LMFDB - Embedded newform 1666.2.b.a.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1666,2,Mod(883,1666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1666, 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("1666.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1666 = 2 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1666.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.3030769767$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 238)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1666.883
Dual form			1666.2.b.a.883.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.00000i q^{3} +1.00000 q^{4} +3.00000i q^{6} -1.00000 q^{8} -6.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.00000i q^{3} +1.00000 q^{4} +3.00000i q^{6} -1.00000 q^{8} -6.00000 q^{9} -3.00000i q^{11} -3.00000i q^{12} -5.00000 q^{13} +1.00000 q^{16} +(-1.00000 + 4.00000i) q^{17} +6.00000 q^{18} -4.00000 q^{19} +3.00000i q^{22} -4.00000i q^{23} +3.00000i q^{24} +5.00000 q^{25} +5.00000 q^{26} +9.00000i q^{27} +4.00000i q^{29} -4.00000i q^{31} -1.00000 q^{32} -9.00000 q^{33} +(1.00000 - 4.00000i) q^{34} -6.00000 q^{36} +8.00000i q^{37} +4.00000 q^{38} +15.0000i q^{39} +4.00000i q^{41} +8.00000 q^{43} -3.00000i q^{44} +4.00000i q^{46} -4.00000 q^{47} -3.00000i q^{48} -5.00000 q^{50} +(12.0000 + 3.00000i) q^{51} -5.00000 q^{52} -1.00000 q^{53} -9.00000i q^{54} +12.0000i q^{57} -4.00000i q^{58} -12.0000 q^{59} +4.00000i q^{62} +1.00000 q^{64} +9.00000 q^{66} -16.0000 q^{67} +(-1.00000 + 4.00000i) q^{68} -12.0000 q^{69} -1.00000i q^{71} +6.00000 q^{72} +8.00000i q^{73} -8.00000i q^{74} -15.0000i q^{75} -4.00000 q^{76} -15.0000i q^{78} -7.00000i q^{79} +9.00000 q^{81} -4.00000i q^{82} +16.0000 q^{83} -8.00000 q^{86} +12.0000 q^{87} +3.00000i q^{88} -9.00000 q^{89} -4.00000i q^{92} -12.0000 q^{93} +4.00000 q^{94} +3.00000i q^{96} +8.00000i q^{97} +18.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{9} - 10 q^{13} + 2 q^{16} - 2 q^{17} + 12 q^{18} - 8 q^{19} + 10 q^{25} + 10 q^{26} - 2 q^{32} - 18 q^{33} + 2 q^{34} - 12 q^{36} + 8 q^{38} + 16 q^{43} - 8 q^{47} - 10 q^{50} + 24 q^{51} - 10 q^{52} - 2 q^{53} - 24 q^{59} + 2 q^{64} + 18 q^{66} - 32 q^{67} - 2 q^{68} - 24 q^{69} + 12 q^{72} - 8 q^{76} + 18 q^{81} + 32 q^{83} - 16 q^{86} + 24 q^{87} - 18 q^{89} - 24 q^{93} + 8 q^{94}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^9 - 10 * q^13 + 2 * q^16 - 2 * q^17 + 12 * q^18 - 8 * q^19 + 10 * q^25 + 10 * q^26 - 2 * q^32 - 18 * q^33 + 2 * q^34 - 12 * q^36 + 8 * q^38 + 16 * q^43 - 8 * q^47 - 10 * q^50 + 24 * q^51 - 10 * q^52 - 2 * q^53 - 24 * q^59 + 2 * q^64 + 18 * q^66 - 32 * q^67 - 2 * q^68 - 24 * q^69 + 12 * q^72 - 8 * q^76 + 18 * q^81 + 32 * q^83 - 16 * q^86 + 24 * q^87 - 18 * q^89 - 24 * q^93 + 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1666\mathbb{Z}\right)^\times$.

$n$	$785$	$885$
$\chi(n)$	$-1$	$1$

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
$2$	−1.00000			−0.707107
$3$		−	3.00000i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$		−	3.00000i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	1.00000			0.500000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$			3.00000i			1.22474i
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$	−6.00000			−2.00000
$10$		0			0
$11$		−	3.00000i		−	0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$		−	3.00000i		−	0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$		−	3.00000i		−	0.866025i
$13$	−5.00000			−1.38675			−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000			−1.38675			−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		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
$18$	6.00000			1.41421
$19$	−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
$21$		0			0
$22$			3.00000i			0.639602i
$23$		−	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
$26$	5.00000			0.980581
$27$			9.00000i			1.73205i
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	−1.00000			−0.176777
$33$	−9.00000			−1.56670
$34$	1.00000	−	4.00000i	0.171499	−	0.685994i
$35$		0			0
$36$	−6.00000			−1.00000
$37$			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
$39$			15.0000i			2.40192i
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$			4.00000i			0.589768i
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		−	3.00000i		−	0.433013i
$49$		0			0
$50$	−5.00000			−0.707107
$51$	12.0000	+	3.00000i	1.68034	+	0.420084i
$52$	−5.00000			−0.693375
$53$	−1.00000			−0.137361			−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000			−0.137361			−0.0686803	+	0.997639i	$0.521879\pi$
$54$		−	9.00000i		−	1.22474i
$55$		0			0
$56$		0			0
$57$			12.0000i			1.58944i
$58$		−	4.00000i		−	0.525226i
$59$	−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
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$			4.00000i			0.508001i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$	9.00000			1.10782
$67$	−16.0000			−1.95471			−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000			−1.95471			−0.977356	+	0.211604i	$0.932131\pi$
$68$	−1.00000	+	4.00000i	−0.121268	+	0.485071i
$69$	−12.0000			−1.44463
$70$		0			0
$71$		−	1.00000i		−	0.118678i	−0.998238	−	0.0593391i	$-0.981101\pi$
