LMFDB - Embedded newform 990.2.f.c.989.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [990,2,Mod(989,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.989"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	990.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-16,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$7.90518980011$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 38x^{14} + 517x^{12} + 3164x^{10} + 9372x^{8} + 14032x^{6} + 10320x^{4} + 3200x^{2} + 256 $ x^16 + 38x^14 + 517x^12 + 3164x^10 + 9372x^8 + 14032x^6 + 10320x^4 + 3200*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			989.16
Root			$-3.96698i$ of defining polynomial
Character	$\chi$	$=$	990.989
Dual form			990.2.f.c.989.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.94442 + 1.10418i) q^{5} -1.41421 q^{7} -1.00000i q^{8} +(-1.10418 + 1.94442i) q^{10} +(0.772087 - 3.22550i) q^{11} +3.62258 q^{13} -1.41421i q^{14} +1.00000 q^{16} +2.00000i q^{17} +6.07263i q^{19} +(-1.94442 - 1.10418i) q^{20} +(3.22550 + 0.772087i) q^{22} +7.77769 q^{23} +(2.56155 + 4.29400i) q^{25} +3.62258i q^{26} +1.41421 q^{28} -5.49966 q^{29} +6.24621 q^{31} +1.00000i q^{32} -2.00000 q^{34} +(-2.74983 - 1.56155i) q^{35} +1.54417i q^{37} -6.07263 q^{38} +(1.10418 - 1.94442i) q^{40} -12.5435 q^{41} +9.45353 q^{43} +(-0.772087 + 3.22550i) q^{44} +7.77769i q^{46} +9.96148 q^{47} -5.00000 q^{49} +(-4.29400 + 2.56155i) q^{50} -3.62258 q^{52} -6.07263 q^{53} +(5.06282 - 5.41922i) q^{55} +1.41421i q^{56} -5.49966i q^{58} +0.794156i q^{59} +9.96148i q^{61} +6.24621i q^{62} -1.00000 q^{64} +(7.04383 + 4.00000i) q^{65} +3.08835i q^{67} -2.00000i q^{68} +(1.56155 - 2.74983i) q^{70} +4.06854 q^{73} -1.54417 q^{74} -6.07263i q^{76} +(-1.09190 + 4.56155i) q^{77} +6.07263i q^{79} +(1.94442 + 1.10418i) q^{80} -12.5435i q^{82} -17.3693i q^{83} +(-2.20837 + 3.88884i) q^{85} +9.45353i q^{86} +(-3.22550 - 0.772087i) q^{88} -11.4878i q^{89} -5.12311 q^{91} -7.77769 q^{92} +9.96148i q^{94} +(-6.70531 + 11.8078i) q^{95} +10.9993i q^{97} -5.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{16} + 8 q^{25} - 32 q^{31} - 32 q^{34} - 80 q^{49} + 24 q^{55} - 16 q^{64} - 8 q^{70} - 16 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 + 8 * q^25 - 32 * q^31 - 32 * q^34 - 80 * q^49 + 24 * q^55 - 16 * q^64 - 8 * q^70 - 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/990\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$551$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.94442	+	1.10418i	0.869572	+	0.493806i
$5$	1.94442	+	1.10418i	0.869572	+	0.493806i
$6$		0			0
$7$	−1.41421			−0.534522			−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421			−0.534522			−0.267261	+	0.963624i	$0.586119\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.10418	+	1.94442i	−0.349174	+	0.614880i
$11$	0.772087	−	3.22550i	0.232793	−	0.972526i
$11$	0.772087	−	3.22550i	0.232793	−	0.972526i
$12$		0			0
$13$	3.62258			1.00472			0.502362	−	0.864657i	$-0.332465\pi$
$13$	3.62258			1.00472			0.502362	+	0.864657i	$0.332465\pi$
$14$		−	1.41421i		−	0.377964i
$15$		0			0
$16$	1.00000			0.250000
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$			6.07263i			1.39316i	0.717480	+	0.696579i	$0.245295\pi$
$19$			6.07263i			1.39316i	−0.717480	+	0.696579i	$0.754705\pi$
$20$	−1.94442	−	1.10418i	−0.434786	−	0.246903i
$21$		0			0
$22$	3.22550	+	0.772087i	0.687680	+	0.164609i
$23$	7.77769			1.62176			0.810880	−	0.585212i	$-0.198989\pi$
$23$	7.77769			1.62176			0.810880	+	0.585212i	$0.198989\pi$
$24$		0			0
$25$	2.56155	+	4.29400i	0.512311	+	0.858800i
$26$			3.62258i			0.710447i
$27$		0			0
$28$	1.41421			0.267261
$29$	−5.49966			−1.02126			−0.510630	−	0.859800i	$-0.670588\pi$
$29$	−5.49966			−1.02126			−0.510630	+	0.859800i	$0.670588\pi$
$30$		0			0
$31$	6.24621			1.12185			0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621			1.12185			0.560926	+	0.827866i	$0.310445\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−2.00000			−0.342997
$35$	−2.74983	−	1.56155i	−0.464806	−	0.263951i
$36$		0			0
$37$			1.54417i			0.253861i	0.991912	+	0.126930i	$0.0405124\pi$
$37$			1.54417i			0.253861i	−0.991912	+	0.126930i	$0.959488\pi$
$38$	−6.07263			−0.985111
$39$		0			0
$40$	1.10418	−	1.94442i	0.174587	−	0.307440i
$41$	−12.5435			−1.95896			−0.979482	−	0.201534i	$-0.935407\pi$
$41$	−12.5435			−1.95896			−0.979482	+	0.201534i	$0.935407\pi$
$42$		0			0
$43$	9.45353			1.44165			0.720825	−	0.693117i	$-0.243763\pi$
$43$	9.45353			1.44165			0.720825	+	0.693117i	$0.243763\pi$
$44$	−0.772087	+	3.22550i	−0.116396	+	0.486263i
$45$		0			0
$46$			7.77769i			1.14676i
$47$	9.96148			1.45303			0.726515	−	0.687150i	$-0.241139\pi$
$47$	9.96148			1.45303			0.726515	+	0.687150i	$0.241139\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$	−4.29400	+	2.56155i	−0.607263	+	0.362258i
$51$		0			0
$52$	−3.62258			−0.502362
$53$	−6.07263			−0.834141			−0.417070	−	0.908874i	$-0.636943\pi$
$53$	−6.07263			−0.834141			−0.417070	+	0.908874i	$0.636943\pi$
$54$		0			0
$55$	5.06282	−	5.41922i	0.682670	−	0.730727i
$56$			1.41421i			0.188982i
$57$		0			0
$58$		−	5.49966i		−	0.722140i
$59$			0.794156i			0.103390i	0.998663	+	0.0516951i	$0.0164624\pi$
$59$			0.794156i			0.103390i	−0.998663	+	0.0516951i	$0.983538\pi$
$60$		0			0
$61$			9.96148i			1.27544i	0.770270	+	0.637718i	$0.220122\pi$
$61$			9.96148i			1.27544i	−0.770270	+	0.637718i	$0.779878\pi$
$62$			6.24621i			0.793270i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	7.04383	+	4.00000i	0.873679	+	0.496139i
$66$		0			0
$67$			3.08835i			0.377302i	0.982044	+	0.188651i	$0.0604114\pi$
$67$			3.08835i			0.377302i	−0.982044	+	0.188651i	$0.939589\pi$
$68$		−	2.00000i		−	0.242536i
$69$		0			0
$70$	1.56155	−	2.74983i	0.186641	−	0.328667i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	4.06854			0.476187			0.238093	−	0.971242i	$-0.423478\pi$
$73$	4.06854			0.476187			0.238093	+	0.971242i	$0.423478\pi$
$74$	−1.54417			−0.179507
$75$		0			0
$76$		−	6.07263i		−	0.696579i
$77$	−1.09190	+	4.56155i	−0.124433	+	0.519837i
$78$		0			0
