LMFDB - Embedded newform 2156.2.q.b.2089.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(901,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("2156.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.q (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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 308)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2089.1
Root			$1.72286 - 0.178197i$ of defining polynomial
Character	$\chi$	$=$	2156.2089
Dual form			2156.2.q.b.901.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{3} +(1.93649 - 1.11803i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(-1.93649 - 1.11803i) q^{3} +(1.93649 - 1.11803i) q^{5} +(1.00000 + 1.73205i) q^{9} +(-2.23861 - 2.44716i) q^{11} +2.82843 q^{13} -5.00000 q^{15} +(2.82843 - 4.89898i) q^{17} +(0.707107 + 1.22474i) q^{19} +(2.50000 + 4.33013i) q^{23} +2.23607i q^{27} -9.48683i q^{29} +(-1.93649 - 1.11803i) q^{31} +(1.59904 + 7.24176i) q^{33} +(-5.50000 - 9.52628i) q^{37} +(-5.47723 - 3.16228i) q^{39} +9.89949 q^{41} -3.16228i q^{43} +(3.87298 + 2.23607i) q^{45} +(3.87298 - 2.23607i) q^{47} +(-10.9545 + 6.32456i) q^{51} +(-4.00000 + 6.92820i) q^{53} +(-7.07107 - 2.23607i) q^{55} -3.16228i q^{57} +(1.93649 + 1.11803i) q^{59} +(2.12132 + 3.67423i) q^{61} +(5.47723 - 3.16228i) q^{65} +(-4.50000 + 7.79423i) q^{67} -11.1803i q^{69} -1.00000 q^{71} +(2.12132 - 3.67423i) q^{73} +(5.47723 - 3.16228i) q^{79} +(5.50000 - 9.52628i) q^{81} -1.41421 q^{83} -12.6491i q^{85} +(-10.6066 + 18.3712i) q^{87} +(-5.80948 + 3.35410i) q^{89} +(2.50000 + 4.33013i) q^{93} +(2.73861 + 1.58114i) q^{95} -15.6525i q^{97} +(2.00000 - 6.32456i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 4 q^{11} - 40 q^{15} + 20 q^{23} - 44 q^{37} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 44 q^{81} + 20 q^{93} + 16 q^{99}+O(q^{100}) $ 8 * q + 8 * q^9 + 4 * q^11 - 40 * q^15 + 20 * q^23 - 44 * q^37 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 44 * q^81 + 20 * q^93 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$	−1.93649	−	1.11803i	−1.11803	−	0.645497i	−0.177136	−	0.984186i	$-0.556683\pi$
$3$	−1.93649	−	1.11803i	−1.11803	−	0.645497i	−0.940898	+	0.338689i	$0.890016\pi$
$4$		0			0
$5$	1.93649	−	1.11803i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$5$	1.93649	−	1.11803i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	−2.23861	−	2.44716i	−0.674967	−	0.737848i
$11$	−2.23861	−	2.44716i	−0.674967	−	0.737848i
$12$		0			0
$13$	2.82843			0.784465			0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843			0.784465			0.392232	+	0.919866i	$0.371703\pi$
$14$		0			0
$15$	−5.00000			−1.29099
$16$		0			0
$17$	2.82843	−	4.89898i	0.685994	−	1.18818i	−0.287129	−	0.957892i	$-0.592701\pi$
$17$	2.82843	−	4.89898i	0.685994	−	1.18818i	0.973123	−	0.230285i	$-0.0739659\pi$
$18$		0			0
$19$	0.707107	+	1.22474i	0.162221	+	0.280976i	0.935665	−	0.352889i	$-0.114801\pi$
$19$	0.707107	+	1.22474i	0.162221	+	0.280976i	−0.773444	+	0.633865i	$0.781467\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	0.999694	+	0.0247559i	$0.00788087\pi$
$23$	2.50000	+	4.33013i	0.521286	+	0.902894i	−0.478407	+	0.878138i	$0.658786\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			2.23607i			0.430331i
$28$		0			0
$29$		−	9.48683i		−	1.76166i	−0.473432	−	0.880830i	$-0.656985\pi$
$29$		−	9.48683i		−	1.76166i	0.473432	−	0.880830i	$-0.343015\pi$
$30$		0			0
$31$	−1.93649	−	1.11803i	−0.347804	−	0.200805i	0.315914	−	0.948788i	$-0.397689\pi$
$31$	−1.93649	−	1.11803i	−0.347804	−	0.200805i	−0.663718	+	0.747983i	$0.731022\pi$
$32$		0			0
$33$	1.59904	+	7.24176i	0.278358	+	1.26063i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.50000	−	9.52628i	−0.904194	−	1.56611i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−5.50000	−	9.52628i	−0.904194	−	1.56611i	−0.0821995	−	0.996616i	$-0.526194\pi$
$38$		0			0
$39$	−5.47723	−	3.16228i	−0.877058	−	0.506370i
