LMFDB - Embedded newform 51.2.d.b.16.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [51,2,Mod(16,51)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(51, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("51.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	51.d (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:	$0.407237050309$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			16.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	51.16
Dual form			51.2.d.b.16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{6} -4.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{6} -4.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +4.00000i q^{11} -1.00000i q^{12} +2.00000 q^{13} -4.00000i q^{14} -1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} -3.00000i q^{24} +5.00000 q^{25} +2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -4.00000i q^{31} +5.00000 q^{32} -4.00000 q^{33} +(1.00000 + 4.00000i) q^{34} +1.00000 q^{36} +8.00000i q^{37} -4.00000 q^{38} +2.00000i q^{39} -8.00000i q^{41} +4.00000 q^{42} -4.00000 q^{43} -4.00000i q^{44} -4.00000i q^{46} -8.00000 q^{47} -1.00000i q^{48} -9.00000 q^{49} +5.00000 q^{50} +(-4.00000 + 1.00000i) q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000i q^{54} +12.0000i q^{56} -4.00000i q^{57} -12.0000 q^{59} -8.00000i q^{61} -4.00000i q^{62} +4.00000i q^{63} +7.00000 q^{64} -4.00000 q^{66} +12.0000 q^{67} +(-1.00000 - 4.00000i) q^{68} +4.00000 q^{69} +12.0000i q^{71} +3.00000 q^{72} +8.00000i q^{74} +5.00000i q^{75} +4.00000 q^{76} +16.0000 q^{77} +2.00000i q^{78} -4.00000i q^{79} +1.00000 q^{81} -8.00000i q^{82} +12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} -12.0000i q^{88} -10.0000 q^{89} -8.00000i q^{91} +4.00000i q^{92} +4.00000 q^{93} -8.00000 q^{94} +5.00000i q^{96} +16.0000i q^{97} -9.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} + 4 q^{13} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 8 q^{19} + 8 q^{21} + 10 q^{25} + 4 q^{26} + 10 q^{32} - 8 q^{33} + 2 q^{34} + 2 q^{36} - 8 q^{38} + 8 q^{42} - 8 q^{43} - 16 q^{47} - 18 q^{49} + 10 q^{50} - 8 q^{51} - 4 q^{52} + 12 q^{53} - 24 q^{59} + 14 q^{64} - 8 q^{66} + 24 q^{67} - 2 q^{68} + 8 q^{69} + 6 q^{72} + 8 q^{76} + 32 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{84} - 8 q^{86} - 20 q^{89} + 8 q^{93} - 16 q^{94} - 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 + 4 * q^13 - 2 * q^16 + 2 * q^17 - 2 * q^18 - 8 * q^19 + 8 * q^21 + 10 * q^25 + 4 * q^26 + 10 * q^32 - 8 * q^33 + 2 * q^34 + 2 * q^36 - 8 * q^38 + 8 * q^42 - 8 * q^43 - 16 * q^47 - 18 * q^49 + 10 * q^50 - 8 * q^51 - 4 * q^52 + 12 * q^53 - 24 * q^59 + 14 * q^64 - 8 * q^66 + 24 * q^67 - 2 * q^68 + 8 * q^69 + 6 * q^72 + 8 * q^76 + 32 * q^77 + 2 * q^81 + 24 * q^83 - 8 * q^84 - 8 * q^86 - 20 * q^89 + 8 * q^93 - 16 * q^94 - 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/51\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	−1.00000			−0.500000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$			1.00000i			0.408248i
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	−3.00000			−1.06066
$9$	−1.00000			−0.333333
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		−	1.00000i		−	0.288675i
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		−	4.00000i		−	1.06904i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$	−1.00000			−0.235702
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$			4.00000i			0.852803i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		−	3.00000i		−	0.612372i
$25$	5.00000			1.00000
$26$	2.00000			0.392232
$27$		−	1.00000i		−	0.192450i
$28$			4.00000i			0.755929i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	5.00000			0.883883
$33$	−4.00000			−0.696311
$34$	1.00000	+	4.00000i	0.171499	+	0.685994i
$35$		0			0
$36$	1.00000			0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	−4.00000			−0.648886
$39$			2.00000i			0.320256i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$	4.00000			0.617213
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		−	4.00000i		−	0.603023i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−9.00000			−1.28571
$50$	5.00000			0.707107
$51$	−4.00000	+	1.00000i	−0.560112	+	0.140028i
$52$	−2.00000			−0.277350
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$			12.0000i			1.60357i
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		−	8.00000i		−	1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$		−	8.00000i		−	1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$		−	4.00000i		−	0.508001i
$63$			4.00000i			0.503953i
$64$	7.00000			0.875000
$65$		0			0
$66$	−4.00000			−0.492366
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.00000	−	4.00000i	−0.121268	−	0.485071i
$69$	4.00000			0.481543
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	3.00000			0.353553
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$			8.00000i			0.929981i
$75$			5.00000i			0.577350i
$76$	4.00000			0.458831
$77$	16.0000			1.82337
$78$			2.00000i			0.226455i
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	8.00000i		−	0.883452i