$79$			6.07263i			0.683225i	0.939841	+	0.341612i	$0.110973\pi$
$79$			6.07263i			0.683225i	−0.939841	+	0.341612i	$0.889027\pi$
$80$	1.94442	+	1.10418i	0.217393	+	0.123452i
$81$		0			0
$82$		−	12.5435i		−	1.38520i
$83$		−	17.3693i		−	1.90653i	−0.302135	−	0.953265i	$-0.597699\pi$
$83$		−	17.3693i		−	1.90653i	0.302135	−	0.953265i	$-0.402301\pi$
$84$		0			0
$85$	−2.20837	+	3.88884i	−0.239531	+	0.421804i
$86$			9.45353i			1.01940i
$87$		0			0
$88$	−3.22550	−	0.772087i	−0.343840	−	0.0823047i
$89$		−	11.4878i		−	1.21771i	−0.793283	−	0.608853i	$-0.791630\pi$
$89$		−	11.4878i		−	1.21771i	0.793283	−	0.608853i	$-0.208370\pi$
$90$		0			0
$91$	−5.12311			−0.537047
$92$	−7.77769			−0.810880
$93$		0			0
$94$			9.96148i			1.02745i
$95$	−6.70531	+	11.8078i	−0.687950	+	1.21145i
$96$		0			0
$97$			10.9993i			1.11681i	0.829568	+	0.558405i	$0.188587\pi$
$97$			10.9993i			1.11681i	−0.829568	+	0.558405i	$0.811413\pi$
$98$		−	5.00000i		−	0.505076i
$99$		0			0
$100$	−2.56155	−	4.29400i	−0.256155	−	0.429400i
$101$	16.4990			1.64171			0.820854	−	0.571138i	$-0.193498\pi$
$101$	16.4990			1.64171			0.820854	+	0.571138i	$0.193498\pi$
$102$		0			0
$103$			3.95548i			0.389745i	0.980829	+	0.194873i	$0.0624293\pi$
$103$			3.95548i			0.389745i	−0.980829	+	0.194873i	$0.937571\pi$
$104$		−	3.62258i		−	0.355223i
$105$		0			0
$106$		−	6.07263i		−	0.589826i
$107$			11.1231i			1.07531i	0.843165	+	0.537656i	$0.180690\pi$
$107$			11.1231i			1.07531i	−0.843165	+	0.537656i	$0.819310\pi$
$108$		0			0
$109$		−	2.18379i		−	0.209169i	−0.994516	−	0.104585i	$-0.966649\pi$
$109$		−	2.18379i		−	0.209169i	0.994516	−	0.104585i	$-0.0333513\pi$
$110$	5.41922	+	5.06282i	0.516702	+	0.482720i
$111$		0			0
$112$	−1.41421			−0.133631
$113$	−8.25643			−0.776699			−0.388350	−	0.921512i	$-0.626955\pi$
$113$	−8.25643			−0.776699			−0.388350	+	0.921512i	$0.626955\pi$
$114$		0			0
$115$	15.1231	+	8.58800i	1.41024	+	0.800835i
$116$	5.49966			0.510630
$117$		0			0
$118$	−0.794156			−0.0731079
$119$		−	2.82843i		−	0.259281i
$120$		0			0
$121$	−9.80776	−	4.98074i	−0.891615	−	0.452794i
$122$	−9.96148			−0.901870
$123$		0			0
$124$	−6.24621			−0.560926
$125$	0.239369	+	11.1778i	0.0214098	+	0.999771i
$126$		0			0
$127$	−9.89949			−0.878438			−0.439219	−	0.898380i	$-0.644745\pi$
$127$	−9.89949			−0.878438			−0.439219	+	0.898380i	$0.644745\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−4.00000	+	7.04383i	−0.350823	+	0.617785i
$131$	1.54417			0.134915			0.0674575	−	0.997722i	$-0.478511\pi$
$131$	1.54417			0.134915			0.0674575	+	0.997722i	$0.478511\pi$
$132$		0			0
$133$		−	8.58800i		−	0.744674i
$134$	−3.08835			−0.266793
$135$		0			0
$136$	2.00000			0.171499
$137$	0.478739			0.0409014			0.0204507	−	0.999791i	$-0.493490\pi$
$137$	0.478739			0.0409014			0.0204507	+	0.999791i	$0.493490\pi$
$138$		0			0
$139$		−	19.4442i		−	1.64924i	−0.565689	−	0.824619i	$-0.691390\pi$
$139$		−	19.4442i		−	1.64924i	0.565689	−	0.824619i	$-0.308610\pi$
$140$	2.74983	+	1.56155i	0.232403	+	0.131975i
$141$		0			0
$142$		0			0
$143$	2.79695	−	11.6847i	0.233893	−	0.977120i
$144$		0			0
$145$	−10.6937	−	6.07263i	−0.888059	−	0.504305i
$146$			4.06854i			0.336715i
$147$		0			0
$148$		−	1.54417i		−	0.126930i
$149$	2.41131			0.197542			0.0987710	−	0.995110i	$-0.468509\pi$
$149$	2.41131			0.197542			0.0987710	+	0.995110i	$0.468509\pi$
$150$		0			0
$151$		−	9.48274i		−	0.771694i	−0.922563	−	0.385847i	$-0.873909\pi$
$151$		−	9.48274i		−	0.771694i	0.922563	−	0.385847i	$-0.126091\pi$
$152$	6.07263			0.492556
$153$		0			0
$154$	−4.56155	−	1.09190i	−0.367580	−	0.0879875i
$155$	12.1453	+	6.89697i	0.975532	+	0.553978i
$156$		0			0
$157$		−	21.1315i		−	1.68648i	−0.537540	−	0.843238i	$-0.680646\pi$
$157$		−	21.1315i		−	1.68648i	0.537540	−	0.843238i	$-0.319354\pi$
$158$	−6.07263			−0.483113
$159$		0			0
$160$	−1.10418	+	1.94442i	−0.0872935	+	0.153720i
$161$	−10.9993			−0.866867
$162$		0			0
$163$			7.91096i			0.619635i	0.950796	+	0.309817i	$0.100268\pi$
$163$			7.91096i			0.619635i	−0.950796	+	0.309817i	$0.899732\pi$
$164$	12.5435			0.979482
$165$		0			0
$166$	17.3693			1.34812
$167$		−	7.36932i		−	0.570255i	−0.958490	−	0.285127i	$-0.907964\pi$
$167$		−	7.36932i		−	0.570255i	0.958490	−	0.285127i	$-0.0920359\pi$
$168$		0			0
$169$	0.123106			0.00946966
$170$	−3.88884	−	2.20837i	−0.298261	−	0.169374i
$171$		0			0
$172$	−9.45353			−0.720825
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		0			0
$175$	−3.62258	−	6.07263i	−0.273842	−	0.459048i
$176$	0.772087	−	3.22550i	0.0581982	−	0.243132i
$177$		0			0
$178$	11.4878			0.861047
$179$		−	10.8677i		−	0.812294i	−0.913808	−	0.406147i	$-0.866872\pi$
$179$		−	10.8677i		−	0.812294i	0.913808	−	0.406147i	$-0.133128\pi$
$180$		0			0
$181$	−9.12311			−0.678115			−0.339058	−	0.940766i	$-0.610108\pi$
$181$	−9.12311			−0.678115			−0.339058	+	0.940766i	$0.610108\pi$
$182$		−	5.12311i		−	0.379750i
$183$		0			0
$184$		−	7.77769i		−	0.573379i
$185$	−1.70505	+	3.00252i	−0.125358	+	0.220750i
$186$		0			0
$187$	6.45101	+	1.54417i	0.471745	+	0.112921i
$188$	−9.96148			−0.726515
$189$		0			0
$190$	−11.8078	−	6.70531i	−0.856625	−	0.486454i
$191$		−	20.1472i		−	1.45780i	−0.684621	−	0.728900i	$-0.740032\pi$
$191$		−	20.1472i		−	1.45780i	0.684621	−	0.728900i	$-0.259968\pi$
$192$		0			0
$193$	17.3188			1.24663			0.623316	−	0.781970i	$-0.285785\pi$
$193$	17.3188			1.24663			0.623316	+	0.781970i	$0.285785\pi$
$194$	−10.9993			−0.789705
$195$		0			0
$196$	5.00000			0.357143
$197$		−	0.876894i		−	0.0624761i	−0.999512	−	0.0312381i	$-0.990055\pi$
$197$		−	0.876894i		−	0.0624761i	0.999512	−	0.0312381i	$-0.00994500\pi$
$198$		0			0
$199$	−16.4924			−1.16912			−0.584558	−	0.811352i	$-0.698732\pi$
$199$	−16.4924			−1.16912			−0.584558	+	0.811352i	$0.698732\pi$
$200$	4.29400	−	2.56155i	0.303632	−	0.181129i
$201$		0			0
$202$			16.4990i			1.16086i
$203$	7.77769			0.545887
$204$		0			0
$205$	−24.3898	−	13.8503i	−1.70346	−	0.967348i
$206$	−3.95548			−0.275591
$207$		0			0
$208$	3.62258			0.251181
$209$	19.5873	+	4.68860i	1.35488	+	0.324317i
$210$		0			0
$211$			3.88884i			0.267719i	0.991000	+	0.133860i	$0.0427371\pi$