$40$		0			0
$41$	9.89949			1.54604			0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949			1.54604			0.773021	+	0.634381i	$0.218745\pi$
$42$		0			0
$43$		−	3.16228i		−	0.482243i	−0.970495	−	0.241121i	$-0.922485\pi$
$43$		−	3.16228i		−	0.482243i	0.970495	−	0.241121i	$-0.0775152\pi$
$44$		0			0
$45$	3.87298	+	2.23607i	0.577350	+	0.333333i
$46$		0			0
$47$	3.87298	−	2.23607i	0.564933	−	0.326164i	−0.190190	−	0.981747i	$-0.560910\pi$
$47$	3.87298	−	2.23607i	0.564933	−	0.326164i	0.755123	+	0.655583i	$0.227577\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−10.9545	+	6.32456i	−1.53393	+	0.885615i
$52$		0			0
$53$	−4.00000	+	6.92820i	−0.549442	+	0.951662i	0.448871	+	0.893597i	$0.351826\pi$
$53$	−4.00000	+	6.92820i	−0.549442	+	0.951662i	−0.998313	+	0.0580651i	$0.981507\pi$
$54$		0			0
$55$	−7.07107	−	2.23607i	−0.953463	−	0.301511i
$56$		0			0
$57$		−	3.16228i		−	0.418854i
$58$		0			0
$59$	1.93649	+	1.11803i	0.252110	+	0.145556i	0.620730	−	0.784024i	$-0.286836\pi$
$59$	1.93649	+	1.11803i	0.252110	+	0.145556i	−0.368620	+	0.929580i	$0.620170\pi$
$60$		0			0
$61$	2.12132	+	3.67423i	0.271607	+	0.470438i	0.969274	−	0.245985i	$-0.0791115\pi$
$61$	2.12132	+	3.67423i	0.271607	+	0.470438i	−0.697666	+	0.716423i	$0.745778\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.47723	−	3.16228i	0.679366	−	0.392232i
$66$		0			0
$67$	−4.50000	+	7.79423i	−0.549762	+	0.952217i	0.448528	+	0.893769i	$0.351948\pi$
$67$	−4.50000	+	7.79423i	−0.549762	+	0.952217i	−0.998290	+	0.0584478i	$0.981385\pi$
$68$		0			0
$69$		−	11.1803i		−	1.34595i
$70$		0			0
$71$	−1.00000			−0.118678			−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000			−0.118678			−0.0593391	+	0.998238i	$0.518899\pi$
$72$		0			0
$73$	2.12132	−	3.67423i	0.248282	−	0.430037i	−0.714767	−	0.699362i	$-0.753467\pi$
$73$	2.12132	−	3.67423i	0.248282	−	0.430037i	0.963049	+	0.269326i	$0.0868008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.47723	−	3.16228i	0.616236	−	0.355784i	−0.159166	−	0.987252i	$-0.550881\pi$
$79$	5.47723	−	3.16228i	0.616236	−	0.355784i	0.775402	+	0.631468i	$0.217547\pi$
$80$		0			0
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$		0			0
$83$	−1.41421			−0.155230			−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421			−0.155230			−0.0776151	+	0.996983i	$0.524731\pi$
$84$		0			0
$85$		−	12.6491i		−	1.37199i
$86$		0			0
$87$	−10.6066	+	18.3712i	−1.13715	+	1.96960i
$88$		0			0
$89$	−5.80948	+	3.35410i	−0.615803	+	0.355534i	−0.775233	−	0.631675i	$-0.782368\pi$
$89$	−5.80948	+	3.35410i	−0.615803	+	0.355534i	0.159430	+	0.987209i	$0.449034\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	2.50000	+	4.33013i	0.259238	+	0.449013i
$94$		0			0
$95$	2.73861	+	1.58114i	0.280976	+	0.162221i
$96$		0			0
$97$		−	15.6525i		−	1.58927i	−0.607089	−	0.794634i	$-0.707663\pi$
$97$		−	15.6525i		−	1.58927i	0.607089	−	0.794634i	$-0.292337\pi$
$98$		0			0
$99$	2.00000	−	6.32456i	0.201008	−	0.635642i
$100$		0			0
$101$	2.12132	−	3.67423i	0.211079	−	0.365600i	−0.740973	−	0.671534i	$-0.765636\pi$
$101$	2.12132	−	3.67423i	0.211079	−	0.365600i	0.952053	+	0.305934i	$0.0989688\pi$
$102$		0			0
$103$	−3.87298	+	2.23607i	−0.381616	+	0.220326i	−0.678521	−	0.734581i	$-0.737379\pi$
$103$	−3.87298	+	2.23607i	−0.381616	+	0.220326i	0.296905	+	0.954907i	$0.404046\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.47723	+	3.16228i	−0.529503	+	0.305709i	−0.740814	−	0.671710i	$-0.765560\pi$
$107$	−5.47723	+	3.16228i	−0.529503	+	0.305709i	0.211311	+	0.977419i	$0.432227\pi$
$108$		0			0
$109$	−13.6931	−	7.90569i	−1.31156	−	0.757228i	−0.329204	−	0.944259i	$-0.606780\pi$
$109$	−13.6931	−	7.90569i	−1.31156	−	0.757228i	−0.982354	+	0.187031i	$0.940114\pi$
$110$		0			0
$111$			24.5967i			2.33462i
$112$		0			0
$113$	−3.00000			−0.282216			−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000			−0.282216			−0.141108	+	0.989994i	$0.545067\pi$
$114$		0			0