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$	−4.00000			−0.436436
$85$		0			0
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	12.0000i		−	1.27920i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$		−	8.00000i		−	0.838628i
$92$			4.00000i			0.417029i
$93$	4.00000			0.414781
$94$	−8.00000			−0.825137
$95$		0			0
$96$			5.00000i			0.510310i
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$	−9.00000			−0.909137
$99$		−	4.00000i		−	0.402015i
$100$	−5.00000			−0.500000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	−4.00000	+	1.00000i	−0.396059	+	0.0990148i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	6.00000			0.582772
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$			1.00000i			0.0962250i
$109$			8.00000i			0.766261i	0.923694	+	0.383131i	$0.125154\pi$
$109$			8.00000i			0.766261i	−0.923694	+	0.383131i	$0.874846\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$			4.00000i			0.377964i
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		−	4.00000i		−	0.374634i
$115$		0			0
$116$		0			0
$117$	−2.00000			−0.184900
$118$	−12.0000			−1.10469
$119$	16.0000	−	4.00000i	1.46672	−	0.366679i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		−	8.00000i		−	0.724286i
$123$	8.00000			0.721336
$124$			4.00000i			0.359211i
$125$		0			0
$126$			4.00000i			0.356348i
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000			−0.265165
$129$		−	4.00000i		−	0.352180i
$130$		0			0
$131$			4.00000i			0.349482i	0.984614	+	0.174741i	$0.0559088\pi$
$131$			4.00000i			0.349482i	−0.984614	+	0.174741i	$0.944091\pi$
$132$	4.00000			0.348155
$133$			16.0000i			1.38738i
$134$	12.0000			1.03664
$135$		0			0
$136$	−3.00000	−	12.0000i	−0.257248	−	1.02899i
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$	4.00000			0.340503
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$			12.0000i			1.00702i
$143$			8.00000i			0.668994i
$144$	1.00000			0.0833333
$145$		0			0
$146$		0			0
$147$		−	9.00000i		−	0.742307i
$148$		−	8.00000i		−	0.657596i
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$			5.00000i			0.408248i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	12.0000			0.973329
$153$	−1.00000	−	4.00000i	−0.0808452	−	0.323381i
$154$	16.0000			1.28932
$155$		0			0
$156$		−	2.00000i		−	0.160128i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		−	4.00000i		−	0.318223i
$159$			6.00000i			0.475831i
$160$		0			0
$161$	−16.0000			−1.26098
$162$	1.00000			0.0785674
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$			8.00000i			0.624695i
$165$		0			0
$166$	12.0000			0.931381
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	−12.0000			−0.925820
$169$	−9.00000			−0.692308
$170$		0			0
$171$	4.00000			0.305888
$172$	4.00000			0.304997
$173$		−	16.0000i		−	1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$		−	16.0000i		−	1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$		0			0
$175$		−	20.0000i		−	1.51186i
$176$		−	4.00000i		−	0.301511i
$177$		−	12.0000i		−	0.901975i
$178$	−10.0000			−0.749532
$179$	4.00000			0.298974			0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000			0.298974			0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$		−	8.00000i		−	0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$		−	8.00000i		−	0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$		−	8.00000i		−	0.592999i
$183$	8.00000			0.591377
$184$			12.0000i			0.884652i
$185$		0			0
$186$	4.00000			0.293294
$187$	−16.0000	+	4.00000i	−1.17004	+	0.292509i
$188$	8.00000			0.583460
$189$	−4.00000			−0.290957
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$			7.00000i			0.505181i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$			16.0000i			1.14873i
$195$		0			0
$196$	9.00000			0.642857
$197$			16.0000i			1.13995i	0.821661	+	0.569976i	$0.193048\pi$
$197$			16.0000i			1.13995i	−0.821661	+	0.569976i	$0.806952\pi$
$198$		−	4.00000i		−	0.284268i
$199$		−	20.0000i		−	1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$		−	20.0000i		−	1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$	−15.0000			−1.06066
$201$			12.0000i			0.846415i
$202$	−6.00000			−0.422159
$203$		0			0
$204$	4.00000	−	1.00000i	0.280056	−	0.0700140i
$205$		0			0
$206$		0			0
$207$			4.00000i			0.278019i
$208$	−2.00000			−0.138675
$209$		−	16.0000i		−	1.10674i
$210$		0			0
$211$			4.00000i			0.275371i	0.990476	+	0.137686i	$0.0439664\pi$
$211$			4.00000i			0.275371i	−0.990476	+	0.137686i	$0.956034\pi$
$212$	−6.00000			−0.412082
$213$	−12.0000			−0.822226
$214$		−	12.0000i		−	0.820303i
$215$		0			0
$216$			3.00000i			0.204124i
$217$	−16.0000			−1.08615
$218$			8.00000i			0.541828i
$219$		0			0
$220$		0			0
$221$	2.00000	+	8.00000i	0.134535	+	0.538138i
$222$	−8.00000			−0.536925
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		−	20.0000i		−	1.33631i
$225$	−5.00000			−0.333333
$226$		−	8.00000i		−	0.532152i
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$			4.00000i			0.264906i
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$			16.0000i			1.05272i
$232$		0			0
