LMFDB - Embedded newform 1568.2.t.d.753.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(177,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("1568.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.t (of order $6$, degree $2$, not 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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.432972864.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{6} + 4x^{5} - 6x^{4} + 8x^{3} + 4x^{2} - 8x + 16 $ x^8 - x^7 + x^6 + 4x^5 - 6x^4 + 8x^3 + 4x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			753.1
Root			$0.630783 - 1.26575i$ of defining polynomial
Character	$\chi$	$=$	1568.753
Dual form			1568.2.t.d.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.61578 - 1.51022i) q^{3} +(1.46890 - 0.848071i) q^{5} +(3.06155 + 5.30277i) q^{9} +O(q^{10})$	$q+(-2.61578 - 1.51022i) q^{3} +(1.46890 - 0.848071i) q^{5} +(3.06155 + 5.30277i) q^{9} +(1.14688 + 0.662153i) q^{11} -1.69614i q^{13} -5.12311 q^{15} +(-1.00000 + 1.73205i) q^{17} +(2.61578 - 1.51022i) q^{19} +(2.56155 + 4.43674i) q^{23} +(-1.06155 + 1.83866i) q^{25} -9.43318i q^{27} +6.04090i q^{29} +(-5.12311 + 8.87348i) q^{31} +(-2.00000 - 3.46410i) q^{33} +(-5.23157 + 3.02045i) q^{37} +(-2.56155 + 4.43674i) q^{39} +4.24621 q^{41} -1.32431i q^{43} +(8.99424 + 5.19283i) q^{45} +(5.23157 - 3.02045i) q^{51} +(-2.29377 - 1.32431i) q^{53} +2.24621 q^{55} -9.12311 q^{57} +(0.322018 + 0.185917i) q^{59} +(-1.46890 + 0.848071i) q^{61} +(-1.43845 - 2.49146i) q^{65} +(9.96029 + 5.75058i) q^{67} -15.4741i q^{69} +8.00000 q^{71} +(3.00000 - 5.19615i) q^{73} +(5.55359 - 3.20636i) q^{75} +(-5.06155 + 8.76687i) q^{81} -5.66906i q^{83} +3.39228i q^{85} +(9.12311 - 15.8017i) q^{87} +(8.12311 + 14.0696i) q^{89} +(26.8019 - 15.4741i) q^{93} +(2.56155 - 4.43674i) q^{95} +12.2462 q^{97} +8.10887i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 8 q^{15} - 8 q^{17} + 4 q^{23} + 8 q^{25} - 8 q^{31} - 16 q^{33} - 4 q^{39} - 32 q^{41} - 48 q^{55} - 40 q^{57} - 28 q^{65} + 64 q^{71} + 24 q^{73} - 24 q^{81} + 40 q^{87} + 32 q^{89} + 4 q^{95} + 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 - 8 * q^15 - 8 * q^17 + 4 * q^23 + 8 * q^25 - 8 * q^31 - 16 * q^33 - 4 * q^39 - 32 * q^41 - 48 * q^55 - 40 * q^57 - 28 * q^65 + 64 * q^71 + 24 * q^73 - 24 * q^81 + 40 * q^87 + 32 * q^89 + 4 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−2.61578	−	1.51022i	−1.51022	−	0.871928i	−0.999929	−	0.0119288i	$-0.996203\pi$
$3$	−2.61578	−	1.51022i	−1.51022	−	0.871928i	−0.510295	−	0.859999i	$-0.670464\pi$
$4$		0			0
$5$	1.46890	−	0.848071i	0.656913	−	0.379269i	−0.134187	−	0.990956i	$-0.542842\pi$
$5$	1.46890	−	0.848071i	0.656913	−	0.379269i	0.791100	+	0.611687i	$0.209509\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	3.06155	+	5.30277i	1.02052	+	1.76759i
$10$		0			0
$11$	1.14688	+	0.662153i	0.345798	+	0.199647i	0.662833	−	0.748767i	$-0.269354\pi$
$11$	1.14688	+	0.662153i	0.345798	+	0.199647i	−0.317035	+	0.948414i	$0.602687\pi$
$12$		0			0
$13$		−	1.69614i		−	0.470425i	−0.971944	−	0.235212i	$-0.924421\pi$
$13$		−	1.69614i		−	0.470425i	0.971944	−	0.235212i	$-0.0755786\pi$
$14$		0			0
$15$	−5.12311			−1.32278
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	2.61578	−	1.51022i	0.600102	−	0.346469i	−0.168980	−	0.985620i	$-0.554047\pi$
$19$	2.61578	−	1.51022i	0.600102	−	0.346469i	0.769082	+	0.639150i	$0.220714\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.56155	+	4.43674i	0.534121	+	0.925124i	0.999205	+	0.0398580i	$0.0126905\pi$
$23$	2.56155	+	4.43674i	0.534121	+	0.925124i	−0.465085	+	0.885266i	$0.653976\pi$
$24$		0			0
$25$	−1.06155	+	1.83866i	−0.212311	+	0.367733i
$26$		0			0
$27$		−	9.43318i		−	1.81542i
$28$		0			0
$29$			6.04090i			1.12177i	0.827895	+	0.560883i	$0.189538\pi$
$29$			6.04090i			1.12177i	−0.827895	+	0.560883i	$0.810462\pi$
$30$		0			0
$31$	−5.12311	+	8.87348i	−0.920137	+	1.59372i	−0.120936	+	0.992660i	$0.538589\pi$
$31$	−5.12311	+	8.87348i	−0.920137	+	1.59372i	−0.799201	+	0.601064i	$0.794744\pi$
$32$		0			0
$33$	−2.00000	−	3.46410i	−0.348155	−	0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.23157	+	3.02045i	−0.860065	+	0.496559i	−0.864034	−	0.503434i	$-0.832070\pi$
$37$	−5.23157	+	3.02045i	−0.860065	+	0.496559i	0.00396926	+	0.999992i	$0.498737\pi$
$38$		0			0
$39$	−2.56155	+	4.43674i	−0.410177	+	0.710447i
$40$		0			0
$41$	4.24621			0.663147			0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621			0.663147			0.331573	+	0.943429i	$0.392421\pi$
$42$		0			0
$43$		−	1.32431i		−	0.201955i	−0.994889	−	0.100977i	$-0.967803\pi$
$43$		−	1.32431i		−	0.201955i	0.994889	−	0.100977i	$-0.0321970\pi$
$44$		0			0
$45$	8.99424	+	5.19283i	1.34078	+	0.774101i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	5.23157	−	3.02045i	0.732566	−	0.422947i
$52$		0			0
$53$	−2.29377	−	1.32431i	−0.315073	−	0.181908i	0.334121	−	0.942530i	$-0.391560\pi$
$53$	−2.29377	−	1.32431i	−0.315073	−	0.181908i	−0.649194	+	0.760623i	$0.724894\pi$
$54$		0			0
$55$	2.24621			0.302879
$56$		0			0
$57$	−9.12311			−1.20838
$58$		0			0
$59$	0.322018	+	0.185917i	0.0419231	+	0.0242043i	0.520815	−	0.853670i	$-0.325628\pi$
$59$	0.322018	+	0.185917i	0.0419231	+	0.0242043i	−0.478892	+	0.877874i	$0.658961\pi$
$60$		0			0
$61$	−1.46890	+	0.848071i	−0.188074	+	0.108584i	−0.591080	−	0.806613i	$-0.701298\pi$
$61$	−1.46890	+	0.848071i	−0.188074	+	0.108584i	0.403007	+	0.915197i	$0.367965\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.43845	−	2.49146i	−0.178417	−	0.309028i
$66$		0			0
$67$	9.96029	+	5.75058i	1.21684	+	0.702545i	0.964241	−	0.265026i	$-0.0853804\pi$
$67$	9.96029	+	5.75058i	1.21684	+	0.702545i	0.252602	+	0.967570i	$0.418714\pi$
$68$		0			0
$69$		−	15.4741i		−	1.86286i
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	−0.635323	−	0.772246i	$-0.719133\pi$
$73$	3.00000	−	5.19615i	0.351123	−	0.608164i	0.986447	+	0.164083i	$0.0524664\pi$
$74$		0			0
$75$	5.55359	−	3.20636i	0.641273	−	0.370239i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	−5.06155	+	8.76687i	−0.562395	+	0.974096i
$82$		0			0
$83$		−	5.66906i		−	0.622260i	−0.950367	−	0.311130i	$-0.899292\pi$