$115$	9.68246	+	5.59017i	0.902894	+	0.521286i
$116$		0			0
$117$	2.82843	+	4.89898i	0.261488	+	0.452911i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.977226	+	10.9565i	−0.0888387	+	0.996046i
$122$		0			0
$123$	−19.1703	−	11.0680i	−1.72853	−	0.997965i
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$		−	15.8114i		−	1.40303i	−0.712653	−	0.701517i	$-0.752506\pi$
$127$		−	15.8114i		−	1.40303i	0.712653	−	0.701517i	$-0.247494\pi$
$128$		0			0
$129$	−3.53553	+	6.12372i	−0.311286	+	0.539164i
$130$		0			0
$131$	−7.07107	−	12.2474i	−0.617802	−	1.07006i	−0.989886	−	0.141865i	$-0.954690\pi$
$131$	−7.07107	−	12.2474i	−0.617802	−	1.07006i	0.372084	−	0.928199i	$-0.378643\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.50000	+	4.33013i	0.215166	+	0.372678i
$136$		0			0
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	−22.6274			−1.91923			−0.959616	−	0.281312i	$-0.909230\pi$
$139$	−22.6274			−1.91923			−0.959616	+	0.281312i	$0.909230\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	−6.33175	−	6.92163i	−0.529488	−	0.578815i
$144$		0			0
$145$	−10.6066	−	18.3712i	−0.880830	−	1.52564i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.47723	+	3.16228i	−0.448712	+	0.259064i	−0.707286	−	0.706928i	$-0.750081\pi$
$149$	−5.47723	+	3.16228i	−0.448712	+	0.259064i	0.258574	+	0.965991i	$0.416747\pi$
$150$		0			0
$151$	−10.9545	−	6.32456i	−0.891461	−	0.514685i	−0.0170406	−	0.999855i	$-0.505424\pi$
$151$	−10.9545	−	6.32456i	−0.891461	−	0.514685i	−0.874420	+	0.485170i	$0.838758\pi$
$152$		0			0
$153$	11.3137			0.914659
$154$		0			0
$155$	−5.00000			−0.401610
$156$		0			0
$157$	−5.80948	−	3.35410i	−0.463647	−	0.267686i	0.249930	−	0.968264i	$-0.419592\pi$
$157$	−5.80948	−	3.35410i	−0.463647	−	0.267686i	−0.713576	+	0.700577i	$0.752926\pi$
$158$		0			0
$159$	15.4919	−	8.94427i	1.22859	−	0.709327i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.00000	−	8.66025i	−0.391630	−	0.678323i	0.601035	−	0.799223i	$-0.294755\pi$
$163$	−5.00000	−	8.66025i	−0.391630	−	0.678323i	−0.992665	+	0.120900i	$0.961422\pi$
$164$		0			0
$165$	11.1931	+	12.2358i	0.871379	+	0.952557i
$166$		0			0
$167$	1.41421			0.109435			0.0547176	−	0.998502i	$-0.482574\pi$
$167$	1.41421			0.109435			0.0547176	+	0.998502i	$0.482574\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	−1.41421	+	2.44949i	−0.108148	+	0.187317i
$172$		0			0
$173$	−6.36396	−	11.0227i	−0.483843	−	0.838041i	0.515985	−	0.856598i	$-0.327426\pi$
$173$	−6.36396	−	11.0227i	−0.483843	−	0.838041i	−0.999828	+	0.0185571i	$0.994093\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.50000	−	4.33013i	−0.187912	−	0.325472i
$178$		0			0
$179$	6.50000	−	11.2583i	0.485833	−	0.841487i	−0.514035	−	0.857769i	$-0.671850\pi$
$179$	6.50000	−	11.2583i	0.485833	−	0.841487i	0.999867	+	0.0162823i	$0.00518305\pi$
$180$		0			0
$181$			20.1246i			1.49585i	0.663783	+	0.747925i	$0.268950\pi$
$181$			20.1246i			1.49585i	−0.663783	+	0.747925i	$0.731050\pi$
$182$		0			0
$183$		−	9.48683i		−	0.701287i
$184$		0			0
$185$	−21.3014	−	12.2984i	−1.56611	−	0.904194i
$186$		0			0
$187$	−18.3204	+	4.04529i	−1.33972	+	0.295821i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	0.915177	−	0.403051i	$-0.132050\pi$
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	−0.806641	+	0.591041i	$0.798717\pi$
$192$		0			0
$193$	13.6931	+	7.90569i	0.985648	+	0.569064i	0.903971	−	0.427594i	$-0.140639\pi$
$193$	13.6931	+	7.90569i	0.985648	+	0.569064i	0.0816776	+	0.996659i	$0.473972\pi$
$194$		0			0
$195$	−14.1421			−1.01274
$196$		0			0
$197$			12.6491i			0.901212i	0.892723	+	0.450606i	$0.148792\pi$
$197$			12.6491i			0.901212i	−0.892723	+	0.450606i	$0.851208\pi$
$198$		0			0
$199$	15.4919	+	8.94427i	1.09819	+	0.634043i	0.935746	−	0.352674i	$-0.114728\pi$
$199$	15.4919	+	8.94427i	1.09819	+	0.634043i	0.162448	+	0.986717i	$0.448061\pi$
$200$		0			0