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	−2.00000			−0.130744
$235$		0			0
$236$	12.0000			0.781133
$237$	4.00000			0.259828
$238$	16.0000	−	4.00000i	1.03713	−	0.259281i
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$		−	16.0000i		−	1.03065i	−0.856995	−	0.515325i	$-0.827671\pi$
$241$		−	16.0000i		−	1.03065i	0.856995	−	0.515325i	$-0.172329\pi$
$242$	−5.00000			−0.321412
$243$			1.00000i			0.0641500i
$244$			8.00000i			0.512148i
$245$		0			0
$246$	8.00000			0.510061
$247$	−8.00000			−0.509028
$248$			12.0000i			0.762001i
$249$			12.0000i			0.760469i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		−	4.00000i		−	0.251976i
$253$	16.0000			1.00591
$254$	8.00000			0.501965
$255$		0			0
$256$	−17.0000			−1.06250
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		−	4.00000i		−	0.249029i
$259$	32.0000			1.98838
$260$		0			0
$261$		0			0
$262$			4.00000i			0.247121i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$	12.0000			0.738549
$265$		0			0
$266$			16.0000i			0.981023i
$267$		−	10.0000i		−	0.611990i
$268$	−12.0000			−0.733017
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−1.00000	−	4.00000i	−0.0606339	−	0.242536i
$273$	8.00000			0.484182
$274$	−10.0000			−0.604122
$275$			20.0000i			1.20605i
$276$	−4.00000			−0.240772
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$			4.00000i			0.239904i
$279$			4.00000i			0.239474i
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		−	8.00000i		−	0.476393i
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		−	12.0000i		−	0.712069i
$285$		0			0
$286$			8.00000i			0.473050i
$287$	−32.0000			−1.88890
$288$	−5.00000			−0.294628
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$	−16.0000			−0.937937
$292$		0			0
$293$	−10.0000			−0.584206			−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000			−0.584206			−0.292103	+	0.956387i	$0.594355\pi$
$294$		−	9.00000i		−	0.524891i
$295$		0			0
$296$		−	24.0000i		−	1.39497i
$297$	4.00000			0.232104
$298$	6.00000			0.347571
$299$		−	8.00000i		−	0.462652i
$300$		−	5.00000i		−	0.288675i
$301$			16.0000i			0.922225i
$302$		0			0
$303$		−	6.00000i		−	0.344691i
$304$	4.00000			0.229416
$305$		0			0
$306$	−1.00000	−	4.00000i	−0.0571662	−	0.228665i
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$	−16.0000			−0.911685
$309$		0			0
$310$		0			0
$311$			12.0000i			0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$			12.0000i			0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$		−	6.00000i		−	0.339683i
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$			4.00000i			0.225018i
$317$		−	32.0000i		−	1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$		−	32.0000i		−	1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$			6.00000i			0.336463i
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$	−16.0000			−0.891645
$323$	−4.00000	−	16.0000i	−0.222566	−	0.890264i
$324$	−1.00000			−0.0555556
$325$	10.0000			0.554700
$326$			20.0000i			1.10770i
$327$	−8.00000			−0.442401
$328$			24.0000i			1.32518i
$329$			32.0000i			1.76422i
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000			−0.658586
$333$		−	8.00000i		−	0.438397i
$334$			12.0000i			0.656611i
$335$		0			0
$336$	−4.00000			−0.218218
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$	−9.00000			−0.489535
$339$	8.00000			0.434500
$340$		0			0
$341$	16.0000			0.866449
$342$	4.00000			0.216295
$343$			8.00000i			0.431959i
$344$	12.0000			0.646997
$345$		0			0
$346$		−	16.0000i		−	0.860165i
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		−	20.0000i		−	1.06904i
$351$		−	2.00000i		−	0.106752i
$352$			20.0000i			1.06600i
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		−	12.0000i		−	0.637793i
$355$		0			0
$356$	10.0000			0.529999
$357$	4.00000	+	16.0000i	0.211702	+	0.846810i
$358$	4.00000			0.211407
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		−	8.00000i		−	0.420471i
$363$		−	5.00000i		−	0.262432i
$364$			8.00000i			0.419314i
$365$		0			0
$366$	8.00000			0.418167
$367$			12.0000i			0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$			12.0000i			0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$			4.00000i			0.208514i
$369$			8.00000i			0.416463i
$370$		0			0
$371$		−	24.0000i		−	1.24602i
$372$	−4.00000			−0.207390
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$	−16.0000	+	4.00000i	−0.827340	+	0.206835i
$375$		0			0
$376$	24.0000			1.23771
$377$		0			0
$378$	−4.00000			−0.205738
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$			8.00000i			0.409852i
$382$	−8.00000			−0.409316
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$		−	3.00000i		−	0.153093i
$385$		0			0
$386$		0			0
$387$	4.00000			0.203331
$388$		−	16.0000i		−	0.812277i
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	16.0000	−	4.00000i	0.809155	−	0.202289i
$392$	27.0000			1.36371
$393$	−4.00000			−0.201773
$394$			16.0000i			0.806068i
$395$		0			0
$396$			4.00000i			0.201008i
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		−	20.0000i		−	1.00251i
$399$	−16.0000			−0.801002
$400$	−5.00000			−0.250000