$83$		−	5.66906i		−	0.622260i	0.950367	−	0.311130i	$-0.100708\pi$
$84$		0			0
$85$			3.39228i			0.367945i
$86$		0			0
$87$	9.12311	−	15.8017i	0.978100	−	1.69412i
$88$		0			0
$89$	8.12311	+	14.0696i	0.861047	+	1.49138i	0.870919	+	0.491427i	$0.163524\pi$
$89$	8.12311	+	14.0696i	0.861047	+	1.49138i	−0.00987147	+	0.999951i	$0.503142\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	26.8019	−	15.4741i	2.77923	−	1.60459i
$94$		0			0
$95$	2.56155	−	4.43674i	0.262810	−	0.455200i
$96$		0			0
$97$	12.2462			1.24341			0.621707	−	0.783250i	$-0.286439\pi$
$97$	12.2462			1.24341			0.621707	+	0.783250i	$0.286439\pi$
$98$		0			0
$99$			8.10887i			0.814972i
$100$		0			0
$101$	−8.99424	−	5.19283i	−0.894960	−	0.516705i	−0.0193984	−	0.999812i	$-0.506175\pi$
$101$	−8.99424	−	5.19283i	−0.894960	−	0.516705i	−0.875562	+	0.483106i	$0.839508\pi$
$102$		0			0
$103$	1.12311	+	1.94528i	0.110663	+	0.191674i	0.916038	−	0.401092i	$-0.131369\pi$
$103$	1.12311	+	1.94528i	0.110663	+	0.191674i	−0.805375	+	0.592766i	$0.798036\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.2541	−	7.07488i	1.18464	−	0.683955i	0.227560	−	0.973764i	$-0.426925\pi$
$107$	12.2541	−	7.07488i	1.18464	−	0.683955i	0.957084	+	0.289809i	$0.0935919\pi$
$108$		0			0
$109$	15.6947	+	9.06134i	1.50328	+	0.867919i	0.999993	+	0.00380035i	$0.00120969\pi$
$109$	15.6947	+	9.06134i	1.50328	+	0.867919i	0.503288	+	0.864119i	$0.332124\pi$
$110$		0			0
$111$	18.2462			1.73185
$112$		0			0
$113$	4.87689			0.458780			0.229390	−	0.973335i	$-0.426327\pi$
$113$	4.87689			0.458780			0.229390	+	0.973335i	$0.426327\pi$
$114$		0			0
$115$	7.52534	+	4.34475i	0.701741	+	0.405150i
$116$		0			0
$117$	8.99424	−	5.19283i	0.831518	−	0.480077i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.62311	−	8.00745i	−0.420282	−	0.727950i
$122$		0			0
$123$	−11.1072	−	6.41273i	−1.00150	−	0.578216i
$124$		0			0
$125$			12.0818i			1.08063i
$126$		0			0
$127$	−13.1231			−1.16449			−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231			−1.16449			−0.582244	+	0.813014i	$0.697825\pi$
$128$		0			0
$129$	−2.00000	+	3.46410i	−0.176090	+	0.304997i
$130$		0			0
$131$	−10.7852	+	6.22681i	−0.942303	+	0.544039i	−0.890682	−	0.454628i	$-0.849772\pi$
$131$	−10.7852	+	6.22681i	−0.942303	+	0.544039i	−0.0516218	+	0.998667i	$0.516439\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−8.00000	−	13.8564i	−0.688530	−	1.19257i
$136$		0			0
$137$	8.12311	−	14.0696i	0.694004	−	1.20205i	−0.276512	−	0.961011i	$-0.589178\pi$
$137$	8.12311	−	14.0696i	0.694004	−	1.20205i	0.970515	−	0.241039i	$-0.0774882\pi$
$138$		0			0
$139$			8.31768i			0.705496i	0.935718	+	0.352748i	$0.114753\pi$
$139$			8.31768i			0.705496i	−0.935718	+	0.352748i	$0.885247\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.12311	−	1.94528i	0.0939188	−	0.162672i
$144$		0			0
$145$	5.12311	+	8.87348i	0.425451	+	0.736902i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.7569	+	7.36520i	−1.04509	+	0.603381i	−0.921270	−	0.388924i	$-0.872847\pi$
$149$	−12.7569	+	7.36520i	−1.04509	+	0.603381i	−0.123817	+	0.992305i	$0.539513\pi$
$150$		0			0
$151$	5.43845	−	9.41967i	0.442575	−	0.766562i	−0.555305	−	0.831647i	$-0.687399\pi$
$151$	5.43845	−	9.41967i	0.442575	−	0.766562i	0.997880	+	0.0650850i	$0.0207319\pi$
$152$		0			0
$153$	−12.2462			−0.990048
$154$		0			0
$155$			17.3790i			1.39592i
$156$		0			0
$157$	7.34451	+	4.24035i	0.586155	+	0.338417i	0.763576	−	0.645718i	$-0.223442\pi$
$157$	7.34451	+	4.24035i	0.586155	+	0.338417i	−0.177420	+	0.984135i	$0.556775\pi$
$158$		0			0
$159$	4.00000	+	6.92820i	0.317221	+	0.549442i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−11.6100	+	6.70305i	−0.909367	+	0.525023i	−0.880227	−	0.474552i	$-0.842610\pi$
$163$	−11.6100	+	6.70305i	−0.909367	+	0.525023i	−0.0291396	+	0.999575i	$0.509277\pi$
$164$		0			0
$165$	−5.87560	−	3.39228i	−0.457415	−	0.264089i
$166$		0			0
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	10.1231			0.778700
$170$		0			0
$171$	16.0167	+	9.24726i	1.22483	+	0.707156i
$172$		0			0
$173$	−16.5196	+	9.53758i	−1.25596	+	0.725129i	−0.972287	−	0.233792i	$-0.924886\pi$
$173$	−16.5196	+	9.53758i	−1.25596	+	0.725129i	−0.283673	+	0.958921i	$0.591553\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.561553	−	0.972638i	−0.0422089	−	0.0731079i
$178$		0			0
$179$	4.08469	+	2.35829i	0.305304	+	0.176267i	0.644823	−	0.764332i	$-0.276931\pi$
$179$	4.08469	+	2.35829i	0.305304	+	0.176267i	−0.339519	+	0.940599i	$0.610264\pi$
$180$		0			0
$181$		−	6.99337i		−	0.519813i	−0.965634	−	0.259906i	$-0.916308\pi$
$181$		−	6.99337i		−	0.519813i	0.965634	−	0.259906i	$-0.0836917\pi$
$182$		0			0
$183$	5.12311			0.378711
$184$		0			0
$185$	−5.12311	+	8.87348i	−0.376658	+	0.652391i
$186$		0			0
$187$	−2.29377	+	1.32431i	−0.167737	+	0.0968429i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	0.995610	+	0.0935936i	$0.0298354\pi$
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	−0.416751	+	0.909021i	$0.636831\pi$
$192$		0			0
$193$	5.56155	−	9.63289i	0.400329	−	0.693391i	−0.593436	−	0.804881i	$-0.702229\pi$
$193$	5.56155	−	9.63289i	0.400329	−	0.693391i	0.993766	+	0.111490i	$0.0355624\pi$
$194$		0			0
$195$			8.68951i			0.622269i
$196$		0			0
$197$		−	21.5150i		−	1.53288i	−0.642317	−	0.766439i	$-0.722027\pi$
$197$		−	21.5150i		−	1.53288i	0.642317	−	0.766439i	$-0.277973\pi$
$198$		0			0
$199$	−9.12311	+	15.8017i	−0.646720	+	1.12015i	0.337182	+	0.941440i	$0.390526\pi$
$199$	−9.12311	+	15.8017i	−0.646720	+	1.12015i	−0.983901	+	0.178712i	$0.942807\pi$
$200$		0			0
$201$	−17.3693	−	30.0845i	−1.22514	−	2.12200i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.23726	−	3.60109i	0.435629	−	0.251511i
$206$		0			0
$207$	−15.6847	+	27.1666i	−1.09016	+	1.88821i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$		−	4.71659i		−	0.324703i	−0.986733	−	0.162352i	$-0.948092\pi$
$211$		−	4.71659i		−	0.324703i	0.986733	−	0.162352i	$-0.0519079\pi$
$212$		0			0
$213$	−20.9263	−	12.0818i	−1.43384	−	0.827831i
$214$		0			0
$215$	−1.12311	−	1.94528i	−0.0765952	−	0.132667i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−15.6947	+	9.06134i	−1.06055	+	0.612309i
$220$		0			0
$221$	2.93780	+	1.69614i	0.197618	+	0.114095i
$222$		0			0