$201$	17.4284	−	10.0623i	1.22931	−	0.709740i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	19.1703	−	11.0680i	1.33891	−	0.773021i
$206$		0			0
$207$	−5.00000	+	8.66025i	−0.347524	+	0.601929i
$208$		0			0
$209$	1.41421	−	4.47214i	0.0978232	−	0.309344i
$210$		0			0
$211$			9.48683i			0.653101i	0.945180	+	0.326550i	$0.105886\pi$
$211$			9.48683i			0.653101i	−0.945180	+	0.326550i	$0.894114\pi$
$212$		0			0
$213$	1.93649	+	1.11803i	0.132686	+	0.0766064i
$214$		0			0
$215$	−3.53553	−	6.12372i	−0.241121	−	0.417635i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−8.21584	+	4.74342i	−0.555175	+	0.320530i
$220$		0			0
$221$	8.00000	−	13.8564i	0.538138	−	0.932083i
$222$		0			0
$223$		−	20.1246i		−	1.34764i	−0.738894	−	0.673822i	$-0.764652\pi$
$223$		−	20.1246i		−	1.34764i	0.738894	−	0.673822i	$-0.235348\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.8492	−	25.7196i	0.985579	−	1.70707i	0.346243	−	0.938145i	$-0.387457\pi$
$227$	14.8492	−	25.7196i	0.985579	−	1.70707i	0.639336	−	0.768928i	$-0.279209\pi$
$228$		0			0
$229$	−9.68246	+	5.59017i	−0.639835	+	0.369409i	−0.784551	−	0.620064i	$-0.787106\pi$
$229$	−9.68246	+	5.59017i	−0.639835	+	0.369409i	0.144716	+	0.989473i	$0.453773\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.21584	+	4.74342i	−0.538237	+	0.310752i	−0.744364	−	0.667774i	$-0.767247\pi$
$233$	−8.21584	+	4.74342i	−0.538237	+	0.310752i	0.206127	+	0.978525i	$0.433914\pi$
$234$		0			0
$235$	5.00000	−	8.66025i	0.326164	−	0.564933i
$236$		0			0
$237$	−14.1421			−0.918630
$238$		0			0
$239$			15.8114i			1.02275i	0.859357	+	0.511377i	$0.170864\pi$
$239$			15.8114i			1.02275i	−0.859357	+	0.511377i	$0.829136\pi$
$240$		0			0
$241$	8.48528	−	14.6969i	0.546585	−	0.946713i	−0.451920	−	0.892058i	$-0.649261\pi$
$241$	8.48528	−	14.6969i	0.546585	−	0.946713i	0.998505	−	0.0546547i	$-0.0174058\pi$
$242$		0			0
$243$	−15.4919	+	8.94427i	−0.993808	+	0.573775i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.00000	+	3.46410i	0.127257	+	0.220416i
$248$		0			0
$249$	2.73861	+	1.58114i	0.173553	+	0.100201i
$250$		0			0
$251$		−	24.5967i		−	1.55253i	−0.630405	−	0.776266i	$-0.717111\pi$
$251$		−	24.5967i		−	1.55253i	0.630405	−	0.776266i	$-0.282889\pi$
$252$		0			0
$253$	5.00000	−	15.8114i	0.314347	−	0.994053i
$254$		0			0
$255$	−14.1421	+	24.4949i	−0.885615	+	1.53393i
$256$		0			0
$257$	−11.6190	+	6.70820i	−0.724770	+	0.418446i	−0.816506	−	0.577337i	$-0.804092\pi$
$257$	−11.6190	+	6.70820i	−0.724770	+	0.418446i	0.0917357	+	0.995783i	$0.470759\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	16.4317	−	9.48683i	1.01710	−	0.587220i
$262$		0			0
$263$	−10.9545	−	6.32456i	−0.675480	−	0.389989i	0.122670	−	0.992448i	$-0.460854\pi$
$263$	−10.9545	−	6.32456i	−0.675480	−	0.389989i	−0.798150	+	0.602459i	$0.794188\pi$
$264$		0			0
$265$			17.8885i			1.09888i
$266$		0			0
$267$	15.0000			0.917985
$268$		0			0
$269$	15.4919	+	8.94427i	0.944560	+	0.545342i	0.891387	−	0.453243i	$-0.149733\pi$
$269$	15.4919	+	8.94427i	0.944560	+	0.545342i	0.0531731	+	0.998585i	$0.483067\pi$
$270$		0			0
$271$	8.48528	+	14.6969i	0.515444	+	0.892775i	0.999839	+	0.0179261i	$0.00570637\pi$
$271$	8.48528	+	14.6969i	0.515444	+	0.892775i	−0.484395	+	0.874849i	$0.660960\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	19.1703	+	11.0680i	1.15183	+	0.665010i	0.949333	−	0.314273i	$-0.101761\pi$
$277$	19.1703	+	11.0680i	1.15183	+	0.665010i	0.202498	+	0.979283i	$0.435094\pi$
$278$		0			0
$279$		−	4.47214i		−	0.267740i
$280$		0			0
$281$		−	3.16228i		−	0.188646i	−0.995542	−	0.0943228i	$-0.969931\pi$
$281$		−	3.16228i		−	0.188646i	0.995542	−	0.0943228i	$-0.0300686\pi$
$282$		0			0
$283$	4.24264	−	7.34847i	0.252199	−	0.436821i	−0.711932	−	0.702248i	$-0.752180\pi$
$283$	4.24264	−	7.34847i	0.252199	−	0.436821i	0.964131	+	0.265427i	$0.0855130\pi$
$284$		0			0
$285$	−3.53553	−	6.12372i	−0.209427	−	0.362738i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.50000	−	12.9904i	−0.441176	−	0.764140i