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$			12.0000i			0.598506i
$403$		−	8.00000i		−	0.398508i
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$	−32.0000			−1.58618
$408$	12.0000	−	3.00000i	0.594089	−	0.148522i
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$		−	10.0000i		−	0.493264i
$412$		0			0
$413$			48.0000i			2.36193i
$414$			4.00000i			0.196589i
$415$		0			0
$416$	10.0000			0.490290
$417$	−4.00000			−0.195881
$418$		−	16.0000i		−	0.782586i
$419$			4.00000i			0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$			4.00000i			0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$			4.00000i			0.194717i
$423$	8.00000			0.388973
$424$	−18.0000			−0.874157
$425$	5.00000	+	20.0000i	0.242536	+	0.970143i
$426$	−12.0000			−0.581402
$427$	−32.0000			−1.54859
$428$			12.0000i			0.580042i
$429$	−8.00000			−0.386244
$430$		0			0
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$			1.00000i			0.0481125i
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−16.0000			−0.768025
$435$		0			0
$436$		−	8.00000i		−	0.383131i
$437$			16.0000i			0.765384i
$438$		0			0
$439$		−	36.0000i		−	1.71819i	−0.511819	−	0.859093i	$-0.671028\pi$
$439$		−	36.0000i		−	1.71819i	0.511819	−	0.859093i	$-0.328972\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$	2.00000	+	8.00000i	0.0951303	+	0.380521i
$443$	−20.0000			−0.950229			−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000			−0.950229			−0.475114	+	0.879924i	$0.657593\pi$
$444$	8.00000			0.379663
$445$		0			0
$446$	16.0000			0.757622
$447$			6.00000i			0.283790i
$448$		−	28.0000i		−	1.32288i
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$	−5.00000			−0.235702
$451$	32.0000			1.50682
$452$			8.00000i			0.376288i
$453$		0			0
$454$		−	12.0000i		−	0.563188i
$455$		0			0
$456$			12.0000i			0.561951i
$457$	38.0000			1.77757			0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000			1.77757			0.888783	+	0.458329i	$0.151552\pi$
$458$	10.0000			0.467269
$459$	4.00000	−	1.00000i	0.186704	−	0.0466760i
$460$		0			0
$461$	−34.0000			−1.58354			−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000			−1.58354			−0.791769	+	0.610821i	$0.790840\pi$
$462$			16.0000i			0.744387i
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$		0			0
$465$		0			0
$466$			8.00000i			0.370593i
$467$	4.00000			0.185098			0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000			0.185098			0.0925490	+	0.995708i	$0.470499\pi$
$468$	2.00000			0.0924500
$469$		−	48.0000i		−	2.21643i
$470$		0			0
$471$		−	2.00000i		−	0.0921551i
$472$	36.0000			1.65703
$473$		−	16.0000i		−	0.735681i
$474$	4.00000			0.183726
$475$	−20.0000			−0.917663
$476$	−16.0000	+	4.00000i	−0.733359	+	0.183340i
$477$	−6.00000			−0.274721
$478$	−8.00000			−0.365911
$479$			12.0000i			0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$			12.0000i			0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$		0			0
$481$			16.0000i			0.729537i
$482$		−	16.0000i		−	0.728780i
$483$		−	16.0000i		−	0.728025i
$484$	5.00000			0.227273
$485$		0			0
$486$			1.00000i			0.0453609i
$487$		−	4.00000i		−	0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$		−	4.00000i		−	0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$			24.0000i			1.08643i
$489$	−20.0000			−0.904431
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$	−8.00000			−0.360668
$493$		0			0
$494$	−8.00000			−0.359937
$495$		0			0
$496$			4.00000i			0.179605i
$497$	48.0000			2.15309
$498$			12.0000i			0.537733i
$499$			36.0000i			1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$			36.0000i			1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$		0			0
$501$	−12.0000			−0.536120
$502$	12.0000			0.535586
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$		−	12.0000i		−	0.534522i
$505$		0			0
$506$	16.0000			0.711287
$507$		−	9.00000i		−	0.399704i
$508$	−8.00000			−0.354943
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$	−11.0000			−0.486136
$513$			4.00000i			0.176604i
$514$	−2.00000			−0.0882162
$515$		0			0
$516$			4.00000i			0.176090i
$517$		−	32.0000i		−	1.40736i
$518$	32.0000			1.40600
$519$	16.0000			0.702322
$520$		0			0
$521$			24.0000i			1.05146i	0.850652	+	0.525730i	$0.176208\pi$
$521$			24.0000i			1.05146i	−0.850652	+	0.525730i	$0.823792\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$		−	4.00000i		−	0.174741i
$525$	20.0000			0.872872
$526$	24.0000			1.04645
$527$	16.0000	−	4.00000i	0.696971	−	0.174243i
$528$	4.00000			0.174078
$529$	7.00000			0.304348
$530$		0			0
$531$	12.0000			0.520756
$532$		−	16.0000i		−	0.693688i
$533$		−	16.0000i		−	0.693037i
$534$		−	10.0000i		−	0.432742i
$535$		0			0
$536$	−36.0000			−1.55496
$537$			4.00000i			0.172613i
$538$		0			0
$539$		−	36.0000i		−	1.55063i
$540$		0			0
$541$			40.0000i			1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$541$			40.0000i			1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$542$		0			0
$543$	8.00000			0.343313
$544$	5.00000	+	20.0000i	0.214373	+	0.857493i
$545$		0			0
$546$	8.00000			0.342368
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$	10.0000			0.427179