$71$		−	1.00000i		−	0.118678i	0.998238	−	0.0593391i	$-0.0188993\pi$
$72$	6.00000			0.707107
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$		−	8.00000i		−	0.929981i
$75$		−	15.0000i		−	1.73205i
$76$	−4.00000			−0.458831
$77$		0			0
$78$		−	15.0000i		−	1.69842i
$79$		−	7.00000i		−	0.787562i	−0.919204	−	0.393781i	$-0.871167\pi$
$79$		−	7.00000i		−	0.787562i	0.919204	−	0.393781i	$-0.128833\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		−	4.00000i		−	0.441726i
$83$	16.0000			1.75623			0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000			1.75623			0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$		0			0
$86$	−8.00000			−0.862662
$87$	12.0000			1.28654
$88$			3.00000i			0.319801i
$89$	−9.00000			−0.953998			−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000			−0.953998			−0.476999	+	0.878904i	$0.658275\pi$
$90$		0			0
$91$		0			0
$92$		−	4.00000i		−	0.417029i
$93$	−12.0000			−1.24434
$94$	4.00000			0.412568
$95$		0			0
$96$			3.00000i			0.306186i
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$			18.0000i			1.80907i
$100$	5.00000			0.500000
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$	−12.0000	−	3.00000i	−1.18818	−	0.297044i
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	5.00000			0.490290
$105$		0			0
$106$	1.00000			0.0971286
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$			9.00000i			0.866025i
$109$			20.0000i			1.91565i	0.287348	+	0.957826i	$0.407226\pi$
$109$			20.0000i			1.91565i	−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$	24.0000			2.27798
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		−	12.0000i		−	1.12390i
$115$		0			0
$116$			4.00000i			0.371391i
$117$	30.0000			2.77350
$118$	12.0000			1.10469
$119$		0			0
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$	12.0000			1.08200
$124$		−	4.00000i		−	0.359211i
$125$		0			0
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000			−0.0883883
$129$		−	24.0000i		−	2.11308i
$130$		0			0
$131$		−	8.00000i		−	0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$		−	8.00000i		−	0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	−9.00000			−0.783349
$133$		0			0
$134$	16.0000			1.38219
$135$		0			0
$136$	1.00000	−	4.00000i	0.0857493	−	0.342997i
$137$	9.00000			0.768922			0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000			0.768922			0.384461	+	0.923141i	$0.374387\pi$
$138$	12.0000			1.02151
$139$			11.0000i			0.933008i	0.884519	+	0.466504i	$0.154487\pi$
$139$			11.0000i			0.933008i	−0.884519	+	0.466504i	$0.845513\pi$
$140$		0			0
$141$			12.0000i			1.01058i
$142$			1.00000i			0.0839181i
$143$			15.0000i			1.25436i
$144$	−6.00000			−0.500000
$145$		0			0
$146$		−	8.00000i		−	0.662085i
$147$		0			0
$148$			8.00000i			0.657596i
$149$	11.0000			0.901155			0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000			0.901155			0.450578	+	0.892737i	$0.351218\pi$
$150$			15.0000i			1.22474i
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$	4.00000			0.324443
$153$	6.00000	−	24.0000i	0.485071	−	1.94029i
$154$		0			0
$155$		0			0
$156$			15.0000i			1.20096i
$157$	−3.00000			−0.239426			−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000			−0.239426			−0.119713	+	0.992809i	$0.538197\pi$
$158$			7.00000i			0.556890i
$159$			3.00000i			0.237915i
$160$		0			0
$161$		0			0
$162$	−9.00000			−0.707107
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	−16.0000			−1.24184
$167$		−	25.0000i		−	1.93456i	−0.253715	−	0.967279i	$-0.581653\pi$
$167$		−	25.0000i		−	1.93456i	0.253715	−	0.967279i	$-0.418347\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	24.0000			1.83533
$172$	8.00000			0.609994
$173$		−	4.00000i		−	0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$		−	4.00000i		−	0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	−12.0000			−0.909718
$175$		0			0
$176$		−	3.00000i		−	0.226134i
$177$			36.0000i			2.70593i
$178$	9.00000			0.674579
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$			4.00000i			0.297318i	0.988889	+	0.148659i	$0.0474956\pi$
$181$			4.00000i			0.297318i	−0.988889	+	0.148659i	$0.952504\pi$
$182$		0			0
$183$		0			0
$184$			4.00000i			0.294884i
$185$		0			0
$186$	12.0000			0.879883
$187$	12.0000	+	3.00000i	0.877527	+	0.219382i
$188$	−4.00000			−0.291730
$189$		0			0
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		−	3.00000i		−	0.216506i
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		−	8.00000i		−	0.574367i
$195$		0			0
$196$		0			0
$197$		−	16.0000i		−	1.13995i	−0.821661	−	0.569976i	$-0.806952\pi$
$197$		−	16.0000i		−	1.13995i	0.821661	−	0.569976i	$-0.193048\pi$
$198$		−	18.0000i		−	1.27920i
$199$			3.00000i			0.212664i	0.994331	+	0.106332i	$0.0339107\pi$
$199$			3.00000i			0.212664i	−0.994331	+	0.106332i	$0.966089\pi$
$200$	−5.00000			−0.353553
$201$			48.0000i			3.38566i
$202$	10.0000			0.703598
$203$		0			0
$204$	12.0000	+	3.00000i	0.840168	+	0.210042i
$205$		0			0
$206$	−8.00000			−0.557386
$207$			24.0000i			1.66812i
$208$	−5.00000			−0.346688
$209$			12.0000i			0.830057i
$210$		0			0
$211$			24.0000i			1.65223i	0.563503	+	0.826114i	$0.309453\pi$
$211$			24.0000i			1.65223i	−0.563503	+	0.826114i	$0.690547\pi$
$212$	−1.00000			−0.0686803
$213$	−3.00000			−0.205557
