	0.967423i	$0.581471\pi$
$80$	33.1662i	0.414578i
$81$	81.0000	1.00000
$82$	−44.0000	−0.536585
$83$	39.7995i	0.479512i	0.970833	+	0.239756i	$0.0770674\pi$
$83$	39.7995i	0.479512i	−0.970833	+	0.239756i	$0.922933\pi$
$84$	168.000	2.00000
$85$	88.0000	1.03529
$86$	− 139.298i	− 1.61975i
$87$	119.398i	1.37240i
$88$	33.0000	0.375000
$89$	119.398i	1.34156i	0.741658	+	0.670778i	$0.234040\pi$
$89$	119.398i	1.34156i	−0.741658	+	0.670778i	$0.765960\pi$
$90$	198.000	2.20000
$91$	−32.0000	−0.351648
$92$	46.4327i	0.504704i
$93$	−78.0000	−0.838710
$94$	−286.000	−3.04255
$95$	− 39.7995i	− 0.418942i
$96$	69.6491i	0.725512i
$97$	62.0000	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$97$	62.0000	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$98$	− 49.7494i	− 0.507647i
$99$	− 29.8496i	− 0.301511i
$100$	133.000	1.33000
$101$	− 106.132i	− 1.05081i	−0.850852	−	0.525406i	$-0.823913\pi$
$101$	− 106.132i	− 1.05081i	0.850852	−	0.525406i	$-0.176087\pi$
$102$	−132.000	−1.29412
$103$	74.0000	0.718447	0.359223	−	0.933252i	$-0.383042\pi$
$103$	74.0000	0.718447	0.359223	+	0.933252i	$0.383042\pi$
$104$	39.7995i	0.382687i
$105$	− 159.198i	− 1.51617i
$106$	198.000	1.86792
$107$	39.7995i	0.371958i	0.982554	+	0.185979i	$0.0595456\pi$
$107$	39.7995i	0.371958i	−0.982554	+	0.185979i	$0.940454\pi$
$108$	−189.000	−1.75000
$109$	−200.000	−1.83486	−0.917431	−	0.397894i	$-0.869741\pi$
$109$	−200.000	−1.83486	−0.917431	+	0.397894i	$0.869741\pi$
$110$	− 72.9657i	− 0.663325i
$111$	90.0000	0.810811
$112$	−40.0000	−0.357143
$113$	39.7995i	0.352208i	0.984372	+	0.176104i	$0.0563495\pi$
$113$	39.7995i	0.352208i	−0.984372	+	0.176104i	$0.943651\pi$
$114$	59.6992i	0.523678i
$115$	44.0000	0.382609
$116$	− 278.596i	− 2.40169i
$117$	36.0000	0.307692
$118$	−220.000	−1.86441
$119$	106.132i	0.891865i
$120$	−198.000	−1.65000
$121$	−11.0000	−0.0909091
$122$	− 39.7995i	− 0.326225i
$123$	− 39.7995i	− 0.323573i
$124$	182.000	1.46774
$125$	39.7995i	0.318396i
$126$	238.797i	1.89521i
$127$	188.000	1.48031	0.740157	−	0.672434i	$-0.234751\pi$
$127$	188.000	1.48031	0.740157	+	0.672434i	$0.234751\pi$
$128$	− 228.847i	− 1.78787i
$129$	126.000	0.976744
$130$	88.0000	0.676923
$131$	− 39.7995i	− 0.303813i	−0.988395	−	0.151906i	$-0.951459\pi$
$131$	− 39.7995i	− 0.303813i	0.988395	−	0.151906i	$-0.0485413\pi$
$132$	69.6491i	0.527645i
$133$	48.0000	0.360902
$134$	− 6.63325i	− 0.0495019i
$135$	179.098i	1.32665i
$136$	132.000	0.970588
$137$	− 106.132i	− 0.774686i	−0.921936	−	0.387343i	$-0.873393\pi$
$137$	− 106.132i	− 0.774686i	0.921936	−	0.387343i	$-0.126607\pi$
$138$	−66.0000	−0.478261
$139$	−74.0000	−0.532374	−0.266187	−	0.963921i	$-0.585764\pi$
$139$	−74.0000	−0.532374	−0.266187	+	0.963921i	$0.585764\pi$
$140$	371.462i	2.65330i
$141$	− 258.697i	− 1.83473i
$142$	198.000	1.39437
$143$	− 13.2665i	− 0.0927727i
$144$	45.0000	0.312500
$145$	−264.000	−1.82069
$146$	245.430i	1.68103i
$147$	45.0000	0.306122
$148$	−210.000	−1.41892
$149$	92.8655i	0.623258i	0.950204	+	0.311629i	$0.100875\pi$
$149$	92.8655i	0.623258i	−0.950204	+	0.311629i	$0.899125\pi$
$150$	189.048i	1.26032i
$151$	−160.000	−1.05960	−0.529801	−	0.848122i	$-0.677734\pi$
$151$	−160.000	−1.05960	−0.529801	+	0.848122i	$0.677734\pi$
$152$	− 59.6992i	− 0.392758i
$153$	− 119.398i	− 0.780382i
$154$	88.0000	0.571429
$155$	− 172.464i	− 1.11267i
$156$	−84.0000	−0.538462
$157$	182.000	1.15924	0.579618	−	0.814888i	$-0.303202\pi$
$157$	182.000	1.15924	0.579618	+	0.814888i	$0.303202\pi$
$158$	132.665i	0.839652i
$159$	179.098i	1.12640i
$160$	−154.000	−0.962500
$161$	53.0660i	0.329602i
$162$	− 268.647i	− 1.65831i
$163$	−290.000	−1.77914	−0.889571	−	0.456798i	$-0.848996\pi$
$163$	−290.000	−1.77914	−0.889571	+	0.456798i	$0.848996\pi$
$164$	92.8655i	0.566253i
$165$	66.0000	0.400000
$166$	132.000	0.795181
$167$	238.797i	1.42992i	0.699164	+	0.714961i	$0.253556\pi$
$167$	238.797i	1.42992i	−0.699164	+	0.714961i	$0.746444\pi$
$168$	− 238.797i	− 1.42141i
$169$	−153.000	−0.905325
$170$	− 291.863i	− 1.71684i
$171$	−54.0000	−0.315789
$172$	−294.000	−1.70930
$173$	− 198.997i	− 1.15027i	−0.818057	−	0.575137i	$-0.804949\pi$
$173$	− 198.997i	− 1.15027i	0.818057	−	0.575137i	$-0.195051\pi$
$174$	396.000	2.27586
$175$	152.000	0.868571
$176$	− 16.5831i	− 0.0942223i
$177$	− 198.997i	− 1.12428i
$178$	396.000	2.22472
$179$	− 198.997i	− 1.11172i	−0.831277	−	0.555859i	$-0.812389\pi$
$179$	− 198.997i	− 1.11172i	0.831277	−	0.555859i	$-0.187611\pi$
$180$	− 417.895i	− 2.32164i
$181$	10.0000	0.0552486	0.0276243	−	0.999618i	$-0.491206\pi$
$181$	10.0000	0.0552486	0.0276243	+	0.999618i	$0.491206\pi$
$182$	106.132i	0.583143i
$183$	36.0000	0.196721
$184$	66.0000	0.358696
$185$	198.997i	1.07566i
$186$	258.697i	1.39084i
$187$	−44.0000	−0.235294
$188$	603.626i	3.21078i
$189$	−216.000	−1.14286
$190$	−132.000	−0.694737
$191$	112.765i	0.590394i	0.955436	+	0.295197i	$0.0953853\pi$
$191$	112.765i	0.590394i	−0.955436	+	0.295197i	$0.904615\pi$
$192$	291.000	1.51562
$193$	298.000	1.54404	0.772021	−	0.635597i	$-0.219246\pi$
$193$	298.000	1.54404	0.772021	+	0.635597i	$0.219246\pi$
$194$	− 205.631i	− 1.05995i
$195$	79.5990i	0.408200i
$196$	−105.000	−0.535714
$197$	− 132.665i	− 0.673426i	−0.941607	−	0.336713i	$-0.890685\pi$
$197$	− 132.665i	− 0.673426i	0.941607	−	0.336713i	$-0.109315\pi$