$549$			8.00000i			0.341432i
$550$			20.0000i			0.852803i
$551$		0			0
$552$	−12.0000			−0.510754
$553$	−16.0000			−0.680389
$554$		−	8.00000i		−	0.339887i
$555$		0			0
$556$		−	4.00000i		−	0.169638i
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$			4.00000i			0.169334i
$559$	−8.00000			−0.338364
$560$		0			0
$561$	−4.00000	−	16.0000i	−0.168880	−	0.675521i
$562$	10.0000			0.421825
$563$	44.0000			1.85438			0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000			1.85438			0.927189	+	0.374593i	$0.122217\pi$
$564$			8.00000i			0.336861i
$565$		0			0
$566$			4.00000i			0.168133i
$567$		−	4.00000i		−	0.167984i
$568$		−	36.0000i		−	1.51053i
$569$	26.0000			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		−	8.00000i		−	0.334497i
$573$		−	8.00000i		−	0.334205i
$574$	−32.0000			−1.33565
$575$		−	20.0000i		−	0.834058i
$576$	−7.00000			−0.291667
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$	−15.0000	+	8.00000i	−0.623918	+	0.332756i
$579$		0			0
$580$		0			0
$581$		−	48.0000i		−	1.99138i
$582$	−16.0000			−0.663221
$583$			24.0000i			0.993978i
$584$		0			0
$585$		0			0
$586$	−10.0000			−0.413096
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$			9.00000i			0.371154i
$589$			16.0000i			0.659269i
$590$		0			0
$591$	−16.0000			−0.658152
$592$		−	8.00000i		−	0.328798i
$593$	−46.0000			−1.88899			−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000			−1.88899			−0.944497	+	0.328521i	$0.893450\pi$
$594$	4.00000			0.164122
$595$		0			0
$596$	−6.00000			−0.245770
$597$	20.0000			0.818546
$598$		−	8.00000i		−	0.327144i
$599$	−8.00000			−0.326871			−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000			−0.326871			−0.163436	+	0.986554i	$0.552258\pi$
$600$		−	15.0000i		−	0.612372i
$601$			16.0000i			0.652654i	0.945257	+	0.326327i	$0.105811\pi$
$601$			16.0000i			0.652654i	−0.945257	+	0.326327i	$0.894189\pi$
$602$			16.0000i			0.652111i
$603$	−12.0000			−0.488678
$604$		0			0
$605$		0			0
$606$		−	6.00000i		−	0.243733i
$607$		−	4.00000i		−	0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$		−	4.00000i		−	0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$	−20.0000			−0.811107
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$	1.00000	+	4.00000i	0.0404226	+	0.161690i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$	−48.0000			−1.93398
$617$		−	24.0000i		−	0.966204i	−0.875564	−	0.483102i	$-0.839510\pi$
$617$		−	24.0000i		−	0.966204i	0.875564	−	0.483102i	$-0.160490\pi$
$618$		0			0
$619$			36.0000i			1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$			36.0000i			1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$		0			0
$621$	−4.00000			−0.160514
$622$			12.0000i			0.481156i
$623$			40.0000i			1.60257i
$624$		−	2.00000i		−	0.0800641i
$625$	25.0000			1.00000
$626$		−	16.0000i		−	0.639489i
$627$	16.0000			0.638978
$628$	2.00000			0.0798087
$629$	−32.0000	+	8.00000i	−1.27592	+	0.318981i
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$			12.0000i			0.477334i
$633$	−4.00000			−0.158986
$634$		−	32.0000i		−	1.27088i
$635$		0			0
$636$		−	6.00000i		−	0.237915i
$637$	−18.0000			−0.713186
$638$		0			0
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$			8.00000i			0.315981i	0.987441	+	0.157991i	$0.0505015\pi$
$641$			8.00000i			0.315981i	−0.987441	+	0.157991i	$0.949498\pi$
$642$	12.0000			0.473602
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	16.0000			0.630488
$645$		0			0
$646$	−4.00000	−	16.0000i	−0.157378	−	0.629512i
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$	−3.00000			−0.117851
$649$		−	48.0000i		−	1.88416i
$650$	10.0000			0.392232
$651$		−	16.0000i		−	0.627089i
$652$		−	20.0000i		−	0.783260i
$653$		−	32.0000i		−	1.25226i	−0.779720	−	0.626128i	$-0.784639\pi$
$653$		−	32.0000i		−	1.25226i	0.779720	−	0.626128i	$-0.215361\pi$
$654$	−8.00000			−0.312825
$655$		0			0
$656$			8.00000i			0.312348i
$657$		0			0
$658$			32.0000i			1.24749i
$659$	−4.00000			−0.155818			−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000			−0.155818			−0.0779089	+	0.996960i	$0.524824\pi$
$660$		0			0
$661$	−42.0000			−1.63361			−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000			−1.63361			−0.816805	+	0.576913i	$0.804257\pi$
$662$	−20.0000			−0.777322
$663$	−8.00000	+	2.00000i	−0.310694	+	0.0776736i
$664$	−36.0000			−1.39707
$665$		0			0
$666$		−	8.00000i		−	0.309994i
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$			16.0000i			0.618596i
$670$		0			0
$671$	32.0000			1.23535
$672$	20.0000			0.771517
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$		−	16.0000i		−	0.616297i
$675$		−	5.00000i		−	0.192450i
$676$	9.00000			0.346154
$677$			48.0000i			1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$			48.0000i			1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	8.00000			0.307238
$679$	64.0000			2.45609
$680$		0			0
$681$	12.0000			0.459841
$682$	16.0000			0.612672
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−4.00000			−0.152944
$685$		0			0
$686$			8.00000i			0.305441i
$687$			10.0000i			0.381524i