$214$		−	3.00000i		−	0.205076i
$215$		0			0
$216$		−	9.00000i		−	0.612372i
$217$		0			0
$218$		−	20.0000i		−	1.35457i
$219$	24.0000			1.62177
$220$		0			0
$221$	5.00000	−	20.0000i	0.336336	−	1.34535i
$222$	−24.0000			−1.61077
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$	−30.0000			−2.00000
$226$			12.0000i			0.798228i
$227$		−	21.0000i		−	1.39382i	−0.717159	−	0.696909i	$-0.754558\pi$
$227$		−	21.0000i		−	1.39382i	0.717159	−	0.696909i	$-0.245442\pi$
$228$			12.0000i			0.794719i
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$		0			0
$232$		−	4.00000i		−	0.262613i
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	−30.0000			−1.96116
$235$		0			0
$236$	−12.0000			−0.781133
$237$	−21.0000			−1.36410
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		−	12.0000i		−	0.772988i	−0.922292	−	0.386494i	$-0.873686\pi$
$241$		−	12.0000i		−	0.772988i	0.922292	−	0.386494i	$-0.126314\pi$
$242$	−2.00000			−0.128565
$243$		0			0
$244$		0			0
$245$		0			0
$246$	−12.0000			−0.765092
$247$	20.0000			1.27257
$248$			4.00000i			0.254000i
$249$		−	48.0000i		−	3.04188i
$250$		0			0
$251$	−16.0000			−1.00991			−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000			−1.00991			−0.504956	+	0.863145i	$0.668491\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$	7.00000			0.436648			0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000			0.436648			0.218324	+	0.975876i	$0.429941\pi$
$258$			24.0000i			1.49417i
$259$		0			0
$260$		0			0
$261$		−	24.0000i		−	1.48556i
$262$			8.00000i			0.494242i
$263$	4.00000			0.246651			0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000			0.246651			0.123325	+	0.992366i	$0.460644\pi$
$264$	9.00000			0.553912
$265$		0			0
$266$		0			0
$267$			27.0000i			1.65237i
$268$	−16.0000			−0.977356
$269$		−	4.00000i		−	0.243884i	−0.992537	−	0.121942i	$-0.961088\pi$
$269$		−	4.00000i		−	0.243884i	0.992537	−	0.121942i	$-0.0389122\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$	−1.00000	+	4.00000i	−0.0606339	+	0.242536i
$273$		0			0
$274$	−9.00000			−0.543710
$275$		−	15.0000i		−	0.904534i
$276$	−12.0000			−0.722315
$277$		−	4.00000i		−	0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$		−	4.00000i		−	0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$		−	11.0000i		−	0.659736i
$279$			24.0000i			1.43684i
$280$		0			0
$281$	7.00000			0.417585			0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000			0.417585			0.208792	+	0.977960i	$0.433047\pi$
$282$		−	12.0000i		−	0.714590i
$283$		−	11.0000i		−	0.653882i	−0.945045	−	0.326941i	$-0.893982\pi$
$283$		−	11.0000i		−	0.653882i	0.945045	−	0.326941i	$-0.106018\pi$
$284$		−	1.00000i		−	0.0593391i
$285$		0			0
$286$		−	15.0000i		−	0.886969i
$287$		0			0
$288$	6.00000			0.353553
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$	24.0000			1.40690
$292$			8.00000i			0.468165i
$293$	−23.0000			−1.34367			−0.671837	−	0.740699i	$-0.734495\pi$
$293$	−23.0000			−1.34367			−0.671837	+	0.740699i	$0.734495\pi$
$294$		0			0
$295$		0			0
$296$		−	8.00000i		−	0.464991i
$297$	27.0000			1.56670
$298$	−11.0000			−0.637213
$299$			20.0000i			1.15663i
$300$		−	15.0000i		−	0.866025i
$301$		0			0
$302$	12.0000			0.690522
$303$			30.0000i			1.72345i
$304$	−4.00000			−0.229416
$305$		0			0
$306$	−6.00000	+	24.0000i	−0.342997	+	1.37199i
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		−	24.0000i		−	1.36531i
$310$		0			0
$311$			21.0000i			1.19080i	0.803429	+	0.595400i	$0.203007\pi$
$311$			21.0000i			1.19080i	−0.803429	+	0.595400i	$0.796993\pi$
$312$		−	15.0000i		−	0.849208i
$313$		−	20.0000i		−	1.13047i	−0.824931	−	0.565233i	$-0.808786\pi$
$313$		−	20.0000i		−	1.13047i	0.824931	−	0.565233i	$-0.191214\pi$
$314$	3.00000			0.169300
$315$		0			0
$316$		−	7.00000i		−	0.393781i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		−	3.00000i		−	0.168232i
$319$	12.0000			0.671871
$320$		0			0
$321$	9.00000			0.502331
$322$		0			0
$323$	4.00000	−	16.0000i	0.222566	−	0.890264i
$324$	9.00000			0.500000
$325$	−25.0000			−1.38675
$326$		−	16.0000i		−	0.886158i
$327$	60.0000			3.31801
$328$		−	4.00000i		−	0.220863i
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$	16.0000			0.878114
$333$		−	48.0000i		−	2.63038i
$334$			25.0000i			1.36794i
$335$		0			0
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	−12.0000			−0.652714
$339$	−36.0000			−1.95525
$340$		0			0
$341$	−12.0000			−0.649836
$342$	−24.0000			−1.29777
$343$		0			0
$344$	−8.00000			−0.431331
$345$		0			0
$346$			4.00000i			0.215041i
$347$			32.0000i			1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$			32.0000i			1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	12.0000			0.643268
$349$	−30.0000			−1.60586			−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000			−1.60586			−0.802932	+	0.596071i	$0.796728\pi$
$350$		0			0
$351$		−	45.0000i		−	2.40192i
$352$			3.00000i			0.159901i
$353$	21.0000			1.11772			0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000			1.11772			0.558859	+	0.829263i	$0.311239\pi$
$354$		−	36.0000i		−	1.91338i
$355$		0			0
$356$	−9.00000			−0.476999
$357$		0			0
$358$	16.0000			0.845626
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		−	4.00000i		−	0.210235i