$211$			3.88884i			0.267719i	−0.991000	+	0.133860i	$0.957263\pi$
$212$	6.07263			0.417070
$213$		0			0
$214$	−11.1231			−0.760360
$215$	18.3817	+	10.4384i	1.25362	+	0.711896i
$216$		0			0
$217$	−8.83348			−0.599655
$218$	2.18379			0.147905
$219$		0			0
$220$	−5.06282	+	5.41922i	−0.341335	+	0.365363i
$221$			7.24517i			0.487363i
$222$		0			0
$223$			3.95548i			0.264879i	0.991191	+	0.132439i	$0.0422810\pi$
$223$			3.95548i			0.264879i	−0.991191	+	0.132439i	$0.957719\pi$
$224$		−	1.41421i		−	0.0944911i
$225$		0			0
$226$		−	8.25643i		−	0.549209i
$227$			5.36932i			0.356374i	0.983997	+	0.178187i	$0.0570232\pi$
$227$			5.36932i			0.356374i	−0.983997	+	0.178187i	$0.942977\pi$
$228$		0			0
$229$	20.2462			1.33791			0.668954	−	0.743304i	$-0.266742\pi$
$229$	20.2462			1.33791			0.668954	+	0.743304i	$0.266742\pi$
$230$	−8.58800	+	15.1231i	−0.566276	+	0.997188i
$231$		0			0
$232$			5.49966i			0.361070i
$233$		−	14.4924i		−	0.949430i	−0.880140	−	0.474715i	$-0.842551\pi$
$233$		−	14.4924i		−	0.949430i	0.880140	−	0.474715i	$-0.157449\pi$
$234$		0			0
$235$	19.3693	+	10.9993i	1.26351	+	0.717516i
$236$		−	0.794156i		−	0.0516951i
$237$		0			0
$238$	2.82843			0.183340
$239$	−16.4990			−1.06723			−0.533615	−	0.845728i	$-0.679167\pi$
$239$	−16.4990			−1.06723			−0.533615	+	0.845728i	$0.679167\pi$
$240$		0			0
$241$			2.18379i			0.140670i	0.997523	+	0.0703352i	$0.0224069\pi$
$241$			2.18379i			0.140670i	−0.997523	+	0.0703352i	$0.977593\pi$
$242$	4.98074	−	9.80776i	0.320174	−	0.630467i
$243$		0			0
$244$		−	9.96148i		−	0.637718i
$245$	−9.72211	−	5.52092i	−0.621123	−	0.352719i
$246$		0			0
$247$			21.9986i			1.39974i
$248$		−	6.24621i		−	0.396635i
$249$		0			0
$250$	−11.1778	+	0.239369i	−0.706945	+	0.0151390i
$251$		−	3.62258i		−	0.228655i	−0.993443	−	0.114328i	$-0.963529\pi$
$251$		−	3.62258i		−	0.228655i	0.993443	−	0.114328i	$-0.0364714\pi$
$252$		0			0
$253$	6.00505	−	25.0870i	0.377534	−	1.57720i
$254$		−	9.89949i		−	0.621150i
$255$		0			0
$256$	1.00000			0.0625000
$257$	3.88884			0.242579			0.121290	−	0.992617i	$-0.461297\pi$
$257$	3.88884			0.242579			0.121290	+	0.992617i	$0.461297\pi$
$258$		0			0
$259$		−	2.18379i		−	0.135694i
$260$	−7.04383	−	4.00000i	−0.436840	−	0.248069i
$261$		0			0
$262$			1.54417i			0.0953994i
$263$		−	15.3693i		−	0.947713i	−0.880602	−	0.473856i	$-0.842862\pi$
$263$		−	15.3693i		−	0.947713i	0.880602	−	0.473856i	$-0.157138\pi$
$264$		0			0
$265$	−11.8078	−	6.70531i	−0.725345	−	0.411904i
$266$	8.58800			0.526564
$267$		0			0
$268$		−	3.08835i		−	0.188651i
$269$			13.8703i			0.845685i	0.906203	+	0.422843i	$0.138968\pi$
$269$			13.8703i			0.845685i	−0.906203	+	0.422843i	$0.861032\pi$
$270$		0			0
$271$		−	17.2604i		−	1.04850i	−0.851566	−	0.524248i	$-0.824346\pi$
$271$		−	17.2604i		−	1.04850i	0.851566	−	0.524248i	$-0.175654\pi$
$272$			2.00000i			0.121268i
$273$		0			0
$274$			0.478739i			0.0289217i
$275$	15.8281	−	4.94696i	0.954468	−	0.298313i
$276$		0			0
$277$	10.8677			0.652980			0.326490	−	0.945201i	$-0.394134\pi$
$277$	10.8677			0.652980			0.326490	+	0.945201i	$0.394134\pi$
$278$	19.4442			1.16619
$279$		0			0
$280$	−1.56155	+	2.74983i	−0.0933206	+	0.164334i
$281$	20.4544			1.22021			0.610105	−	0.792321i	$-0.291127\pi$
$281$	20.4544			1.22021			0.610105	+	0.792321i	$0.291127\pi$
$282$		0			0
$283$	−13.5221			−0.803804			−0.401902	−	0.915683i	$-0.631651\pi$
$283$	−13.5221			−0.803804			−0.401902	+	0.915683i	$0.631651\pi$
$284$		0			0
$285$		0			0
$286$	11.6847	+	2.79695i	0.690928	+	0.165387i
$287$	17.7392			1.04711
$288$		0			0
$289$	13.0000			0.764706
$290$	6.07263	−	10.6937i	0.356597	−	0.627953i
$291$		0			0
$292$	−4.06854			−0.238093
$293$			11.6155i			0.678586i	0.940681	+	0.339293i	$0.110188\pi$
$293$			11.6155i			0.678586i	−0.940681	+	0.339293i	$0.889812\pi$
$294$		0			0
$295$	−0.876894	+	1.54417i	−0.0510548	+	0.0899053i
$296$	1.54417			0.0897533
$297$		0			0
$298$			2.41131i			0.139683i
$299$	28.1753			1.62942
$300$		0			0
$301$	−13.3693			−0.770595
$302$	9.48274			0.545670
$303$		0			0
$304$			6.07263i			0.348289i
$305$	−10.9993	+	19.3693i	−0.629819	+	1.10908i
$306$		0			0
$307$	−10.6937			−0.610319			−0.305159	−	0.952301i	$-0.598710\pi$
$307$	−10.6937			−0.610319			−0.305159	+	0.952301i	$0.598710\pi$
$308$	1.09190	−	4.56155i	0.0622165	−	0.259919i
$309$		0			0
$310$	−6.89697	+	12.1453i	−0.391722	+	0.689805i
$311$			4.41674i			0.250450i	0.992128	+	0.125225i	$0.0399653\pi$
$311$			4.41674i			0.250450i	−0.992128	+	0.125225i	$0.960035\pi$
$312$		0			0
$313$			16.4990i			0.932577i	0.884633	+	0.466288i	$0.154409\pi$
$313$			16.4990i			0.932577i	−0.884633	+	0.466288i	$0.845591\pi$
$314$	21.1315			1.19252
$315$		0			0
$316$		−	6.07263i		−	0.341612i
$317$	−27.2219			−1.52893			−0.764467	−	0.644663i	$-0.776998\pi$
$317$	−27.2219			−1.52893			−0.764467	+	0.644663i	$0.776998\pi$
$318$		0			0
$319$	−4.24621	+	17.7392i	−0.237742	+	0.993203i
$320$	−1.94442	−	1.10418i	−0.108696	−	0.0617258i
$321$		0			0
$322$		−	10.9993i		−	0.612968i
$323$	−12.1453			−0.675781
$324$		0			0
$325$	9.27944	+	15.5554i	0.514731	+	0.862857i
$326$	−7.91096			−0.438148
$327$		0			0
$328$			12.5435i			0.692598i
$329$	−14.0877			−0.776678
$330$		0			0
$331$	−7.36932			−0.405054			−0.202527	−	0.979277i	$-0.564915\pi$
$331$	−7.36932			−0.405054			−0.202527	+	0.979277i	$0.564915\pi$
$332$			17.3693i			0.953265i
$333$		0			0
$334$	7.36932			0.403231
$335$	−3.41011	+	6.00505i	−0.186314	+	0.328091i
$336$		0			0
$337$	−16.0786			−0.875859			−0.437930	−	0.899009i	$-0.644288\pi$
$337$	−16.0786			−0.875859			−0.437930	+	0.899009i	$0.644288\pi$
$338$			0.123106i			0.00669606i
$339$		0			0
$340$	2.20837	−	3.88884i	0.119766	−	0.210902i
$341$	4.82262	−	20.1472i	0.261159	−	1.09103i
$342$		0			0
$343$	16.9706			0.916324
$344$		−	9.45353i		−	0.509700i
$345$		0			0
$346$	18.0000			0.967686
$347$		−	23.6155i		−	1.26775i	−0.773436	−	0.633874i	$-0.781464\pi$
$347$		−	23.6155i		−	1.26775i	0.773436	−	0.633874i	$-0.218536\pi$
$348$		0			0
$349$		−	29.8844i		−	1.59968i	−0.600215	−	0.799839i	$-0.704918\pi$