$688$	4.00000			0.152499
$689$	12.0000			0.457164
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$			16.0000i			0.608229i
$693$	−16.0000			−0.607790
$694$			4.00000i			0.151838i
$695$		0			0
$696$		0			0
$697$	32.0000	−	8.00000i	1.21209	−	0.303022i
$698$	−2.00000			−0.0757011
$699$	−8.00000			−0.302588
$700$			20.0000i			0.755929i
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		−	2.00000i		−	0.0754851i
$703$		−	32.0000i		−	1.20690i
$704$			28.0000i			1.05529i
$705$		0			0
$706$	18.0000			0.677439
$707$			24.0000i			0.902613i
$708$			12.0000i			0.450988i
$709$		−	40.0000i		−	1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$		−	40.0000i		−	1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$		0			0
$711$			4.00000i			0.150012i
$712$	30.0000			1.12430
$713$	−16.0000			−0.599205
$714$	4.00000	+	16.0000i	0.149696	+	0.598785i
$715$		0			0
$716$	−4.00000			−0.149487
$717$		−	8.00000i		−	0.298765i
$718$	32.0000			1.19423
$719$		−	36.0000i		−	1.34257i	−0.741198	−	0.671287i	$-0.765742\pi$
$719$		−	36.0000i		−	1.34257i	0.741198	−	0.671287i	$-0.234258\pi$
$720$		0			0
$721$		0			0
$722$	−3.00000			−0.111648
$723$	16.0000			0.595046
$724$			8.00000i			0.297318i
$725$		0			0
$726$		−	5.00000i		−	0.185567i
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$			24.0000i			0.889499i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−4.00000	−	16.0000i	−0.147945	−	0.591781i
$732$	−8.00000			−0.295689
$733$	−34.0000			−1.25582			−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000			−1.25582			−0.627909	+	0.778287i	$0.716089\pi$
$734$			12.0000i			0.442928i
$735$		0			0
$736$		−	20.0000i		−	0.737210i
$737$			48.0000i			1.76810i
$738$			8.00000i			0.294484i
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		−	8.00000i		−	0.293887i
$742$		−	24.0000i		−	0.881068i
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	−12.0000			−0.439941
$745$		0			0
$746$	26.0000			0.951928
$747$	−12.0000			−0.439057
$748$	16.0000	−	4.00000i	0.585018	−	0.146254i
$749$	−48.0000			−1.75388
$750$		0			0
$751$		−	20.0000i		−	0.729810i	−0.931045	−	0.364905i	$-0.881101\pi$
$751$		−	20.0000i		−	0.729810i	0.931045	−	0.364905i	$-0.118899\pi$
$752$	8.00000			0.291730
$753$			12.0000i			0.437304i
$754$		0			0
$755$		0			0
$756$	4.00000			0.145479
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$			20.0000i			0.726433i
$759$			16.0000i			0.580763i
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$			8.00000i			0.289809i
$763$	32.0000			1.15848
$764$	8.00000			0.289430
$765$		0			0
$766$	−16.0000			−0.578103
$767$	−24.0000			−0.866590
$768$		−	17.0000i		−	0.613435i
$769$	−34.0000			−1.22607			−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000			−1.22607			−0.613036	+	0.790055i	$0.710052\pi$
$770$		0			0
$771$		−	2.00000i		−	0.0720282i
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000			0.143777
$775$		−	20.0000i		−	0.718421i
$776$		−	48.0000i		−	1.72310i
$777$			32.0000i			1.14799i
$778$	−6.00000			−0.215110
$779$			32.0000i			1.14652i
$780$		0			0
$781$	−48.0000			−1.71758
$782$	16.0000	−	4.00000i	0.572159	−	0.143040i
$783$		0			0
$784$	9.00000			0.321429
$785$		0			0
$786$	−4.00000			−0.142675
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$		−	16.0000i		−	0.569976i
$789$			24.0000i			0.854423i
$790$		0			0
$791$	−32.0000			−1.13779
$792$			12.0000i			0.426401i
$793$		−	16.0000i		−	0.568177i
$794$		−	8.00000i		−	0.283909i
$795$		0			0
$796$			20.0000i			0.708881i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$	−16.0000			−0.566394
$799$	−8.00000	−	32.0000i	−0.283020	−	1.13208i
$800$	25.0000			0.883883
$801$	10.0000			0.353333
$802$		−	24.0000i		−	0.847469i
$803$		0			0
$804$		−	12.0000i		−	0.423207i
$805$		0			0
$806$		−	8.00000i		−	0.281788i
$807$		0			0
$808$	18.0000			0.633238
$809$		−	40.0000i		−	1.40633i	−0.711029	−	0.703163i	$-0.751771\pi$
$809$		−	40.0000i		−	1.40633i	0.711029	−	0.703163i	$-0.248229\pi$
$810$		0			0
$811$			4.00000i			0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$			4.00000i			0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$		0			0
$813$		0			0
$814$	−32.0000			−1.12160
$815$		0			0
$816$	4.00000	−	1.00000i	0.140028	−	0.0350070i
$817$	16.0000			0.559769
$818$	10.0000			0.349642
$819$			8.00000i			0.279543i
$820$		0			0
$821$			32.0000i			1.11681i	0.829569	+	0.558404i	$0.188586\pi$
$821$			32.0000i			1.11681i	−0.829569	+	0.558404i	$0.811414\pi$
$822$		−	10.0000i		−	0.348790i
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$	−20.0000			−0.696311
$826$			48.0000i			1.67013i
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		−	4.00000i		−	0.139010i
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$	14.0000			0.485363
$833$	−9.00000	−	36.0000i	−0.311832	−	1.24733i
$834$	−4.00000			−0.138509
$835$		0			0
$836$			16.0000i			0.553372i
$837$	−4.00000			−0.138260
$838$			4.00000i			0.138178i
$839$			44.0000i			1.51905i	0.650479	+	0.759524i	$0.274568\pi$
$839$			44.0000i			1.51905i	−0.650479	+	0.759524i	$0.725432\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	−22.0000			−0.758170
