LMFDB - Embedded newform 392.2.i.b.177.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [392,2,Mod(177,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.177"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-1,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$3.13013575923$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	392.177
Dual form			392.2.i.b.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} +6.00000 q^{13} +1.00000 q^{15} +(-2.50000 - 4.33013i) q^{17} +(0.500000 - 0.866025i) q^{19} +(3.50000 - 6.06218i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.00000 q^{27} +2.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(-1.50000 + 2.59808i) q^{37} +(-3.00000 - 5.19615i) q^{39} +2.00000 q^{41} -4.00000 q^{43} +(1.00000 + 1.73205i) q^{45} +(2.50000 - 4.33013i) q^{47} +(-2.50000 + 4.33013i) q^{51} +(0.500000 + 0.866025i) q^{53} +3.00000 q^{55} -1.00000 q^{57} +(7.50000 + 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(4.50000 + 7.79423i) q^{67} -7.00000 q^{69} +(3.50000 + 6.06218i) q^{73} +(2.00000 - 3.46410i) q^{75} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +5.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} +(3.50000 - 6.06218i) q^{89} +(-2.50000 + 4.33013i) q^{93} +(0.500000 + 0.866025i) q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{9} - 3 q^{11} + 12 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} - 10 q^{27} + 4 q^{29} - 5 q^{31} - 3 q^{33} - 3 q^{37} - 6 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^9 - 3 * q^11 + 12 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 7 * q^23 + 4 * q^25 - 10 * q^27 + 4 * q^29 - 5 * q^31 - 3 * q^33 - 3 * q^37 - 6 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 5 * q^47 - 5 * q^51 + q^53 + 6 * q^55 - 2 * q^57 + 15 * q^59 - 5 * q^61 - 6 * q^65 + 9 * q^67 - 14 * q^69 + 7 * q^73 + 4 * q^75 - q^79 - q^81 - 24 * q^83 + 10 * q^85 - 2 * q^87 + 7 * q^89 - 5 * q^93 + q^95 + 4 * q^97 - 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	0.684819	−	0.728714i	$-0.259881\pi$
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−2.50000	−	4.33013i	−0.606339	−	1.05021i	−0.991838	−	0.127502i	$-0.959304\pi$
$17$	−2.50000	−	4.33013i	−0.606339	−	1.05021i	0.385499	−	0.922708i	$-0.374029\pi$
$18$		0			0
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	0.500000	−	0.866025i	0.114708	−	0.198680i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	−0.227167	−	0.973856i	$-0.572946\pi$
$23$	3.50000	−	6.06218i	0.729800	−	1.26405i	0.956967	−	0.290196i	$-0.0937204\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$	−1.50000	+	2.59808i	−0.261116	+	0.452267i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	−0.962580	−	0.270998i	$-0.912646\pi$
$37$	−1.50000	+	2.59808i	−0.246598	+	0.427121i	0.715981	+	0.698119i	$0.245980\pi$
$38$		0			0
$39$	−3.00000	−	5.19615i	−0.480384	−	0.832050i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	1.00000	+	1.73205i	0.149071	+	0.258199i
$46$		0			0
$47$	2.50000	−	4.33013i	0.364662	−	0.631614i	−0.624059	−	0.781377i	$-0.714518\pi$
$47$	2.50000	−	4.33013i	0.364662	−	0.631614i	0.988722	+	0.149763i	$0.0478510\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.50000	+	4.33013i	−0.350070	+	0.606339i
$52$		0			0
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	0.898321	−	0.439340i	$-0.144788\pi$
$53$	0.500000	+	0.866025i	0.0686803	+	0.118958i	−0.829640	+	0.558298i	$0.811454\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$	−1.00000			−0.132453
$58$		0			0
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.675178	+	0.737655i	$0.264067\pi$
$59$	7.50000	+	12.9904i	0.976417	+	1.69120i	0.301239	+	0.953549i	$0.402600\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	0.998290	+	0.0584478i	$0.0186151\pi$
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	−0.448528	+	0.893769i	$0.648052\pi$
$68$		0			0
$69$	−7.00000			−0.842701
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	0.994850	−	0.101361i	$-0.0323196\pi$
$73$	3.50000	+	6.06218i	0.409644	+	0.709524i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$	2.00000	−	3.46410i	0.230940	−	0.400000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	5.00000			0.542326
$86$		0			0
$87$	−1.00000	−	1.73205i	−0.107211	−	0.185695i
$88$		0			0
$89$	3.50000	−	6.06218i	0.370999	−	0.642590i	−0.618720	−	0.785611i	$-0.712349\pi$
$89$	3.50000	−	6.06218i	0.370999	−	0.642590i	0.989720	+	0.143022i	$0.0456819\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.50000	+	4.33013i	−0.259238	+	0.449013i
$94$		0			0
$95$	0.500000	+	0.866025i	0.0512989	+	0.0888523i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	0.930953	−	0.365140i	$-0.118979\pi$
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	−0.781697	+	0.623658i	$0.785646\pi$
$102$		0			0
$103$	−7.50000	+	12.9904i	−0.738997	+	1.27998i	0.213950	+	0.976845i	$0.431367\pi$
$103$	−7.50000	+	12.9904i	−0.738997	+	1.27998i	−0.952947	+	0.303136i	$0.901966\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.50000	+	7.79423i	−0.435031	+	0.753497i	−0.997298	−	0.0734594i	$-0.976596\pi$
$107$	−4.50000	+	7.79423i	−0.435031	+	0.753497i	0.562267	+	0.826956i	$0.309929\pi$
$108$		0			0
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	0.960558	−	0.278078i	$-0.0896974\pi$