$363$		−	6.00000i		−	0.314918i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			7.00000i			0.365397i	0.983169	+	0.182699i	$0.0584832\pi$
$367$			7.00000i			0.365397i	−0.983169	+	0.182699i	$0.941517\pi$
$368$		−	4.00000i		−	0.208514i
$369$		−	24.0000i		−	1.24939i
$370$		0			0
$371$		0			0
$372$	−12.0000			−0.622171
$373$	−19.0000			−0.983783			−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000			−0.983783			−0.491891	+	0.870657i	$0.663694\pi$
$374$	−12.0000	−	3.00000i	−0.620505	−	0.155126i
$375$		0			0
$376$	4.00000			0.206284
$377$		−	20.0000i		−	1.03005i
$378$		0			0
$379$		−	1.00000i		−	0.0513665i	−0.999670	−	0.0256833i	$-0.991824\pi$
$379$		−	1.00000i		−	0.0513665i	0.999670	−	0.0256833i	$-0.00817614\pi$
$380$		0			0
$381$		−	24.0000i		−	1.22956i
$382$	12.0000			0.613973
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$			3.00000i			0.153093i
$385$		0			0
$386$		−	4.00000i		−	0.203595i
$387$	−48.0000			−2.43998
$388$			8.00000i			0.406138i
$389$	−15.0000			−0.760530			−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000			−0.760530			−0.380265	+	0.924878i	$0.624167\pi$
$390$		0			0
$391$	16.0000	+	4.00000i	0.809155	+	0.202289i
$392$		0			0
$393$	−24.0000			−1.21064
$394$			16.0000i			0.806068i
$395$		0			0
$396$			18.0000i			0.904534i
$397$			32.0000i			1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$			32.0000i			1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$		−	3.00000i		−	0.150376i
$399$		0			0
$400$	5.00000			0.250000
$401$		−	20.0000i		−	0.998752i	−0.866385	−	0.499376i	$-0.833563\pi$
$401$		−	20.0000i		−	0.998752i	0.866385	−	0.499376i	$-0.166437\pi$
$402$		−	48.0000i		−	2.39402i
$403$			20.0000i			0.996271i
$404$	−10.0000			−0.497519
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$	−12.0000	−	3.00000i	−0.594089	−	0.148522i
$409$	−19.0000			−0.939490			−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000			−0.939490			−0.469745	+	0.882802i	$0.655654\pi$
$410$		0			0
$411$		−	27.0000i		−	1.33181i
$412$	8.00000			0.394132
$413$		0			0
$414$		−	24.0000i		−	1.17954i
$415$		0			0
$416$	5.00000			0.245145
$417$	33.0000			1.61602
$418$		−	12.0000i		−	0.586939i
$419$		−	5.00000i		−	0.244266i	−0.992514	−	0.122133i	$-0.961027\pi$
$419$		−	5.00000i		−	0.244266i	0.992514	−	0.122133i	$-0.0389734\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		−	24.0000i		−	1.16830i
$423$	24.0000			1.16692
$424$	1.00000			0.0485643
$425$	−5.00000	+	20.0000i	−0.242536	+	0.970143i
$426$	3.00000			0.145350
$427$		0			0
$428$			3.00000i			0.145010i
$429$	45.0000			2.17262
$430$		0			0
$431$		−	21.0000i		−	1.01153i	−0.862670	−	0.505767i	$-0.831209\pi$
$431$		−	21.0000i		−	1.01153i	0.862670	−	0.505767i	$-0.168791\pi$
$432$			9.00000i			0.433013i
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$			20.0000i			0.957826i
$437$			16.0000i			0.765384i
$438$	−24.0000			−1.14676
$439$			1.00000i			0.0477274i	0.999715	+	0.0238637i	$0.00759677\pi$
$439$			1.00000i			0.0477274i	−0.999715	+	0.0238637i	$0.992403\pi$
$440$		0			0
$441$		0			0
$442$	−5.00000	+	20.0000i	−0.237826	+	0.951303i
$443$	−16.0000			−0.760183			−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000			−0.760183			−0.380091	+	0.924949i	$0.624107\pi$
$444$	24.0000			1.13899
$445$		0			0
$446$	12.0000			0.568216
$447$		−	33.0000i		−	1.56085i
$448$		0			0
$449$		−	4.00000i		−	0.188772i	−0.995536	−	0.0943858i	$-0.969911\pi$
$449$		−	4.00000i		−	0.188772i	0.995536	−	0.0943858i	$-0.0300887\pi$
$450$	30.0000			1.41421
$451$	12.0000			0.565058
$452$		−	12.0000i		−	0.564433i
$453$			36.0000i			1.69143i
$454$			21.0000i			0.985579i
$455$		0			0
$456$		−	12.0000i		−	0.561951i
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$	26.0000			1.21490
$459$	−36.0000	−	9.00000i	−1.68034	−	0.420084i
$460$		0			0
$461$	1.00000			0.0465746			0.0232873	−	0.999729i	$-0.492587\pi$
$461$	1.00000			0.0465746			0.0232873	+	0.999729i	$0.492587\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$			4.00000i			0.185695i
$465$		0			0
$466$			16.0000i			0.741186i
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$	30.0000			1.38675
$469$		0			0
$470$		0			0
$471$			9.00000i			0.414698i
$472$	12.0000			0.552345
$473$		−	24.0000i		−	1.10352i
$474$	21.0000			0.964562
$475$	−20.0000			−0.917663
$476$		0			0
$477$	6.00000			0.274721
$478$	16.0000			0.731823
$479$			36.0000i			1.64488i	0.568850	+	0.822441i	$0.307388\pi$
$479$			36.0000i			1.64488i	−0.568850	+	0.822441i	$0.692612\pi$
$480$		0			0
$481$		−	40.0000i		−	1.82384i
$482$			12.0000i			0.546585i
$483$		0			0
$484$	2.00000			0.0909091
$485$		0			0
$486$		0			0
$487$		−	19.0000i		−	0.860972i	−0.902597	−	0.430486i	$-0.858342\pi$
$487$		−	19.0000i		−	0.860972i	0.902597	−	0.430486i	$-0.141658\pi$
$488$		0			0
$489$	48.0000			2.17064
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$	12.0000			0.541002
$493$	−16.0000	−	4.00000i	−0.720604	−	0.180151i
$494$	−20.0000			−0.899843
$495$		0			0
$496$		−	4.00000i		−	0.179605i
$497$		0			0
$498$			48.0000i			2.15093i
$499$		−	25.0000i		−	1.11915i	−0.828778	−	0.559577i	$-0.810964\pi$
$499$		−	25.0000i		−	1.11915i	0.828778	−	0.559577i	$-0.189036\pi$
$500$		0			0
$501$	−75.0000			−3.35075
$502$	16.0000			0.714115