$349$		−	29.8844i		−	1.59968i	0.600215	−	0.799839i	$-0.295082\pi$
$350$	6.07263	−	3.62258i	0.324596	−	0.193635i
$351$		0			0
$352$	3.22550	+	0.772087i	0.171920	+	0.0411524i
$353$	−16.0341			−0.853410			−0.426705	−	0.904391i	$-0.640326\pi$
$353$	−16.0341			−0.853410			−0.426705	+	0.904391i	$0.640326\pi$
$354$		0			0
$355$		0			0
$356$			11.4878i			0.608853i
$357$		0			0
$358$	10.8677			0.574378
$359$	−21.9986			−1.16104			−0.580521	−	0.814245i	$-0.697151\pi$
$359$	−21.9986			−1.16104			−0.580521	+	0.814245i	$0.697151\pi$
$360$		0			0
$361$	−17.8769			−0.940889
$362$		−	9.12311i		−	0.479500i
$363$		0			0
$364$	5.12311			0.268524
$365$	7.91096	+	4.49242i	0.414079	+	0.235144i
$366$		0			0
$367$		−	7.04383i		−	0.367685i	−0.982956	−	0.183842i	$-0.941146\pi$
$367$		−	7.04383i		−	0.367685i	0.982956	−	0.183842i	$-0.0588536\pi$
$368$	7.77769			0.405440
$369$		0			0
$370$	−3.00252	−	1.70505i	−0.156094	−	0.0886415i
$371$	8.58800			0.445867
$372$		0			0
$373$	−24.1180			−1.24878			−0.624390	−	0.781112i	$-0.714653\pi$
$373$	−24.1180			−1.24878			−0.624390	+	0.781112i	$0.714653\pi$
$374$	−1.54417	+	6.45101i	−0.0798473	+	0.333574i
$375$		0			0
$376$		−	9.96148i		−	0.513724i
$377$	−19.9230			−1.02608
$378$		0			0
$379$	0.492423			0.0252940			0.0126470	−	0.999920i	$-0.495974\pi$
$379$	0.492423			0.0252940			0.0126470	+	0.999920i	$0.495974\pi$
$380$	6.70531	−	11.8078i	0.343975	−	0.605726i
$381$		0			0
$382$	20.1472			1.03082
$383$	−22.1067			−1.12960			−0.564801	−	0.825227i	$-0.691047\pi$
$383$	−22.1067			−1.12960			−0.564801	+	0.825227i	$0.691047\pi$
$384$		0			0
$385$	−7.15990	+	7.66393i	−0.364902	+	0.390590i
$386$			17.3188i			0.881502i
$387$		0			0
$388$		−	10.9993i		−	0.558405i
$389$			36.1495i			1.83285i	0.400204	+	0.916426i	$0.368939\pi$
$389$			36.1495i			1.83285i	−0.400204	+	0.916426i	$0.631061\pi$
$390$		0			0
$391$			15.5554i			0.786669i
$392$			5.00000i			0.252538i
$393$		0			0
$394$	0.876894			0.0441773
$395$	−6.70531	+	11.8078i	−0.337381	+	0.594113i
$396$		0			0
$397$		−	23.5428i		−	1.18158i	−0.806826	−	0.590790i	$-0.798816\pi$
$397$		−	23.5428i		−	1.18158i	0.806826	−	0.590790i	$-0.201184\pi$
$398$		−	16.4924i		−	0.826690i
$399$		0			0
$400$	2.56155	+	4.29400i	0.128078	+	0.214700i
$401$			9.89949i			0.494357i	0.968970	+	0.247179i	$0.0795034\pi$
$401$			9.89949i			0.494357i	−0.968970	+	0.247179i	$0.920497\pi$
$402$		0			0
$403$	22.6274			1.12715
$404$	−16.4990			−0.820854
$405$		0			0
$406$			7.77769i			0.386000i
$407$	4.98074	+	1.19224i	0.246886	+	0.0590969i
$408$		0			0
$409$			18.9655i			0.937783i	0.883256	+	0.468891i	$0.155346\pi$
$409$			18.9655i			0.937783i	−0.883256	+	0.468891i	$0.844654\pi$
$410$	13.8503	−	24.3898i	0.684019	−	1.20453i
$411$		0			0
$412$		−	3.95548i		−	0.194873i
$413$		−	1.12311i		−	0.0552644i
$414$		0			0
$415$	19.1789	−	33.7733i	0.941457	−	1.65787i
$416$			3.62258i			0.177612i
$417$		0			0
$418$	−4.68860	+	19.5873i	−0.229327	+	0.958047i
$419$		−	13.3480i		−	0.652091i	−0.945354	−	0.326046i	$-0.894284\pi$
$419$		−	13.3480i		−	0.652091i	0.945354	−	0.326046i	$-0.105716\pi$
$420$		0			0
$421$	−14.8769			−0.725055			−0.362528	−	0.931973i	$-0.618086\pi$
$421$	−14.8769			−0.725055			−0.362528	+	0.931973i	$0.618086\pi$
$422$	−3.88884			−0.189306
$423$		0			0
$424$			6.07263i			0.294913i
$425$	−8.58800	+	5.12311i	−0.416579	+	0.248507i
$426$		0			0
$427$		−	14.0877i		−	0.681750i
$428$		−	11.1231i		−	0.537656i
$429$		0			0
$430$	−10.4384	+	18.3817i	−0.503387	+	0.886442i
$431$	−36.7633			−1.77083			−0.885413	−	0.464804i	$-0.846125\pi$
$431$	−36.7633			−1.77083			−0.885413	+	0.464804i	$0.846125\pi$
$432$		0			0
$433$		−	5.49966i		−	0.264297i	−0.991230	−	0.132148i	$-0.957812\pi$
$433$		−	5.49966i		−	0.264297i	0.991230	−	0.132148i	$-0.0421875\pi$
$434$		−	8.83348i		−	0.424020i
$435$		0			0
$436$			2.18379i			0.104585i
$437$			47.2311i			2.25937i
$438$		0			0
$439$			6.07263i			0.289831i	0.989444	+	0.144916i	$0.0462910\pi$
$439$			6.07263i			0.289831i	−0.989444	+	0.144916i	$0.953709\pi$
$440$	−5.41922	−	5.06282i	−0.258351	−	0.241360i
$441$		0			0
$442$	−7.24517			−0.344617
$443$	12.1453			0.577039			0.288520	−	0.957474i	$-0.406837\pi$
$443$	12.1453			0.577039			0.288520	+	0.957474i	$0.406837\pi$
$444$		0			0
$445$	12.6847	−	22.3371i	0.601310	−	1.05888i
$446$	−3.95548			−0.187298
$447$		0			0
$448$	1.41421			0.0668153
$449$			21.9096i			1.03398i	0.855992	+	0.516989i	$0.172947\pi$
$449$			21.9096i			1.03398i	−0.855992	+	0.516989i	$0.827053\pi$
$450$		0			0
$451$	−9.68466	+	40.4591i	−0.456033	+	1.90514i
$452$	8.25643			0.388350
$453$		0			0
$454$	−5.36932			−0.251995
$455$	−9.96148	−	5.65685i	−0.467001	−	0.265197i
$456$		0			0
$457$	12.0101			0.561809			0.280904	−	0.959736i	$-0.409366\pi$
$457$	12.0101			0.561809			0.280904	+	0.959736i	$0.409366\pi$
$458$			20.2462i			0.946043i
$459$		0			0
$460$	−15.1231	−	8.58800i	−0.705118	−	0.400418i
$461$	−14.7647			−0.687660			−0.343830	−	0.939032i	$-0.611724\pi$
$461$	−14.7647			−0.687660			−0.343830	+	0.939032i	$0.611724\pi$
$462$		0			0
$463$		−	22.8658i		−	1.06266i	−0.847164	−	0.531331i	$-0.821692\pi$
$463$		−	22.8658i		−	1.06266i	0.847164	−	0.531331i	$-0.178308\pi$
$464$	−5.49966			−0.255315
$465$		0			0
$466$	14.4924			0.671349
$467$	−12.1453			−0.562016			−0.281008	−	0.959705i	$-0.590669\pi$
$467$	−12.1453			−0.562016			−0.281008	+	0.959705i	$0.590669\pi$
$468$		0			0
$469$		−	4.36758i		−	0.201676i
$470$	−10.9993	+	19.3693i	−0.507360	+	0.893440i
$471$		0			0
$472$	0.794156			0.0365540
$473$	7.29895	−	30.4924i	0.335606	−	1.40204i
$474$		0			0
$475$	−26.0759	+	15.5554i	−1.19644	+	0.713730i
$476$			2.82843i			0.129641i
$477$		0			0
$478$		−	16.4990i		−	0.754645i
$479$	27.4983			1.25643			0.628214	−	0.778040i	$-0.283786\pi$
$479$	27.4983			1.25643			0.628214	+	0.778040i	$0.283786\pi$
$480$		0			0
$481$			5.59390i			0.255060i
$482$	−2.18379			−0.0994690
$483$		0			0
$484$	9.80776	+	4.98074i	0.445807	+	0.226397i
$485$	−12.1453	+	21.3873i	−0.551488	+	0.971147i
$486$		0			0
$487$			41.3958i			1.87582i	0.346873	+	0.937912i	$0.387244\pi$
$487$			41.3958i			1.87582i	−0.346873	+	0.937912i	$0.612756\pi$