$109$	2.50000	+	4.33013i	0.239457	+	0.414751i	−0.721102	+	0.692829i	$0.756364\pi$
$110$		0			0
$111$	3.00000			0.284747
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	3.50000	+	6.06218i	0.326377	+	0.565301i
$116$		0			0
$117$	6.00000	−	10.3923i	0.554700	−	0.960769i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$	−1.00000	−	1.73205i	−0.0901670	−	0.156174i
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	2.00000	+	3.46410i	0.176090	+	0.304997i
$130$		0			0
$131$	2.50000	−	4.33013i	0.218426	−	0.378325i	−0.735901	−	0.677089i	$-0.763241\pi$
$131$	2.50000	−	4.33013i	0.218426	−	0.378325i	0.954327	+	0.298764i	$0.0965744\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.50000	−	4.33013i	0.215166	−	0.372678i
$136$		0			0
$137$	−5.50000	−	9.52628i	−0.469897	−	0.813885i	0.529511	−	0.848303i	$-0.322376\pi$
$137$	−5.50000	−	9.52628i	−0.469897	−	0.813885i	−0.999408	+	0.0344182i	$0.989042\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$	−5.00000			−0.421076
$142$		0			0
$143$	−9.00000	−	15.5885i	−0.752618	−	1.30357i
$144$		0			0
$145$	−1.00000	+	1.73205i	−0.0830455	+	0.143839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	−0.273377	−	0.961907i	$-0.588141\pi$
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	0.969724	−	0.244202i	$-0.0785259\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$		0			0
$153$	−10.0000			−0.808452
$154$		0			0
$155$	5.00000			0.401610
$156$		0			0
$157$	−4.50000	−	7.79423i	−0.359139	−	0.622047i	0.628678	−	0.777666i	$-0.283596\pi$
$157$	−4.50000	−	7.79423i	−0.359139	−	0.622047i	−0.987817	+	0.155618i	$0.950263\pi$
$158$		0			0
$159$	0.500000	−	0.866025i	0.0396526	−	0.0686803i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	0.490825	+	0.871258i	$0.336695\pi$
$163$	−6.50000	+	11.2583i	−0.509119	+	0.881820i	−0.999944	+	0.0105623i	$0.996638\pi$
$164$		0			0
$165$	−1.50000	−	2.59808i	−0.116775	−	0.202260i
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$	−1.00000	−	1.73205i	−0.0764719	−	0.132453i
$172$		0			0
$173$	−6.50000	+	11.2583i	−0.494186	+	0.855955i	−0.999978	−	0.00670064i	$-0.997867\pi$
$173$	−6.50000	+	11.2583i	−0.494186	+	0.855955i	0.505792	+	0.862656i	$0.331200\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.50000	−	12.9904i	0.563735	−	0.976417i
$178$		0			0
$179$	6.50000	+	11.2583i	0.485833	+	0.841487i	0.999867	−	0.0162823i	$-0.00518305\pi$
$179$	6.50000	+	11.2583i	0.485833	+	0.841487i	−0.514035	+	0.857769i	$0.671850\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	5.00000			0.369611
$184$		0			0
$185$	−1.50000	−	2.59808i	−0.110282	−	0.191014i
$186$		0			0
$187$	−7.50000	+	12.9904i	−0.548454	+	0.949951i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.50000	−	9.52628i	0.397966	−	0.689297i	−0.595509	−	0.803349i	$-0.703050\pi$
$191$	5.50000	−	9.52628i	0.397966	−	0.689297i	0.993475	+	0.114051i	$0.0363829\pi$
$192$		0			0
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	0.806976	−	0.590584i	$-0.201102\pi$
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	−0.914949	+	0.403570i	$0.867769\pi$
$194$		0			0
$195$	6.00000			0.429669
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	0.538227	−	0.842800i	$-0.319094\pi$
$199$	−6.50000	−	11.2583i	−0.460773	−	0.798082i	−0.999000	+	0.0447181i	$0.985761\pi$
$200$		0			0
$201$	4.50000	−	7.79423i	0.317406	−	0.549762i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.00000	+	1.73205i	−0.0698430	+	0.120972i
$206$		0			0
$207$	−7.00000	−	12.1244i	−0.486534	−	0.842701i
$208$		0			0
$209$	−3.00000			−0.207514
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.00000	−	3.46410i	0.136399	−	0.236250i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3.50000	−	6.06218i	0.236508	−	0.409644i
$220$		0			0
$221$	−15.0000	−	25.9808i	−1.00901	−	1.74766i
$222$		0			0
$223$	24.0000			1.60716			0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000			1.60716			0.803579	+	0.595198i	$0.202926\pi$
$224$		0			0
$225$	8.00000			0.533333
$226$		0			0
$227$	5.50000	+	9.52628i	0.365048	+	0.632281i	0.988784	−	0.149354i	$-0.0477193\pi$
$227$	5.50000	+	9.52628i	0.365048	+	0.632281i	−0.623736	+	0.781635i	$0.714386\pi$
$228$		0			0
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	−0.182939	−	0.983124i	$-0.558561\pi$
$229$	11.5000	−	19.9186i	0.759941	−	1.31626i	0.942880	−	0.333133i	$-0.108106\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.50000	+	9.52628i	−0.360317	+	0.624087i	−0.988013	−	0.154371i	$-0.950665\pi$
$233$	−5.50000	+	9.52628i	−0.360317	+	0.624087i	0.627696	+	0.778459i	$0.283998\pi$
$234$		0			0
$235$	2.50000	+	4.33013i	0.163082	+	0.282466i
$236$		0			0
$237$	1.00000			0.0649570
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$		0			0
$243$	−8.00000	+	13.8564i	−0.513200	+	0.888889i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.00000	−	5.19615i	0.190885	−	0.330623i
$248$		0			0
$249$	6.00000	+	10.3923i	0.380235	+	0.658586i
$250$		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
$253$	−21.0000			−1.32026
$254$		0			0
$255$	−2.50000	−	4.33013i	−0.156556	−	0.271163i
$256$		0			0
$257$	−10.5000	+	18.1865i	−0.654972	+	1.13444i	0.326929	+	0.945049i	$0.393986\pi$
$257$	−10.5000	+	18.1865i	−0.654972	+	1.13444i	−0.981901	+	0.189396i	$0.939347\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2.00000	−	3.46410i	0.123797	−	0.214423i