$223$	5.75379			0.385302			0.192651	−	0.981267i	$-0.438291\pi$
$223$	5.75379			0.385302			0.192651	+	0.981267i	$0.438291\pi$
$224$		0			0
$225$	−13.0000			−0.866667
$226$		0			0
$227$	−8.49139	−	4.90251i	−0.563593	−	0.325391i	0.190993	−	0.981591i	$-0.438829\pi$
$227$	−8.49139	−	4.90251i	−0.563593	−	0.325391i	−0.754586	+	0.656201i	$0.772163\pi$
$228$		0			0
$229$	22.3952	−	12.9299i	1.47992	−	0.854429i	0.480174	−	0.877173i	$-0.340574\pi$
$229$	22.3952	−	12.9299i	1.47992	−	0.854429i	0.999741	+	0.0227441i	$0.00724029\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.12311	+	14.0696i	0.532162	+	0.921732i	0.999295	+	0.0375449i	$0.0119537\pi$
$233$	8.12311	+	14.0696i	0.532162	+	0.921732i	−0.467133	+	0.884187i	$0.654713\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.6155			1.13945			0.569727	−	0.821834i	$-0.307049\pi$
$239$	17.6155			1.13945			0.569727	+	0.821834i	$0.307049\pi$
$240$		0			0
$241$	1.87689	−	3.25088i	0.120901	−	0.209407i	−0.799222	−	0.601036i	$-0.794755\pi$
$241$	1.87689	−	3.25088i	0.120901	−	0.209407i	0.920123	+	0.391629i	$0.128088\pi$
$242$		0			0
$243$	1.97175	−	1.13839i	0.126488	−	0.0730277i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.56155	−	4.43674i	−0.162988	−	0.282303i
$248$		0			0
$249$	−8.56155	+	14.8290i	−0.542566	+	0.939753i
$250$		0			0
$251$		−	10.9663i		−	0.692186i	−0.938200	−	0.346093i	$-0.887508\pi$
$251$		−	10.9663i		−	0.692186i	0.938200	−	0.346093i	$-0.112492\pi$
$252$		0			0
$253$			6.78456i			0.426542i
$254$		0			0
$255$	5.12311	−	8.87348i	0.320821	−	0.555679i
$256$		0			0
$257$	−11.2462	−	19.4790i	−0.701519	−	1.21507i	−0.967933	−	0.251208i	$-0.919172\pi$
$257$	−11.2462	−	19.4790i	−0.701519	−	1.21507i	0.266414	−	0.963859i	$-0.414161\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−32.0335	+	18.4945i	−1.98282	+	1.14478i
$262$		0			0
$263$	6.24621	−	10.8188i	0.385158	−	0.667113i	−0.606633	−	0.794982i	$-0.707480\pi$
$263$	6.24621	−	10.8188i	0.385158	−	0.667113i	0.991791	+	0.127869i	$0.0408137\pi$
$264$		0			0
$265$	−4.49242			−0.275967
$266$		0			0
$267$		−	49.0708i		−	3.00309i
$268$		0			0
$269$	−10.2823	−	5.93649i	−0.626923	−	0.361954i	0.152636	−	0.988282i	$-0.451224\pi$
$269$	−10.2823	−	5.93649i	−0.626923	−	0.361954i	−0.779560	+	0.626328i	$0.784557\pi$
$270$		0			0
$271$	5.12311	+	8.87348i	0.311207	+	0.539025i	0.978624	−	0.205658i	$-0.0659336\pi$
$271$	5.12311	+	8.87348i	0.311207	+	0.539025i	−0.667417	+	0.744684i	$0.732600\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.43495	+	1.40582i	−0.146833	+	0.0847742i
$276$		0			0
$277$	−2.29377	−	1.32431i	−0.137819	−	0.0795699i	0.429505	−	0.903064i	$-0.358688\pi$
$277$	−2.29377	−	1.32431i	−0.137819	−	0.0795699i	−0.567324	+	0.823495i	$0.692021\pi$
$278$		0			0
$279$	−62.7386			−3.75606
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−18.9545	−	10.9434i	−1.12673	−	0.650518i	−0.183619	−	0.982997i	$-0.558781\pi$
$283$	−18.9545	−	10.9434i	−1.12673	−	0.650518i	−0.943110	+	0.332480i	$0.892115\pi$
$284$		0			0
$285$	−13.4009	+	7.73704i	−0.793803	+	0.458303i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	−32.0335	−	18.4945i	−1.87783	−	1.08417i
$292$		0			0
$293$			10.3857i			0.606736i	0.952873	+	0.303368i	$0.0981112\pi$
$293$			10.3857i			0.606736i	−0.952873	+	0.303368i	$0.901889\pi$
$294$		0			0
$295$	0.630683			0.0367198
$296$		0			0
$297$	6.24621	−	10.8188i	0.362442	−	0.627768i
$298$		0			0
$299$	7.52534	−	4.34475i	0.435201	−	0.251264i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	15.6847	+	27.1666i	0.901060	+	1.56068i
$304$		0			0
$305$	−1.43845	+	2.49146i	−0.0823652	+	0.142661i
$306$		0			0
$307$			25.2791i			1.44275i	0.692543	+	0.721377i	$0.256490\pi$
$307$			25.2791i			1.44275i	−0.692543	+	0.721377i	$0.743510\pi$
$308$		0			0
$309$		−	6.78456i		−	0.385960i
$310$		0			0
$311$	−6.24621	+	10.8188i	−0.354190	+	0.613475i	−0.986979	−	0.160849i	$-0.948577\pi$
$311$	−6.24621	+	10.8188i	−0.354190	+	0.613475i	0.632789	+	0.774324i	$0.281910\pi$
$312$		0			0
$313$	10.3693	+	17.9602i	0.586108	+	1.01517i	0.994736	+	0.102468i	$0.0326741\pi$
$313$	10.3693	+	17.9602i	0.586108	+	1.01517i	−0.408628	+	0.912701i	$0.633993\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.58184	−	2.06798i	0.201176	−	0.116149i	−0.396028	−	0.918238i	$-0.629612\pi$
$317$	3.58184	−	2.06798i	0.201176	−	0.116149i	0.597204	+	0.802089i	$0.296278\pi$
$318$		0			0
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$		0			0
$321$	−42.7386			−2.38544
$322$		0			0
$323$			6.04090i			0.336124i
$324$		0			0
$325$	3.11863	+	1.80054i	0.172991	+	0.0998762i
$326$		0			0
$327$	−27.3693	−	47.4050i	−1.51353	−	2.62151i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.8981	+	7.44672i	−0.708943	+	0.409309i	−0.810670	−	0.585504i	$-0.800897\pi$
$331$	−12.8981	+	7.44672i	−0.708943	+	0.409309i	0.101726	+	0.994812i	$0.467563\pi$
$332$		0			0
$333$	−32.0335	−	18.4945i	−1.75542	−	1.01349i
$334$		0			0
$335$	19.5076			1.06581
$336$		0			0
$337$	−0.876894			−0.0477675			−0.0238837	−	0.999715i	$-0.507603\pi$
$337$	−0.876894			−0.0477675			−0.0238837	+	0.999715i	$0.507603\pi$
$338$		0			0
$339$	−12.7569	−	7.36520i	−0.692860	−	0.400023i
$340$		0			0
$341$	−11.7512	+	6.78456i	−0.636364	+	0.367405i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−13.1231	−	22.7299i	−0.706524	−	1.22374i
$346$		0			0
$347$	7.02249	+	4.05444i	0.376987	+	0.217654i	0.676507	−	0.736437i	$-0.263493\pi$
$347$	7.02249	+	4.05444i	0.376987	+	0.217654i	−0.299520	+	0.954090i	$0.596826\pi$
$348$		0			0
$349$			27.3471i			1.46385i	0.681383	+	0.731927i	$0.261379\pi$
$349$			27.3471i			1.46385i	−0.681383	+	0.731927i	$0.738621\pi$
$350$		0			0
$351$	−16.0000			−0.854017
$352$		0			0
$353$	−3.87689	+	6.71498i	−0.206346	+	0.357402i	−0.950561	−	0.310538i	$-0.899491\pi$
$353$	−3.87689	+	6.71498i	−0.206346	+	0.357402i	0.744215	+	0.667941i	$0.232824\pi$
$354$		0			0
$355$	11.7512	−	6.78456i	0.623689	−	0.360087i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.6847	−	27.1666i	−0.827805	−	1.43380i	−0.899757	−	0.436392i	$-0.856256\pi$
$359$	−15.6847	−	27.1666i	−0.827805	−	1.43380i	0.0719522	−	0.997408i	$-0.477077\pi$
$360$		0			0
$361$	−4.93845	+	8.55364i	−0.259918	+	0.450192i