$488$	9.96148			0.450935
$489$		0			0
$490$	5.52092	−	9.72211i	0.249410	−	0.439200i
$491$	−43.1301			−1.94643			−0.973217	−	0.229887i	$-0.926164\pi$
$491$	−43.1301			−1.94643			−0.973217	+	0.229887i	$0.926164\pi$
$492$		0			0
$493$		−	10.9993i		−	0.495384i
$494$	−21.9986			−0.989765
$495$		0			0
$496$	6.24621			0.280463
$497$		0			0
$498$		0			0
$499$	−14.8769			−0.665981			−0.332991	−	0.942930i	$-0.608058\pi$
$499$	−14.8769			−0.665981			−0.332991	+	0.942930i	$0.608058\pi$
$500$	−0.239369	−	11.1778i	−0.0107049	−	0.499885i
$501$		0			0
$502$	3.62258			0.161684
$503$		−	3.50758i		−	0.156395i	−0.996938	−	0.0781976i	$-0.975084\pi$
$503$		−	3.50758i		−	0.156395i	0.996938	−	0.0781976i	$-0.0249165\pi$
$504$		0			0
$505$	32.0810	+	18.2179i	1.42758	+	0.810686i
$506$	25.0870	+	6.00505i	1.11525	+	0.266957i
$507$		0			0
$508$	9.89949			0.439219
$509$			38.4342i			1.70357i	0.523895	+	0.851783i	$0.324478\pi$
$509$			38.4342i			1.70357i	−0.523895	+	0.851783i	$0.675522\pi$
$510$		0			0
$511$	−5.75379			−0.254533
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			3.88884i			0.171530i
$515$	−4.36758	+	7.69113i	−0.192459	+	0.338911i
$516$		0			0
$517$	7.69113	−	32.1308i	0.338255	−	1.41311i
$518$	2.18379			0.0959503
$519$		0			0
$520$	4.00000	−	7.04383i	0.175412	−	0.308892i
$521$		−	28.1102i		−	1.23153i	−0.787930	−	0.615765i	$-0.788847\pi$
$521$		−	28.1102i		−	1.23153i	0.787930	−	0.615765i	$-0.211153\pi$
$522$		0			0
$523$	22.3556			0.977540			0.488770	−	0.872413i	$-0.337446\pi$
$523$	22.3556			0.977540			0.488770	+	0.872413i	$0.337446\pi$
$524$	−1.54417			−0.0674575
$525$		0			0
$526$	15.3693			0.670134
$527$			12.4924i			0.544178i
$528$		0			0
$529$	37.4924			1.63011
$530$	6.70531	−	11.8078i	0.291260	−	0.512896i
$531$		0			0
$532$			8.58800i			0.372337i
$533$	−45.4398			−1.96822
$534$		0			0
$535$	−12.2820	+	21.6280i	−0.530996	+	0.935060i
$536$	3.08835			0.133396
$537$		0			0
$538$	−13.8703			−0.597990
$539$	−3.86043	+	16.1275i	−0.166281	+	0.694662i
$540$		0			0
$541$			6.55137i			0.281666i	0.990033	+	0.140833i	$0.0449780\pi$
$541$			6.55137i			0.281666i	−0.990033	+	0.140833i	$0.955022\pi$
$542$	17.2604			0.741399
$543$		0			0
$544$	−2.00000			−0.0857493
$545$	2.41131	−	4.24621i	0.103289	−	0.181888i
$546$		0			0
$547$	8.75714			0.374428			0.187214	−	0.982319i	$-0.440054\pi$
$547$	8.75714			0.374428			0.187214	+	0.982319i	$0.440054\pi$
$548$	−0.478739			−0.0204507
$549$		0			0
$550$	4.94696	+	15.8281i	0.210939	+	0.674911i
$551$		−	33.3974i		−	1.42278i
$552$		0			0
$553$		−	8.58800i		−	0.365199i
$554$			10.8677i			0.461726i
$555$		0			0
$556$			19.4442i			0.824619i
$557$			6.49242i			0.275093i	0.990495	+	0.137546i	$0.0439216\pi$
$557$			6.49242i			0.275093i	−0.990495	+	0.137546i	$0.956078\pi$
$558$		0			0
$559$	34.2462			1.44846
$560$	−2.74983	−	1.56155i	−0.116201	−	0.0659877i
$561$		0			0
$562$			20.4544i			0.862819i
$563$			38.2462i			1.61189i	0.591993	+	0.805943i	$0.298341\pi$
$563$			38.2462i			1.61189i	−0.591993	+	0.805943i	$0.701659\pi$
$564$		0			0
$565$	−16.0540	−	9.11662i	−0.675396	−	0.383539i
$566$		−	13.5221i		−	0.568375i
$567$		0			0
$568$		0			0
$569$	10.8092			0.453146			0.226573	−	0.973994i	$-0.427248\pi$
$569$	10.8092			0.453146			0.226573	+	0.973994i	$0.427248\pi$
$570$		0			0
$571$			2.66253i			0.111423i	0.998447	+	0.0557117i	$0.0177428\pi$
$571$			2.66253i			0.111423i	−0.998447	+	0.0557117i	$0.982257\pi$
$572$	−2.79695	+	11.6847i	−0.116946	+	0.488560i
$573$		0			0
$574$			17.7392i			0.740418i
$575$	19.9230	+	33.3974i	0.830845	+	1.39277i
$576$		0			0
$577$			6.17669i			0.257139i	0.991700	+	0.128570i	$0.0410386\pi$
$577$			6.17669i			0.257139i	−0.991700	+	0.128570i	$0.958961\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$	10.6937	+	6.07263i	0.444030	+	0.252152i
$581$			24.5639i			1.01908i
$582$		0			0
$583$	−4.68860	+	19.5873i	−0.194182	+	0.811224i
$584$		−	4.06854i		−	0.168358i
$585$		0			0
$586$	−11.6155			−0.479833
$587$	12.1453			0.501289			0.250644	−	0.968079i	$-0.419357\pi$
$587$	12.1453			0.501289			0.250644	+	0.968079i	$0.419357\pi$
$588$		0			0
$589$			37.9310i			1.56292i
$590$	−1.54417	−	0.876894i	−0.0635726	−	0.0361012i
$591$		0			0
$592$			1.54417i			0.0634651i
$593$			28.7386i			1.18015i	0.807347	+	0.590077i	$0.200903\pi$
$593$			28.7386i			1.18015i	−0.807347	+	0.590077i	$0.799097\pi$
$594$		0			0
$595$	3.12311	−	5.49966i	0.128035	−	0.225464i
$596$	−2.41131			−0.0987710
$597$		0			0
$598$			28.1753i			1.15217i
$599$		−	17.3188i		−	0.707625i	−0.935316	−	0.353813i	$-0.884885\pi$
$599$		−	17.3188i		−	0.707625i	0.935316	−	0.353813i	$-0.115115\pi$
$600$		0			0
$601$		−	1.22631i		−	0.0500224i	−0.999687	−	0.0250112i	$-0.992038\pi$
$601$		−	1.22631i		−	0.0500224i	0.999687	−	0.0250112i	$-0.00796214\pi$
$602$		−	13.3693i		−	0.544893i
$603$		0			0
$604$			9.48274i			0.385847i
$605$	−13.5708	−	20.5142i	−0.551730	−	0.834022i
$606$		0			0
$607$	28.8066			1.16922			0.584611	−	0.811314i	$-0.301247\pi$
$607$	28.8066			1.16922			0.584611	+	0.811314i	$0.301247\pi$
$608$	−6.07263			−0.246278
$609$		0			0
$610$	−19.3693	−	10.9993i	−0.784241	−	0.445349i
$611$	36.0863			1.45989
$612$		0			0
$613$	−27.4901			−1.11032			−0.555158	−	0.831745i	$-0.687342\pi$
$613$	−27.4901			−1.11032			−0.555158	+	0.831745i	$0.687342\pi$
$614$		−	10.6937i		−	0.431561i
$615$		0			0
$616$	4.56155	+	1.09190i	0.183790	+	0.0439937i
$617$	3.88884			0.156559			0.0782795	−	0.996931i	$-0.475057\pi$
$617$	3.88884			0.156559			0.0782795	+	0.996931i	$0.475057\pi$
$618$		0			0
$619$	22.7386			0.913943			0.456971	−	0.889481i	$-0.348934\pi$
$619$	22.7386			0.913943			0.456971	+	0.889481i	$0.348934\pi$
$620$	−12.1453	−	6.89697i	−0.487766	−	0.276989i
$621$		0			0
$622$	−4.41674			−0.177095
$623$			16.2462i			0.650891i
$624$		0			0
$625$	−11.8769	+	21.9986i	−0.475076	+	0.879945i
$626$	−16.4990			−0.659431
$627$		0			0
$628$			21.1315i			0.843238i
$629$	−3.08835			−0.123140
$630$		0			0
$631$	22.2462			0.885608			0.442804	−	0.896619i	$-0.353984\pi$
$631$	22.2462			0.885608			0.442804	+	0.896619i	$0.353984\pi$