$262$		0			0
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	0.970758	−	0.240059i	$-0.0771668\pi$
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	−0.693276	+	0.720672i	$0.743833\pi$
$264$		0			0
$265$	−1.00000			−0.0614295
$266$		0			0
$267$	−7.00000			−0.428393
$268$		0			0
$269$	−4.50000	−	7.79423i	−0.274370	−	0.475223i	0.695606	−	0.718423i	$-0.255136\pi$
$269$	−4.50000	−	7.79423i	−0.274370	−	0.475223i	−0.969976	+	0.243201i	$0.921803\pi$
$270$		0			0
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	−0.952531	−	0.304443i	$-0.901530\pi$
$271$	−3.50000	+	6.06218i	−0.212610	+	0.368251i	0.739921	+	0.672694i	$0.234863\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	−	10.3923i	0.361814	−	0.626680i
$276$		0			0
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	0.605575	−	0.795788i	$-0.292943\pi$
$283$	−6.50000	−	11.2583i	−0.386385	−	0.669238i	−0.991960	+	0.126550i	$0.959610\pi$
$284$		0			0
$285$	0.500000	−	0.866025i	0.0296174	−	0.0512989i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$		0			0
$291$	−1.00000	−	1.73205i	−0.0586210	−	0.101535i
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−15.0000			−0.873334
$296$		0			0
$297$	7.50000	+	12.9904i	0.435194	+	0.753778i
$298$		0			0
$299$	21.0000	−	36.3731i	1.21446	−	2.10351i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.50000	−	2.59808i	0.0861727	−	0.149256i
$304$		0			0
$305$	−2.50000	−	4.33013i	−0.143150	−	0.247942i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$	15.0000			0.853320
$310$		0			0
$311$	7.50000	+	12.9904i	0.425286	+	0.736617i	0.996447	−	0.0842210i	$-0.0268402\pi$
$311$	7.50000	+	12.9904i	0.425286	+	0.736617i	−0.571161	+	0.820838i	$0.693507\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	−0.879810	−	0.475325i	$-0.842331\pi$
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	0.851549	+	0.524276i	$0.175664\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.50000	+	2.59808i	−0.0842484	+	0.145922i	−0.905071	−	0.425261i	$-0.860182\pi$
$317$	−1.50000	+	2.59808i	−0.0842484	+	0.145922i	0.820822	+	0.571184i	$0.193516\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$		0			0
$321$	9.00000			0.502331
$322$		0			0
$323$	−5.00000			−0.278207
$324$		0			0
$325$	12.0000	+	20.7846i	0.665640	+	1.15292i
$326$		0			0
$327$	2.50000	−	4.33013i	0.138250	−	0.239457i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.5000	+	25.1147i	−0.796992	+	1.38043i	0.124574	+	0.992210i	$0.460243\pi$
$331$	−14.5000	+	25.1147i	−0.796992	+	1.38043i	−0.921567	+	0.388221i	$0.873090\pi$
$332$		0			0
$333$	3.00000	+	5.19615i	0.164399	+	0.284747i
$334$		0			0
$335$	−9.00000			−0.491723
$336$		0			0
$337$	−10.0000			−0.544735			−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000			−0.544735			−0.272367	+	0.962193i	$0.587807\pi$
$338$		0			0
$339$	9.00000	+	15.5885i	0.488813	+	0.846649i
$340$		0			0
$341$	−7.50000	+	12.9904i	−0.406148	+	0.703469i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.50000	−	6.06218i	0.188434	−	0.326377i
$346$		0			0
$347$	−7.50000	−	12.9904i	−0.402621	−	0.697360i	0.591420	−	0.806363i	$-0.298567\pi$
$347$	−7.50000	−	12.9904i	−0.402621	−	0.697360i	−0.994041	+	0.109003i	$0.965234\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	−30.0000			−1.60128
$352$		0			0
$353$	1.50000	+	2.59808i	0.0798369	+	0.138282i	0.903179	−	0.429263i	$-0.141227\pi$
$353$	1.50000	+	2.59808i	0.0798369	+	0.138282i	−0.823343	+	0.567545i	$0.807893\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.5000	+	18.1865i	−0.554169	+	0.959849i	0.443799	+	0.896126i	$0.353630\pi$
$359$	−10.5000	+	18.1865i	−0.554169	+	0.959849i	−0.997968	+	0.0637221i	$0.979703\pi$
$360$		0			0
$361$	9.00000	+	15.5885i	0.473684	+	0.820445i
$362$		0			0
$363$	−2.00000			−0.104973
$364$		0			0
$365$	−7.00000			−0.366397
$366$		0			0
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	0.992776	+	0.119982i	$0.0382835\pi$
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	−0.392481	+	0.919760i	$0.628383\pi$
$368$		0			0
$369$	2.00000	−	3.46410i	0.104116	−	0.180334i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	−0.972556	−	0.232671i	$-0.925254\pi$
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	0.687776	+	0.725923i	$0.258587\pi$
$374$		0			0
$375$	4.50000	+	7.79423i	0.232379	+	0.402492i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	24.0000			1.23280			0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000			1.23280			0.616399	+	0.787434i	$0.288591\pi$
$380$		0			0
$381$	−4.00000	−	6.92820i	−0.204926	−	0.354943i
$382$		0			0
$383$	−1.50000	+	2.59808i	−0.0766464	+	0.132755i	−0.901801	−	0.432151i	$-0.857755\pi$
$383$	−1.50000	+	2.59808i	−0.0766464	+	0.132755i	0.825155	+	0.564907i	$0.191088\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000	+	6.92820i	−0.203331	+	0.352180i
$388$		0			0
$389$	14.5000	+	25.1147i	0.735179	+	1.27337i	0.954645	+	0.297747i	$0.0962353\pi$
$389$	14.5000	+	25.1147i	0.735179	+	1.27337i	−0.219465	+	0.975620i	$0.570431\pi$
$390$		0			0
$391$	−35.0000			−1.77003
$392$		0			0
$393$	−5.00000			−0.252217
$394$		0			0
$395$	−0.500000	−	0.866025i	−0.0251577	−	0.0435745i
$396$		0			0
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	−0.996569	−	0.0827707i	$-0.973623\pi$