$362$		0			0
$363$			27.9277i			1.46582i
$364$		0			0
$365$		−	10.1768i		−	0.532680i
$366$		0			0
$367$	5.12311	−	8.87348i	0.267424	−	0.463192i	−0.700772	−	0.713385i	$-0.747161\pi$
$367$	5.12311	−	8.87348i	0.267424	−	0.463192i	0.968196	+	0.250194i	$0.0804943\pi$
$368$		0			0
$369$	13.0000	+	22.5167i	0.676753	+	1.17217i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−12.7569	+	7.36520i	−0.660528	+	0.381356i	−0.792478	−	0.609901i	$-0.791209\pi$
$373$	−12.7569	+	7.36520i	−0.660528	+	0.381356i	0.131950	+	0.991256i	$0.457876\pi$
$374$		0			0
$375$	18.2462	−	31.6034i	0.942230	−	1.63199i
$376$		0			0
$377$	10.2462			0.527707
$378$		0			0
$379$			36.4084i			1.87017i	0.354418	+	0.935087i	$0.384679\pi$
$379$			36.4084i			1.87017i	−0.354418	+	0.935087i	$0.615321\pi$
$380$		0			0
$381$	34.3272	+	19.8188i	1.75864	+	1.01535i
$382$		0			0
$383$	−2.24621	−	3.89055i	−0.114776	−	0.198798i	0.802914	−	0.596095i	$-0.203282\pi$
$383$	−2.24621	−	3.89055i	−0.114776	−	0.198798i	−0.917690	+	0.397297i	$0.869948\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.02249	−	4.05444i	0.356973	−	0.206099i
$388$		0			0
$389$	−21.5703	−	12.4536i	−1.09366	−	0.631424i	−0.159110	−	0.987261i	$-0.550863\pi$
$389$	−21.5703	−	12.4536i	−1.09366	−	0.631424i	−0.934548	+	0.355837i	$0.884196\pi$
$390$		0			0
$391$	−10.2462			−0.518173
$392$		0			0
$393$	37.6155			1.89745
$394$		0			0
$395$		0			0
$396$		0			0
$397$	17.8076	−	10.2812i	0.893740	−	0.516001i	0.0185761	−	0.999827i	$-0.494087\pi$
$397$	17.8076	−	10.2812i	0.893740	−	0.516001i	0.875164	+	0.483826i	$0.160753\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.438447	+	0.759413i	0.0218950	+	0.0379233i	0.876765	−	0.480919i	$-0.159697\pi$
$401$	0.438447	+	0.759413i	0.0218950	+	0.0379233i	−0.854870	+	0.518842i	$0.826363\pi$
$402$		0			0
$403$	15.0507	+	8.68951i	0.749727	+	0.432855i
$404$		0			0
$405$			17.1702i			0.853195i
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	0.341994	+	0.939702i	$0.388898\pi$
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	−0.984803	+	0.173675i	$0.944436\pi$
$410$		0			0
$411$	−42.4966	+	24.5354i	−2.09620	+	1.21024i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.80776	−	8.32729i	−0.236004	−	0.408771i
$416$		0			0
$417$	12.5616	−	21.7572i	0.615142	−	1.06546i
$418$		0			0
$419$		−	27.9277i		−	1.36436i	−0.731185	−	0.682179i	$-0.761033\pi$
$419$		−	27.9277i		−	1.36436i	0.731185	−	0.682179i	$-0.238967\pi$
$420$		0			0
$421$			26.8122i			1.30675i	0.757036	+	0.653373i	$0.226647\pi$
$421$			26.8122i			1.30675i	−0.757036	+	0.653373i	$0.773353\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.12311	−	3.67733i	−0.102986	−	0.178377i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−5.87560	+	3.39228i	−0.283677	+	0.163781i
$430$		0			0
$431$	8.80776	−	15.2555i	0.424255	−	0.734831i	−0.572095	−	0.820187i	$-0.693869\pi$
$431$	8.80776	−	15.2555i	0.424255	−	0.734831i	0.996351	+	0.0853557i	$0.0272027\pi$
$432$		0			0
$433$	−18.4924			−0.888689			−0.444345	−	0.895856i	$-0.646563\pi$
$433$	−18.4924			−0.888689			−0.444345	+	0.895856i	$0.646563\pi$
$434$		0			0
$435$		−	30.9481i		−	1.48385i
$436$		0			0
$437$	13.4009	+	7.73704i	0.641054	+	0.370113i
$438$		0			0
$439$	11.3693	+	19.6922i	0.542628	+	0.939859i	0.998752	+	0.0499429i	$0.0159039\pi$
$439$	11.3693	+	19.6922i	0.542628	+	0.939859i	−0.456124	+	0.889916i	$0.650763\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.37845	−	3.68260i	0.303049	−	0.174966i	−0.340763	−	0.940149i	$-0.610685\pi$
$443$	6.37845	−	3.68260i	0.303049	−	0.174966i	0.643812	+	0.765184i	$0.277352\pi$
$444$		0			0
$445$	23.8641	+	13.7779i	1.13127	+	0.653137i
$446$		0			0
$447$	44.4924			2.10442
$448$		0			0
$449$	16.7386			0.789945			0.394972	−	0.918693i	$-0.370754\pi$
$449$	16.7386			0.789945			0.394972	+	0.918693i	$0.370754\pi$
$450$		0			0
$451$	4.86991	+	2.81164i	0.229315	+	0.132395i
$452$		0			0
$453$	−28.4516	+	16.4265i	−1.33677	+	0.771786i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.68466	−	15.0423i	−0.406251	−	0.703648i	0.588215	−	0.808705i	$-0.299831\pi$
$457$	−8.68466	−	15.0423i	−0.406251	−	0.703648i	−0.994466	+	0.105057i	$0.966498\pi$
$458$		0			0
$459$	16.3387	+	9.43318i	0.762627	+	0.440303i
$460$		0			0
$461$			24.3724i			1.13514i	0.823327	+	0.567568i	$0.192115\pi$
$461$			24.3724i			1.13514i	−0.823327	+	0.567568i	$0.807885\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	26.2462	−	45.4598i	1.21714	−	2.10815i
$466$		0			0
$467$	2.61578	−	1.51022i	0.121044	−	0.0698848i	−0.438255	−	0.898850i	$-0.644403\pi$
$467$	2.61578	−	1.51022i	0.121044	−	0.0698848i	0.559300	+	0.828966i	$0.311070\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.8078	−	22.1837i	−0.590151	−	1.02217i
$472$		0			0
$473$	0.876894	−	1.51883i	0.0403196	−	0.0698357i
$474$		0			0
$475$			6.41273i			0.294236i
$476$		0			0
$477$		−	16.2177i		−	0.742559i
$478$		0			0
$479$	5.12311	−	8.87348i	0.234081	−	0.405440i	−0.724924	−	0.688828i	$-0.758125\pi$
$479$	5.12311	−	8.87348i	0.234081	−	0.405440i	0.959005	+	0.283389i	$0.0914587\pi$
$480$		0			0
$481$	5.12311	+	8.87348i	0.233594	+	0.404596i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.9885	−	10.3857i	0.816815	−	0.471588i
$486$		0			0
$487$	0.315342	−	0.546188i	0.0142895	−	0.0247501i	−0.858792	−	0.512324i	$-0.828785\pi$
$487$	0.315342	−	0.546188i	0.0142895	−	0.0247501i	0.873082	+	0.487574i	$0.162118\pi$
$488$		0			0
$489$	40.4924			1.83113
$490$		0			0
$491$		−	34.1774i		−	1.54240i	−0.636590	−	0.771202i	$-0.719656\pi$
$491$		−	34.1774i		−	1.54240i	0.636590	−	0.771202i	$-0.280344\pi$
$492$		0			0
$493$	−10.4631	−	6.04090i	−0.471236	−	0.272068i
$494$		0			0
$495$	6.87689	+	11.9111i	0.309093	+	0.535366i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7.66652	−	4.42627i	0.343201	−	0.198147i	−0.318486	−	0.947928i	$-0.603174\pi$
$499$	7.66652	−	4.42627i	0.343201	−	0.198147i	0.661687	+	0.749781i	$0.269841\pi$
$500$		0			0
$501$	20.9263	+	12.0818i	0.934917	+	0.539775i
$502$		0			0
$503$	−13.7538			−0.613251			−0.306626	−	0.951830i	$-0.599200\pi$
$503$	−13.7538			−0.613251			−0.306626	+	0.951830i	$0.599200\pi$
$504$		0			0
$505$	−17.6155			−0.783881