$843$			10.0000i			0.344418i
$844$		−	4.00000i		−	0.137686i
$845$		0			0
$846$	8.00000			0.275046
$847$			20.0000i			0.687208i
$848$	−6.00000			−0.206041
$849$	−4.00000			−0.137280
$850$	5.00000	+	20.0000i	0.171499	+	0.685994i
$851$	32.0000			1.09695
$852$	12.0000			0.411113
$853$		−	24.0000i		−	0.821744i	−0.911693	−	0.410872i	$-0.865224\pi$
$853$		−	24.0000i		−	0.821744i	0.911693	−	0.410872i	$-0.134776\pi$
$854$	−32.0000			−1.09502
$855$		0			0
$856$			36.0000i			1.23045i
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$	−8.00000			−0.273115
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$		−	32.0000i		−	1.09056i
$862$		−	20.0000i		−	0.681203i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		−	5.00000i		−	0.170103i
$865$		0			0
$866$	−2.00000			−0.0679628
$867$	−8.00000	−	15.0000i	−0.271694	−	0.509427i
$868$	16.0000			0.543075
$869$	16.0000			0.542763
$870$		0			0
$871$	24.0000			0.813209
$872$		−	24.0000i		−	0.812743i
$873$		−	16.0000i		−	0.541518i
$874$			16.0000i			0.541208i
$875$		0			0
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$		−	36.0000i		−	1.21494i
$879$		−	10.0000i		−	0.337292i
$880$		0			0
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$	9.00000			0.303046
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$	−2.00000	−	8.00000i	−0.0672673	−	0.269069i
$885$		0			0
$886$	−20.0000			−0.671913
$887$		−	4.00000i		−	0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$		−	4.00000i		−	0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$	24.0000			0.805387
$889$		−	32.0000i		−	1.07325i
$890$		0			0
$891$			4.00000i			0.134005i
$892$	−16.0000			−0.535720
$893$	32.0000			1.07084
$894$			6.00000i			0.200670i
$895$		0			0
$896$			12.0000i			0.400892i
$897$	8.00000			0.267112
$898$			8.00000i			0.266963i
$899$		0			0
$900$	5.00000			0.166667
$901$	6.00000	+	24.0000i	0.199889	+	0.799556i
$902$	32.0000			1.06548
$903$	−16.0000			−0.532447
$904$			24.0000i			0.798228i
$905$		0			0
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$			12.0000i			0.398234i
$909$	6.00000			0.199007
$910$		0			0
$911$			28.0000i			0.927681i	0.885919	+	0.463841i	$0.153529\pi$
$911$			28.0000i			0.927681i	−0.885919	+	0.463841i	$0.846471\pi$
$912$			4.00000i			0.132453i
$913$			48.0000i			1.58857i
$914$	38.0000			1.25693
$915$		0			0
$916$	−10.0000			−0.330409
$917$	16.0000			0.528367
$918$	4.00000	−	1.00000i	0.132020	−	0.0330049i
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		−	12.0000i		−	0.395413i
$922$	−34.0000			−1.11973
$923$			24.0000i			0.789970i
$924$		−	16.0000i		−	0.526361i
$925$			40.0000i			1.31519i
$926$	−40.0000			−1.31448
$927$		0			0
$928$		0			0
$929$			24.0000i			0.787414i	0.919236	+	0.393707i	$0.128808\pi$
$929$			24.0000i			0.787414i	−0.919236	+	0.393707i	$0.871192\pi$
$930$		0			0
$931$	36.0000			1.17985
$932$		−	8.00000i		−	0.262049i
$933$	−12.0000			−0.392862
$934$	4.00000			0.130884
$935$		0			0
$936$	6.00000			0.196116
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		−	48.0000i		−	1.56726i
$939$	16.0000			0.522140
$940$		0			0
$941$		−	48.0000i		−	1.56476i	−0.622804	−	0.782378i	$-0.714007\pi$
$941$		−	48.0000i		−	1.56476i	0.622804	−	0.782378i	$-0.285993\pi$
$942$		−	2.00000i		−	0.0651635i
$943$	−32.0000			−1.04206
$944$	12.0000			0.390567
$945$		0			0
$946$		−	16.0000i		−	0.520205i
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	−4.00000			−0.129914
$949$		0			0
$950$	−20.0000			−0.648886
$951$	32.0000			1.03767
$952$	−48.0000	+	12.0000i	−1.55569	+	0.388922i
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$	−6.00000			−0.194257
$955$		0			0
$956$	8.00000			0.258738
$957$		0			0
$958$			12.0000i			0.387702i
$959$			40.0000i			1.29167i
$960$		0			0
$961$	15.0000			0.483871
$962$			16.0000i			0.515861i
$963$			12.0000i			0.386695i
$964$			16.0000i			0.515325i
$965$		0			0
$966$		−	16.0000i		−	0.514792i
$967$	24.0000			0.771788			0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000			0.771788			0.385894	+	0.922543i	$0.373893\pi$
$968$	15.0000			0.482118
$969$	16.0000	−	4.00000i	0.513994	−	0.128499i
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		−	1.00000i		−	0.0320750i
$973$	16.0000			0.512936
$974$		−	4.00000i		−	0.128168i
$975$			10.0000i			0.320256i
$976$			8.00000i			0.256074i
$977$	−46.0000			−1.47167			−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000			−1.47167			−0.735835	+	0.677161i	$0.763210\pi$
$978$	−20.0000			−0.639529
$979$		−	40.0000i		−	1.27841i
$980$		0			0
$981$		−	8.00000i		−	0.255420i
$982$	−20.0000			−0.638226
$983$		−	20.0000i		−	0.637901i	−0.947771	−	0.318950i	$-0.896670\pi$
$983$		−	20.0000i		−	0.637901i	0.947771	−	0.318950i	$-0.103330\pi$
$984$	−24.0000			−0.765092
$985$		0			0
$986$		0			0
$987$	−32.0000			−1.01857
$988$	8.00000			0.254514
$989$			16.0000i			0.508770i
$990$		0			0
$991$			12.0000i			0.381193i	0.981669	+	0.190596i	$0.0610421\pi$
$991$			12.0000i			0.381193i	−0.981669	+	0.190596i	$0.938958\pi$
$992$		−	20.0000i		−	0.635001i
$993$		−	20.0000i		−	0.634681i
$994$	48.0000			1.52247