$198$	−99.0000	−0.500000
$199$	−42.0000	−0.211055	−0.105528	−	0.994416i	$-0.533653\pi$
$199$	−42.0000	−0.211055	−0.105528	+	0.994416i	$0.533653\pi$
$200$	− 189.048i	− 0.945238i
$201$	6.00000	0.0298507
$202$	−352.000	−1.74257
$203$	− 318.396i	− 1.56845i
$204$	278.596i	1.36567i
$205$	88.0000	0.429268
$206$	− 245.430i	− 1.19141i
$207$	− 59.6992i	− 0.288402i
$208$	20.0000	0.0961538
$209$	19.8997i	0.0952141i
$210$	−528.000	−2.51429
$211$	246.000	1.16588	0.582938	−	0.812516i	$-0.301903\pi$
$211$	246.000	1.16588	0.582938	+	0.812516i	$0.301903\pi$
$212$	− 417.895i	− 1.97120i
$213$	179.098i	0.840834i
$214$	132.000	0.616822
$215$	278.596i	1.29580i
$216$	268.647i	1.24373i
$217$	208.000	0.958525
$218$	663.325i	3.04278i
$219$	−222.000	−1.01370
$220$	−154.000	−0.700000
$221$	− 53.0660i	− 0.240118i
$222$	− 298.496i	− 1.34458i
$223$	−302.000	−1.35426	−0.677130	−	0.735863i	$-0.736777\pi$
$223$	−302.000	−1.35426	−0.677130	+	0.735863i	$0.736777\pi$
$224$	− 185.731i	− 0.829156i
$225$	−171.000	−0.760000
$226$	132.000	0.584071
$227$	198.997i	0.876641i	0.898819	+	0.438320i	$0.144426\pi$
$227$	198.997i	0.876641i	−0.898819	+	0.438320i	$0.855574\pi$
$228$	126.000	0.552632
$229$	150.000	0.655022	0.327511	−	0.944847i	$-0.393790\pi$
$229$	150.000	0.655022	0.327511	+	0.944847i	$0.393790\pi$
$230$	− 145.931i	− 0.634485i
$231$	79.5990i	0.344584i
$232$	−396.000	−1.70690
$233$	437.794i	1.87895i	0.342623	+	0.939473i	$0.388685\pi$
$233$	437.794i	1.87895i	−0.342623	+	0.939473i	$0.611315\pi$
$234$	− 119.398i	− 0.510250i
$235$	572.000	2.43404
$236$	464.327i	1.96749i
$237$	−120.000	−0.506329
$238$	352.000	1.47899
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	99.4987i	0.414578i
$241$	130.000	0.539419	0.269710	−	0.962942i	$-0.413072\pi$
$241$	130.000	0.539419	0.269710	+	0.962942i	$0.413072\pi$
$242$	36.4829i	0.150756i
$243$	243.000	1.00000
$244$	−84.0000	−0.344262
$245$	99.4987i	0.406117i
$246$	−132.000	−0.536585
$247$	−24.0000	−0.0971660
$248$	− 258.697i	− 1.04313i
$249$	119.398i	0.479512i
$250$	132.000	0.528000
$251$	− 291.863i	− 1.16280i	−0.813618	−	0.581400i	$-0.802505\pi$
$251$	− 291.863i	− 1.16280i	0.813618	−	0.581400i	$-0.197495\pi$
$252$	504.000	2.00000
$253$	−22.0000	−0.0869565
$254$	− 623.525i	− 2.45482i
$255$	264.000	1.03529
$256$	−371.000	−1.44922
$257$	− 490.860i	− 1.90996i	−0.296665	−	0.954981i	$-0.595875\pi$
$257$	− 490.860i	− 1.90996i	0.296665	−	0.954981i	$-0.404125\pi$
$258$	− 417.895i	− 1.61975i
$259$	−240.000	−0.926641
$260$	− 185.731i	− 0.714350i
$261$	358.195i	1.37240i
$262$	−132.000	−0.503817
$263$	− 225.530i	− 0.857530i	−0.903416	−	0.428765i	$-0.858949\pi$
$263$	− 225.530i	− 0.857530i	0.903416	−	0.428765i	$-0.141051\pi$
$264$	99.0000	0.375000
$265$	−396.000	−1.49434
$266$	− 159.198i	− 0.598489i
$267$	358.195i	1.34156i
$268$	−14.0000	−0.0522388
$269$	− 72.9657i	− 0.271248i	−0.990760	−	0.135624i	$-0.956696\pi$
$269$	− 72.9657i	− 0.271248i	0.990760	−	0.135624i	$-0.0433039\pi$
$270$	594.000	2.20000
$271$	−448.000	−1.65314	−0.826568	−	0.562836i	$-0.809710\pi$
$271$	−448.000	−1.65314	−0.826568	+	0.562836i	$0.809710\pi$
$272$	− 66.3325i	− 0.243869i
$273$	−96.0000	−0.351648
$274$	−352.000	−1.28467
$275$	63.0159i	0.229149i
$276$	139.298i	0.504704i
$277$	−260.000	−0.938628	−0.469314	−	0.883031i	$-0.655499\pi$
$277$	−260.000	−0.938628	−0.469314	+	0.883031i	$0.655499\pi$
$278$	245.430i	0.882843i
$279$	−234.000	−0.838710
$280$	528.000	1.88571
$281$	198.997i	0.708176i	0.935212	+	0.354088i	$0.115209\pi$
$281$	198.997i	0.708176i	−0.935212	+	0.354088i	$0.884791\pi$
$282$	−858.000	−3.04255
$283$	−50.0000	−0.176678	−0.0883392	−	0.996090i	$-0.528156\pi$
$283$	−50.0000	−0.176678	−0.0883392	+	0.996090i	$0.528156\pi$
$284$	− 417.895i	− 1.47146i
$285$	− 119.398i	− 0.418942i
$286$	−44.0000	−0.153846
$287$	106.132i	0.369798i
$288$	208.947i	0.725512i
$289$	113.000	0.391003
$290$	875.589i	3.01927i
$291$	186.000	0.639175
$292$	518.000	1.77397
$293$	− 477.594i	− 1.63001i	−0.579451	−	0.815007i	$-0.696733\pi$
$293$	− 477.594i	− 1.63001i	0.579451	−	0.815007i	$-0.303267\pi$
$294$	− 149.248i	− 0.507647i
$295$	440.000	1.49153
$296$	298.496i	1.00843i
$297$	− 89.5489i	− 0.301511i
$298$	308.000	1.03356
$299$	− 26.5330i	− 0.0887391i
$300$	399.000	1.33000
$301$	−336.000	−1.11628
$302$	530.660i	1.75715i
$303$	− 318.396i	− 1.05081i
$304$	−30.0000	−0.0986842
$305$	79.5990i	0.260980i
$306$	−396.000	−1.29412
$307$	86.0000	0.280130	0.140065	−	0.990142i	$-0.455269\pi$
$307$	86.0000	0.280130	0.140065	+	0.990142i	$0.455269\pi$
$308$	− 185.731i	− 0.603023i
$309$	222.000	0.718447
$310$	−572.000	−1.84516
$311$	− 19.8997i	− 0.0639863i	−0.999488	−	0.0319932i	$-0.989815\pi$
$311$	− 19.8997i	− 0.0639863i	0.999488	−	0.0319932i	$-0.0101855\pi$
$312$	119.398i	0.382687i
$313$	98.0000	0.313099	0.156550	−	0.987670i	$-0.449963\pi$
$313$	98.0000	0.313099	0.156550	+	0.987670i	$0.449963\pi$
$314$	− 603.626i	− 1.92237i
$315$	− 477.594i	− 1.51617i
$316$	280.000	0.886076
$317$	− 311.763i	− 0.983479i	−0.870742	−	0.491739i	$-0.836361\pi$
$317$	− 311.763i	− 0.983479i	0.870742	−	0.491739i	$-0.163639\pi$
$318$	594.000	1.86792
$319$	132.000	0.413793
$320$	643.425i	2.01070i
$321$	119.398i	0.371958i
$322$	176.000	0.546584
$323$	79.5990i	0.246437i