$290$		0			0
$291$	−17.5000	+	30.3109i	−1.02587	+	1.77686i
$292$		0			0
$293$	15.5563			0.908812			0.454406	−	0.890795i	$-0.349852\pi$
$293$	15.5563			0.908812			0.454406	+	0.890795i	$0.349852\pi$
$294$		0			0
$295$	5.00000			0.291111
$296$		0			0
$297$	5.47203	−	5.00569i	0.317519	−	0.290460i
$298$		0			0
$299$	7.07107	+	12.2474i	0.408930	+	0.708288i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−8.21584	+	4.74342i	−0.471988	+	0.272502i
$304$		0			0
$305$	8.21584	+	4.74342i	0.470438	+	0.271607i
$306$		0			0
$307$	19.7990			1.12999			0.564994	−	0.825095i	$-0.308878\pi$
$307$	19.7990			1.12999			0.564994	+	0.825095i	$0.308878\pi$
$308$		0			0
$309$	10.0000			0.568880
$310$		0			0
$311$	−19.3649	−	11.1803i	−1.09808	−	0.633979i	−0.162367	−	0.986730i	$-0.551913\pi$
$311$	−19.3649	−	11.1803i	−1.09808	−	0.633979i	−0.935717	+	0.352752i	$0.885246\pi$
$312$		0			0
$313$	17.4284	−	10.0623i	0.985113	−	0.568755i	0.0813030	−	0.996689i	$-0.474092\pi$
$313$	17.4284	−	10.0623i	0.985113	−	0.568755i	0.903810	+	0.427934i	$0.140759\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.50000	+	12.9904i	0.421242	+	0.729612i	0.996061	−	0.0886679i	$-0.0282610\pi$
$317$	7.50000	+	12.9904i	0.421242	+	0.729612i	−0.574819	+	0.818280i	$0.694928\pi$
$318$		0			0
$319$	−23.2158	+	21.2373i	−1.29984	+	1.18906i
$320$		0			0
$321$	14.1421			0.789337
$322$		0			0
$323$	8.00000			0.445132
$324$		0			0
$325$		0			0
$326$		0			0
$327$	17.6777	+	30.6186i	0.977577	+	1.69321i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	0.879440	−	0.476011i	$-0.157918\pi$
$331$	0.500000	+	0.866025i	0.0274825	+	0.0476011i	−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	11.0000	−	19.0526i	0.602796	−	1.04407i
$334$		0			0
$335$			20.1246i			1.09952i
$336$		0			0
$337$			25.2982i			1.37808i	0.724722	+	0.689041i	$0.241968\pi$
$337$			25.2982i			1.37808i	−0.724722	+	0.689041i	$0.758032\pi$
$338$		0			0
$339$	5.80948	+	3.35410i	0.315527	+	0.182170i
$340$		0			0
$341$	1.59904	+	7.24176i	0.0865930	+	0.392163i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−12.5000	−	21.6506i	−0.672977	−	1.16563i
$346$		0			0
$347$	2.73861	+	1.58114i	0.147016	+	0.0848800i	0.571704	−	0.820460i	$-0.306283\pi$
$347$	2.73861	+	1.58114i	0.147016	+	0.0848800i	−0.424687	+	0.905340i	$0.639616\pi$
$348$		0			0
$349$	−22.6274			−1.21122			−0.605609	−	0.795762i	$-0.707070\pi$
$349$	−22.6274			−1.21122			−0.605609	+	0.795762i	$0.707070\pi$
$350$		0			0
$351$			6.32456i			0.337580i
$352$		0			0
$353$	9.68246	+	5.59017i	0.515345	+	0.297535i	0.735028	−	0.678037i	$-0.237169\pi$
$353$	9.68246	+	5.59017i	0.515345	+	0.297535i	−0.219683	+	0.975571i	$0.570502\pi$
$354$		0			0
$355$	−1.93649	+	1.11803i	−0.102778	+	0.0593391i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−21.9089	+	12.6491i	−1.15631	+	0.667595i	−0.950416	−	0.310980i	$-0.899343\pi$
$359$	−21.9089	+	12.6491i	−1.15631	+	0.667595i	−0.205891	+	0.978575i	$0.566009\pi$
$360$		0			0
$361$	8.50000	−	14.7224i	0.447368	−	0.774865i
$362$		0			0
$363$	14.1421	−	20.1246i	0.742270	−	1.05627i
$364$		0			0
$365$		−	9.48683i		−	0.496564i
$366$		0			0
$367$	5.80948	+	3.35410i	0.303252	+	0.175083i	0.643903	−	0.765107i	$-0.277314\pi$
$367$	5.80948	+	3.35410i	0.303252	+	0.175083i	−0.340651	+	0.940190i	$0.610647\pi$
$368$		0			0
$369$	9.89949	+	17.1464i	0.515347	+	0.892607i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	30.1247	−	17.3925i	1.55980	−	0.900551i	0.562525	−	0.826780i	$-0.309830\pi$
$373$	30.1247	−	17.3925i	1.55980	−	0.900551i	0.997275	−	0.0737704i	$-0.0235032\pi$
$374$		0			0
$375$	12.5000	−	21.6506i	0.645497	−	1.11803i
$376$		0			0
$377$		−	26.8328i		−	1.38196i
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$		0			0
$381$	−17.6777	+	30.6186i	−0.905654	+	1.56864i
$382$		0			0
$383$	13.5554	−	7.82624i	0.692651	−	0.399902i	−0.111954	−	0.993713i	$-0.535711\pi$