$397$	−8.50000	+	14.7224i	−0.426603	+	0.738898i	0.569966	+	0.821668i	$0.306956\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.5000	+	23.3827i	−0.674158	+	1.16768i	0.302556	+	0.953131i	$0.402160\pi$
$401$	−13.5000	+	23.3827i	−0.674158	+	1.16768i	−0.976714	+	0.214544i	$0.931173\pi$
$402$		0			0
$403$	−15.0000	−	25.9808i	−0.747203	−	1.29419i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	9.00000			0.446113
$408$		0			0
$409$	13.5000	+	23.3827i	0.667532	+	1.15620i	0.978592	+	0.205809i	$0.0659826\pi$
$409$	13.5000	+	23.3827i	0.667532	+	1.15620i	−0.311060	+	0.950390i	$0.600684\pi$
$410$		0			0
$411$	−5.50000	+	9.52628i	−0.271295	+	0.469897i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$	−6.00000	−	10.3923i	−0.293821	−	0.508913i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$	−5.00000	−	8.66025i	−0.243108	−	0.421076i
$424$		0			0
$425$	10.0000	−	17.3205i	0.485071	−	0.840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−9.00000	+	15.5885i	−0.434524	+	0.752618i
$430$		0			0
$431$	−1.50000	−	2.59808i	−0.0722525	−	0.125145i	0.827636	−	0.561266i	$-0.189685\pi$
$431$	−1.50000	−	2.59808i	−0.0722525	−	0.125145i	−0.899888	+	0.436121i	$0.856352\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$	2.00000			0.0958927
$436$		0			0
$437$	−3.50000	−	6.06218i	−0.167428	−	0.289993i
$438$		0			0
$439$	10.5000	−	18.1865i	0.501138	−	0.867996i	−0.498861	−	0.866682i	$-0.666248\pi$
$439$	10.5000	−	18.1865i	0.501138	−	0.867996i	0.999999	−	0.00131415i	$-0.000418308\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.5000	−	26.8468i	0.736427	−	1.27553i	−0.217667	−	0.976023i	$-0.569845\pi$
$443$	15.5000	−	26.8468i	0.736427	−	1.27553i	0.954094	−	0.299506i	$-0.0968220\pi$
$444$		0			0
$445$	3.50000	+	6.06218i	0.165916	+	0.287375i
$446$		0			0
$447$	−17.0000			−0.804072
$448$		0			0
$449$	−26.0000			−1.22702			−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000			−1.22702			−0.613508	+	0.789689i	$0.710242\pi$
$450$		0			0
$451$	−3.00000	−	5.19615i	−0.141264	−	0.244677i
$452$		0			0
$453$	2.50000	−	4.33013i	0.117460	−	0.203447i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	−0.595818	−	0.803120i	$-0.703172\pi$
$457$	8.50000	−	14.7224i	0.397613	−	0.688686i	0.993431	+	0.114433i	$0.0365053\pi$
$458$		0			0
$459$	12.5000	+	21.6506i	0.583450	+	1.01057i
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	−2.50000	−	4.33013i	−0.115935	−	0.200805i
$466$		0			0
$467$	−1.50000	+	2.59808i	−0.0694117	+	0.120225i	−0.898642	−	0.438682i	$-0.855446\pi$
$467$	−1.50000	+	2.59808i	−0.0694117	+	0.120225i	0.829231	+	0.558906i	$0.188779\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.50000	+	7.79423i	−0.207349	+	0.359139i
$472$		0			0
$473$	6.00000	+	10.3923i	0.275880	+	0.477839i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$	2.00000			0.0915737
$478$		0			0
$479$	15.5000	+	26.8468i	0.708213	+	1.22666i	0.965519	+	0.260331i	$0.0838317\pi$
$479$	15.5000	+	26.8468i	0.708213	+	1.22666i	−0.257306	+	0.966330i	$0.582835\pi$
$480$		0			0
$481$	−9.00000	+	15.5885i	−0.410365	+	0.710772i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00000	+	1.73205i	−0.0454077	+	0.0786484i
$486$		0			0
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	0.775764	−	0.631023i	$-0.217365\pi$
$487$	−3.50000	−	6.06218i	−0.158600	−	0.274703i	−0.934364	+	0.356320i	$0.884031\pi$
$488$		0			0
$489$	13.0000			0.587880
$490$		0			0
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$	−5.00000	−	8.66025i	−0.225189	−	0.390038i
$494$		0			0
$495$	3.00000	−	5.19615i	0.134840	−	0.233550i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$	−6.00000	−	10.3923i	−0.268060	−	0.464294i
$502$		0			0
$503$	32.0000			1.42681			0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000			1.42681			0.713405	+	0.700752i	$0.247152\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$		0			0
$507$	−11.5000	−	19.9186i	−0.510733	−	0.884615i
$508$		0			0
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	−0.948353	−	0.317217i	$-0.897252\pi$
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	0.748894	+	0.662690i	$0.230585\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−2.50000	+	4.33013i	−0.110378	+	0.191180i
$514$		0			0
$515$	−7.50000	−	12.9904i	−0.330489	−	0.572425i
$516$		0			0
$517$	−15.0000			−0.659699
$518$		0			0
$519$	13.0000			0.570637
$520$		0			0
$521$	−16.5000	−	28.5788i	−0.722878	−	1.25206i	−0.959841	−	0.280543i	$-0.909485\pi$
$521$	−16.5000	−	28.5788i	−0.722878	−	1.25206i	0.236963	−	0.971519i	$-0.423848\pi$
$522$		0			0
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	−0.932345	−	0.361569i	$-0.882241\pi$
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	0.779301	+	0.626650i	$0.215574\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.5000	+	21.6506i	−0.544509	+	0.943116i
$528$		0			0
$529$	−13.0000	−	22.5167i	−0.565217	−	0.978985i
$530$		0			0
$531$	30.0000			1.30189
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$	−4.50000	−	7.79423i	−0.194552	−	0.336974i
$536$		0			0
$537$	6.50000	−	11.2583i	0.280496	−	0.485833i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.0315385	−	0.999503i	$-0.489959\pi$
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.849825	−	0.527064i	$-0.176707\pi$