$506$		0			0
$507$	−26.4799	−	15.2882i	−1.17601	−	0.678971i
$508$		0			0
$509$	−26.6211	+	15.3697i	−1.17996	+	0.681249i	−0.956005	−	0.293351i	$-0.905229\pi$
$509$	−26.6211	+	15.3697i	−1.17996	+	0.681249i	−0.223953	+	0.974600i	$0.571896\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−14.2462	−	24.6752i	−0.628986	−	1.08944i
$514$		0			0
$515$	3.29946	+	1.90495i	0.145392	+	0.0839419i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	57.6155			2.52904
$520$		0			0
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	−0.517866	−	0.855462i	$-0.673273\pi$
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	0.999785	+	0.0207541i	$0.00660670\pi$
$522$		0			0
$523$	35.6549	−	20.5854i	1.55908	−	0.900136i	0.561736	−	0.827316i	$-0.310134\pi$
$523$	35.6549	−	20.5854i	1.55908	−	0.900136i	0.997345	−	0.0728196i	$-0.0231997\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.2462	−	17.7470i	−0.446332	−	0.773070i
$528$		0			0
$529$	−1.62311	+	2.81130i	−0.0705698	+	0.122230i
$530$		0			0
$531$			2.27678i			0.0988038i
$532$		0			0
$533$		−	7.20217i		−	0.311961i
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$		0			0
$537$	−7.12311	−	12.3376i	−0.307385	−	0.532406i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−11.4688	+	6.62153i	−0.493084	+	0.284682i	−0.725853	−	0.687850i	$-0.758555\pi$
$541$	−11.4688	+	6.62153i	−0.493084	+	0.284682i	0.232769	+	0.972532i	$0.425221\pi$
$542$		0			0
$543$	−10.5616	+	18.2931i	−0.453240	+	0.785034i
$544$		0			0
$545$	30.7386			1.31670
$546$		0			0
$547$		−	9.59621i		−	0.410304i	−0.978730	−	0.205152i	$-0.934231\pi$
$547$		−	9.59621i		−	0.410304i	0.978730	−	0.205152i	$-0.0657689\pi$
$548$		0			0
$549$	−8.99424	−	5.19283i	−0.383865	−	0.221624i
$550$		0			0
$551$	9.12311	+	15.8017i	0.388657	+	0.673174i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	26.8019	−	15.4741i	1.13768	−	0.656838i
$556$		0			0
$557$	2.29377	+	1.32431i	0.0971900	+	0.0561127i	0.547807	−	0.836605i	$-0.315463\pi$
$557$	2.29377	+	1.32431i	0.0971900	+	0.0561127i	−0.450617	+	0.892717i	$0.648796\pi$
$558$		0			0
$559$	−2.24621			−0.0950046
$560$		0			0
$561$	8.00000			0.337760
$562$		0			0
$563$	25.8358	+	14.9163i	1.08885	+	0.628648i	0.933270	−	0.359175i	$-0.116942\pi$
$563$	25.8358	+	14.9163i	1.08885	+	0.628648i	0.155580	+	0.987823i	$0.450275\pi$
$564$		0			0
$565$	7.16368	−	4.13595i	0.301378	−	0.174001i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.68466	+	11.5782i	0.280235	+	0.485382i	0.971443	−	0.237275i	$-0.0762541\pi$
$569$	6.68466	+	11.5782i	0.280235	+	0.485382i	−0.691207	+	0.722657i	$0.742921\pi$
$570$		0			0
$571$	−8.02818	−	4.63507i	−0.335969	−	0.193972i	0.322519	−	0.946563i	$-0.395470\pi$
$571$	−8.02818	−	4.63507i	−0.335969	−	0.193972i	−0.658488	+	0.752591i	$0.728804\pi$
$572$		0			0
$573$		−	48.3272i		−	2.01890i
$574$		0			0
$575$	−10.8769			−0.453598
$576$		0			0
$577$	−3.87689	+	6.71498i	−0.161397	+	0.279548i	−0.935370	−	0.353671i	$-0.884933\pi$
$577$	−3.87689	+	6.71498i	−0.161397	+	0.279548i	0.773973	+	0.633219i	$0.218267\pi$
$578$		0			0
$579$	−29.0956	+	16.7984i	−1.20917	+	0.698117i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.75379	−	3.03765i	−0.0726345	−	0.125807i
$584$		0			0
$585$	8.80776	−	15.2555i	0.364156	−	0.630737i
$586$		0			0
$587$			21.8868i			0.903365i	0.892179	+	0.451683i	$0.149176\pi$
$587$			21.8868i			0.903365i	−0.892179	+	0.451683i	$0.850824\pi$
$588$		0			0
$589$			30.9481i			1.27520i
$590$		0			0
$591$	−32.4924	+	56.2785i	−1.33656	+	2.31499i
$592$		0			0
$593$	−9.00000	−	15.5885i	−0.369586	−	0.640141i	0.619915	−	0.784669i	$-0.287167\pi$
$593$	−9.00000	−	15.5885i	−0.369586	−	0.640141i	−0.989501	+	0.144528i	$0.953834\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	47.7282	−	27.5559i	1.95338	−	1.12779i
$598$		0			0
$599$	−6.24621	+	10.8188i	−0.255213	+	0.442042i	−0.964953	−	0.262421i	$-0.915479\pi$
$599$	−6.24621	+	10.8188i	−0.255213	+	0.442042i	0.709740	+	0.704464i	$0.248812\pi$
$600$		0			0
$601$	−16.2462			−0.662697			−0.331348	−	0.943508i	$-0.607504\pi$
$601$	−16.2462			−0.662697			−0.331348	+	0.943508i	$0.607504\pi$
$602$		0			0
$603$			70.4228i			2.86784i
$604$		0			0
$605$	−13.5818	−	7.84144i	−0.552178	−	0.318800i
$606$		0			0
$607$	20.4924	+	35.4939i	0.831762	+	1.44065i	0.896640	+	0.442760i	$0.146001\pi$
$607$	20.4924	+	35.4939i	0.831762	+	1.44065i	−0.0648782	+	0.997893i	$0.520666\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.93211	+	1.11550i	0.0780371	+	0.0450547i	0.538511	−	0.842619i	$-0.318987\pi$
$613$	1.93211	+	1.11550i	0.0780371	+	0.0450547i	−0.460474	+	0.887673i	$0.652320\pi$
$614$		0			0
$615$	−21.7538			−0.877197
$616$		0			0
$617$	−29.3693			−1.18236			−0.591182	−	0.806538i	$-0.701339\pi$
$617$	−29.3693			−1.18236			−0.591182	+	0.806538i	$0.701339\pi$
$618$		0			0
$619$	17.0224	+	9.82790i	0.684189	+	0.395017i	0.801431	−	0.598087i	$-0.204072\pi$
$619$	17.0224	+	9.82790i	0.684189	+	0.395017i	−0.117243	+	0.993103i	$0.537405\pi$
$620$		0			0
$621$	41.8526	−	24.1636i	1.67949	−	0.969651i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.93845	+	8.55364i	0.197538	+	0.342146i
$626$		0			0
$627$	−10.4631	−	6.04090i	−0.417858	−	0.241250i
$628$		0			0
$629$		−	12.0818i		−	0.481733i
$630$		0			0
$631$	3.50758			0.139634			0.0698172	−	0.997560i	$-0.477758\pi$
$631$	3.50758			0.139634			0.0698172	+	0.997560i	$0.477758\pi$
$632$		0			0
$633$	−7.12311	+	12.3376i	−0.283118	+	0.490375i
$634$		0			0
$635$	−19.2765	+	11.1293i	−0.764966	+	0.441654i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	24.4924	+	42.4221i	0.968905	+	1.67819i
$640$		0			0
$641$	2.68466	−	4.64996i	0.106038	−	0.183663i	−0.808124	−	0.589012i	$-0.799517\pi$
$641$	2.68466	−	4.64996i	0.106038	−	0.183663i	0.914162	+	0.405350i	$0.132850\pi$
$642$		0			0
$643$		−	5.66906i		−	0.223566i	−0.993733	−	0.111783i	$-0.964344\pi$
$643$		−	5.66906i		−	0.223566i	0.993733	−	0.111783i	$-0.0356562\pi$
$644$		0			0
$645$			6.78456i			0.267142i
$646$		0			0
$647$	21.6155	−	37.4392i	0.849794	−	1.47189i	−0.0315977	−	0.999501i	$-0.510060\pi$
$647$	21.6155	−	37.4392i	0.849794	−	1.47189i	0.881392	−	0.472386i	$-0.156607\pi$
$648$		0			0