$632$	6.07263			0.241556
$633$		0			0
$634$		−	27.2219i		−	1.08112i
$635$	−19.2488	−	10.9309i	−0.763865	−	0.433778i
$636$		0			0
$637$	−18.1129			−0.717660
$638$	−17.7392	−	4.24621i	−0.702300	−	0.168109i
$639$		0			0
$640$	1.10418	−	1.94442i	0.0436467	−	0.0768600i
$641$		−	31.6350i		−	1.24951i	−0.780822	−	0.624754i	$-0.785199\pi$
$641$		−	31.6350i		−	1.24951i	0.780822	−	0.624754i	$-0.214801\pi$
$642$		0			0
$643$		−	14.0877i		−	0.555563i	−0.960644	−	0.277782i	$-0.910401\pi$
$643$		−	14.0877i		−	0.555563i	0.960644	−	0.277782i	$-0.0895991\pi$
$644$	10.9993			0.433434
$645$		0			0
$646$		−	12.1453i		−	0.477849i
$647$	13.3716			0.525691			0.262846	−	0.964838i	$-0.415339\pi$
$647$	13.3716			0.525691			0.262846	+	0.964838i	$0.415339\pi$
$648$		0			0
$649$	2.56155	+	0.613157i	0.100550	+	0.0240685i
$650$	−15.5554	+	9.27944i	−0.610132	+	0.363969i
$651$		0			0
$652$		−	7.91096i		−	0.309817i
$653$	16.0341			0.627463			0.313732	−	0.949512i	$-0.398421\pi$
$653$	16.0341			0.627463			0.313732	+	0.949512i	$0.398421\pi$
$654$		0			0
$655$	3.00252	+	1.70505i	0.117318	+	0.0666219i
$656$	−12.5435			−0.489741
$657$		0			0
$658$		−	14.0877i		−	0.549194i
$659$	−23.5428			−0.917097			−0.458549	−	0.888669i	$-0.651630\pi$
$659$	−23.5428			−0.917097			−0.458549	+	0.888669i	$0.651630\pi$
$660$		0			0
$661$	−48.7386			−1.89571			−0.947857	−	0.318697i	$-0.896755\pi$
$661$	−48.7386			−1.89571			−0.947857	+	0.318697i	$0.896755\pi$
$662$		−	7.36932i		−	0.286417i
$663$		0			0
$664$	−17.3693			−0.674060
$665$	9.48274	−	16.6987i	0.367725	−	0.647548i
$666$		0			0
$667$	−42.7746			−1.65624
$668$			7.36932i			0.285127i
$669$		0			0
$670$	−6.00505	−	3.41011i	−0.231995	−	0.131744i
$671$	32.1308	+	7.69113i	1.24040	+	0.296913i
$672$		0			0
$673$	34.2893			1.32176			0.660878	−	0.750493i	$-0.270184\pi$
$673$	34.2893			1.32176			0.660878	+	0.750493i	$0.270184\pi$
$674$		−	16.0786i		−	0.619326i
$675$		0			0
$676$	−0.123106			−0.00473483
$677$		−	23.6155i		−	0.907618i	−0.891099	−	0.453809i	$-0.850065\pi$
$677$		−	23.6155i		−	0.907618i	0.891099	−	0.453809i	$-0.149935\pi$
$678$		0			0
$679$		−	15.5554i		−	0.596961i
$680$	3.88884	+	2.20837i	0.149130	+	0.0846871i
$681$		0			0
$682$	20.1472	+	4.82262i	0.771476	+	0.184668i
$683$	51.0337			1.95275			0.976375	−	0.216082i	$-0.0693278\pi$
$683$	51.0337			1.95275			0.976375	+	0.216082i	$0.0693278\pi$
$684$		0			0
$685$	0.930870	+	0.528616i	0.0355667	+	0.0201974i
$686$			16.9706i			0.647939i
$687$		0			0
$688$	9.45353			0.360413
$689$	−21.9986			−0.838081
$690$		0			0
$691$	−25.6155			−0.974461			−0.487230	−	0.873274i	$-0.661993\pi$
$691$	−25.6155			−0.974461			−0.487230	+	0.873274i	$0.661993\pi$
$692$			18.0000i			0.684257i
$693$		0			0
$694$	23.6155			0.896433
$695$	21.4700	−	37.8078i	0.814404	−	1.43413i
$696$		0			0
$697$		−	25.0870i		−	0.950237i
$698$	29.8844			1.13114
$699$		0			0
$700$	3.62258	+	6.07263i	0.136921	+	0.229524i
$701$	−39.8517			−1.50518			−0.752588	−	0.658491i	$-0.771195\pi$
$701$	−39.8517			−1.50518			−0.752588	+	0.658491i	$0.771195\pi$
$702$		0			0
$703$	−9.37720			−0.353668
$704$	−0.772087	+	3.22550i	−0.0290991	+	0.121566i
$705$		0			0
$706$		−	16.0341i		−	0.603452i
$707$	−23.3331			−0.877530
$708$		0			0
$709$	22.0000			0.826227			0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000			0.826227			0.413114	+	0.910679i	$0.364441\pi$
$710$		0			0
$711$		0			0
$712$	−11.4878			−0.430524
$713$	48.5811			1.81938
$714$		0			0
$715$	18.3405	−	19.6316i	0.685895	−	0.734179i
$716$			10.8677i			0.406147i
$717$		0			0
$718$		−	21.9986i		−	0.820981i
$719$			26.6960i			0.995591i	0.867294	+	0.497796i	$0.165857\pi$
$719$			26.6960i			0.995591i	−0.867294	+	0.497796i	$0.834143\pi$
$720$		0			0
$721$		−	5.59390i		−	0.208328i
$722$		−	17.8769i		−	0.665309i
$723$		0			0
$724$	9.12311			0.339058
$725$	−14.0877	−	23.6155i	−0.523202	−	0.877059i
$726$		0			0
$727$		−	38.3075i		−	1.42075i	−0.703825	−	0.710373i	$-0.748526\pi$
$727$		−	38.3075i		−	1.42075i	0.703825	−	0.710373i	$-0.251474\pi$
$728$			5.12311i			0.189875i
$729$		0			0
$730$	−4.49242	+	7.91096i	−0.166272	+	0.292798i
$731$			18.9071i			0.699303i
$732$		0			0
$733$	−4.31897			−0.159525			−0.0797625	−	0.996814i	$-0.525416\pi$
$733$	−4.31897			−0.159525			−0.0797625	+	0.996814i	$0.525416\pi$
$734$	7.04383			0.259992
$735$		0			0
$736$			7.77769i			0.286689i
$737$	9.96148	+	2.38447i	0.366936	+	0.0878332i
$738$		0			0
$739$		−	25.0381i		−	0.921042i	−0.887649	−	0.460521i	$-0.847663\pi$
$739$		−	25.0381i		−	0.921042i	0.887649	−	0.460521i	$-0.152337\pi$
$740$	1.70505	−	3.00252i	0.0626790	−	0.110375i
$741$		0			0
$742$			8.58800i			0.315275i
$743$			14.6307i			0.536748i	0.963315	+	0.268374i	$0.0864862\pi$
$743$			14.6307i			0.536748i	−0.963315	+	0.268374i	$0.913514\pi$
$744$		0			0
$745$	4.68860	+	2.66253i	0.171777	+	0.0975475i
$746$		−	24.1180i		−	0.883021i
$747$		0			0
$748$	−6.45101	−	1.54417i	−0.235872	−	0.0564606i
$749$		−	15.7304i		−	0.574778i
$750$		0			0
$751$	10.2462			0.373890			0.186945	−	0.982370i	$-0.440141\pi$
$751$	10.2462			0.373890			0.186945	+	0.982370i	$0.440141\pi$
$752$	9.96148			0.363258
$753$		0			0
$754$		−	19.9230i		−	0.725551i
$755$	10.4707	−	18.4384i	0.381068	−	0.671044i
$756$		0			0
$757$			43.8071i			1.59220i	0.605166	+	0.796099i	$0.293107\pi$
$757$			43.8071i			1.59220i	−0.605166	+	0.796099i	$0.706893\pi$
$758$			0.492423i			0.0178856i
$759$		0			0
$760$	11.8078	+	6.70531i	0.428313	+	0.243227i
$761$	21.8085			0.790558			0.395279	−	0.918561i	$-0.370648\pi$
$761$	21.8085			0.790558			0.395279	+	0.918561i	$0.370648\pi$
$762$		0			0
$763$			3.08835i			0.111806i
$764$			20.1472i			0.728900i
$765$		0			0
$766$		−	22.1067i		−	0.798749i
$767$			2.87689i			0.103879i
$768$		0			0
$769$		−	45.4398i		−	1.63860i	−0.573364	−	0.819301i	$-0.694362\pi$
$769$		−	45.4398i		−	1.63860i	0.573364	−	0.819301i	$-0.305638\pi$
$770$	−7.66393	−	7.15990i	−0.276189	−	0.258025i
$771$		0			0
$772$	−17.3188			−0.623316
$773$	−42.7773			−1.53859			−0.769296	−	0.638893i	$-0.779393\pi$
$773$	−42.7773			−1.53859			−0.769296	+	0.638893i	$0.779393\pi$