$542$		0			0
$543$	−5.00000	−	8.66025i	−0.214571	−	0.371647i
$544$		0			0
$545$	−5.00000			−0.214176
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$	5.00000	+	8.66025i	0.213395	+	0.369611i
$550$		0			0
$551$	1.00000	−	1.73205i	0.0426014	−	0.0737878i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.50000	+	2.59808i	−0.0636715	+	0.110282i
$556$		0			0
$557$	2.50000	+	4.33013i	0.105928	+	0.183473i	0.914117	−	0.405450i	$-0.132885\pi$
$557$	2.50000	+	4.33013i	0.105928	+	0.183473i	−0.808189	+	0.588924i	$0.799552\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	15.0000			0.633300
$562$		0			0
$563$	−20.5000	−	35.5070i	−0.863972	−	1.49644i	−0.868064	−	0.496452i	$-0.834636\pi$
$563$	−20.5000	−	35.5070i	−0.863972	−	1.49644i	0.00409232	−	0.999992i	$-0.498697\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	16.5000	−	28.5788i	0.691716	−	1.19809i	−0.279559	−	0.960128i	$-0.590188\pi$
$569$	16.5000	−	28.5788i	0.691716	−	1.19809i	0.971275	−	0.237959i	$-0.0764783\pi$
$570$		0			0
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	0.969378	−	0.245573i	$-0.0789761\pi$
$571$	6.50000	+	11.2583i	0.272017	+	0.471146i	−0.697362	+	0.716720i	$0.745643\pi$
$572$		0			0
$573$	−11.0000			−0.459532
$574$		0			0
$575$	28.0000			1.16768
$576$		0			0
$577$	−16.5000	−	28.5788i	−0.686904	−	1.18975i	−0.972834	−	0.231502i	$-0.925636\pi$
$577$	−16.5000	−	28.5788i	−0.686904	−	1.18975i	0.285930	−	0.958250i	$-0.407697\pi$
$578$		0			0
$579$	−1.50000	+	2.59808i	−0.0623379	+	0.107972i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.50000	−	2.59808i	0.0621237	−	0.107601i
$584$		0			0
$585$	6.00000	+	10.3923i	0.248069	+	0.429669i
$586$		0			0
$587$	20.0000			0.825488			0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000			0.825488			0.412744	+	0.910847i	$0.364570\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$	3.00000	+	5.19615i	0.123404	+	0.213741i
$592$		0			0
$593$	−22.5000	+	38.9711i	−0.923964	+	1.60035i	−0.130746	+	0.991416i	$0.541737\pi$
$593$	−22.5000	+	38.9711i	−0.923964	+	1.60035i	−0.793219	+	0.608937i	$0.791596\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−6.50000	+	11.2583i	−0.266027	+	0.460773i
$598$		0			0
$599$	8.50000	+	14.7224i	0.347301	+	0.601542i	0.985769	−	0.168106i	$-0.0537650\pi$
$599$	8.50000	+	14.7224i	0.347301	+	0.601542i	−0.638468	+	0.769648i	$0.720432\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$	18.0000			0.733017
$604$		0			0
$605$	1.00000	+	1.73205i	0.0406558	+	0.0704179i
$606$		0			0
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	−0.703429	−	0.710766i	$-0.748349\pi$
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	0.967256	+	0.253804i	$0.0816819\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.0000	−	25.9808i	0.606835	−	1.05107i
$612$		0			0
$613$	−21.5000	−	37.2391i	−0.868377	−	1.50407i	−0.863655	−	0.504084i	$-0.831830\pi$
$613$	−21.5000	−	37.2391i	−0.868377	−	1.50407i	−0.00472215	−	0.999989i	$-0.501503\pi$
$614$		0			0
$615$	2.00000			0.0806478
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	−4.50000	−	7.79423i	−0.180870	−	0.313276i	0.761307	−	0.648392i	$-0.224558\pi$
$619$	−4.50000	−	7.79423i	−0.180870	−	0.313276i	−0.942177	+	0.335115i	$0.891225\pi$
$620$		0			0
$621$	−17.5000	+	30.3109i	−0.702251	+	1.21633i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$	1.50000	+	2.59808i	0.0599042	+	0.103757i
$628$		0			0
$629$	15.0000			0.598089
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$	2.00000	+	3.46410i	0.0794929	+	0.137686i
$634$		0			0
$635$	−4.00000	+	6.92820i	−0.158735	+	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.50000	+	14.7224i	0.335730	+	0.581501i	0.983625	−	0.180229i	$-0.0576838\pi$
$641$	8.50000	+	14.7224i	0.335730	+	0.581501i	−0.647895	+	0.761730i	$0.724350\pi$
$642$		0			0
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$		0			0
$645$	−4.00000			−0.157500
$646$		0			0
$647$	7.50000	+	12.9904i	0.294855	+	0.510705i	0.974951	−	0.222419i	$-0.0713952\pi$
$647$	7.50000	+	12.9904i	0.294855	+	0.510705i	−0.680096	+	0.733123i	$0.738062\pi$
$648$		0			0
$649$	22.5000	−	38.9711i	0.883202	−	1.52975i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.5000	+	30.3109i	−0.684828	+	1.18616i	0.288663	+	0.957431i	$0.406789\pi$
$653$	−17.5000	+	30.3109i	−0.684828	+	1.18616i	−0.973491	+	0.228726i	$0.926544\pi$
$654$		0			0
$655$	2.50000	+	4.33013i	0.0976831	+	0.169192i
$656$		0			0
$657$	14.0000			0.546192
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$	11.5000	+	19.9186i	0.447298	+	0.774743i	0.998209	−	0.0598209i	$-0.0190530\pi$
$661$	11.5000	+	19.9186i	0.447298	+	0.774743i	−0.550911	+	0.834564i	$0.685720\pi$
$662$		0			0
$663$	−15.0000	+	25.9808i	−0.582552	+	1.00901i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.00000	−	12.1244i	0.271041	−	0.469457i
$668$		0			0
$669$	−12.0000	−	20.7846i	−0.463947	−	0.803579i
$670$		0			0
$671$	15.0000			0.579069
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$	−10.0000	−	17.3205i	−0.384900	−	0.666667i
$676$		0			0
$677$	19.5000	−	33.7750i	0.749446	−	1.29808i	−0.198643	−	0.980072i	$-0.563653\pi$
$677$	19.5000	−	33.7750i	0.749446	−	1.29808i	0.948089	−	0.318006i	$-0.103013\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	5.50000	−	9.52628i	0.210760	−	0.365048i