$503$			1.00000i			0.0445878i	0.999751	+	0.0222939i	$0.00709696\pi$
$503$			1.00000i			0.0445878i	−0.999751	+	0.0222939i	$0.992903\pi$
$504$		0			0
$505$		0			0
$506$	12.0000			0.533465
$507$		−	36.0000i		−	1.59882i
$508$	8.00000			0.354943
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		−	36.0000i		−	1.58944i
$514$	−7.00000			−0.308757
$515$		0			0
$516$		−	24.0000i		−	1.05654i
$517$			12.0000i			0.527759i
$518$		0			0
$519$	−12.0000			−0.526742
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$			24.0000i			1.05045i
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		−	8.00000i		−	0.349482i
$525$		0			0
$526$	−4.00000			−0.174408
$527$	16.0000	+	4.00000i	0.696971	+	0.174243i
$528$	−9.00000			−0.391675
$529$	7.00000			0.304348
$530$		0			0
$531$	72.0000			3.12453
$532$		0			0
$533$		−	20.0000i		−	0.866296i
$534$		−	27.0000i		−	1.16840i
$535$		0			0
$536$	16.0000			0.691095
$537$			48.0000i			2.07135i
$538$			4.00000i			0.172452i
$539$		0			0
$540$		0			0
$541$		−	32.0000i		−	1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$		−	32.0000i		−	1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$	12.0000			0.515444
$543$	12.0000			0.514969
$544$	1.00000	−	4.00000i	0.0428746	−	0.171499i
$545$		0			0
$546$		0			0
$547$			9.00000i			0.384812i	0.981315	+	0.192406i	$0.0616291\pi$
$547$			9.00000i			0.384812i	−0.981315	+	0.192406i	$0.938371\pi$
$548$	9.00000			0.384461
$549$		0			0
$550$			15.0000i			0.639602i
$551$		−	16.0000i		−	0.681623i
$552$	12.0000			0.510754
$553$		0			0
$554$			4.00000i			0.169944i
$555$		0			0
$556$			11.0000i			0.466504i
$557$	−39.0000			−1.65248			−0.826242	−	0.563316i	$-0.809525\pi$
$557$	−39.0000			−1.65248			−0.826242	+	0.563316i	$0.809525\pi$
$558$		−	24.0000i		−	1.01600i
$559$	−40.0000			−1.69182
$560$		0			0
$561$	9.00000	−	36.0000i	0.379980	−	1.51992i
$562$	−7.00000			−0.295277
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$			12.0000i			0.505291i
$565$		0			0
$566$			11.0000i			0.462364i
$567$		0			0
$568$			1.00000i			0.0419591i
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$		−	16.0000i		−	0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$			15.0000i			0.627182i
$573$			36.0000i			1.50392i
$574$		0			0
$575$		−	20.0000i		−	0.834058i
$576$	−6.00000			−0.250000
$577$	−17.0000			−0.707719			−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000			−0.707719			−0.353860	+	0.935299i	$0.615131\pi$
$578$	15.0000	+	8.00000i	0.623918	+	0.332756i
$579$	12.0000			0.498703
$580$		0			0
$581$		0			0
$582$	−24.0000			−0.994832
$583$			3.00000i			0.124247i
$584$		−	8.00000i		−	0.331042i
$585$		0			0
$586$	23.0000			0.950121
$587$	−8.00000			−0.330195			−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000			−0.330195			−0.165098	+	0.986277i	$0.552794\pi$
$588$		0			0
$589$			16.0000i			0.659269i
$590$		0			0
$591$	−48.0000			−1.97446
$592$			8.00000i			0.328798i
$593$	9.00000			0.369586			0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000			0.369586			0.184793	+	0.982777i	$0.440839\pi$
$594$	−27.0000			−1.10782
$595$		0			0
$596$	11.0000			0.450578
$597$	9.00000			0.368345
$598$		−	20.0000i		−	0.817861i
$599$	−40.0000			−1.63436			−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000			−1.63436			−0.817178	+	0.576386i	$0.804463\pi$
$600$			15.0000i			0.612372i
$601$		−	8.00000i		−	0.326327i	−0.986599	−	0.163163i	$-0.947830\pi$
$601$		−	8.00000i		−	0.326327i	0.986599	−	0.163163i	$-0.0521698\pi$
$602$		0			0
$603$	96.0000			3.90942
$604$	−12.0000			−0.488273
$605$		0			0
$606$		−	30.0000i		−	1.21867i
$607$		−	37.0000i		−	1.50178i	−0.660425	−	0.750892i	$-0.729624\pi$
$607$		−	37.0000i		−	1.50178i	0.660425	−	0.750892i	$-0.270376\pi$
$608$	4.00000			0.162221
$609$		0			0
$610$		0			0
$611$	20.0000			0.809113
$612$	6.00000	−	24.0000i	0.242536	−	0.970143i
$613$	27.0000			1.09052			0.545260	−	0.838267i	$-0.316431\pi$
$613$	27.0000			1.09052			0.545260	+	0.838267i	$0.316431\pi$
$614$	−16.0000			−0.645707
$615$		0			0
$616$		0			0
$617$		−	48.0000i		−	1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$617$		−	48.0000i		−	1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$618$			24.0000i			0.965422i
$619$		−	7.00000i		−	0.281354i	−0.990056	−	0.140677i	$-0.955072\pi$
$619$		−	7.00000i		−	0.281354i	0.990056	−	0.140677i	$-0.0449279\pi$
$620$		0			0
$621$	36.0000			1.44463
$622$		−	21.0000i		−	0.842023i
$623$		0			0
$624$			15.0000i			0.600481i
$625$	25.0000			1.00000
$626$			20.0000i			0.799361i
$627$	36.0000			1.43770
$628$	−3.00000			−0.119713
$629$	−32.0000	−	8.00000i	−1.27592	−	0.318981i
$630$		0			0
$631$	−36.0000			−1.43314			−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000			−1.43314			−0.716569	+	0.697517i	$0.754288\pi$
$632$			7.00000i			0.278445i
$633$	72.0000			2.86174
$634$		0			0
$635$		0			0
$636$			3.00000i			0.118958i
$637$		0			0
$638$	−12.0000			−0.475085
$639$			6.00000i			0.237356i
$640$		0			0
$641$			12.0000i			0.473972i	0.971513	+	0.236986i	$0.0761595\pi$
$641$			12.0000i			0.473972i	−0.971513	+	0.236986i	$0.923841\pi$
$642$	−9.00000			−0.355202
$643$			43.0000i			1.69575i	0.530193	+	0.847877i	$0.322120\pi$
$643$			43.0000i			1.69575i	−0.530193	+	0.847877i	$0.677880\pi$