$324$	−567.000	−1.75000
$325$	−76.0000	−0.233846
$326$	961.821i	2.95037i
$327$	−600.000	−1.83486
$328$	132.000	0.402439
$329$	689.858i	2.09683i
$330$	− 218.897i	− 0.663325i
$331$	−218.000	−0.658610	−0.329305	−	0.944224i	$-0.606814\pi$
$331$	−218.000	−0.658610	−0.329305	+	0.944224i	$0.606814\pi$
$332$	− 278.596i	− 0.839146i
$333$	270.000	0.810811
$334$	792.000	2.37126
$335$	13.2665i	0.0396015i
$336$	−120.000	−0.357143
$337$	278.000	0.824926	0.412463	−	0.910974i	$-0.364669\pi$
$337$	278.000	0.824926	0.412463	+	0.910974i	$0.364669\pi$
$338$	507.444i	1.50131i
$339$	119.398i	0.352208i
$340$	−616.000	−1.81176
$341$	86.2322i	0.252880i
$342$	179.098i	0.523678i
$343$	272.000	0.793003
$344$	417.895i	1.21481i
$345$	132.000	0.382609
$346$	−660.000	−1.90751
$347$	− 331.662i	− 0.955800i	−0.878414	−	0.477900i	$-0.841398\pi$
$347$	− 331.662i	− 0.955800i	0.878414	−	0.477900i	$-0.158602\pi$
$348$	− 835.789i	− 2.40169i
$349$	324.000	0.928367	0.464183	−	0.885739i	$-0.346348\pi$
$349$	324.000	0.928367	0.464183	+	0.885739i	$0.346348\pi$
$350$	− 504.127i	− 1.44036i
$351$	108.000	0.307692
$352$	77.0000	0.218750
$353$	− 397.995i	− 1.12746i	−0.825958	−	0.563732i	$-0.809365\pi$
$353$	− 397.995i	− 1.12746i	0.825958	−	0.563732i	$-0.190635\pi$
$354$	−660.000	−1.86441
$355$	−396.000	−1.11549
$356$	− 835.789i	− 2.34772i
$357$	318.396i	0.891865i
$358$	−660.000	−1.84358
$359$	305.129i	0.849943i	0.905207	+	0.424971i	$0.139716\pi$
$359$	305.129i	0.849943i	−0.905207	+	0.424971i	$0.860284\pi$
$360$	−594.000	−1.65000
$361$	−325.000	−0.900277
$362$	− 33.1662i	− 0.0916195i
$363$	−33.0000	−0.0909091
$364$	224.000	0.615385
$365$	− 490.860i	− 1.34482i
$366$	− 119.398i	− 0.326225i
$367$	−278.000	−0.757493	−0.378747	−	0.925500i	$-0.623645\pi$
$367$	−278.000	−0.757493	−0.378747	+	0.925500i	$0.623645\pi$
$368$	− 33.1662i	− 0.0901257i
$369$	− 119.398i	− 0.323573i
$370$	660.000	1.78378
$371$	− 477.594i	− 1.28732i
$372$	546.000	1.46774
$373$	−68.0000	−0.182306	−0.0911528	−	0.995837i	$-0.529055\pi$
$373$	−68.0000	−0.182306	−0.0911528	+	0.995837i	$0.529055\pi$
$374$	145.931i	0.390191i
$375$	119.398i	0.318396i
$376$	858.000	2.28191
$377$	159.198i	0.422276i
$378$	716.391i	1.89521i
$379$	670.000	1.76781	0.883905	−	0.467666i	$-0.154905\pi$
$379$	670.000	1.76781	0.883905	+	0.467666i	$0.154905\pi$
$380$	278.596i	0.733149i
$381$	564.000	1.48031
$382$	374.000	0.979058
$383$	33.1662i	0.0865959i	0.999062	+	0.0432980i	$0.0137865\pi$
$383$	33.1662i	0.0865959i	−0.999062	+	0.0432980i	$0.986214\pi$
$384$	− 686.541i	− 1.78787i
$385$	−176.000	−0.457143
$386$	− 988.354i	− 2.56050i
$387$	378.000	0.976744
$388$	−434.000	−1.11856
$389$	417.895i	1.07428i	0.843493	+	0.537140i	$0.180495\pi$
$389$	417.895i	1.07428i	−0.843493	+	0.537140i	$0.819505\pi$
$390$	264.000	0.676923
$391$	−88.0000	−0.225064
$392$	149.248i	0.380735i
$393$	− 119.398i	− 0.303813i
$394$	−440.000	−1.11675
$395$	− 265.330i	− 0.671721i
$396$	208.947i	0.527645i
$397$	−86.0000	−0.216625	−0.108312	−	0.994117i	$-0.534545\pi$
$397$	−86.0000	−0.216625	−0.108312	+	0.994117i	$0.534545\pi$
$398$	139.298i	0.349996i
$399$	144.000	0.360902
$400$	−95.0000	−0.237500
$401$	252.063i	0.628587i	0.949326	+	0.314294i	$0.101768\pi$
$401$	252.063i	0.628587i	−0.949326	+	0.314294i	$0.898232\pi$
$402$	− 19.8997i	− 0.0495019i
$403$	−104.000	−0.258065
$404$	742.924i	1.83892i
$405$	537.293i	1.32665i
$406$	−1056.00	−2.60099
$407$	− 99.4987i	− 0.244469i
$408$	396.000	0.970588
$409$	510.000	1.24694	0.623472	−	0.781846i	$-0.285722\pi$
$409$	510.000	1.24694	0.623472	+	0.781846i	$0.285722\pi$
$410$	− 291.863i	− 0.711861i
$411$	− 318.396i	− 0.774686i
$412$	−518.000	−1.25728
$413$	530.660i	1.28489i
$414$	−198.000	−0.478261
$415$	−264.000	−0.636145
$416$	92.8655i	0.223234i
$417$	−222.000	−0.532374
$418$	66.0000	0.157895
$419$	530.660i	1.26649i	0.773951	+	0.633246i	$0.218278\pi$
$419$	530.660i	1.26649i	−0.773951	+	0.633246i	$0.781722\pi$
$420$	1114.39i	2.65330i
$421$	−170.000	−0.403800	−0.201900	−	0.979406i	$-0.564712\pi$
$421$	−170.000	−0.403800	−0.201900	+	0.979406i	$0.564712\pi$
$422$	− 815.890i	− 1.93339i
$423$	− 776.090i	− 1.83473i
$424$	−594.000	−1.40094
$425$	252.063i	0.593091i
$426$	594.000	1.39437
$427$	−96.0000	−0.224824
$428$	− 278.596i	− 0.650926i
$429$	− 39.7995i	− 0.0927727i
$430$	924.000	2.14884
$431$	− 278.596i	− 0.646396i	−0.946331	−	0.323198i	$-0.895242\pi$
$431$	− 278.596i	− 0.646396i	0.946331	−	0.323198i	$-0.104758\pi$
$432$	135.000	0.312500
$433$	−542.000	−1.25173	−0.625866	−	0.779931i	$-0.715254\pi$
$433$	−542.000	−1.25173	−0.625866	+	0.779931i	$0.715254\pi$
$434$	− 689.858i	− 1.58953i
$435$	−792.000	−1.82069
$436$	1400.00	3.21101
$437$	39.7995i	0.0910744i
$438$	736.291i	1.68103i
$439$	328.000	0.747153	0.373576	−	0.927599i	$-0.378131\pi$
$439$	328.000	0.747153	0.373576	+	0.927599i	$0.378131\pi$
$440$	218.897i	0.497494i
$441$	135.000	0.306122
$442$	−176.000	−0.398190
$443$	− 132.665i	− 0.299470i	−0.988726	−	0.149735i	$-0.952158\pi$
$443$	− 132.665i	− 0.299470i	0.988726	−	0.149735i	$-0.0478420\pi$
$444$	−630.000	−1.41892
$445$	−792.000	−1.77978
$446$	1001.62i	2.24579i
$447$	278.596i	0.623258i
$448$	−776.000	−1.73214
$449$	451.061i	1.00459i	0.864696	+	0.502295i	$0.167511\pi$
$449$	451.061i	1.00459i	−0.864696	+	0.502295i	$0.832489\pi$