$774$		0			0
$775$	16.0000	+	26.8212i	0.574737	+	0.963447i
$776$	10.9993			0.394852
$777$		0			0
$778$	−36.1495			−1.29602
$779$		−	76.1720i		−	2.72915i
$780$		0			0
$781$		0			0
$782$	−15.5554			−0.556259
$783$		0			0
$784$	−5.00000			−0.178571
$785$	23.3331	−	41.0885i	0.832793	−	1.46651i
$786$		0			0
$787$	4.14488			0.147749			0.0738744	−	0.997268i	$-0.476464\pi$
$787$	4.14488			0.147749			0.0738744	+	0.997268i	$0.476464\pi$
$788$			0.876894i			0.0312381i
$789$		0			0
$790$	−11.8078	−	6.70531i	−0.420101	−	0.238564i
$791$	11.6763			0.415163
$792$		0			0
$793$			36.0863i			1.28146i
$794$	23.5428			0.835503
$795$		0			0
$796$	16.4924			0.584558
$797$	−10.4402			−0.369811			−0.184906	−	0.982756i	$-0.559198\pi$
$797$	−10.4402			−0.369811			−0.184906	+	0.982756i	$0.559198\pi$
$798$		0			0
$799$			19.9230i			0.704824i
$800$	−4.29400	+	2.56155i	−0.151816	+	0.0905646i
$801$		0			0
$802$	−9.89949			−0.349563
$803$	3.14127	−	13.1231i	0.110853	−	0.463104i
$804$		0			0
$805$	−21.3873	−	12.1453i	−0.753803	−	0.428065i
$806$			22.6274i			0.797017i
$807$		0			0
$808$		−	16.4990i		−	0.580432i
$809$	21.8085			0.766747			0.383373	−	0.923593i	$-0.374762\pi$
$809$	21.8085			0.766747			0.383373	+	0.923593i	$0.374762\pi$
$810$		0			0
$811$		−	22.8543i		−	0.802524i	−0.915963	−	0.401262i	$-0.868572\pi$
$811$		−	22.8543i		−	0.802524i	0.915963	−	0.401262i	$-0.131428\pi$
$812$	−7.77769			−0.272943
$813$		0			0
$814$	−1.19224	+	4.98074i	−0.0417878	+	0.174575i
$815$	−8.73516	+	15.3823i	−0.305979	+	0.538817i
$816$		0			0
$817$			57.4079i			2.00845i
$818$	−18.9655			−0.663112
$819$		0			0
$820$	24.3898	+	13.8503i	0.851730	+	0.483674i
$821$	−5.49966			−0.191939			−0.0959697	−	0.995384i	$-0.530595\pi$
$821$	−5.49966			−0.191939			−0.0959697	+	0.995384i	$0.530595\pi$
$822$		0			0
$823$			10.1322i			0.353185i	0.984284	+	0.176593i	$0.0565075\pi$
$823$			10.1322i			0.353185i	−0.984284	+	0.176593i	$0.943492\pi$
$824$	3.95548			0.137796
$825$		0			0
$826$	1.12311			0.0390778
$827$		−	29.3693i		−	1.02127i	−0.859797	−	0.510636i	$-0.829410\pi$
$827$		−	29.3693i		−	1.02127i	0.859797	−	0.510636i	$-0.170590\pi$
$828$		0			0
$829$	−32.7386			−1.13706			−0.568530	−	0.822663i	$-0.692488\pi$
$829$	−32.7386			−1.13706			−0.568530	+	0.822663i	$0.692488\pi$
$830$	33.7733	+	19.1789i	1.17229	+	0.665710i
$831$		0			0
$832$	−3.62258			−0.125590
$833$		−	10.0000i		−	0.346479i
$834$		0			0
$835$	8.13709	−	14.3291i	0.281595	−	0.495878i
$836$	−19.5873	−	4.68860i	−0.677441	−	0.162159i
$837$		0			0
$838$	13.3480			0.461098
$839$			2.82843i			0.0976481i	0.998807	+	0.0488241i	$0.0155474\pi$
$839$			2.82843i			0.0976481i	−0.998807	+	0.0488241i	$0.984453\pi$
$840$		0			0
$841$	1.24621			0.0429728
$842$		−	14.8769i		−	0.512692i
$843$		0			0
$844$		−	3.88884i		−	0.133860i
$845$	0.239369	+	0.135931i	0.00823455	+	0.00467618i
$846$		0			0
$847$	13.8703	+	7.04383i	0.476588	+	0.242029i
$848$	−6.07263			−0.208535
$849$		0			0
$850$	−5.12311	−	8.58800i	−0.175721	−	0.294566i
$851$			12.0101i			0.411701i
$852$		0			0
$853$	−9.62763			−0.329644			−0.164822	−	0.986323i	$-0.552705\pi$
$853$	−9.62763			−0.329644			−0.164822	+	0.986323i	$0.552705\pi$
$854$	14.0877			0.482070
$855$		0			0
$856$	11.1231			0.380180
$857$		−	26.4924i		−	0.904964i	−0.891774	−	0.452482i	$-0.850539\pi$
$857$		−	26.4924i		−	0.904964i	0.891774	−	0.452482i	$-0.149461\pi$
$858$		0			0
$859$	−12.9848			−0.443037			−0.221519	−	0.975156i	$-0.571101\pi$
$859$	−12.9848			−0.443037			−0.221519	+	0.975156i	$0.571101\pi$
$860$	−18.3817	−	10.4384i	−0.626809	−	0.355948i
$861$		0			0
$862$		−	36.7633i		−	1.25216i
$863$	16.5129			0.562104			0.281052	−	0.959693i	$-0.409317\pi$
$863$	16.5129			0.562104			0.281052	+	0.959693i	$0.409317\pi$
$864$		0			0
$865$	19.8753	−	34.9996i	0.675781	−	1.19002i
$866$	5.49966			0.186886
$867$		0			0
$868$	8.83348			0.299828
$869$	19.5873	+	4.68860i	0.664454	+	0.159050i
$870$		0			0
$871$			11.1878i			0.379084i
$872$	−2.18379			−0.0739525
$873$		0			0
$874$	−47.2311			−1.59761
$875$	−0.338519	−	15.8078i	−0.0114440	−	0.534400i
$876$		0			0
$877$	40.3921			1.36395			0.681973	−	0.731378i	$-0.261122\pi$
$877$	40.3921			1.36395			0.681973	+	0.731378i	$0.261122\pi$
$878$	−6.07263			−0.204942
$879$		0			0
$880$	5.06282	−	5.41922i	0.170667	−	0.182682i
$881$		−	12.1842i		−	0.410496i	−0.978710	−	0.205248i	$-0.934200\pi$
$881$		−	12.1842i		−	0.410496i	0.978710	−	0.205248i	$-0.0658001\pi$
$882$		0			0
$883$			47.0856i			1.58456i	0.610160	+	0.792278i	$0.291105\pi$
$883$			47.0856i			1.58456i	−0.610160	+	0.792278i	$0.708895\pi$
$884$		−	7.24517i		−	0.243681i
$885$		0			0
$886$			12.1453i			0.408028i
$887$			10.0000i			0.335767i	0.985807	+	0.167884i	$0.0536933\pi$
$887$			10.0000i			0.335767i	−0.985807	+	0.167884i	$0.946307\pi$
$888$		0			0
$889$	14.0000			0.469545
$890$	22.3371	+	12.6847i	0.748743	+	0.425191i
$891$		0			0
$892$		−	3.95548i		−	0.132439i
$893$			60.4924i			2.02430i
$894$		0			0
$895$	12.0000	−	21.1315i	0.401116	−	0.706348i
$896$			1.41421i			0.0472456i
$897$		0			0
$898$	−21.9096			−0.731133
$899$	−34.3520			−1.14570
$900$		0			0
$901$		−	12.1453i		−	0.404618i
$902$	−40.4591	−	9.68466i	−1.34714	−	0.322464i
$903$		0			0
$904$			8.25643i			0.274605i
$905$	−17.7392	−	10.0736i	−0.589670	−	0.334858i
$906$		0			0
$907$		−	6.17669i		−	0.205094i	−0.994728	−	0.102547i	$-0.967301\pi$
$907$		−	6.17669i		−	0.205094i	0.994728	−	0.102547i	$-0.0326992\pi$
$908$		−	5.36932i		−	0.178187i
$909$		0			0
$910$	5.65685	−	9.96148i	0.187523	−	0.330220i
$911$		−	10.4218i		−	0.345289i	−0.984984	−	0.172645i	$-0.944769\pi$
$911$		−	10.4218i		−	0.345289i	0.984984	−	0.172645i	$-0.0552312\pi$
$912$		0			0
$913$	−56.0248	−	13.4106i	−1.85415	−	0.443827i
$914$			12.0101i			0.397259i
$915$		0			0
$916$	−20.2462			−0.668954
$917$	−2.18379			−0.0721151
$918$		0			0
$919$		−	8.52526i		−	0.281222i	−0.990065	−	0.140611i	$-0.955093\pi$
$919$		−	8.52526i		−	0.281222i	0.990065	−	0.140611i	$-0.0449068\pi$
$920$	8.58800	−	15.1231i	0.283138	−	0.498594i
$921$		0			0
$922$		−	14.7647i		−	0.486249i
$923$		0			0