$649$	0.246211	+	0.426450i	0.00966464	+	0.0167396i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.4459	+	15.8459i	−1.07404	+	0.620098i	−0.929283	−	0.369369i	$-0.879574\pi$
$653$	−27.4459	+	15.8459i	−1.07404	+	0.620098i	−0.144759	+	0.989467i	$0.546241\pi$
$654$		0			0
$655$	−10.5616	+	18.2931i	−0.412674	+	0.714772i
$656$		0			0
$657$	36.7386			1.43331
$658$		0			0
$659$		−	6.62153i		−	0.257938i	−0.991649	−	0.128969i	$-0.958833\pi$
$659$		−	6.62153i		−	0.257938i	0.991649	−	0.128969i	$-0.0411668\pi$
$660$		0			0
$661$	38.7339	+	22.3630i	1.50657	+	0.869821i	0.999971	+	0.00764267i	$0.00243276\pi$
$661$	38.7339	+	22.3630i	1.50657	+	0.869821i	0.506604	+	0.862179i	$0.330901\pi$
$662$		0			0
$663$	−5.12311	−	8.87348i	−0.198965	−	0.344617i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−26.8019	+	15.4741i	−1.03777	+	0.599159i
$668$		0			0
$669$	−15.0507	−	8.68951i	−0.581893	−	0.335956i
$670$		0			0
$671$	−2.24621			−0.0867140
$672$		0			0
$673$	−38.9848			−1.50276			−0.751378	−	0.659872i	$-0.770610\pi$
$673$	−38.9848			−1.50276			−0.751378	+	0.659872i	$0.770610\pi$
$674$		0			0
$675$	17.3444	+	10.0138i	0.667588	+	0.385432i
$676$		0			0
$677$	14.8698	−	8.58511i	0.571494	−	0.329952i	−0.186252	−	0.982502i	$-0.559634\pi$
$677$	14.8698	−	8.58511i	0.571494	−	0.329952i	0.757746	+	0.652550i	$0.226301\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	14.8078	+	25.6478i	0.567435	+	0.982826i
$682$		0			0
$683$	−0.502848	−	0.290319i	−0.0192409	−	0.0111088i	0.490349	−	0.871526i	$-0.336869\pi$
$683$	−0.502848	−	0.290319i	−0.0192409	−	0.0111088i	−0.509590	+	0.860418i	$0.670203\pi$
$684$		0			0
$685$		−	27.5559i		−	1.05286i
$686$		0			0
$687$	−78.1080			−2.98000
$688$		0			0
$689$	−2.24621	+	3.89055i	−0.0855738	+	0.148218i
$690$		0			0
$691$	33.6435	−	19.4241i	1.27986	−	0.738928i	0.303038	−	0.952978i	$-0.401999\pi$
$691$	33.6435	−	19.4241i	1.27986	−	0.738928i	0.976823	+	0.214051i	$0.0686657\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	7.05398	+	12.2178i	0.267573	+	0.463449i
$696$		0			0
$697$	−4.24621	+	7.35465i	−0.160837	+	0.278577i
$698$		0			0
$699$		−	49.0708i		−	1.85603i
$700$		0			0
$701$			2.23100i			0.0842639i	0.999112	+	0.0421319i	$0.0134150\pi$
$701$			2.23100i			0.0842639i	−0.999112	+	0.0421319i	$0.986585\pi$
$702$		0			0
$703$	−9.12311	+	15.8017i	−0.344084	+	0.595972i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	24.8698	−	14.3586i	0.934004	−	0.539247i	0.0459283	−	0.998945i	$-0.485375\pi$
$709$	24.8698	−	14.3586i	0.934004	−	0.539247i	0.888076	+	0.459697i	$0.152042\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−52.4924			−1.96586
$714$		0			0
$715$		−	3.80989i		−	0.142482i
$716$		0			0
$717$	−46.0784	−	26.6034i	−1.72083	−	0.993522i
$718$		0			0
$719$	−26.2462	−	45.4598i	−0.978819	−	1.69536i	−0.666711	−	0.745317i	$-0.732298\pi$
$719$	−26.2462	−	45.4598i	−0.978819	−	1.69536i	−0.312108	−	0.950047i	$-0.601035\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−9.81910	+	5.66906i	−0.365176	+	0.210835i
$724$		0			0
$725$	−11.1072	−	6.41273i	−0.412510	−	0.238163i
$726$		0			0
$727$	16.9848			0.629933			0.314967	−	0.949103i	$-0.398007\pi$
$727$	16.9848			0.629933			0.314967	+	0.949103i	$0.398007\pi$
$728$		0			0
$729$	23.4924			0.870090
$730$		0			0
$731$	2.29377	+	1.32431i	0.0848380	+	0.0489813i
$732$		0			0
$733$	25.3330	−	14.6260i	0.935695	−	0.540224i	0.0470868	−	0.998891i	$-0.485006\pi$
$733$	25.3330	−	14.6260i	0.935695	−	0.540224i	0.888608	+	0.458667i	$0.151673\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.61553	+	13.1905i	0.280522	+	0.485878i
$738$		0			0
$739$	−18.4913	−	10.6760i	−0.680214	−	0.392722i	0.119721	−	0.992808i	$-0.461800\pi$
$739$	−18.4913	−	10.6760i	−0.680214	−	0.392722i	−0.799936	+	0.600086i	$0.795133\pi$
$740$		0			0
$741$			15.4741i			0.568454i
$742$		0			0
$743$	−0.630683			−0.0231375			−0.0115688	−	0.999933i	$-0.503683\pi$
$743$	−0.630683			−0.0231375			−0.0115688	+	0.999933i	$0.503683\pi$
$744$		0			0
$745$	−12.4924	+	21.6375i	−0.457687	+	0.792737i
$746$		0			0
$747$	30.0617	−	17.3561i	1.09990	−	0.635028i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.31534	+	7.47439i	0.157469	+	0.272744i	0.933955	−	0.357390i	$-0.116333\pi$
$751$	4.31534	+	7.47439i	0.157469	+	0.272744i	−0.776486	+	0.630134i	$0.783000\pi$
$752$		0			0
$753$	−16.5616	+	28.6855i	−0.603537	+	1.04536i
$754$		0			0
$755$		−	18.4487i		−	0.671419i
$756$		0			0
$757$		−	26.3946i		−	0.959328i	−0.877452	−	0.479664i	$-0.840759\pi$
$757$		−	26.3946i		−	0.959328i	0.877452	−	0.479664i	$-0.159241\pi$
$758$		0			0
$759$	10.2462	−	17.7470i	0.371914	−	0.644174i
$760$		0			0
$761$	−4.36932	−	7.56788i	−0.158388	−	0.274335i	0.775900	−	0.630856i	$-0.217296\pi$
$761$	−4.36932	−	7.56788i	−0.158388	−	0.274335i	−0.934287	+	0.356521i	$0.883963\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−17.9885	+	10.3857i	−0.650375	+	0.375494i
$766$		0			0
$767$	0.315342	−	0.546188i	0.0113863	−	0.0197217i
$768$		0			0
$769$	−40.2462			−1.45132			−0.725658	−	0.688056i	$-0.758464\pi$
$769$	−40.2462			−1.45132			−0.725658	+	0.688056i	$0.758464\pi$
$770$		0			0
$771$			67.9372i			2.44670i
$772$		0			0
$773$	−1.46890	−	0.848071i	−0.0528327	−	0.0305030i	0.473351	−	0.880874i	$-0.343044\pi$
$773$	−1.46890	−	0.848071i	−0.0528327	−	0.0305030i	−0.526184	+	0.850371i	$0.676378\pi$
$774$		0			0
$775$	−10.8769	−	18.8393i	−0.390710	−	0.676729i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11.1072	−	6.41273i	0.397956	−	0.229760i
$780$		0			0
$781$	9.17507	+	5.29723i	0.328310	+	0.189550i
$782$		0			0
$783$	56.9848			2.03647
$784$		0			0
$785$	14.3845			0.513404
$786$		0			0
$787$	9.13543	+	5.27434i	0.325643	+	0.188010i	0.653905	−	0.756577i	$-0.273130\pi$
$787$	9.13543	+	5.27434i	0.325643	+	0.188010i	−0.328262	+	0.944587i	$0.606463\pi$
$788$		0			0
$789$	−32.6775	+	18.8664i	−1.16335	+	0.671660i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.43845	+	2.49146i	0.0510808	+	0.0884745i
$794$		0			0
$795$	11.7512	+	6.78456i	0.416772	+	0.240624i
$796$		0			0
$797$		−	25.8597i		−	0.915998i	−0.888953	−	0.457999i	$-0.848566\pi$
$797$		−	25.8597i		−	0.915998i	0.888953	−	0.457999i	$-0.151434\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−49.7386	+	86.1498i	−1.75743	+	3.04395i