$383$	13.5554	−	7.82624i	0.692651	−	0.399902i	0.804604	+	0.593811i	$0.202377\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.47723	−	3.16228i	0.278423	−	0.160748i
$388$		0			0
$389$	8.50000	−	14.7224i	0.430967	−	0.746457i	−0.565990	−	0.824412i	$-0.691506\pi$
$389$	8.50000	−	14.7224i	0.430967	−	0.746457i	0.996957	+	0.0779554i	$0.0248392\pi$
$390$		0			0
$391$	28.2843			1.43040
$392$		0			0
$393$			31.6228i			1.59516i
$394$		0			0
$395$	7.07107	−	12.2474i	0.355784	−	0.616236i
$396$		0			0
$397$	15.4919	−	8.94427i	0.777518	−	0.448900i	−0.0580320	−	0.998315i	$-0.518483\pi$
$397$	15.4919	−	8.94427i	0.777518	−	0.448900i	0.835550	+	0.549415i	$0.185149\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	−5.47723	−	3.16228i	−0.272840	−	0.157524i
$404$		0			0
$405$		−	24.5967i		−	1.22222i
$406$		0			0
$407$	−11.0000	+	34.7851i	−0.545250	+	1.72423i
$408$		0			0
$409$	−2.82843	+	4.89898i	−0.139857	+	0.242239i	−0.927442	−	0.373966i	$-0.877998\pi$
$409$	−2.82843	+	4.89898i	−0.139857	+	0.242239i	0.787586	+	0.616205i	$0.211331\pi$
$410$		0			0
$411$	17.4284	−	10.0623i	0.859681	−	0.496337i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.73861	+	1.58114i	−0.134433	+	0.0776151i
$416$		0			0
$417$	43.8178	+	25.2982i	2.14577	+	1.23886i
$418$		0			0
$419$			31.3050i			1.52935i	0.644418	+	0.764673i	$0.277100\pi$
$419$			31.3050i			1.52935i	−0.644418	+	0.764673i	$0.722900\pi$
$420$		0			0
$421$	−36.0000			−1.75453			−0.877266	−	0.480004i	$-0.840635\pi$
$421$	−36.0000			−1.75453			−0.877266	+	0.480004i	$0.840635\pi$
$422$		0			0
$423$	7.74597	+	4.47214i	0.376622	+	0.217443i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.52277	+	20.4828i	0.218362	+	0.988918i
$430$		0			0
$431$	13.6931	+	7.90569i	0.659572	+	0.380804i	0.792114	−	0.610373i	$-0.208981\pi$
$431$	13.6931	+	7.90569i	0.659572	+	0.380804i	−0.132542	+	0.991177i	$0.542314\pi$
$432$		0			0
$433$			24.5967i			1.18204i	0.806655	+	0.591022i	$0.201275\pi$
$433$			24.5967i			1.18204i	−0.806655	+	0.591022i	$0.798725\pi$
$434$		0			0
$435$			47.4342i			2.27429i
$436$		0			0
$437$	−3.53553	+	6.12372i	−0.169128	+	0.292937i
$438$		0			0
$439$	13.4350	+	23.2702i	0.641219	+	1.11062i	0.985161	+	0.171633i	$0.0549043\pi$
$439$	13.4350	+	23.2702i	0.641219	+	1.11062i	−0.343942	+	0.938991i	$0.611762\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	−0.985119	−	0.171871i	$-0.945019\pi$
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	0.343715	−	0.939074i	$-0.388315\pi$
$444$		0			0
$445$	−7.50000	+	12.9904i	−0.355534	+	0.615803i
$446$		0			0
$447$	14.1421			0.668900
$448$		0			0
$449$	37.0000			1.74614			0.873069	−	0.487597i	$-0.162126\pi$
$449$	37.0000			1.74614			0.873069	+	0.487597i	$0.162126\pi$
$450$		0			0
$451$	−22.1611	−	24.2257i	−1.04353	−	1.14074i
$452$		0			0
$453$	14.1421	+	24.4949i	0.664455	+	1.15087i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	16.4317	−	9.48683i	0.768641	−	0.443775i	−0.0637483	−	0.997966i	$-0.520305\pi$
$457$	16.4317	−	9.48683i	0.768641	−	0.443775i	0.832390	+	0.554191i	$0.186972\pi$
$458$		0			0
$459$	10.9545	+	6.32456i	0.511310	+	0.295205i
$460$		0			0
$461$	−9.89949			−0.461065			−0.230533	−	0.973065i	$-0.574047\pi$
$461$	−9.89949			−0.461065			−0.230533	+	0.973065i	$0.574047\pi$
$462$		0			0
$463$	−7.00000			−0.325318			−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000			−0.325318			−0.162659	+	0.986682i	$0.552007\pi$
$464$		0			0
$465$	9.68246	+	5.59017i	0.449013	+	0.259238i
$466$		0			0
$467$	−5.80948	+	3.35410i	−0.268830	+	0.155209i	−0.628356	−	0.777926i	$-0.716272\pi$
$467$	−5.80948	+	3.35410i	−0.268830	+	0.155209i	0.359526	+	0.933135i	$0.382939\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	7.50000	+	12.9904i	0.345582	+	0.598565i
$472$		0			0
$473$	−7.73861	+	7.07912i	−0.355822	+	0.325498i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.0000			−0.732590
$478$		0			0
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	−0.998530	−	0.0542078i	$-0.982737\pi$