$924$		0			0
$925$	−6.63068	+	3.95548i	−0.218016	+	0.130055i
$926$	22.8658			0.751415
$927$		0			0
$928$		−	5.49966i		−	0.180535i
$929$			24.7380i			0.811628i	0.913956	+	0.405814i	$0.133012\pi$
$929$			24.7380i			0.811628i	−0.913956	+	0.405814i	$0.866988\pi$
$930$		0			0
$931$		−	30.3632i		−	0.995113i
$932$			14.4924i			0.474715i
$933$		0			0
$934$		−	12.1453i		−	0.397405i
$935$	10.8384	+	10.1256i	0.354455	+	0.331144i
$936$		0			0
$937$	−47.1913			−1.54167			−0.770837	−	0.637032i	$-0.780162\pi$
$937$	−47.1913			−1.54167			−0.770837	+	0.637032i	$0.780162\pi$
$938$	4.36758			0.142607
$939$		0			0
$940$	−19.3693	−	10.9993i	−0.631757	−	0.358758i
$941$	2.41131			0.0786064			0.0393032	−	0.999227i	$-0.487486\pi$
$941$	2.41131			0.0786064			0.0393032	+	0.999227i	$0.487486\pi$
$942$		0			0
$943$	−97.5593			−3.17697
$944$			0.794156i			0.0258476i
$945$		0			0
$946$	30.4924	+	7.29895i	0.991394	+	0.237309i
$947$	15.5554			0.505482			0.252741	−	0.967534i	$-0.418668\pi$
$947$	15.5554			0.505482			0.252741	+	0.967534i	$0.418668\pi$
$948$		0			0
$949$	14.7386			0.478436
$950$	−15.5554	−	26.0759i	−0.504683	−	0.846014i
$951$		0			0
$952$	−2.82843			−0.0916698
$953$			48.2462i			1.56285i	0.624000	+	0.781424i	$0.285506\pi$
$953$			48.2462i			1.56285i	−0.624000	+	0.781424i	$0.714494\pi$
$954$		0			0
$955$	22.2462	−	39.1746i	0.719870	−	1.26766i
$956$	16.4990			0.533615
$957$		0			0
$958$			27.4983i			0.888429i
$959$	−0.677039			−0.0218627
$960$		0			0
$961$	8.01515			0.258553
$962$	−5.59390			−0.180354
$963$		0			0
$964$		−	2.18379i		−	0.0703352i
$965$	33.6750	+	19.1231i	1.08404	+	0.615595i
$966$		0			0
$967$	16.7965			0.540138			0.270069	−	0.962841i	$-0.412954\pi$
$967$	16.7965			0.540138			0.270069	+	0.962841i	$0.412954\pi$
$968$	−4.98074	+	9.80776i	−0.160087	+	0.315233i
$969$		0			0
$970$	−21.3873	−	12.1453i	−0.686705	−	0.389961i
$971$			7.14740i			0.229371i	0.993402	+	0.114685i	$0.0365860\pi$
$971$			7.14740i			0.229371i	−0.993402	+	0.114685i	$0.963414\pi$
$972$		0			0
$973$			27.4983i			0.881554i
$974$	−41.3958			−1.32641
$975$		0			0
$976$			9.96148i			0.318859i
$977$	59.2901			1.89686			0.948430	−	0.316988i	$-0.102671\pi$
$977$	59.2901			1.89686			0.948430	+	0.316988i	$0.102671\pi$
$978$		0			0
$979$	−37.0540	−	8.86958i	−1.18425	−	0.283473i
$980$	9.72211	+	5.52092i	0.310561	+	0.176359i
$981$		0			0
$982$		−	43.1301i		−	1.37634i
$983$	13.3716			0.426487			0.213244	−	0.976999i	$-0.431597\pi$
$983$	13.3716			0.426487			0.213244	+	0.976999i	$0.431597\pi$
$984$		0			0
$985$	0.968253	−	1.70505i	0.0308511	−	0.0543275i
$986$	10.9993			0.350289
$987$		0			0
$988$		−	21.9986i		−	0.699869i
$989$	73.5266			2.33801
$990$		0			0
$991$	−12.4924			−0.396835			−0.198417	−	0.980118i	$-0.563580\pi$
$991$	−12.4924			−0.396835			−0.198417	+	0.980118i	$0.563580\pi$
$992$			6.24621i			0.198317i
$993$		0			0
$994$		0			0
$995$	−32.0682	−	18.2107i	−1.01663	−	0.577317i
$996$		0			0
$997$	48.3337			1.53074			0.765372	−	0.643589i	$-0.222555\pi$
$997$	48.3337			1.53074			0.765372	+	0.643589i	$0.222555\pi$
$998$		−	14.8769i		−	0.470920i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	990.2.f.c.989.16	yes	16
3.2	odd	2	inner	990.2.f.c.989.1	✓	16
5.2	odd	4		4950.2.d.g.4751.3		8
5.3	odd	4		4950.2.d.n.4751.7		8
5.4	even	2	inner	990.2.f.c.989.2	yes	16
11.10	odd	2	inner	990.2.f.c.989.8	yes	16
15.2	even	4		4950.2.d.n.4751.2		8
15.8	even	4		4950.2.d.g.4751.6		8
15.14	odd	2	inner	990.2.f.c.989.15	yes	16
33.32	even	2	inner	990.2.f.c.989.9	yes	16
55.32	even	4		4950.2.d.n.4751.6		8
55.43	even	4		4950.2.d.g.4751.2		8
55.54	odd	2	inner	990.2.f.c.989.10	yes	16
165.32	odd	4		4950.2.d.g.4751.7		8
165.98	odd	4		4950.2.d.n.4751.3		8
165.164	even	2	inner	990.2.f.c.989.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.f.c.989.1	✓	16	3.2	odd	2	inner
990.2.f.c.989.2	yes	16	5.4	even	2	inner
990.2.f.c.989.7	yes	16	165.164	even	2	inner
990.2.f.c.989.8	yes	16	11.10	odd	2	inner
990.2.f.c.989.9	yes	16	33.32	even	2	inner
990.2.f.c.989.10	yes	16	55.54	odd	2	inner
990.2.f.c.989.15	yes	16	15.14	odd	2	inner
990.2.f.c.989.16	yes	16	1.1	even	1	trivial
4950.2.d.g.4751.2		8	55.43	even	4
4950.2.d.g.4751.3		8	5.2	odd	4
4950.2.d.g.4751.6		8	15.8	even	4
4950.2.d.g.4751.7		8	165.32	odd	4
4950.2.d.n.4751.2		8	15.2	even	4
4950.2.d.n.4751.3		8	165.98	odd	4
4950.2.d.n.4751.6		8	55.32	even	4
4950.2.d.n.4751.7		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	990.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.94442 + 1.10418i) q^{5} -1.41421 q^{7} -1.00000i q^{8} +(-1.10418 + 1.94442i) q^{10} +(0.772087 - 3.22550i) q^{11} +3.62258 q^{13} -1.41421i q^{14} +1.00000 q^{16} +2.00000i q^{17} +6.07263i q^{19} +(-1.94442 - 1.10418i) q^{20} +(3.22550 + 0.772087i) q^{22} +7.77769 q^{23} +(2.56155 + 4.29400i) q^{25} +3.62258i q^{26} +1.41421 q^{28} -5.49966 q^{29} +6.24621 q^{31} +1.00000i q^{32} -2.00000 q^{34} +(-2.74983 - 1.56155i) q^{35} +1.54417i q^{37} -6.07263 q^{38} +(1.10418 - 1.94442i) q^{40} -12.5435 q^{41} +9.45353 q^{43} +(-0.772087 + 3.22550i) q^{44} +7.77769i q^{46} +9.96148 q^{47} -5.00000 q^{49} +(-4.29400 + 2.56155i) q^{50} -3.62258 q^{52} -6.07263 q^{53} +(5.06282 - 5.41922i) q^{55} +1.41421i q^{56} -5.49966i q^{58} +0.794156i q^{59} +9.96148i q^{61} +6.24621i q^{62} -1.00000 q^{64} +(7.04383 + 4.00000i) q^{65} +3.08835i q^{67} -2.00000i q^{68} +(1.56155 - 2.74983i) q^{70} +4.06854 q^{73} -1.54417 q^{74} -6.07263i q^{76} +(-1.09190 + 4.56155i) q^{77} +6.07263i q^{79} +(1.94442 + 1.10418i) q^{80} -12.5435i q^{82} -17.3693i q^{83} +(-2.20837 + 3.88884i) q^{85} +9.45353i q^{86} +(-3.22550 - 0.772087i) q^{88} -11.4878i q^{89} -5.12311 q^{91} -7.77769 q^{92} +9.96148i q^{94} +(-6.70531 + 11.8078i) q^{95} +10.9993i q^{97} -5.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{16} + 8 q^{25} - 32 q^{31} - 32 q^{34} - 80 q^{49} + 24 q^{55} - 16 q^{64} - 8 q^{70} - 16 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 + 8 * q^25 - 32 * q^31 - 32 * q^34 - 80 * q^49 + 24 * q^55 - 16 * q^64 - 8 * q^70 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more