$450$	567.143i	1.26032i
$451$	−44.0000	−0.0975610
$452$	− 278.596i	− 0.616364i
$453$	−480.000	−1.05960
$454$	660.000	1.45374
$455$	− 212.264i	− 0.466514i
$456$	− 179.098i	− 0.392758i
$457$	342.000	0.748359	0.374179	−	0.927356i	$-0.377924\pi$
$457$	342.000	0.748359	0.374179	+	0.927356i	$0.377924\pi$
$458$	− 497.494i	− 1.08623i
$459$	− 358.195i	− 0.780382i
$460$	−308.000	−0.669565
$461$	79.5990i	0.172666i	0.996266	+	0.0863330i	$0.0275149\pi$
$461$	79.5990i	0.172666i	−0.996266	+	0.0863330i	$0.972485\pi$
$462$	264.000	0.571429
$463$	−86.0000	−0.185745	−0.0928726	−	0.995678i	$-0.529605\pi$
$463$	−86.0000	−0.185745	−0.0928726	+	0.995678i	$0.529605\pi$
$464$	198.997i	0.428874i
$465$	− 517.393i	− 1.11267i
$466$	1452.00	3.11588
$467$	596.992i	1.27836i	0.769059	+	0.639178i	$0.220725\pi$
$467$	596.992i	1.27836i	−0.769059	+	0.639178i	$0.779275\pi$
$468$	−252.000	−0.538462
$469$	−16.0000	−0.0341151
$470$	− 1897.11i	− 4.03640i
$471$	546.000	1.15924
$472$	660.000	1.39831
$473$	− 139.298i	− 0.294499i
$474$	397.995i	0.839652i
$475$	114.000	0.240000
$476$	− 742.924i	− 1.56076i
$477$	537.293i	1.12640i
$478$	0	0
$479$	− 225.530i	− 0.470836i	−0.971894	−	0.235418i	$-0.924354\pi$
$479$	− 225.530i	− 0.470836i	0.971894	−	0.235418i	$-0.0756459\pi$
$480$	−462.000	−0.962500
$481$	120.000	0.249480
$482$	− 431.161i	− 0.894525i
$483$	159.198i	0.329602i
$484$	77.0000	0.159091
$485$	411.261i	0.847962i
$486$	− 805.940i	− 1.65831i
$487$	446.000	0.915811	0.457906	−	0.889001i	$-0.348600\pi$
$487$	446.000	0.915811	0.457906	+	0.889001i	$0.348600\pi$
$488$	119.398i	0.244669i
$489$	−870.000	−1.77914
$490$	330.000	0.673469
$491$	703.124i	1.43203i	0.698087	+	0.716013i	$0.254035\pi$
$491$	703.124i	1.43203i	−0.698087	+	0.716013i	$0.745965\pi$
$492$	278.596i	0.566253i
$493$	528.000	1.07099
$494$	79.5990i	0.161132i
$495$	198.000	0.400000
$496$	−130.000	−0.262097
$497$	− 477.594i	− 0.960954i
$498$	396.000	0.795181
$499$	−58.0000	−0.116232	−0.0581162	−	0.998310i	$-0.518509\pi$
$499$	−58.0000	−0.116232	−0.0581162	+	0.998310i	$0.518509\pi$
$500$	− 278.596i	− 0.557193i
$501$	716.391i	1.42992i
$502$	−968.000	−1.92829
$503$	504.127i	1.00224i	0.865378	+	0.501120i	$0.167079\pi$
$503$	504.127i	1.00224i	−0.865378	+	0.501120i	$0.832921\pi$
$504$	− 716.391i	− 1.42141i
$505$	704.000	1.39406
$506$	72.9657i	0.144201i
$507$	−459.000	−0.905325
$508$	−1316.00	−2.59055
$509$	− 736.291i	− 1.44654i	−0.690563	−	0.723272i	$-0.742637\pi$
$509$	− 736.291i	− 1.44654i	0.690563	−	0.723272i	$-0.257363\pi$
$510$	− 875.589i	− 1.71684i
$511$	592.000	1.15851
$512$	315.079i	0.615389i
$513$	−162.000	−0.315789
$514$	−1628.00	−3.16732
$515$	490.860i	0.953127i
$516$	−882.000	−1.70930
$517$	−286.000	−0.553191
$518$	795.990i	1.53666i
$519$	− 596.992i	− 1.15027i
$520$	−264.000	−0.507692
$521$	− 278.596i	− 0.534734i	−0.963595	−	0.267367i	$-0.913846\pi$
$521$	− 278.596i	− 0.534734i	0.963595	−	0.267367i	$-0.0861536\pi$
$522$	1188.00	2.27586
$523$	−142.000	−0.271511	−0.135755	−	0.990742i	$-0.543346\pi$
$523$	−142.000	−0.271511	−0.135755	+	0.990742i	$0.543346\pi$
$524$	278.596i	0.531673i
$525$	456.000	0.868571
$526$	−748.000	−1.42205
$527$	344.929i	0.654514i
$528$	− 49.7494i	− 0.0942223i
$529$	485.000	0.916824
$530$	1313.38i	2.47808i
$531$	− 596.992i	− 1.12428i
$532$	−336.000	−0.631579
$533$	− 53.0660i	− 0.0995610i
$534$	1188.00	2.22472
$535$	−264.000	−0.493458
$536$	19.8997i	0.0371264i
$537$	− 596.992i	− 1.11172i
$538$	−242.000	−0.449814
$539$	− 49.7494i	− 0.0922994i
$540$	− 1253.68i	− 2.32164i
$541$	400.000	0.739372	0.369686	−	0.929157i	$-0.379465\pi$
$541$	400.000	0.739372	0.369686	+	0.929157i	$0.379465\pi$
$542$	1485.85i	2.74142i
$543$	30.0000	0.0552486
$544$	308.000	0.566176
$545$	− 1326.65i	− 2.43422i
$546$	318.396i	0.583143i
$547$	170.000	0.310786	0.155393	−	0.987853i	$-0.450336\pi$
$547$	170.000	0.310786	0.155393	+	0.987853i	$0.450336\pi$
$548$	742.924i	1.35570i
$549$	108.000	0.196721
$550$	209.000	0.380000
$551$	− 238.797i	− 0.433388i
$552$	198.000	0.358696
$553$	320.000	0.578662
$554$	862.322i	1.55654i
$555$	596.992i	1.07566i
$556$	518.000	0.931655
$557$	− 849.056i	− 1.52434i	−0.647378	−	0.762169i	$-0.724135\pi$
$557$	− 849.056i	− 1.52434i	0.647378	−	0.762169i	$-0.275865\pi$
$558$	776.090i	1.39084i
$559$	168.000	0.300537
$560$	− 265.330i	− 0.473804i
$561$	−132.000	−0.235294
$562$	660.000	1.17438
$563$	703.124i	1.24889i	0.781069	+	0.624444i	$0.214675\pi$
$563$	703.124i	1.24889i	−0.781069	+	0.624444i	$0.785325\pi$
$564$	1810.88i	3.21078i
$565$	−264.000	−0.467257
$566$	165.831i	0.292988i
$567$	−648.000	−1.14286
$568$	−594.000	−1.04577
$569$	119.398i	0.209839i	0.994481	+	0.104920i	$0.0334585\pi$
$569$	119.398i	0.209839i	−0.994481	+	0.104920i	$0.966541\pi$
$570$	−396.000	−0.694737
$571$	−706.000	−1.23643	−0.618214	−	0.786010i	$-0.712143\pi$
$571$	−706.000	−1.23643	−0.618214	+	0.786010i	$0.712143\pi$
$572$	92.8655i	0.162352i
$573$	338.296i	0.590394i
$574$	352.000	0.613240
$575$	126.032i	0.219186i
$576$	873.000	1.51562
$577$	−738.000	−1.27903	−0.639515	−	0.768779i	$-0.720865\pi$
$577$	−738.000	−1.27903	−0.639515	+	0.768779i	$0.720865\pi$
$578$	− 374.779i	− 0.648406i
$579$	894.000	1.54404
$580$	1848.00	3.18621
$581$	− 318.396i	− 0.548014i
$582$	− 616.892i	− 1.05995i