$644$		0			0
$645$		0			0
$646$	−4.00000	+	16.0000i	−0.157378	+	0.629512i
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$	−9.00000			−0.353553
$649$			36.0000i			1.41312i
$650$	25.0000			0.980581
$651$		0			0
$652$			16.0000i			0.626608i
$653$			36.0000i			1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$			36.0000i			1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	−60.0000			−2.34619
$655$		0			0
$656$			4.00000i			0.156174i
$657$		−	48.0000i		−	1.87266i
$658$		0			0
$659$	−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
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$	4.00000			0.155464
$663$	−60.0000	−	15.0000i	−2.33021	−	0.582552i
$664$	−16.0000			−0.620920
$665$		0			0
$666$			48.0000i			1.85996i
$667$	16.0000			0.619522
$668$		−	25.0000i		−	0.967279i
$669$			36.0000i			1.39184i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			4.00000i			0.154189i	0.997024	+	0.0770943i	$0.0245643\pi$
$673$			4.00000i			0.154189i	−0.997024	+	0.0770943i	$0.975436\pi$
$674$		−	8.00000i		−	0.308148i
$675$			45.0000i			1.73205i
$676$	12.0000			0.461538
$677$		−	20.0000i		−	0.768662i	−0.923195	−	0.384331i	$-0.874432\pi$
$677$		−	20.0000i		−	0.768662i	0.923195	−	0.384331i	$-0.125568\pi$
$678$	36.0000			1.38257
$679$		0			0
$680$		0			0
$681$	−63.0000			−2.41417
$682$	12.0000			0.459504
$683$			9.00000i			0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$			9.00000i			0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	24.0000			0.917663
$685$		0			0
$686$		0			0
$687$			78.0000i			2.97589i
$688$	8.00000			0.304997
$689$	5.00000			0.190485
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		−	4.00000i		−	0.152057i
$693$		0			0
$694$		−	32.0000i		−	1.21470i
$695$		0			0
$696$	−12.0000			−0.454859
$697$	−16.0000	−	4.00000i	−0.606043	−	0.151511i
$698$	30.0000			1.13552
$699$	−48.0000			−1.81553
$700$		0			0
$701$	1.00000			0.0377695			0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000			0.0377695			0.0188847	+	0.999822i	$0.493988\pi$
$702$			45.0000i			1.69842i
$703$		−	32.0000i		−	1.20690i
$704$		−	3.00000i		−	0.113067i
$705$		0			0
$706$	−21.0000			−0.790345
$707$		0			0
$708$			36.0000i			1.35296i
$709$			48.0000i			1.80268i	0.433114	+	0.901339i	$0.357415\pi$
$709$			48.0000i			1.80268i	−0.433114	+	0.901339i	$0.642585\pi$
$710$		0			0
$711$			42.0000i			1.57512i
$712$	9.00000			0.337289
$713$	−16.0000			−0.599205
$714$		0			0
$715$		0			0
$716$	−16.0000			−0.597948
$717$			48.0000i			1.79259i
$718$		0			0
$719$		−	9.00000i		−	0.335643i	−0.985817	−	0.167822i	$-0.946327\pi$
$719$		−	9.00000i		−	0.335643i	0.985817	−	0.167822i	$-0.0536733\pi$
$720$		0			0
$721$		0			0
$722$	3.00000			0.111648
$723$	−36.0000			−1.33885
$724$			4.00000i			0.148659i
$725$			20.0000i			0.742781i
$726$			6.00000i			0.222681i
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−8.00000	+	32.0000i	−0.295891	+	1.18356i
$732$		0			0
$733$	29.0000			1.07114			0.535570	−	0.844491i	$-0.320097\pi$
$733$	29.0000			1.07114			0.535570	+	0.844491i	$0.320097\pi$
$734$		−	7.00000i		−	0.258375i
$735$		0			0
$736$			4.00000i			0.147442i
$737$			48.0000i			1.76810i
$738$			24.0000i			0.883452i
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$		−	60.0000i		−	2.20416i
$742$		0			0
$743$			33.0000i			1.21065i	0.795977	+	0.605326i	$0.206957\pi$
$743$			33.0000i			1.21065i	−0.795977	+	0.605326i	$0.793043\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$	19.0000			0.695639
$747$	−96.0000			−3.51246
$748$	12.0000	+	3.00000i	0.438763	+	0.109691i
$749$		0			0
$750$		0			0
$751$		−	13.0000i		−	0.474377i	−0.971464	−	0.237188i	$-0.923774\pi$
$751$		−	13.0000i		−	0.474377i	0.971464	−	0.237188i	$-0.0762259\pi$
$752$	−4.00000			−0.145865
$753$			48.0000i			1.74922i
$754$			20.0000i			0.728357i
$755$		0			0
$756$		0			0
$757$	−39.0000			−1.41748			−0.708740	−	0.705470i	$-0.750736\pi$
$757$	−39.0000			−1.41748			−0.708740	+	0.705470i	$0.750736\pi$
$758$			1.00000i			0.0363216i
$759$			36.0000i			1.30672i
$760$		0			0
$761$	15.0000			0.543750			0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000			0.543750			0.271875	+	0.962333i	$0.412356\pi$
$762$			24.0000i			0.869428i
$763$		0			0
$764$	−12.0000			−0.434145
$765$		0			0
$766$	24.0000			0.867155
$767$	60.0000			2.16647
$768$		−	3.00000i		−	0.108253i
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$		0			0
$771$		−	21.0000i		−	0.756297i
$772$			4.00000i			0.143963i
$773$	−1.00000			−0.0359675			−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000			−0.0359675			−0.0179838	+	0.999838i	$0.505725\pi$
$774$	48.0000			1.72532
$775$		−	20.0000i		−	0.718421i
$776$		−	8.00000i		−	0.287183i
$777$		0			0
$778$	15.0000			0.537776
$779$		−	16.0000i		−	0.573259i
$780$		0			0
$781$	−3.00000			−0.107348
$782$	−16.0000	−	4.00000i	−0.572159	−	0.143040i
$783$	−36.0000			−1.28654
$784$		0			0
$785$		0			0
$786$	24.0000			0.856052
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$		−	16.0000i		−	0.569976i
$789$		−	12.0000i		−	0.427211i
$790$		0			0
$791$		0			0
$792$		−	18.0000i		−	0.639602i
$793$		0			0
$794$		−	32.0000i		−	1.13564i
$795$		0			0
$796$			3.00000i			0.106332i
$797$	9.00000			0.318796			0.159398	−	0.987214i	$-0.449045\pi$