$802$		0			0
$803$	6.88130	−	3.97292i	0.242836	−	0.140201i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	17.9309	+	31.0572i	0.631197	+	1.09326i
$808$		0			0
$809$	−18.9309	+	32.7892i	−0.665574	+	1.15281i	0.313555	+	0.949570i	$0.398480\pi$
$809$	−18.9309	+	32.7892i	−0.665574	+	1.15281i	−0.979129	+	0.203238i	$0.934853\pi$
$810$		0			0
$811$		−	15.8459i		−	0.556425i	−0.960520	−	0.278213i	$-0.910258\pi$
$811$		−	15.8459i		−	0.556425i	0.960520	−	0.278213i	$-0.0897420\pi$
$812$		0			0
$813$		−	30.9481i		−	1.08540i
$814$		0			0
$815$	−11.3693	+	19.6922i	−0.398250	+	0.689789i
$816$		0			0
$817$	−2.00000	−	3.46410i	−0.0699711	−	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.0956	−	16.7984i	1.01545	−	0.586268i	0.102664	−	0.994716i	$-0.467263\pi$
$821$	29.0956	−	16.7984i	1.01545	−	0.586268i	0.912781	+	0.408449i	$0.133930\pi$
$822$		0			0
$823$	−16.4924	+	28.5657i	−0.574890	+	0.995738i	0.421164	+	0.906985i	$0.361622\pi$
$823$	−16.4924	+	28.5657i	−0.574890	+	0.995738i	−0.996054	+	0.0887536i	$0.971712\pi$
$824$		0			0
$825$	8.49242			0.295668
$826$		0			0
$827$		−	6.20393i		−	0.215732i	−0.994165	−	0.107866i	$-0.965598\pi$
$827$		−	6.20393i		−	0.215732i	0.994165	−	0.107866i	$-0.0344017\pi$
$828$		0			0
$829$	−40.3837	−	23.3155i	−1.40258	−	0.809781i	−0.407925	−	0.913015i	$-0.633748\pi$
$829$	−40.3837	−	23.3155i	−1.40258	−	0.809781i	−0.994657	+	0.103234i	$0.967081\pi$
$830$		0			0
$831$	4.00000	+	6.92820i	0.138758	+	0.240337i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−11.7512	+	6.78456i	−0.406667	+	0.234790i
$836$		0			0
$837$	83.7051	+	48.3272i	2.89327	+	1.67043i
$838$		0			0
$839$	6.73863			0.232643			0.116322	−	0.993212i	$-0.462890\pi$
$839$	6.73863			0.232643			0.116322	+	0.993212i	$0.462890\pi$
$840$		0			0
$841$	−7.49242			−0.258359
$842$		0			0
$843$	15.6947	+	9.06134i	0.540554	+	0.312089i
$844$		0			0
$845$	14.8698	−	8.58511i	0.511538	−	0.295337i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	33.0540	+	57.2512i	1.13441	+	1.96485i
$850$		0			0
$851$	−26.8019	−	15.4741i	−0.918757	−	0.530444i
$852$		0			0
$853$		−	25.4421i		−	0.871121i	−0.900159	−	0.435561i	$-0.856550\pi$
$853$		−	25.4421i		−	0.871121i	0.900159	−	0.435561i	$-0.143450\pi$
$854$		0			0
$855$	31.3693			1.07281
$856$		0			0
$857$	23.4924	−	40.6901i	0.802486	−	1.38995i	−0.115489	−	0.993309i	$-0.536844\pi$
$857$	23.4924	−	40.6901i	0.802486	−	1.38995i	0.917975	−	0.396638i	$-0.129823\pi$
$858$		0			0
$859$	−24.5478	+	14.1727i	−0.837559	+	0.483565i	−0.856434	−	0.516257i	$-0.827325\pi$
$859$	−24.5478	+	14.1727i	−0.837559	+	0.483565i	0.0188750	+	0.999822i	$0.493992\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.2462	+	31.6034i	0.621108	+	1.07579i	0.989280	+	0.146033i	$0.0466506\pi$
$863$	18.2462	+	31.6034i	0.621108	+	1.07579i	−0.368171	+	0.929758i	$0.620016\pi$
$864$		0			0
$865$	−16.1771	+	28.0195i	−0.550037	+	0.952692i
$866$		0			0
$867$		−	39.2658i		−	1.33354i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	9.75379	−	16.8941i	0.330495	−	0.572433i
$872$		0			0
$873$	37.4924	+	64.9388i	1.26893	+	2.19784i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−30.7454	+	17.7509i	−1.03820	+	0.599404i	−0.919322	−	0.393505i	$-0.871262\pi$
$877$	−30.7454	+	17.7509i	−1.03820	+	0.599404i	−0.118875	+	0.992909i	$0.537929\pi$
$878$		0			0
$879$	15.6847	−	27.1666i	0.529030	−	0.916308i
$880$		0			0
$881$	44.2462			1.49069			0.745346	−	0.666677i	$-0.232284\pi$
$881$	44.2462			1.49069			0.745346	+	0.666677i	$0.232284\pi$
$882$		0			0
$883$		−	4.71659i		−	0.158726i	−0.996846	−	0.0793629i	$-0.974711\pi$
$883$		−	4.71659i		−	0.158726i	0.996846	−	0.0793629i	$-0.0252886\pi$
$884$		0			0
$885$	−1.64973	−	0.952473i	−0.0554551	−	0.0320170i
$886$		0			0
$887$	−14.2462	−	24.6752i	−0.478341	−	0.828511i	0.521351	−	0.853343i	$-0.325428\pi$
$887$	−14.2462	−	24.6752i	−0.478341	−	0.828511i	−0.999692	+	0.0248317i	$0.992095\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−11.6100	+	6.70305i	−0.388950	+	0.224561i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	8.00000			0.267411
$896$		0			0
$897$	−26.2462			−0.876335
$898$		0			0
$899$	−53.6038	−	30.9481i	−1.78779	−	1.03218i
$900$		0			0
$901$	4.58753	−	2.64861i	0.152833	−	0.0882381i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.93087	−	10.2726i	−0.197149	−	0.341472i
$906$		0			0
$907$	44.6492	+	25.7782i	1.48255	+	0.855951i	0.999804	−	0.0198082i	$-0.00630555\pi$
$907$	44.6492	+	25.7782i	1.48255	+	0.855951i	0.482748	+	0.875760i	$0.339639\pi$
$908$		0			0
$909$		−	63.5924i		−	2.10923i
$910$		0			0
$911$	−11.8617			−0.392997			−0.196498	−	0.980504i	$-0.562957\pi$
$911$	−11.8617			−0.392997			−0.196498	+	0.980504i	$0.562957\pi$
$912$		0			0
$913$	3.75379	−	6.50175i	0.124232	−	0.215177i
$914$		0			0
$915$	7.52534	−	4.34475i	0.248780	−	0.143633i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	0.924427	−	0.381358i	$-0.124544\pi$
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	−0.792480	+	0.609898i	$0.791210\pi$
$920$		0			0
$921$	38.1771	−	66.1246i	1.25798	−	2.17888i
$922$		0			0
$923$		−	13.5691i		−	0.446633i
$924$		0			0
$925$		−	12.8255i		−	0.421699i
$926$		0			0
$927$	−6.87689	+	11.9111i	−0.225867	+	0.391213i
$928$		0			0
$929$	1.24621	+	2.15850i	0.0408869	+	0.0708181i	0.885745	−	0.464173i	$-0.153648\pi$
$929$	1.24621	+	2.15850i	0.0408869	+	0.0708181i	−0.844858	+	0.534991i	$0.820315\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	32.6775	−	18.8664i	1.06981	−	0.617657i
$934$		0			0
$935$	−2.24621	+	3.89055i	−0.0734590	+	0.127235i
$936$		0			0
$937$	34.9848			1.14291			0.571453	−	0.820635i	$-0.306380\pi$
$937$	34.9848			1.14291			0.571453	+	0.820635i	$0.306380\pi$
$938$		0			0
$939$		−	62.6400i		−	2.04418i
$940$		0			0
$941$	26.6211	+	15.3697i	0.867821	+	0.501037i	0.866624	−	0.498962i	$-0.166285\pi$
$941$	26.6211	+	15.3697i	0.867821	+	0.501037i	0.00119780	+	0.999999i	$0.499619\pi$
$942$		0			0
$943$	10.8769	+	18.8393i	0.354200	+	0.613493i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	36.1181	−	20.8528i	1.17368	−	0.677625i	0.219137	−	0.975694i	$-0.429676\pi$