$583$	198.000	0.339623
$584$	− 736.291i	− 1.26077i
$585$	238.797i	0.408200i
$586$	−1584.00	−2.70307
$587$	941.921i	1.60464i	0.596897	+	0.802318i	$0.296400\pi$
$587$	941.921i	1.60464i	−0.596897	+	0.802318i	$0.703600\pi$
$588$	−315.000	−0.535714
$589$	156.000	0.264856
$590$	− 1459.31i	− 2.47342i
$591$	− 397.995i	− 0.673426i
$592$	150.000	0.253378
$593$	− 543.926i	− 0.917245i	−0.888631	−	0.458623i	$-0.848343\pi$
$593$	− 543.926i	− 0.917245i	0.888631	−	0.458623i	$-0.151657\pi$
$594$	−297.000	−0.500000
$595$	−704.000	−1.18319
$596$	− 650.058i	− 1.09070i
$597$	−126.000	−0.211055
$598$	−88.0000	−0.147157
$599$	46.4327i	0.0775171i	0.999249	+	0.0387586i	$0.0123403\pi$
$599$	46.4327i	0.0775171i	−0.999249	+	0.0387586i	$0.987660\pi$
$600$	− 567.143i	− 0.945238i
$601$	542.000	0.901830	0.450915	−	0.892567i	$-0.351098\pi$
$601$	542.000	0.901830	0.450915	+	0.892567i	$0.351098\pi$
$602$	1114.39i	1.85114i
$603$	18.0000	0.0298507
$604$	1120.00	1.85430
$605$	− 72.9657i	− 0.120605i
$606$	−1056.00	−1.74257
$607$	−700.000	−1.15321	−0.576606	−	0.817022i	$-0.695623\pi$
$607$	−700.000	−1.15321	−0.576606	+	0.817022i	$0.695623\pi$
$608$	− 139.298i	− 0.229109i
$609$	− 955.188i	− 1.56845i
$610$	264.000	0.432787
$611$	− 344.929i	− 0.564532i
$612$	835.789i	1.36567i
$613$	764.000	1.24633	0.623165	−	0.782091i	$-0.285847\pi$
$613$	764.000	1.24633	0.623165	+	0.782091i	$0.285847\pi$
$614$	− 285.230i	− 0.464544i
$615$	264.000	0.429268
$616$	−264.000	−0.428571
$617$	− 39.7995i	− 0.0645049i	−0.999480	−	0.0322524i	$-0.989732\pi$
$617$	− 39.7995i	− 0.0645049i	0.999480	−	0.0322524i	$-0.0102680\pi$
$618$	− 736.291i	− 1.19141i
$619$	−742.000	−1.19871	−0.599354	−	0.800484i	$-0.704576\pi$
$619$	−742.000	−1.19871	−0.599354	+	0.800484i	$0.704576\pi$
$620$	1207.25i	1.94718i
$621$	− 179.098i	− 0.288402i
$622$	−66.0000	−0.106109
$623$	− 955.188i	− 1.53321i
$624$	60.0000	0.0961538
$625$	−739.000	−1.18240
$626$	− 325.029i	− 0.519216i
$627$	59.6992i	0.0952141i
$628$	−1274.00	−2.02866
$629$	− 397.995i	− 0.632742i
$630$	−1584.00	−2.51429
$631$	−410.000	−0.649762	−0.324881	−	0.945755i	$-0.605324\pi$
$631$	−410.000	−0.649762	−0.324881	+	0.945755i	$0.605324\pi$
$632$	− 397.995i	− 0.629739i
$633$	738.000	1.16588
$634$	−1034.00	−1.63091
$635$	1247.05i	1.96386i
$636$	− 1253.68i	− 1.97120i
$637$	60.0000	0.0941915
$638$	− 437.794i	− 0.686198i
$639$	537.293i	0.840834i
$640$	1518.00	2.37188
$641$	756.190i	1.17970i	0.807511	+	0.589852i	$0.200814\pi$
$641$	756.190i	1.17970i	−0.807511	+	0.589852i	$0.799186\pi$
$642$	396.000	0.616822
$643$	890.000	1.38414	0.692068	−	0.721832i	$-0.256700\pi$
$643$	890.000	1.38414	0.692068	+	0.721832i	$0.256700\pi$
$644$	− 371.462i	− 0.576804i
$645$	835.789i	1.29580i
$646$	264.000	0.408669
$647$	− 484.227i	− 0.748419i	−0.927344	−	0.374210i	$-0.877914\pi$
$647$	− 484.227i	− 0.748419i	0.927344	−	0.374210i	$-0.122086\pi$
$648$	805.940i	1.24373i
$649$	−220.000	−0.338983
$650$	252.063i	0.387790i
$651$	624.000	0.958525
$652$	2030.00	3.11350
$653$	− 391.362i	− 0.599329i	−0.954045	−	0.299664i	$-0.903125\pi$
$653$	− 391.362i	− 0.599329i	0.954045	−	0.299664i	$-0.0968747\pi$
$654$	1989.97i	3.04278i
$655$	264.000	0.403053
$656$	− 66.3325i	− 0.101117i
$657$	−666.000	−1.01370
$658$	2288.00	3.47720
$659$	384.728i	0.583806i	0.956448	+	0.291903i	$0.0942885\pi$
$659$	384.728i	0.583806i	−0.956448	+	0.291903i	$0.905711\pi$
$660$	−462.000	−0.700000
$661$	−746.000	−1.12859	−0.564297	−	0.825572i	$-0.690853\pi$
$661$	−746.000	−1.12859	−0.564297	+	0.825572i	$0.690853\pi$
$662$	723.024i	1.09218i
$663$	− 159.198i	− 0.240118i
$664$	−396.000	−0.596386
$665$	318.396i	0.478791i
$666$	− 895.489i	− 1.34458i
$667$	264.000	0.395802
$668$	− 1671.58i	− 2.50236i
$669$	−906.000	−1.35426
$670$	44.0000	0.0656716
$671$	− 39.7995i	− 0.0593137i
$672$	− 557.193i	− 0.829156i
$673$	−634.000	−0.942051	−0.471025	−	0.882120i	$-0.656116\pi$
$673$	−634.000	−0.942051	−0.471025	+	0.882120i	$0.656116\pi$
$674$	− 922.022i	− 1.36798i
$675$	−513.000	−0.760000
$676$	1071.00	1.58432
$677$	− 795.990i	− 1.17576i	−0.808948	−	0.587880i	$-0.799963\pi$
$677$	− 795.990i	− 1.17576i	0.808948	−	0.587880i	$-0.200037\pi$
$678$	396.000	0.584071
$679$	−496.000	−0.730486
$680$	875.589i	1.28763i
$681$	596.992i	0.876641i
$682$	286.000	0.419355
$683$	451.061i	0.660411i	0.943909	+	0.330206i	$0.107118\pi$
$683$	451.061i	0.660411i	−0.943909	+	0.330206i	$0.892882\pi$
$684$	378.000	0.552632
$685$	704.000	1.02774
$686$	− 902.122i	− 1.31505i
$687$	450.000	0.655022
$688$	210.000	0.305233
$689$	238.797i	0.346585i
$690$	− 437.794i	− 0.634485i
$691$	458.000	0.662808	0.331404	−	0.943489i	$-0.392478\pi$
$691$	458.000	0.662808	0.331404	+	0.943489i	$0.392478\pi$
$692$	1392.98i	2.01298i
$693$	238.797i	0.344584i
$694$	−1100.00	−1.58501
$695$	− 490.860i	− 0.706274i
$696$	−1188.00	−1.70690
$697$	−176.000	−0.252511
$698$	− 1074.59i	− 1.53952i
$699$	1313.38i	1.87895i
$700$	−1064.00	−1.52000
$701$	− 504.127i	− 0.719154i	−0.933115	−	0.359577i	$-0.882921\pi$
$701$	− 504.127i	− 0.719154i	0.933115	−	0.359577i	$-0.117079\pi$
$702$	− 358.195i	− 0.510250i
$703$	−180.000	−0.256046
$704$	− 321.713i	− 0.456978i
$705$	1716.00	2.43404
$706$	−1320.00	−1.86969
$707$	849.056i	1.20093i
$708$	1392.98i	1.96749i
$709$	−562.000	−0.792666	−0.396333	−	0.918107i	$-0.629717\pi$