$797$	9.00000			0.318796			0.159398	+	0.987214i	$0.449045\pi$
$798$		0			0
$799$	4.00000	−	16.0000i	0.141510	−	0.566039i
$800$	−5.00000			−0.176777
$801$	54.0000			1.90800
$802$			20.0000i			0.706225i
$803$	24.0000			0.846942
$804$			48.0000i			1.69283i
$805$		0			0
$806$		−	20.0000i		−	0.704470i
$807$	−12.0000			−0.422420
$808$	10.0000			0.351799
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			33.0000i			1.15879i	0.815048	+	0.579393i	$0.196710\pi$
$811$			33.0000i			1.15879i	−0.815048	+	0.579393i	$0.803290\pi$
$812$		0			0
$813$			36.0000i			1.26258i
$814$	−24.0000			−0.841200
$815$		0			0
$816$	12.0000	+	3.00000i	0.420084	+	0.105021i
$817$	−32.0000			−1.11954
$818$	19.0000			0.664319
$819$		0			0
$820$		0			0
$821$		−	48.0000i		−	1.67521i	−0.546275	−	0.837606i	$-0.683955\pi$
$821$		−	48.0000i		−	1.67521i	0.546275	−	0.837606i	$-0.316045\pi$
$822$			27.0000i			0.941733i
$823$			15.0000i			0.522867i	0.965221	+	0.261434i	$0.0841952\pi$
$823$			15.0000i			0.522867i	−0.965221	+	0.261434i	$0.915805\pi$
$824$	−8.00000			−0.278693
$825$	−45.0000			−1.56670
$826$		0			0
$827$		−	43.0000i		−	1.49526i	−0.664117	−	0.747628i	$-0.731193\pi$
$827$		−	43.0000i		−	1.49526i	0.664117	−	0.747628i	$-0.268807\pi$
$828$			24.0000i			0.834058i
$829$	−7.00000			−0.243120			−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000			−0.243120			−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$	−12.0000			−0.416275
$832$	−5.00000			−0.173344
$833$		0			0
$834$	−33.0000			−1.14270
$835$		0			0
$836$			12.0000i			0.415029i
$837$	36.0000			1.24434
$838$			5.00000i			0.172722i
$839$			12.0000i			0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$			12.0000i			0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$	−10.0000			−0.344623
$843$		−	21.0000i		−	0.723278i
$844$			24.0000i			0.826114i
$845$		0			0
$846$	−24.0000			−0.825137
$847$		0			0
$848$	−1.00000			−0.0343401
$849$	−33.0000			−1.13256
$850$	5.00000	−	20.0000i	0.171499	−	0.685994i
$851$	32.0000			1.09695
$852$	−3.00000			−0.102778
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$		0			0
$855$		0			0
$856$		−	3.00000i		−	0.102538i
$857$			24.0000i			0.819824i	0.912125	+	0.409912i	$0.134441\pi$
$857$			24.0000i			0.819824i	−0.912125	+	0.409912i	$0.865559\pi$
$858$	−45.0000			−1.53627
$859$	12.0000			0.409435			0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000			0.409435			0.204717	+	0.978821i	$0.434372\pi$
$860$		0			0
$861$		0			0
$862$			21.0000i			0.715263i
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		−	9.00000i		−	0.306186i
$865$		0			0
$866$	−34.0000			−1.15537
$867$	−24.0000	+	45.0000i	−0.815083	+	1.52828i
$868$		0			0
$869$	−21.0000			−0.712376
$870$		0			0
$871$	80.0000			2.71070
$872$		−	20.0000i		−	0.677285i
$873$		−	48.0000i		−	1.62455i
$874$		−	16.0000i		−	0.541208i
$875$		0			0
$876$	24.0000			0.810885
$877$			12.0000i			0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$			12.0000i			0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		−	1.00000i		−	0.0337484i
$879$			69.0000i			2.32731i
$880$		0			0
$881$		−	12.0000i		−	0.404290i	−0.979356	−	0.202145i	$-0.935209\pi$
$881$		−	12.0000i		−	0.404290i	0.979356	−	0.202145i	$-0.0647913\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$	5.00000	−	20.0000i	0.168168	−	0.672673i
$885$		0			0
$886$	16.0000			0.537531
$887$		−	13.0000i		−	0.436497i	−0.975893	−	0.218249i	$-0.929966\pi$
$887$		−	13.0000i		−	0.436497i	0.975893	−	0.218249i	$-0.0700344\pi$
$888$	−24.0000			−0.805387
$889$		0			0
$890$		0			0
$891$		−	27.0000i		−	0.904534i
$892$	−12.0000			−0.401790
$893$	16.0000			0.535420
$894$			33.0000i			1.10369i
$895$		0			0
$896$		0			0
$897$	60.0000			2.00334
$898$			4.00000i			0.133482i
$899$	16.0000			0.533630
$900$	−30.0000			−1.00000
$901$	1.00000	−	4.00000i	0.0333148	−	0.133259i
$902$	−12.0000			−0.399556
$903$		0			0
$904$			12.0000i			0.399114i
$905$		0			0
$906$		−	36.0000i		−	1.19602i
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		−	21.0000i		−	0.696909i
$909$	60.0000			1.99007
$910$		0			0
$911$			36.0000i			1.19273i	0.802712	+	0.596367i	$0.203390\pi$
$911$			36.0000i			1.19273i	−0.802712	+	0.596367i	$0.796610\pi$
$912$			12.0000i			0.397360i
$913$		−	48.0000i		−	1.58857i
$914$	10.0000			0.330771
$915$		0			0
$916$	−26.0000			−0.859064
$917$		0			0
$918$	36.0000	+	9.00000i	1.18818	+	0.297044i
$919$	44.0000			1.45143			0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000			1.45143			0.725713	+	0.687998i	$0.241510\pi$
$920$		0			0
$921$		−	48.0000i		−	1.58165i
$922$	−1.00000			−0.0329332
$923$			5.00000i			0.164577i
$924$		0			0
$925$			40.0000i			1.31519i
$926$	−20.0000			−0.657241
$927$	−48.0000			−1.57653
$928$		−	4.00000i		−	0.131306i
$929$		−	40.0000i		−	1.31236i	−0.754606	−	0.656179i	$-0.772172\pi$
$929$		−	40.0000i		−	1.31236i	0.754606	−	0.656179i	$-0.227828\pi$
$930$		0			0
$931$		0			0
$932$		−	16.0000i		−	0.524097i
$933$	63.0000			2.06253
$934$	−24.0000			−0.785304
$935$		0			0
$936$	−30.0000			−0.980581
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$	−60.0000			−1.95803
$940$		0			0
$941$			60.0000i			1.95594i	0.208736	+	0.977972i	$0.433065\pi$
$941$			60.0000i			1.95594i	−0.208736	+	0.977972i	$0.566935\pi$