$479$	−9.89949	+	17.1464i	−0.452319	+	0.783440i	0.546210	+	0.837648i	$0.316070\pi$
$480$		0			0
$481$	−15.5563	−	26.9444i	−0.709308	−	1.22856i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.5000	−	30.3109i	−0.794634	−	1.37635i
$486$		0			0
$487$	10.5000	−	18.1865i	0.475800	−	0.824110i	−0.523815	−	0.851832i	$-0.675492\pi$
$487$	10.5000	−	18.1865i	0.475800	−	0.824110i	0.999616	+	0.0277214i	$0.00882512\pi$
$488$		0			0
$489$			22.3607i			1.01118i
$490$		0			0
$491$			9.48683i			0.428135i	0.976819	+	0.214067i	$0.0686712\pi$
$491$			9.48683i			0.428135i	−0.976819	+	0.214067i	$0.931329\pi$
$492$		0			0
$493$	−46.4758	−	26.8328i	−2.09316	−	1.20849i
$494$		0			0
$495$	−3.19808	−	14.4835i	−0.143743	−	0.650986i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.00000	−	6.92820i	−0.179065	−	0.310149i	0.762496	−	0.646993i	$-0.223974\pi$
$499$	−4.00000	−	6.92820i	−0.179065	−	0.310149i	−0.941560	+	0.336844i	$0.890640\pi$
$500$		0			0
$501$	−2.73861	−	1.58114i	−0.122352	−	0.0706401i
$502$		0			0
$503$	32.5269			1.45030			0.725152	−	0.688589i	$-0.241770\pi$
$503$	32.5269			1.45030			0.725152	+	0.688589i	$0.241770\pi$
$504$		0			0
$505$		−	9.48683i		−	0.422159i
$506$		0			0
$507$	9.68246	+	5.59017i	0.430013	+	0.248268i
$508$		0			0
$509$	−1.93649	+	1.11803i	−0.0858335	+	0.0495560i	−0.542302	−	0.840183i	$-0.682447\pi$
$509$	−1.93649	+	1.11803i	−0.0858335	+	0.0495560i	0.456469	+	0.889739i	$0.349114\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−2.73861	+	1.58114i	−0.120913	+	0.0698090i
$514$		0			0
$515$	−5.00000	+	8.66025i	−0.220326	+	0.381616i
$516$		0			0
$517$	−14.1421	−	4.47214i	−0.621970	−	0.196684i
$518$		0			0
$519$			28.4605i			1.24928i
$520$		0			0
$521$	1.93649	+	1.11803i	0.0848392	+	0.0489820i	0.541819	−	0.840495i	$-0.317736\pi$
$521$	1.93649	+	1.11803i	0.0848392	+	0.0489820i	−0.456980	+	0.889477i	$0.651069\pi$
$522$		0			0
$523$	−1.41421	−	2.44949i	−0.0618392	−	0.107109i	0.833448	−	0.552597i	$-0.186363\pi$
$523$	−1.41421	−	2.44949i	−0.0618392	−	0.107109i	−0.895288	+	0.445489i	$0.853030\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.9545	+	6.32456i	−0.477183	+	0.275502i
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$			4.47214i			0.194074i
$532$		0			0
$533$	28.0000			1.21281
$534$		0			0
$535$	−7.07107	+	12.2474i	−0.305709	+	0.529503i
$536$		0			0
$537$	−25.1744	+	14.5344i	−1.08636	+	0.627207i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	27.3861	−	15.8114i	1.17742	−	0.679785i	0.222005	−	0.975046i	$-0.428740\pi$
$541$	27.3861	−	15.8114i	1.17742	−	0.679785i	0.955417	+	0.295261i	$0.0954066\pi$
$542$		0			0
$543$	22.5000	−	38.9711i	0.965567	−	1.67241i
$544$		0			0
$545$	−35.3553			−1.51446
$546$		0			0
$547$		−	44.2719i		−	1.89293i	−0.322808	−	0.946465i	$-0.604627\pi$
$547$		−	44.2719i		−	1.89293i	0.322808	−	0.946465i	$-0.395373\pi$
$548$		0			0
$549$	−4.24264	+	7.34847i	−0.181071	+	0.313625i
$550$		0			0
$551$	11.6190	−	6.70820i	0.494984	−	0.285779i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	27.5000	+	47.6314i	1.16731	+	2.02184i
$556$		0			0
$557$	10.9545	+	6.32456i	0.464155	+	0.267980i	0.713790	−	0.700360i	$-0.246977\pi$
$557$	10.9545	+	6.32456i	0.464155	+	0.267980i	−0.249635	+	0.968340i	$0.580311\pi$
$558$		0			0
$559$		−	8.94427i		−	0.378302i
$560$		0			0
$561$	40.0000	+	12.6491i	1.68880	+	0.534046i
$562$		0			0
$563$	19.7990	−	34.2929i	0.834428	−	1.44527i	−0.0600674	−	0.998194i	$-0.519132\pi$
$563$	19.7990	−	34.2929i	0.834428	−	1.44527i	0.894495	−	0.447077i	$-0.147535\pi$
$564$		0			0
$565$	−5.80948	+	3.35410i	−0.244406	+	0.141108i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.21584	+	4.74342i	−0.344426	+	0.198854i	−0.662227	−	0.749303i	$-0.730389\pi$
$569$	−8.21584	+	4.74342i	−0.344426	+	0.198854i	0.317802	+	0.948157i	$0.397055\pi$
$570$		0			0
$571$	10.9545	+	6.32456i	0.458430	+	0.264674i	0.711384	−	0.702804i	$-0.248069\pi$