$709$	−562.000	−0.792666	−0.396333	+	0.918107i	$0.629717\pi$
$710$	1313.38i	1.84984i
$711$	−360.000	−0.506329
$712$	−1188.00	−1.66854
$713$	172.464i	0.241886i
$714$	1056.00	1.47899
$715$	88.0000	0.123077
$716$	1392.98i	1.94551i
$717$	0	0
$718$	1012.00	1.40947
$719$	− 338.296i	− 0.470509i	−0.971934	−	0.235254i	$-0.924408\pi$
$719$	− 338.296i	− 0.470509i	0.971934	−	0.235254i	$-0.0755923\pi$
$720$	298.496i	0.414578i
$721$	−592.000	−0.821082
$722$	1077.90i	1.49294i
$723$	390.000	0.539419
$724$	−70.0000	−0.0966851
$725$	− 756.190i	− 1.04302i
$726$	109.449i	0.150756i
$727$	−42.0000	−0.0577717	−0.0288858	−	0.999583i	$-0.509196\pi$
$727$	−42.0000	−0.0577717	−0.0288858	+	0.999583i	$0.509196\pi$
$728$	− 318.396i	− 0.437357i
$729$	729.000	1.00000
$730$	−1628.00	−2.23014
$731$	− 557.193i	− 0.762234i
$732$	−252.000	−0.344262
$733$	−624.000	−0.851296	−0.425648	−	0.904889i	$-0.639954\pi$
$733$	−624.000	−0.851296	−0.425648	+	0.904889i	$0.639954\pi$
$734$	922.022i	1.25616i
$735$	298.496i	0.406117i
$736$	154.000	0.209239
$737$	− 6.63325i	− 0.00900034i
$738$	−396.000	−0.536585
$739$	686.000	0.928281	0.464141	−	0.885761i	$-0.346363\pi$
$739$	686.000	0.928281	0.464141	+	0.885761i	$0.346363\pi$
$740$	− 1392.98i	− 1.88241i
$741$	−72.0000	−0.0971660
$742$	−1584.00	−2.13477
$743$	862.322i	1.16060i	0.814404	+	0.580298i	$0.197064\pi$
$743$	862.322i	1.16060i	−0.814404	+	0.580298i	$0.802936\pi$
$744$	− 776.090i	− 1.04313i
$745$	−616.000	−0.826846
$746$	225.530i	0.302320i
$747$	358.195i	0.479512i
$748$	308.000	0.411765
$749$	− 318.396i	− 0.425095i
$750$	396.000	0.528000
$751$	94.0000	0.125166	0.0625832	−	0.998040i	$-0.480066\pi$
$751$	94.0000	0.125166	0.0625832	+	0.998040i	$0.480066\pi$
$752$	− 431.161i	− 0.573353i
$753$	− 875.589i	− 1.16280i
$754$	528.000	0.700265
$755$	− 1061.32i	− 1.40572i
$756$	1512.00	2.00000
$757$	1118.00	1.47688	0.738441	−	0.674318i	$-0.235562\pi$
$757$	1118.00	1.47688	0.738441	+	0.674318i	$0.235562\pi$
$758$	− 2222.14i	− 2.93158i
$759$	−66.0000	−0.0869565
$760$	396.000	0.521053
$761$	1154.19i	1.51667i	0.651865	+	0.758335i	$0.273987\pi$
$761$	1154.19i	1.51667i	−0.651865	+	0.758335i	$0.726013\pi$
$762$	− 1870.58i	− 2.45482i
$763$	1600.00	2.09699
$764$	− 789.357i	− 1.03319i
$765$	792.000	1.03529
$766$	110.000	0.143603
$767$	− 265.330i	− 0.345932i
$768$	−1113.00	−1.44922
$769$	1274.00	1.65670	0.828349	−	0.560213i	$-0.189281\pi$
$769$	1274.00	1.65670	0.828349	+	0.560213i	$0.189281\pi$
$770$	583.726i	0.758086i
$771$	− 1472.58i	− 1.90996i
$772$	−2086.00	−2.70207
$773$	935.288i	1.20995i	0.796246	+	0.604973i	$0.206816\pi$
$773$	935.288i	1.20995i	−0.796246	+	0.604973i	$0.793184\pi$
$774$	− 1253.68i	− 1.61975i
$775$	494.000	0.637419
$776$	616.892i	0.794964i
$777$	−720.000	−0.926641
$778$	1386.00	1.78149
$779$	79.5990i	0.102181i
$780$	− 557.193i	− 0.714350i
$781$	198.000	0.253521
$782$	291.863i	0.373226i
$783$	1074.59i	1.37240i
$784$	75.0000	0.0956633
$785$	1207.25i	1.53790i
$786$	−396.000	−0.503817
$787$	298.000	0.378653	0.189327	−	0.981914i	$-0.439370\pi$
$787$	298.000	0.378653	0.189327	+	0.981914i	$0.439370\pi$
$788$	928.655i	1.17850i
$789$	− 676.591i	− 0.857530i
$790$	−880.000	−1.11392
$791$	− 318.396i	− 0.402523i
$792$	297.000	0.375000
$793$	48.0000	0.0605296
$794$	285.230i	0.359231i
$795$	−1188.00	−1.49434
$796$	294.000	0.369347
$797$	− 524.027i	− 0.657499i	−0.944417	−	0.328750i	$-0.893373\pi$
$797$	− 524.027i	− 0.657499i	0.944417	−	0.328750i	$-0.106627\pi$
$798$	− 477.594i	− 0.598489i
$799$	−1144.00	−1.43179
$800$	− 441.111i	− 0.551389i
$801$	1074.59i	1.34156i
$802$	836.000	1.04239
$803$	245.430i	0.305642i
$804$	−42.0000	−0.0522388
$805$	−352.000	−0.437267
$806$	344.929i	0.427952i
$807$	− 218.897i	− 0.271248i
$808$	1056.00	1.30693
$809$	915.388i	1.13151i	0.824575	+	0.565753i	$0.191414\pi$
$809$	915.388i	1.13151i	−0.824575	+	0.565753i	$0.808586\pi$
$810$	1782.00	2.20000
$811$	−182.000	−0.224414	−0.112207	−	0.993685i	$-0.535792\pi$
$811$	−182.000	−0.224414	−0.112207	+	0.993685i	$0.535792\pi$
$812$	2228.77i	2.74479i
$813$	−1344.00	−1.65314
$814$	−330.000	−0.405405
$815$	− 1923.64i	− 2.36030i
$816$	− 198.997i	− 0.243869i
$817$	−252.000	−0.308446
$818$	− 1691.48i	− 2.06782i
$819$	−288.000	−0.351648
$820$	−616.000	−0.751220
$821$	822.523i	1.00185i	0.865489	+	0.500927i	$0.167008\pi$
$821$	822.523i	1.00185i	−0.865489	+	0.500927i	$0.832992\pi$
$822$	−1056.00	−1.28467
$823$	−246.000	−0.298906	−0.149453	−	0.988769i	$-0.547751\pi$
$823$	−246.000	−0.298906	−0.149453	+	0.988769i	$0.547751\pi$
$824$	736.291i	0.893557i
$825$	189.048i	0.229149i
$826$	1760.00	2.13075
$827$	− 543.926i	− 0.657710i	−0.944380	−	0.328855i	$-0.893337\pi$
$827$	− 543.926i	− 0.657710i	0.944380	−	0.328855i	$-0.106663\pi$
$828$	417.895i	0.504704i
$829$	250.000	0.301568	0.150784	−	0.988567i	$-0.451820\pi$
$829$	250.000	0.301568	0.150784	+	0.988567i	$0.451820\pi$
$830$	875.589i	1.05493i
$831$	−780.000	−0.938628
$832$	388.000	0.466346
$833$	− 198.997i	− 0.238893i
$834$	736.291i	0.882843i
$835$	−1584.00	−1.89701
$836$	− 139.298i	− 0.166625i
$837$	−702.000	−0.838710
$838$	1760.00	2.10024
$839$	− 470.961i	− 0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$839$	− 470.961i	− 0.561336i	0.959805	−	0.280668i	$-0.0905559\pi$