$682$		0			0
$683$	−13.5000	−	23.3827i	−0.516563	−	0.894714i	−0.999815	−	0.0192323i	$-0.993878\pi$
$683$	−13.5000	−	23.3827i	−0.516563	−	0.894714i	0.483252	−	0.875481i	$-0.339456\pi$
$684$		0			0
$685$	11.0000			0.420288
$686$		0			0
$687$	−23.0000			−0.877505
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	−0.814548	−	0.580097i	$-0.803015\pi$
$691$	2.50000	−	4.33013i	0.0951045	−	0.164726i	0.909652	+	0.415371i	$0.136348\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000	+	10.3923i	−0.227593	+	0.394203i
$696$		0			0
$697$	−5.00000	−	8.66025i	−0.189389	−	0.328031i
$698$		0			0
$699$	11.0000			0.416058
$700$		0			0
$701$	−50.0000			−1.88847			−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000			−1.88847			−0.944237	+	0.329267i	$0.893198\pi$
$702$		0			0
$703$	1.50000	+	2.59808i	0.0565736	+	0.0979883i
$704$		0			0
$705$	2.50000	−	4.33013i	0.0941554	−	0.163082i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	−0.454076	−	0.890963i	$-0.650030\pi$
$709$	14.5000	−	25.1147i	0.544559	−	0.943204i	0.998635	−	0.0522406i	$-0.0166363\pi$
$710$		0			0
$711$	1.00000	+	1.73205i	0.0375029	+	0.0649570i
$712$		0			0
$713$	−35.0000			−1.31076
$714$		0			0
$715$	18.0000			0.673162
$716$		0			0
$717$	−10.0000	−	17.3205i	−0.373457	−	0.646846i
$718$		0			0
$719$	−19.5000	+	33.7750i	−0.727227	+	1.25959i	0.230823	+	0.972996i	$0.425858\pi$
$719$	−19.5000	+	33.7750i	−0.727227	+	1.25959i	−0.958051	+	0.286599i	$0.907475\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−8.50000	+	14.7224i	−0.316118	+	0.547533i
$724$		0			0
$725$	4.00000	+	6.92820i	0.148556	+	0.257307i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	10.0000	+	17.3205i	0.369863	+	0.640622i
$732$		0			0
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	−0.635515	−	0.772088i	$-0.719212\pi$
$733$	9.50000	−	16.4545i	0.350891	−	0.607760i	0.986406	+	0.164328i	$0.0525456\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.5000	−	23.3827i	0.497279	−	0.861312i
$738$		0			0
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	0.908337	−	0.418238i	$-0.137352\pi$
$739$	2.50000	+	4.33013i	0.0919640	+	0.159286i	−0.816373	+	0.577524i	$0.804019\pi$
$740$		0			0
$741$	−6.00000			−0.220416
$742$		0			0
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$		0			0
$745$	8.50000	+	14.7224i	0.311416	+	0.539388i
$746$		0			0
$747$	−12.0000	+	20.7846i	−0.439057	+	0.760469i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.5000	+	45.8993i	−0.966999	+	1.67489i	−0.262852	+	0.964836i	$0.584663\pi$
$751$	−26.5000	+	45.8993i	−0.966999	+	1.67489i	−0.704146	+	0.710055i	$0.748670\pi$
$752$		0			0
$753$	8.00000	+	13.8564i	0.291536	+	0.504956i
$754$		0			0
$755$	−5.00000			−0.181969
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$	10.5000	+	18.1865i	0.381126	+	0.660129i
$760$		0			0
$761$	1.50000	−	2.59808i	0.0543750	−	0.0941802i	−0.837557	−	0.546350i	$-0.816017\pi$
$761$	1.50000	−	2.59808i	0.0543750	−	0.0941802i	0.891932	+	0.452170i	$0.149350\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	5.00000	−	8.66025i	0.180775	−	0.313112i
$766$		0			0
$767$	45.0000	+	77.9423i	1.62486	+	2.81433i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	21.0000			0.756297
$772$		0			0
$773$	9.50000	+	16.4545i	0.341691	+	0.591827i	0.984747	−	0.173993i	$-0.0556670\pi$
$773$	9.50000	+	16.4545i	0.341691	+	0.591827i	−0.643056	+	0.765819i	$0.722334\pi$
$774$		0			0
$775$	10.0000	−	17.3205i	0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.00000	−	1.73205i	0.0358287	−	0.0620572i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−10.0000			−0.357371
$784$		0			0
$785$	9.00000			0.321224
$786$		0			0
$787$	−8.50000	−	14.7224i	−0.302992	−	0.524798i	0.673820	−	0.738896i	$-0.264652\pi$
$787$	−8.50000	−	14.7224i	−0.302992	−	0.524798i	−0.976812	+	0.214097i	$0.931319\pi$
$788$		0			0
$789$	4.50000	−	7.79423i	0.160204	−	0.277482i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−15.0000	+	25.9808i	−0.532666	+	0.922604i
$794$		0			0
$795$	0.500000	+	0.866025i	0.0177332	+	0.0307148i
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−25.0000			−0.884436
$800$		0			0
$801$	−7.00000	−	12.1244i	−0.247333	−	0.428393i
$802$		0			0
$803$	10.5000	−	18.1865i	0.370537	−	0.641789i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.50000	+	7.79423i	−0.158408	+	0.274370i
$808$		0			0
$809$	−19.5000	−	33.7750i	−0.685583	−	1.18747i	−0.973253	−	0.229736i	$-0.926214\pi$
$809$	−19.5000	−	33.7750i	−0.685583	−	1.18747i	0.287670	−	0.957730i	$-0.407120\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$		0			0
$813$	7.00000			0.245501
$814$		0			0
$815$	−6.50000	−	11.2583i	−0.227685	−	0.394362i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	−0.776753	−	0.629805i	$-0.783135\pi$
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	0.933804	+	0.357785i	$0.116468\pi$
$822$		0			0
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−23.5000	−	40.7032i	−0.819159	−	1.41882i	0.0871445	−	0.996196i	$-0.472226\pi$
$824$		0			0
$825$	−12.0000			−0.417786
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	0.945589	−	0.325362i	$-0.105486\pi$
$829$	5.50000	+	9.52628i	0.191023	+	0.330861i	−0.754567	+	0.656223i	$0.772153\pi$
$830$		0			0