$995$		0			0
$996$		−	12.0000i		−	0.380235i
$997$			8.00000i			0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$			8.00000i			0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$			36.0000i			1.13956i
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.2.d.b.16.2	yes	2
3.2	odd	2		153.2.d.a.118.1		2
4.3	odd	2		816.2.c.c.577.1		2
5.2	odd	4		1275.2.d.d.424.2		2
5.3	odd	4		1275.2.d.b.424.1		2
5.4	even	2		1275.2.g.a.526.1		2
8.3	odd	2		3264.2.c.d.577.2		2
8.5	even	2		3264.2.c.e.577.1		2
12.11	even	2		2448.2.c.j.577.2		2
17.2	even	8		867.2.e.d.829.2		4
17.3	odd	16		867.2.h.d.688.1		8
17.4	even	4		867.2.a.b.1.1		1
17.5	odd	16		867.2.h.d.757.1		8
17.6	odd	16		867.2.h.d.712.2		8
17.7	odd	16		867.2.h.d.733.2		8
17.8	even	8		867.2.e.d.616.2		4
17.9	even	8		867.2.e.d.616.1		4
17.10	odd	16		867.2.h.d.733.1		8
17.11	odd	16		867.2.h.d.712.1		8
17.12	odd	16		867.2.h.d.757.2		8
17.13	even	4		867.2.a.a.1.1		1
17.14	odd	16		867.2.h.d.688.2		8
17.15	even	8		867.2.e.d.829.1		4
17.16	even	2	inner	51.2.d.b.16.1	✓	2
51.38	odd	4		2601.2.a.j.1.1		1
51.47	odd	4		2601.2.a.i.1.1		1
51.50	odd	2		153.2.d.a.118.2		2
68.67	odd	2		816.2.c.c.577.2		2
85.33	odd	4		1275.2.d.d.424.1		2
85.67	odd	4		1275.2.d.b.424.2		2
85.84	even	2		1275.2.g.a.526.2		2
136.67	odd	2		3264.2.c.d.577.1		2
136.101	even	2		3264.2.c.e.577.2		2
204.203	even	2		2448.2.c.j.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.b.16.1	✓	2	17.16	even	2	inner
51.2.d.b.16.2	yes	2	1.1	even	1	trivial
153.2.d.a.118.1		2	3.2	odd	2
153.2.d.a.118.2		2	51.50	odd	2
816.2.c.c.577.1		2	4.3	odd	2
816.2.c.c.577.2		2	68.67	odd	2
867.2.a.a.1.1		1	17.13	even	4
867.2.a.b.1.1		1	17.4	even	4
867.2.e.d.616.1		4	17.9	even	8
867.2.e.d.616.2		4	17.8	even	8
867.2.e.d.829.1		4	17.15	even	8
867.2.e.d.829.2		4	17.2	even	8
867.2.h.d.688.1		8	17.3	odd	16
867.2.h.d.688.2		8	17.14	odd	16
867.2.h.d.712.1		8	17.11	odd	16
867.2.h.d.712.2		8	17.6	odd	16
867.2.h.d.733.1		8	17.10	odd	16
867.2.h.d.733.2		8	17.7	odd	16
867.2.h.d.757.1		8	17.5	odd	16
867.2.h.d.757.2		8	17.12	odd	16
1275.2.d.b.424.1		2	5.3	odd	4
1275.2.d.b.424.2		2	85.67	odd	4
1275.2.d.d.424.1		2	85.33	odd	4
1275.2.d.d.424.2		2	5.2	odd	4
1275.2.g.a.526.1		2	5.4	even	2
1275.2.g.a.526.2		2	85.84	even	2
2448.2.c.j.577.1		2	204.203	even	2
2448.2.c.j.577.2		2	12.11	even	2
2601.2.a.i.1.1		1	51.47	odd	4
2601.2.a.j.1.1		1	51.38	odd	4
3264.2.c.d.577.1		2	136.67	odd	2
3264.2.c.d.577.2		2	8.3	odd	2
3264.2.c.e.577.1		2	8.5	even	2
3264.2.c.e.577.2		2	136.101	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:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	51.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{6} -4.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000i q^{6} -4.00000i q^{7} -3.00000 q^{8} -1.00000 q^{9} +4.00000i q^{11} -1.00000i q^{12} +2.00000 q^{13} -4.00000i q^{14} -1.00000 q^{16} +(1.00000 + 4.00000i) q^{17} -1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} -3.00000i q^{24} +5.00000 q^{25} +2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -4.00000i q^{31} +5.00000 q^{32} -4.00000 q^{33} +(1.00000 + 4.00000i) q^{34} +1.00000 q^{36} +8.00000i q^{37} -4.00000 q^{38} +2.00000i q^{39} -8.00000i q^{41} +4.00000 q^{42} -4.00000 q^{43} -4.00000i q^{44} -4.00000i q^{46} -8.00000 q^{47} -1.00000i q^{48} -9.00000 q^{49} +5.00000 q^{50} +(-4.00000 + 1.00000i) q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000i q^{54} +12.0000i q^{56} -4.00000i q^{57} -12.0000 q^{59} -8.00000i q^{61} -4.00000i q^{62} +4.00000i q^{63} +7.00000 q^{64} -4.00000 q^{66} +12.0000 q^{67} +(-1.00000 - 4.00000i) q^{68} +4.00000 q^{69} +12.0000i q^{71} +3.00000 q^{72} +8.00000i q^{74} +5.00000i q^{75} +4.00000 q^{76} +16.0000 q^{77} +2.00000i q^{78} -4.00000i q^{79} +1.00000 q^{81} -8.00000i q^{82} +12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} -12.0000i q^{88} -10.0000 q^{89} -8.00000i q^{91} +4.00000i q^{92} +4.00000 q^{93} -8.00000 q^{94} +5.00000i q^{96} +16.0000i q^{97} -9.00000 q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{9} + 4 q^{13} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 8 q^{19} + 8 q^{21} + 10 q^{25} + 4 q^{26} + 10 q^{32} - 8 q^{33} + 2 q^{34} + 2 q^{36} - 8 q^{38} + 8 q^{42} - 8 q^{43} - 16 q^{47} - 18 q^{49} + 10 q^{50} - 8 q^{51} - 4 q^{52} + 12 q^{53} - 24 q^{59} + 14 q^{64} - 8 q^{66} + 24 q^{67} - 2 q^{68} + 8 q^{69} + 6 q^{72} + 8 q^{76} + 32 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{84} - 8 q^{86} - 20 q^{89} + 8 q^{93} - 16 q^{94} - 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^9 + 4 * q^13 - 2 * q^16 + 2 * q^17 - 2 * q^18 - 8 * q^19 + 8 * q^21 + 10 * q^25 + 4 * q^26 + 10 * q^32 - 8 * q^33 + 2 * q^34 + 2 * q^36 - 8 * q^38 + 8 * q^42 - 8 * q^43 - 16 * q^47 - 18 * q^49 + 10 * q^50 - 8 * q^51 - 4 * q^52 + 12 * q^53 - 24 * q^59 + 14 * q^64 - 8 * q^66 + 24 * q^67 - 2 * q^68 + 8 * q^69 + 6 * q^72 + 8 * q^76 + 32 * q^77 + 2 * q^81 + 24 * q^83 - 8 * q^84 - 8 * q^86 - 20 * q^89 + 8 * q^93 - 16 * q^94 - 18 * q^98

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