$840$	1584.00	1.88571
$841$	−743.000	−0.883472
$842$	563.826i	0.669627i
$843$	596.992i	0.708176i
$844$	−1722.00	−2.04028
$845$	− 1014.89i	− 1.20105i
$846$	−2574.00	−3.04255
$847$	88.0000	0.103896
$848$	298.496i	0.352000i
$849$	−150.000	−0.176678
$850$	836.000	0.983529
$851$	− 198.997i	− 0.233840i
$852$	− 1253.68i	− 1.47146i
$853$	56.0000	0.0656506	0.0328253	−	0.999461i	$-0.489549\pi$
$853$	56.0000	0.0656506	0.0328253	+	0.999461i	$0.489549\pi$
$854$	318.396i	0.372829i
$855$	− 358.195i	− 0.418942i
$856$	−396.000	−0.462617
$857$	252.063i	0.294123i	0.989127	+	0.147062i	$0.0469815\pi$
$857$	252.063i	0.294123i	−0.989127	+	0.147062i	$0.953018\pi$
$858$	−132.000	−0.153846
$859$	1278.00	1.48778	0.743888	−	0.668304i	$-0.232979\pi$
$859$	1278.00	1.48778	0.743888	+	0.668304i	$0.232979\pi$
$860$	− 1950.18i	− 2.26765i
$861$	318.396i	0.369798i
$862$	−924.000	−1.07193
$863$	1134.29i	1.31435i	0.753737	+	0.657176i	$0.228249\pi$
$863$	1134.29i	1.31435i	−0.753737	+	0.657176i	$0.771751\pi$
$864$	626.842i	0.725512i
$865$	1320.00	1.52601
$866$	1797.61i	2.07576i
$867$	339.000	0.391003
$868$	−1456.00	−1.67742
$869$	132.665i	0.152664i
$870$	2626.77i	3.01927i
$871$	8.00000	0.00918485
$872$	− 1989.97i	− 2.28208i
$873$	558.000	0.639175
$874$	132.000	0.151030
$875$	− 318.396i	− 0.363881i
$876$	1554.00	1.77397
$877$	456.000	0.519954	0.259977	−	0.965615i	$-0.416285\pi$
$877$	456.000	0.519954	0.259977	+	0.965615i	$0.416285\pi$
$878$	− 1087.85i	− 1.23901i
$879$	− 1432.78i	− 1.63001i
$880$	110.000	0.125000
$881$	610.259i	0.692689i	0.938107	+	0.346344i	$0.112577\pi$
$881$	610.259i	0.692689i	−0.938107	+	0.346344i	$0.887423\pi$
$882$	− 447.744i	− 0.507647i
$883$	−1094.00	−1.23896	−0.619479	−	0.785013i	$-0.712656\pi$
$883$	−1094.00	−1.23896	−0.619479	+	0.785013i	$0.712656\pi$
$884$	371.462i	0.420206i
$885$	1320.00	1.49153
$886$	−440.000	−0.496614
$887$	− 145.931i	− 0.164523i	−0.996611	−	0.0822613i	$-0.973786\pi$
$887$	− 145.931i	− 0.164523i	0.996611	−	0.0822613i	$-0.0262142\pi$
$888$	895.489i	1.00843i
$889$	−1504.00	−1.69179
$890$	2626.77i	2.95142i
$891$	− 268.647i	− 0.301511i
$892$	2114.00	2.36996
$893$	517.393i	0.579388i
$894$	924.000	1.03356
$895$	1320.00	1.47486
$896$	1830.78i	2.04328i
$897$	− 79.5990i	− 0.0887391i
$898$	1496.00	1.66592
$899$	− 1034.79i	− 1.15104i
$900$	1197.00	1.33000
$901$	792.000	0.879023
$902$	145.931i	0.161787i
$903$	−1008.00	−1.11628
$904$	−396.000	−0.438053
$905$	66.3325i	0.0732956i
$906$	1591.98i	1.75715i
$907$	186.000	0.205072	0.102536	−	0.994729i	$-0.467304\pi$
$907$	186.000	0.205072	0.102536	+	0.994729i	$0.467304\pi$
$908$	− 1392.98i	− 1.53412i
$909$	− 955.188i	− 1.05081i
$910$	−704.000	−0.773626
$911$	− 855.689i	− 0.939286i	−0.882857	−	0.469643i	$-0.844383\pi$
$911$	− 855.689i	− 0.939286i	0.882857	−	0.469643i	$-0.155617\pi$
$912$	−90.0000	−0.0986842
$913$	132.000	0.144578
$914$	− 1134.29i	− 1.24101i
$915$	238.797i	0.260980i
$916$	−1050.00	−1.14629
$917$	318.396i	0.347215i
$918$	−1188.00	−1.29412
$919$	−428.000	−0.465724	−0.232862	−	0.972510i	$-0.574809\pi$
$919$	−428.000	−0.465724	−0.232862	+	0.972510i	$0.574809\pi$
$920$	437.794i	0.475864i
$921$	258.000	0.280130
$922$	264.000	0.286334
$923$	238.797i	0.258718i
$924$	− 557.193i	− 0.603023i
$925$	−570.000	−0.616216
$926$	285.230i	0.308023i
$927$	666.000	0.718447
$928$	−924.000	−0.995690
$929$	− 636.792i	− 0.685460i	−0.939434	−	0.342730i	$-0.888648\pi$
$929$	− 636.792i	− 0.685460i	0.939434	−	0.342730i	$-0.111352\pi$
$930$	−1716.00	−1.84516
$931$	−90.0000	−0.0966702
$932$	− 3064.56i	− 3.28816i
$933$	− 59.6992i	− 0.0639863i
$934$	1980.00	2.11991
$935$	− 291.863i	− 0.312153i
$936$	358.195i	0.382687i
$937$	290.000	0.309498	0.154749	−	0.987954i	$-0.450543\pi$
$937$	290.000	0.309498	0.154749	+	0.987954i	$0.450543\pi$
$938$	53.0660i	0.0565736i
$939$	294.000	0.313099
$940$	−4004.00	−4.25957
$941$	− 875.589i	− 0.930488i	−0.885183	−	0.465244i	$-0.845967\pi$
$941$	− 875.589i	− 0.930488i	0.885183	−	0.465244i	$-0.154033\pi$
$942$	− 1810.88i	− 1.92237i
$943$	−88.0000	−0.0933192
$944$	− 331.662i	− 0.351337i
$945$	− 1432.78i	− 1.51617i
$946$	−462.000	−0.488372
$947$	79.5990i	0.0840538i	0.999116	+	0.0420269i	$0.0133815\pi$
$947$	79.5990i	0.0840538i	−0.999116	+	0.0420269i	$0.986618\pi$
$948$	840.000	0.886076
$949$	−296.000	−0.311907
$950$	− 378.095i	− 0.397995i
$951$	− 935.288i	− 0.983479i
$952$	−1056.00	−1.10924
$953$	782.723i	0.821326i	0.911787	+	0.410663i	$0.134703\pi$
$953$	782.723i	0.821326i	−0.911787	+	0.410663i	$0.865297\pi$
$954$	1782.00	1.86792
$955$	−748.000	−0.783246
$956$	0	0
$957$	396.000	0.413793
$958$	−748.000	−0.780793
$959$	849.056i	0.885356i
$960$	1930.28i	2.01070i
$961$	−285.000	−0.296566
$962$	− 397.995i	− 0.413716i
$963$	358.195i	0.371958i
$964$	−910.000	−0.943983
$965$	1976.71i	2.04840i
$966$	528.000	0.546584
$967$	460.000	0.475698	0.237849	−	0.971302i	$-0.423558\pi$
$967$	460.000	0.475698	0.237849	+	0.971302i	$0.423558\pi$
$968$	− 109.449i	− 0.113067i
$969$	238.797i	0.246437i
$970$	1364.00	1.40619
$971$	1167.45i	1.20232i	0.799129	+	0.601160i	$0.205294\pi$
$971$	1167.45i	1.20232i	−0.799129	+	0.601160i	$0.794706\pi$
$972$	−1701.00	−1.75000
$973$	592.000	0.608428
$974$	− 1479.21i	− 1.51870i
$975$	−228.000	−0.233846
$976$	60.0000	0.0614754