$831$	8.50000	−	14.7224i	0.294862	−	0.510716i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$	12.5000	+	21.6506i	0.432063	+	0.748355i
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−3.00000	−	5.19615i	−0.103325	−	0.178965i
$844$		0			0
$845$	−11.5000	+	19.9186i	−0.395612	+	0.685220i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−6.50000	+	11.2583i	−0.223079	+	0.386385i
$850$		0			0
$851$	10.5000	+	18.1865i	0.359935	+	0.623426i
$852$		0			0
$853$	−34.0000			−1.16414			−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000			−1.16414			−0.582069	+	0.813139i	$0.697757\pi$
$854$		0			0
$855$	2.00000			0.0683986
$856$		0			0
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	0.978984	+	0.203938i	$0.0653741\pi$
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	−0.312877	+	0.949794i	$0.601293\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	−0.857369	−	0.514701i	$-0.827903\pi$
$859$	0.500000	−	0.866025i	0.0170598	−	0.0295484i	0.874429	+	0.485153i	$0.161236\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.50000	+	14.7224i	−0.289343	+	0.501157i	−0.973653	−	0.228034i	$-0.926770\pi$
$863$	−8.50000	+	14.7224i	−0.289343	+	0.501157i	0.684310	+	0.729191i	$0.260104\pi$
$864$		0			0
$865$	−6.50000	−	11.2583i	−0.221007	−	0.382795i
$866$		0			0
$867$	8.00000			0.271694
$868$		0			0
$869$	3.00000			0.101768
$870$		0			0
$871$	27.0000	+	46.7654i	0.914860	+	1.58458i
$872$		0			0
$873$	2.00000	−	3.46410i	0.0676897	−	0.117242i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	−0.992225	−	0.124459i	$-0.960280\pi$
$877$	−11.5000	+	19.9186i	−0.388327	+	0.672603i	0.603897	+	0.797062i	$0.293614\pi$
$878$		0			0
$879$	3.00000	+	5.19615i	0.101187	+	0.175262i
$880$		0			0
$881$	14.0000			0.471672			0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000			0.471672			0.235836	+	0.971793i	$0.424217\pi$
$882$		0			0
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$	7.50000	+	12.9904i	0.252110	+	0.436667i
$886$		0			0
$887$	−1.50000	+	2.59808i	−0.0503651	+	0.0872349i	−0.890109	−	0.455748i	$-0.849372\pi$
$887$	−1.50000	+	2.59808i	−0.0503651	+	0.0872349i	0.839744	+	0.542983i	$0.182705\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.50000	+	2.59808i	−0.0502519	+	0.0870388i
$892$		0			0
$893$	−2.50000	−	4.33013i	−0.0836593	−	0.144902i
$894$		0			0
$895$	−13.0000			−0.434542
$896$		0			0
$897$	−42.0000			−1.40234
$898$		0			0
$899$	−5.00000	−	8.66025i	−0.166759	−	0.288836i
$900$		0			0
$901$	2.50000	−	4.33013i	0.0832871	−	0.144257i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.00000	+	8.66025i	−0.166206	+	0.287877i
$906$		0			0
$907$	0.500000	+	0.866025i	0.0166022	+	0.0287559i	0.874207	−	0.485553i	$-0.161382\pi$
$907$	0.500000	+	0.866025i	0.0166022	+	0.0287559i	−0.857605	+	0.514309i	$0.828048\pi$
$908$		0			0
$909$	6.00000			0.199007
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$		0			0
$913$	18.0000	+	31.1769i	0.595713	+	1.03181i
$914$		0			0
$915$	−2.50000	+	4.33013i	−0.0826475	+	0.143150i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	−0.760939	−	0.648824i	$-0.775261\pi$
$919$	5.50000	−	9.52628i	0.181428	−	0.314243i	0.942367	+	0.334581i	$0.108595\pi$
$920$		0			0
$921$	2.00000	+	3.46410i	0.0659022	+	0.114146i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−12.0000			−0.394558
$926$		0			0
$927$	15.0000	+	25.9808i	0.492665	+	0.853320i
$928$		0			0
$929$	3.50000	−	6.06218i	0.114831	−	0.198894i	−0.802881	−	0.596139i	$-0.796701\pi$
$929$	3.50000	−	6.06218i	0.114831	−	0.198894i	0.917712	+	0.397246i	$0.130034\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	7.50000	−	12.9904i	0.245539	−	0.425286i
$934$		0			0
$935$	−7.50000	−	12.9904i	−0.245276	−	0.424831i
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$	1.00000			0.0326338
$940$		0			0
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	−0.999018	−	0.0443125i	$-0.985890\pi$
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	0.461133	−	0.887331i	$-0.347443\pi$
$942$		0			0
$943$	7.00000	−	12.1244i	0.227951	−	0.394823i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.5000	+	32.0429i	−0.601169	+	1.04126i	0.391475	+	0.920189i	$0.371965\pi$
$947$	−18.5000	+	32.0429i	−0.601169	+	1.04126i	−0.992644	+	0.121067i	$0.961368\pi$
$948$		0			0
$949$	21.0000	+	36.3731i	0.681689	+	1.18072i
$950$		0			0
$951$	3.00000			0.0972817
$952$		0			0
$953$	22.0000			0.712650			0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000			0.712650			0.356325	+	0.934362i	$0.384030\pi$
$954$		0			0
$955$	5.50000	+	9.52628i	0.177976	+	0.308263i
$956$		0			0
$957$	−3.00000	+	5.19615i	−0.0969762	+	0.167968i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$	9.00000	+	15.5885i	0.290021	+	0.502331i
$964$		0			0
$965$	3.00000			0.0965734
$966$		0			0
$967$	48.0000			1.54358			0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000			1.54358			0.771788	+	0.635880i	$0.219363\pi$
$968$		0			0
$969$	2.50000	+	4.33013i	0.0803116	+	0.139104i
$970$		0			0
$971$	−17.5000	+	30.3109i	−0.561602	+	0.972723i	0.435755	+	0.900065i	$0.356481\pi$
$971$	−17.5000	+	30.3109i	−0.561602	+	0.972723i	−0.997357	+	0.0726575i	$0.976852\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	12.0000	−	20.7846i	0.384308	−	0.665640i
$976$		0			0
$977$	−1.50000	−	2.59808i	−0.0479893	−	0.0831198i	0.841033	−	0.540984i	$-0.181948\pi$