$942$		−	9.00000i		−	0.293236i
$943$	16.0000			0.521032
$944$	−12.0000			−0.390567
$945$		0			0
$946$			24.0000i			0.780307i
$947$		−	1.00000i		−	0.0324956i	−0.999868	−	0.0162478i	$-0.994828\pi$
$947$		−	1.00000i		−	0.0324956i	0.999868	−	0.0162478i	$-0.00517207\pi$
$948$	−21.0000			−0.682048
$949$		−	40.0000i		−	1.29845i
$950$	20.0000			0.648886
$951$		0			0
$952$		0			0
$953$	−27.0000			−0.874616			−0.437308	−	0.899312i	$-0.644068\pi$
$953$	−27.0000			−0.874616			−0.437308	+	0.899312i	$0.644068\pi$
$954$	−6.00000			−0.194257
$955$		0			0
$956$	−16.0000			−0.517477
$957$		−	36.0000i		−	1.16371i
$958$		−	36.0000i		−	1.16311i
$959$		0			0
$960$		0			0
$961$	15.0000			0.483871
$962$			40.0000i			1.28965i
$963$		−	18.0000i		−	0.580042i
$964$		−	12.0000i		−	0.386494i
$965$		0			0
$966$		0			0
$967$	−4.00000			−0.128631			−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000			−0.128631			−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−2.00000			−0.0642824
$969$	−48.0000	−	12.0000i	−1.54198	−	0.385496i
$970$		0			0
$971$	4.00000			0.128366			0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000			0.128366			0.0641831	+	0.997938i	$0.479556\pi$
$972$		0			0
$973$		0			0
$974$			19.0000i			0.608799i
$975$			75.0000i			2.40192i
$976$		0			0
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$	−48.0000			−1.53487
$979$			27.0000i			0.862924i
$980$		0			0
$981$		−	120.000i		−	3.83131i
$982$	36.0000			1.14881
$983$		−	23.0000i		−	0.733586i	−0.930303	−	0.366793i	$-0.880456\pi$
$983$		−	23.0000i		−	0.733586i	0.930303	−	0.366793i	$-0.119544\pi$
$984$	−12.0000			−0.382546
$985$		0			0
$986$	16.0000	+	4.00000i	0.509544	+	0.127386i
$987$		0			0
$988$	20.0000			0.636285
$989$		−	32.0000i		−	1.01754i
$990$		0			0
$991$		−	47.0000i		−	1.49300i	−0.665383	−	0.746502i	$-0.731732\pi$
$991$		−	47.0000i		−	1.49300i	0.665383	−	0.746502i	$-0.268268\pi$
$992$			4.00000i			0.127000i
$993$			12.0000i			0.380808i
$994$		0			0
$995$		0			0
$996$		−	48.0000i		−	1.52094i
$997$		−	16.0000i		−	0.506725i	−0.967371	−	0.253363i	$-0.918463\pi$
$997$		−	16.0000i		−	0.506725i	0.967371	−	0.253363i	$-0.0815366\pi$
$998$			25.0000i			0.791361i
$999$	−72.0000			−2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1666.2.b.a.883.1		2
7.3	odd	6		238.2.j.a.135.2	yes	4
7.5	odd	6		238.2.j.a.67.1	✓	4
7.6	odd	2		1666.2.b.b.883.2		2
17.16	even	2	inner	1666.2.b.a.883.2		2
119.33	odd	6		238.2.j.a.67.2	yes	4
119.101	odd	6		238.2.j.a.135.1	yes	4
119.118	odd	2		1666.2.b.b.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.j.a.67.1	✓	4	7.5	odd	6
238.2.j.a.67.2	yes	4	119.33	odd	6
238.2.j.a.135.1	yes	4	119.101	odd	6
238.2.j.a.135.2	yes	4	7.3	odd	6
1666.2.b.a.883.1		2	1.1	even	1	trivial
1666.2.b.a.883.2		2	17.16	even	2	inner
1666.2.b.b.883.1		2	119.118	odd	2
1666.2.b.b.883.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1666.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.00000i q^{3} +1.00000 q^{4} +3.00000i q^{6} -1.00000 q^{8} -6.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.00000i q^{3} +1.00000 q^{4} +3.00000i q^{6} -1.00000 q^{8} -6.00000 q^{9} -3.00000i q^{11} -3.00000i q^{12} -5.00000 q^{13} +1.00000 q^{16} +(-1.00000 + 4.00000i) q^{17} +6.00000 q^{18} -4.00000 q^{19} +3.00000i q^{22} -4.00000i q^{23} +3.00000i q^{24} +5.00000 q^{25} +5.00000 q^{26} +9.00000i q^{27} +4.00000i q^{29} -4.00000i q^{31} -1.00000 q^{32} -9.00000 q^{33} +(1.00000 - 4.00000i) q^{34} -6.00000 q^{36} +8.00000i q^{37} +4.00000 q^{38} +15.0000i q^{39} +4.00000i q^{41} +8.00000 q^{43} -3.00000i q^{44} +4.00000i q^{46} -4.00000 q^{47} -3.00000i q^{48} -5.00000 q^{50} +(12.0000 + 3.00000i) q^{51} -5.00000 q^{52} -1.00000 q^{53} -9.00000i q^{54} +12.0000i q^{57} -4.00000i q^{58} -12.0000 q^{59} +4.00000i q^{62} +1.00000 q^{64} +9.00000 q^{66} -16.0000 q^{67} +(-1.00000 + 4.00000i) q^{68} -12.0000 q^{69} -1.00000i q^{71} +6.00000 q^{72} +8.00000i q^{73} -8.00000i q^{74} -15.0000i q^{75} -4.00000 q^{76} -15.0000i q^{78} -7.00000i q^{79} +9.00000 q^{81} -4.00000i q^{82} +16.0000 q^{83} -8.00000 q^{86} +12.0000 q^{87} +3.00000i q^{88} -9.00000 q^{89} -4.00000i q^{92} -12.0000 q^{93} +4.00000 q^{94} +3.00000i q^{96} +8.00000i q^{97} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{9} - 10 q^{13} + 2 q^{16} - 2 q^{17} + 12 q^{18} - 8 q^{19} + 10 q^{25} + 10 q^{26} - 2 q^{32} - 18 q^{33} + 2 q^{34} - 12 q^{36} + 8 q^{38} + 16 q^{43} - 8 q^{47} - 10 q^{50} + 24 q^{51} - 10 q^{52} - 2 q^{53} - 24 q^{59} + 2 q^{64} + 18 q^{66} - 32 q^{67} - 2 q^{68} - 24 q^{69} + 12 q^{72} - 8 q^{76} + 18 q^{81} + 32 q^{83} - 16 q^{86} + 24 q^{87} - 18 q^{89} - 24 q^{93} + 8 q^{94}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^9 - 10 * q^13 + 2 * q^16 - 2 * q^17 + 12 * q^18 - 8 * q^19 + 10 * q^25 + 10 * q^26 - 2 * q^32 - 18 * q^33 + 2 * q^34 - 12 * q^36 + 8 * q^38 + 16 * q^43 - 8 * q^47 - 10 * q^50 + 24 * q^51 - 10 * q^52 - 2 * q^53 - 24 * q^59 + 2 * q^64 + 18 * q^66 - 32 * q^67 - 2 * q^68 - 24 * q^69 + 12 * q^72 - 8 * q^76 + 18 * q^81 + 32 * q^83 - 16 * q^86 + 24 * q^87 - 18 * q^89 - 24 * q^93 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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