$947$	36.1181	−	20.8528i	1.17368	−	0.677625i	0.954544	+	0.298069i	$0.0963424\pi$
$948$		0			0
$949$	−8.81341	−	5.08842i	−0.286095	−	0.165177i
$950$		0			0
$951$	−12.4924			−0.405095
$952$		0			0
$953$	−17.5076			−0.567126			−0.283563	−	0.958954i	$-0.591517\pi$
$953$	−17.5076			−0.567126			−0.283563	+	0.958954i	$0.591517\pi$
$954$		0			0
$955$	23.5024	+	13.5691i	0.760520	+	0.439087i
$956$		0			0
$957$	20.9263	−	12.0818i	0.676450	−	0.390549i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−36.9924	−	64.0728i	−1.19330	−	2.06686i
$962$		0			0
$963$	75.0329	+	43.3203i	2.41790	+	1.39598i
$964$		0			0
$965$		−	18.8664i		−	0.607329i
$966$		0			0
$967$	10.8769			0.349777			0.174889	−	0.984588i	$-0.444043\pi$
$967$	10.8769			0.349777			0.174889	+	0.984588i	$0.444043\pi$
$968$		0			0
$969$	9.12311	−	15.8017i	0.293076	−	0.507623i
$970$		0			0
$971$	−9.49709	+	5.48314i	−0.304776	+	0.175962i	−0.644586	−	0.764531i	$-0.722970\pi$
$971$	−9.49709	+	5.48314i	−0.304776	+	0.175962i	0.339810	+	0.940494i	$0.389637\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−5.43845	−	9.41967i	−0.174170	−	0.301671i
$976$		0			0
$977$	−11.8769	+	20.5714i	−0.379976	+	0.658137i	−0.991058	−	0.133430i	$-0.957401\pi$
$977$	−11.8769	+	20.5714i	−0.379976	+	0.658137i	0.611083	+	0.791567i	$0.290734\pi$
$978$		0			0
$979$			21.5150i			0.687621i
$980$		0			0
$981$			110.967i			3.54291i
$982$		0			0
$983$	−17.1231	+	29.6581i	−0.546142	+	0.945946i	0.452392	+	0.891819i	$0.350571\pi$
$983$	−17.1231	+	29.6581i	−0.546142	+	0.945946i	−0.998534	+	0.0541268i	$0.982762\pi$
$984$		0			0
$985$	−18.2462	−	31.6034i	−0.581373	−	1.00697i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.87560	−	3.39228i	0.186833	−	0.107868i
$990$		0			0
$991$	−18.2462	+	31.6034i	−0.579610	+	1.00391i	0.415914	+	0.909404i	$0.363462\pi$
$991$	−18.2462	+	31.6034i	−0.579610	+	1.00391i	−0.995524	+	0.0945100i	$0.969872\pi$
$992$		0			0
$993$	44.9848			1.42755
$994$		0			0
$995$			30.9481i			0.981122i
$996$		0			0
$997$	28.6324	+	16.5309i	0.906799	+	0.523540i	0.879400	−	0.476084i	$-0.157944\pi$
$997$	28.6324	+	16.5309i	0.906799	+	0.523540i	0.0273989	+	0.999625i	$0.491278\pi$
$998$		0			0
$999$	28.4924	+	49.3503i	0.901460	+	1.56138i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.t.d.753.1		8
4.3	odd	2		392.2.p.f.165.3		8
7.2	even	3	inner	1568.2.t.d.177.4		8
7.3	odd	6		1568.2.b.d.785.1		4
7.4	even	3		224.2.b.b.113.4		4
7.5	odd	6		1568.2.t.e.177.1		8
7.6	odd	2		1568.2.t.e.753.4		8
8.3	odd	2		392.2.p.f.165.1		8
8.5	even	2	inner	1568.2.t.d.753.4		8
21.11	odd	6		2016.2.c.c.1009.2		4
28.3	even	6		392.2.b.c.197.4		4
28.11	odd	6		56.2.b.b.29.4	yes	4
28.19	even	6		392.2.p.e.373.1		8
28.23	odd	6		392.2.p.f.373.1		8
28.27	even	2		392.2.p.e.165.3		8
56.3	even	6		392.2.b.c.197.3		4
56.5	odd	6		1568.2.t.e.177.4		8
56.11	odd	6		56.2.b.b.29.3	✓	4
56.13	odd	2		1568.2.t.e.753.1		8
56.19	even	6		392.2.p.e.373.3		8
56.27	even	2		392.2.p.e.165.1		8
56.37	even	6	inner	1568.2.t.d.177.1		8
56.45	odd	6		1568.2.b.d.785.4		4
56.51	odd	6		392.2.p.f.373.3		8
56.53	even	6		224.2.b.b.113.1		4
84.11	even	6		504.2.c.d.253.1		4
112.11	odd	12		1792.2.a.x.1.1		4
112.53	even	12		1792.2.a.v.1.4		4
112.67	odd	12		1792.2.a.x.1.4		4
112.109	even	12		1792.2.a.v.1.1		4
168.11	even	6		504.2.c.d.253.2		4
168.53	odd	6		2016.2.c.c.1009.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.b.29.3	✓	4	56.11	odd	6
56.2.b.b.29.4	yes	4	28.11	odd	6
224.2.b.b.113.1		4	56.53	even	6
224.2.b.b.113.4		4	7.4	even	3
392.2.b.c.197.3		4	56.3	even	6
392.2.b.c.197.4		4	28.3	even	6
392.2.p.e.165.1		8	56.27	even	2
392.2.p.e.165.3		8	28.27	even	2
392.2.p.e.373.1		8	28.19	even	6
392.2.p.e.373.3		8	56.19	even	6
392.2.p.f.165.1		8	8.3	odd	2
392.2.p.f.165.3		8	4.3	odd	2
392.2.p.f.373.1		8	28.23	odd	6
392.2.p.f.373.3		8	56.51	odd	6
504.2.c.d.253.1		4	84.11	even	6
504.2.c.d.253.2		4	168.11	even	6
1568.2.b.d.785.1		4	7.3	odd	6
1568.2.b.d.785.4		4	56.45	odd	6
1568.2.t.d.177.1		8	56.37	even	6	inner
1568.2.t.d.177.4		8	7.2	even	3	inner
1568.2.t.d.753.1		8	1.1	even	1	trivial
1568.2.t.d.753.4		8	8.5	even	2	inner
1568.2.t.e.177.1		8	7.5	odd	6
1568.2.t.e.177.4		8	56.5	odd	6
1568.2.t.e.753.1		8	56.13	odd	2
1568.2.t.e.753.4		8	7.6	odd	2
1792.2.a.v.1.1		4	112.109	even	12
1792.2.a.v.1.4		4	112.53	even	12
1792.2.a.x.1.1		4	112.11	odd	12
1792.2.a.x.1.4		4	112.67	odd	12
2016.2.c.c.1009.2		4	21.11	odd	6
2016.2.c.c.1009.3		4	168.53	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.61578 - 1.51022i) q^{3} +(1.46890 - 0.848071i) q^{5} +(3.06155 + 5.30277i) q^{9} +O(q^{10})\)	\(q+(-2.61578 - 1.51022i) q^{3} +(1.46890 - 0.848071i) q^{5} +(3.06155 + 5.30277i) q^{9} +(1.14688 + 0.662153i) q^{11} -1.69614i q^{13} -5.12311 q^{15} +(-1.00000 + 1.73205i) q^{17} +(2.61578 - 1.51022i) q^{19} +(2.56155 + 4.43674i) q^{23} +(-1.06155 + 1.83866i) q^{25} -9.43318i q^{27} +6.04090i q^{29} +(-5.12311 + 8.87348i) q^{31} +(-2.00000 - 3.46410i) q^{33} +(-5.23157 + 3.02045i) q^{37} +(-2.56155 + 4.43674i) q^{39} +4.24621 q^{41} -1.32431i q^{43} +(8.99424 + 5.19283i) q^{45} +(5.23157 - 3.02045i) q^{51} +(-2.29377 - 1.32431i) q^{53} +2.24621 q^{55} -9.12311 q^{57} +(0.322018 + 0.185917i) q^{59} +(-1.46890 + 0.848071i) q^{61} +(-1.43845 - 2.49146i) q^{65} +(9.96029 + 5.75058i) q^{67} -15.4741i q^{69} +8.00000 q^{71} +(3.00000 - 5.19615i) q^{73} +(5.55359 - 3.20636i) q^{75} +(-5.06155 + 8.76687i) q^{81} -5.66906i q^{83} +3.39228i q^{85} +(9.12311 - 15.8017i) q^{87} +(8.12311 + 14.0696i) q^{89} +(26.8019 - 15.4741i) q^{93} +(2.56155 - 4.43674i) q^{95} +12.2462 q^{97} +8.10887i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 8 q^{15} - 8 q^{17} + 4 q^{23} + 8 q^{25} - 8 q^{31} - 16 q^{33} - 4 q^{39} - 32 q^{41} - 48 q^{55} - 40 q^{57} - 28 q^{65} + 64 q^{71} + 24 q^{73} - 24 q^{81} + 40 q^{87} + 32 q^{89} + 4 q^{95} + 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 - 8 * q^15 - 8 * q^17 + 4 * q^23 + 8 * q^25 - 8 * q^31 - 16 * q^33 - 4 * q^39 - 32 * q^41 - 48 * q^55 - 40 * q^57 - 28 * q^65 + 64 * q^71 + 24 * q^73 - 24 * q^81 + 40 * q^87 + 32 * q^89 + 4 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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