$977$	− 1313.38i	− 1.34430i	−0.740414	−	0.672151i	$-0.765370\pi$
$977$	− 1313.38i	− 1.34430i	0.740414	−	0.672151i	$-0.234630\pi$
$978$	2885.46i	2.95037i
$979$	396.000	0.404494
$980$	− 696.491i	− 0.710705i
$981$	−1800.00	−1.83486
$982$	2332.00	2.37475
$983$	− 417.895i	− 0.425122i	−0.977148	−	0.212561i	$-0.931820\pi$
$983$	− 417.895i	− 0.425122i	0.977148	−	0.212561i	$-0.0681804\pi$
$984$	396.000	0.402439
$985$	880.000	0.893401
$986$	− 1751.18i	− 1.77604i
$987$	2069.57i	2.09683i
$988$	168.000	0.170040
$989$	− 278.596i	− 0.281695i
$990$	− 656.692i	− 0.663325i
$991$	838.000	0.845610	0.422805	−	0.906221i	$-0.361045\pi$
$991$	838.000	0.845610	0.422805	+	0.906221i	$0.361045\pi$
$992$	− 603.626i	− 0.608494i
$993$	−654.000	−0.658610
$994$	−1584.00	−1.59356
$995$	− 278.596i	− 0.279996i
$996$	− 835.789i	− 0.839146i
$997$	−52.0000	−0.0521565	−0.0260782	−	0.999660i	$-0.508302\pi$
$997$	−52.0000	−0.0521565	−0.0260782	+	0.999660i	$0.508302\pi$
$998$	192.364i	0.192750i
$999$	810.000	0.810811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.b.a.23.1	✓	2
3.2	odd	2	inner	33.3.b.a.23.2	yes	2
4.3	odd	2		528.3.i.a.353.2		2
11.2	odd	10		363.3.h.d.323.2		8
11.3	even	5		363.3.h.e.251.1		8
11.4	even	5		363.3.h.e.269.2		8
11.5	even	5		363.3.h.e.245.2		8
11.6	odd	10		363.3.h.d.245.1		8
11.7	odd	10		363.3.h.d.269.1		8
11.8	odd	10		363.3.h.d.251.2		8
11.9	even	5		363.3.h.e.323.1		8
11.10	odd	2		363.3.b.d.122.2		2
12.11	even	2		528.3.i.a.353.1		2
33.2	even	10		363.3.h.d.323.1		8
33.5	odd	10		363.3.h.e.245.1		8
33.8	even	10		363.3.h.d.251.1		8
33.14	odd	10		363.3.h.e.251.2		8
33.17	even	10		363.3.h.d.245.2		8
33.20	odd	10		363.3.h.e.323.2		8
33.26	odd	10		363.3.h.e.269.1		8
33.29	even	10		363.3.h.d.269.2		8
33.32	even	2		363.3.b.d.122.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.b.a.23.1	✓	2	1.1	even	1	trivial
33.3.b.a.23.2	yes	2	3.2	odd	2	inner
363.3.b.d.122.1		2	33.32	even	2
363.3.b.d.122.2		2	11.10	odd	2
363.3.h.d.245.1		8	11.6	odd	10
363.3.h.d.245.2		8	33.17	even	10
363.3.h.d.251.1		8	33.8	even	10
363.3.h.d.251.2		8	11.8	odd	10
363.3.h.d.269.1		8	11.7	odd	10
363.3.h.d.269.2		8	33.29	even	10
363.3.h.d.323.1		8	33.2	even	10
363.3.h.d.323.2		8	11.2	odd	10
363.3.h.e.245.1		8	33.5	odd	10
363.3.h.e.245.2		8	11.5	even	5
363.3.h.e.251.1		8	11.3	even	5
363.3.h.e.251.2		8	33.14	odd	10
363.3.h.e.269.1		8	33.26	odd	10
363.3.h.e.269.2		8	11.4	even	5
363.3.h.e.323.1		8	11.9	even	5
363.3.h.e.323.2		8	33.20	odd	10
528.3.i.a.353.1		2	12.11	even	2
528.3.i.a.353.2		2	4.3	odd	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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	33.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.31662i q^{2} +3.00000 q^{3} -7.00000 q^{4} +6.63325i q^{5} -9.94987i q^{6} -8.00000 q^{7} +9.94987i q^{8} +9.00000 q^{9} +22.0000 q^{10} -3.31662i q^{11} -21.0000 q^{12} +4.00000 q^{13} +26.5330i q^{14} +19.8997i q^{15} +5.00000 q^{16} -13.2665i q^{17} -29.8496i q^{18} -6.00000 q^{19} -46.4327i q^{20} -24.0000 q^{21} -11.0000 q^{22} -6.63325i q^{23} +29.8496i q^{24} -19.0000 q^{25} -13.2665i q^{26} +27.0000 q^{27} +56.0000 q^{28} +39.7995i q^{29} +66.0000 q^{30} -26.0000 q^{31} +23.2164i q^{32} -9.94987i q^{33} -44.0000 q^{34} -53.0660i q^{35} -63.0000 q^{36} +30.0000 q^{37} +19.8997i q^{38} +12.0000 q^{39} -66.0000 q^{40} -13.2665i q^{41} +79.5990i q^{42} +42.0000 q^{43} +23.2164i q^{44} +59.6992i q^{45} -22.0000 q^{46} -86.2322i q^{47} +15.0000 q^{48} +15.0000 q^{49} +63.0159i q^{50} -39.7995i q^{51} -28.0000 q^{52} +59.6992i q^{53} -89.5489i q^{54} +22.0000 q^{55} -79.5990i q^{56} -18.0000 q^{57} +132.000 q^{58} -66.3325i q^{59} -139.298i q^{60} +12.0000 q^{61} +86.2322i q^{62} -72.0000 q^{63} +97.0000 q^{64} +26.5330i q^{65} -33.0000 q^{66} +2.00000 q^{67} +92.8655i q^{68} -19.8997i q^{69} -176.000 q^{70} +59.6992i q^{71} +89.5489i q^{72} -74.0000 q^{73} -99.4987i q^{74} -57.0000 q^{75} +42.0000 q^{76} +26.5330i q^{77} -39.7995i q^{78} -40.0000 q^{79} +33.1662i q^{80} +81.0000 q^{81} -44.0000 q^{82} +39.7995i q^{83} +168.000 q^{84} +88.0000 q^{85} -139.298i q^{86} +119.398i q^{87} +33.0000 q^{88} +119.398i q^{89} +198.000 q^{90} -32.0000 q^{91} +46.4327i q^{92} -78.0000 q^{93} -286.000 q^{94} -39.7995i q^{95} +69.6491i q^{96} +62.0000 q^{97} -49.7494i q^{98} -29.8496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 14 q^{4} - 16 q^{7} + 18 q^{9} + 44 q^{10} - 42 q^{12} + 8 q^{13} + 10 q^{16} - 12 q^{19} - 48 q^{21} - 22 q^{22} - 38 q^{25} + 54 q^{27} + 112 q^{28} + 132 q^{30} - 52 q^{31} - 88 q^{34}+ \cdots + 124 q^{97}+O(q^{100}) \) 2 * q + 6 * q^3 - 14 * q^4 - 16 * q^7 + 18 * q^9 + 44 * q^10 - 42 * q^12 + 8 * q^13 + 10 * q^16 - 12 * q^19 - 48 * q^21 - 22 * q^22 - 38 * q^25 + 54 * q^27 + 112 * q^28 + 132 * q^30 - 52 * q^31 - 88 * q^34 - 126 * q^36 + 60 * q^37 + 24 * q^39 - 132 * q^40 + 84 * q^43 - 44 * q^46 + 30 * q^48 + 30 * q^49 - 56 * q^52 + 44 * q^55 - 36 * q^57 + 264 * q^58 + 24 * q^61 - 144 * q^63 + 194 * q^64 - 66 * q^66 + 4 * q^67 - 352 * q^70 - 148 * q^73 - 114 * q^75 + 84 * q^76 - 80 * q^79 + 162 * q^81 - 88 * q^82 + 336 * q^84 + 176 * q^85 + 66 * q^88 + 396 * q^90 - 64 * q^91 - 156 * q^93 - 572 * q^94 + 124 * q^97

Introduction

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