$977$	−1.50000	−	2.59808i	−0.0479893	−	0.0831198i	−0.889022	+	0.457864i	$0.848615\pi$
$978$		0			0
$979$	−21.0000			−0.671163
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−10.5000	−	18.1865i	−0.334898	−	0.580060i	0.648567	−	0.761157i	$-0.275369\pi$
$983$	−10.5000	−	18.1865i	−0.334898	−	0.580060i	−0.983465	+	0.181097i	$0.942035\pi$
$984$		0			0
$985$	3.00000	−	5.19615i	0.0955879	−	0.165563i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.0000	+	24.2487i	−0.445174	+	0.771064i
$990$		0			0
$991$	−7.50000	−	12.9904i	−0.238245	−	0.412653i	0.721966	−	0.691929i	$-0.243239\pi$
$991$	−7.50000	−	12.9904i	−0.238245	−	0.412653i	−0.960211	+	0.279276i	$0.909906\pi$
$992$		0			0
$993$	29.0000			0.920287
$994$		0			0
$995$	13.0000			0.412128
$996$		0			0
$997$	−20.5000	−	35.5070i	−0.649242	−	1.12452i	−0.983304	−	0.181968i	$-0.941753\pi$
$997$	−20.5000	−	35.5070i	−0.649242	−	1.12452i	0.334063	−	0.942551i	$-0.391580\pi$
$998$		0			0
$999$	7.50000	−	12.9904i	0.237289	−	0.410997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.i.b.177.1		2
3.2	odd	2		3528.2.s.q.3313.1		2
4.3	odd	2		784.2.i.h.177.1		2
7.2	even	3		392.2.a.e.1.1		1
7.3	odd	6		56.2.i.b.25.1	yes	2
7.4	even	3	inner	392.2.i.b.361.1		2
7.5	odd	6		392.2.a.c.1.1		1
7.6	odd	2		56.2.i.b.9.1	✓	2
21.2	odd	6		3528.2.a.j.1.1		1
21.5	even	6		3528.2.a.p.1.1		1
21.11	odd	6		3528.2.s.q.361.1		2
21.17	even	6		504.2.s.c.361.1		2
21.20	even	2		504.2.s.c.289.1		2
28.3	even	6		112.2.i.a.81.1		2
28.11	odd	6		784.2.i.h.753.1		2
28.19	even	6		784.2.a.h.1.1		1
28.23	odd	6		784.2.a.c.1.1		1
28.27	even	2		112.2.i.a.65.1		2
35.3	even	12		1400.2.bh.a.249.2		4
35.9	even	6		9800.2.a.s.1.1		1
35.13	even	4		1400.2.bh.a.849.1		4
35.17	even	12		1400.2.bh.a.249.1		4
35.19	odd	6		9800.2.a.be.1.1		1
35.24	odd	6		1400.2.q.d.1201.1		2
35.27	even	4		1400.2.bh.a.849.2		4
35.34	odd	2		1400.2.q.d.401.1		2
56.3	even	6		448.2.i.d.193.1		2
56.5	odd	6		3136.2.a.u.1.1		1
56.13	odd	2		448.2.i.b.65.1		2
56.19	even	6		3136.2.a.j.1.1		1
56.27	even	2		448.2.i.d.65.1		2
56.37	even	6		3136.2.a.i.1.1		1
56.45	odd	6		448.2.i.b.193.1		2
56.51	odd	6		3136.2.a.t.1.1		1
84.23	even	6		7056.2.a.u.1.1		1
84.47	odd	6		7056.2.a.bj.1.1		1
84.59	odd	6		1008.2.s.g.865.1		2
84.83	odd	2		1008.2.s.g.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.b.9.1	✓	2	7.6	odd	2
56.2.i.b.25.1	yes	2	7.3	odd	6
112.2.i.a.65.1		2	28.27	even	2
112.2.i.a.81.1		2	28.3	even	6
392.2.a.c.1.1		1	7.5	odd	6
392.2.a.e.1.1		1	7.2	even	3
392.2.i.b.177.1		2	1.1	even	1	trivial
392.2.i.b.361.1		2	7.4	even	3	inner
448.2.i.b.65.1		2	56.13	odd	2
448.2.i.b.193.1		2	56.45	odd	6
448.2.i.d.65.1		2	56.27	even	2
448.2.i.d.193.1		2	56.3	even	6
504.2.s.c.289.1		2	21.20	even	2
504.2.s.c.361.1		2	21.17	even	6
784.2.a.c.1.1		1	28.23	odd	6
784.2.a.h.1.1		1	28.19	even	6
784.2.i.h.177.1		2	4.3	odd	2
784.2.i.h.753.1		2	28.11	odd	6
1008.2.s.g.289.1		2	84.83	odd	2
1008.2.s.g.865.1		2	84.59	odd	6
1400.2.q.d.401.1		2	35.34	odd	2
1400.2.q.d.1201.1		2	35.24	odd	6
1400.2.bh.a.249.1		4	35.17	even	12
1400.2.bh.a.249.2		4	35.3	even	12
1400.2.bh.a.849.1		4	35.13	even	4
1400.2.bh.a.849.2		4	35.27	even	4
3136.2.a.i.1.1		1	56.37	even	6
3136.2.a.j.1.1		1	56.19	even	6
3136.2.a.t.1.1		1	56.51	odd	6
3136.2.a.u.1.1		1	56.5	odd	6
3528.2.a.j.1.1		1	21.2	odd	6
3528.2.a.p.1.1		1	21.5	even	6
3528.2.s.q.361.1		2	21.11	odd	6
3528.2.s.q.3313.1		2	3.2	odd	2
7056.2.a.u.1.1		1	84.23	even	6
7056.2.a.bj.1.1		1	84.47	odd	6
9800.2.a.s.1.1		1	35.9	even	6
9800.2.a.be.1.1		1	35.19	odd	6

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} +6.00000 q^{13} +1.00000 q^{15} +(-2.50000 - 4.33013i) q^{17} +(0.500000 - 0.866025i) q^{19} +(3.50000 - 6.06218i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.00000 q^{27} +2.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(-1.50000 + 2.59808i) q^{37} +(-3.00000 - 5.19615i) q^{39} +2.00000 q^{41} -4.00000 q^{43} +(1.00000 + 1.73205i) q^{45} +(2.50000 - 4.33013i) q^{47} +(-2.50000 + 4.33013i) q^{51} +(0.500000 + 0.866025i) q^{53} +3.00000 q^{55} -1.00000 q^{57} +(7.50000 + 12.9904i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(4.50000 + 7.79423i) q^{67} -7.00000 q^{69} +(3.50000 + 6.06218i) q^{73} +(2.00000 - 3.46410i) q^{75} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000 q^{83} +5.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} +(3.50000 - 6.06218i) q^{89} +(-2.50000 + 4.33013i) q^{93} +(0.500000 + 0.866025i) q^{95} +2.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{9} - 3 q^{11} + 12 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 7 q^{23} + 4 q^{25} - 10 q^{27} + 4 q^{29} - 5 q^{31} - 3 q^{33} - 3 q^{37} - 6 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 12 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^9 - 3 * q^11 + 12 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 7 * q^23 + 4 * q^25 - 10 * q^27 + 4 * q^29 - 5 * q^31 - 3 * q^33 - 3 * q^37 - 6 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 5 * q^47 - 5 * q^51 + q^53 + 6 * q^55 - 2 * q^57 + 15 * q^59 - 5 * q^61 - 6 * q^65 + 9 * q^67 - 14 * q^69 + 7 * q^73 + 4 * q^75 - q^79 - q^81 - 24 * q^83 + 10 * q^85 - 2 * q^87 + 7 * q^89 - 5 * q^93 + q^95 + 4 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more