$571$	10.9545	+	6.32456i	0.458430	+	0.264674i	−0.252954	+	0.967478i	$0.581402\pi$
$572$		0			0
$573$		−	6.70820i		−	0.280239i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	25.1744	+	14.5344i	1.04802	+	0.605077i	0.922095	−	0.386963i	$-0.126476\pi$
$577$	25.1744	+	14.5344i	1.04802	+	0.605077i	0.125928	+	0.992039i	$0.459809\pi$
$578$		0			0
$579$	−17.6777	−	30.6186i	−0.734659	−	1.27247i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	25.9089	−	5.72091i	1.07304	−	0.236936i
$584$		0			0
$585$	10.9545	+	6.32456i	0.452911	+	0.261488i
$586$		0			0
$587$		−	13.4164i		−	0.553754i	−0.960905	−	0.276877i	$-0.910700\pi$
$587$		−	13.4164i		−	0.553754i	0.960905	−	0.276877i	$-0.0892995\pi$
$588$		0			0
$589$		−	3.16228i		−	0.130299i
$590$		0			0
$591$	14.1421	−	24.4949i	0.581730	−	1.00759i
$592$		0			0
$593$	−0.707107	−	1.22474i	−0.0290374	−	0.0502942i	0.851142	−	0.524936i	$-0.175911\pi$
$593$	−0.707107	−	1.22474i	−0.0290374	−	0.0502942i	−0.880179	+	0.474642i	$0.842578\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.0000	−	34.6410i	−0.818546	−	1.41776i
$598$		0			0
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	−0.962175	−	0.272433i	$-0.912172\pi$
$599$	−6.00000	+	10.3923i	−0.245153	+	0.424618i	0.717021	+	0.697051i	$0.245505\pi$
$600$		0			0
$601$	21.2132			0.865305			0.432652	−	0.901561i	$-0.357578\pi$
$601$	21.2132			0.865305			0.432652	+	0.901561i	$0.357578\pi$
$602$		0			0
$603$	−18.0000			−0.733017
$604$		0			0
$605$	10.3574	+	22.3098i	0.421086	+	0.907021i
$606$		0			0
$607$	−12.7279	−	22.0454i	−0.516610	−	0.894795i	−0.999814	−	0.0192875i	$-0.993860\pi$
$607$	−12.7279	−	22.0454i	−0.516610	−	0.894795i	0.483204	−	0.875508i	$-0.339473\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.9545	−	6.32456i	0.443170	−	0.255864i
$612$		0			0
$613$	16.4317	+	9.48683i	0.663669	+	0.383170i	0.793674	−	0.608344i	$-0.208166\pi$
$613$	16.4317	+	9.48683i	0.663669	+	0.383170i	−0.130004	+	0.991513i	$0.541499\pi$
$614$		0			0
$615$	−49.4975			−1.99593
$616$		0			0
$617$	−24.0000			−0.966204			−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000			−0.966204			−0.483102	+	0.875564i	$0.660490\pi$
$618$		0			0
$619$	−25.1744	−	14.5344i	−1.01184	−	0.584189i	−0.100113	−	0.994976i	$-0.531920\pi$
$619$	−25.1744	−	14.5344i	−1.01184	−	0.584189i	−0.911731	+	0.410788i	$0.865254\pi$
$620$		0			0
$621$	−9.68246	+	5.59017i	−0.388544	+	0.224326i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	12.5000	+	21.6506i	0.500000	+	0.866025i
$626$		0			0
$627$	−7.73861	+	7.07912i	−0.309050	+	0.282713i
$628$		0			0
$629$	−62.2254			−2.48109
$630$		0			0
$631$	13.0000			0.517522			0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000			0.517522			0.258761	+	0.965941i	$0.416686\pi$
$632$		0			0
$633$	10.6066	−	18.3712i	0.421575	−	0.730189i
$634$		0			0
$635$	−17.6777	−	30.6186i	−0.701517	−	1.21506i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.00000	−	1.73205i	−0.0395594	−	0.0685189i
$640$		0			0
$641$	5.50000	−	9.52628i	0.217237	−	0.376265i	−0.736725	−	0.676192i	$-0.763629\pi$
$641$	5.50000	−	9.52628i	0.217237	−	0.376265i	0.953962	+	0.299927i	$0.0969622\pi$
$642$		0			0
$643$			11.1803i			0.440910i	0.975397	+	0.220455i	$0.0707541\pi$
$643$			11.1803i			0.440910i	−0.975397	+	0.220455i	$0.929246\pi$
$644$		0			0
$645$			15.8114i			0.622573i
$646$		0			0
$647$	32.9204	+	19.0066i	1.29423	+	0.747226i	0.979402	−	0.201922i	$-0.0647187\pi$
$647$	32.9204	+	19.0066i	1.29423	+	0.747226i	0.314831	+	0.949148i	$0.398052\pi$
$648$		0			0
$649$	−1.59904	−	7.24176i	−0.0627679	−	0.284264i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.5000	−	23.3827i	−0.528296	−	0.915035i	−0.999456	−	0.0329874i	$-0.989498\pi$
$653$	−13.5000	−	23.3827i	−0.528296	−	0.915035i	0.471160	−	0.882048i	$-0.343835\pi$
$654$		0			0
$655$	−27.3861	−	15.8114i	−1.07006	−	0.617802i
$656$		0			0
$657$	8.48528			0.331042
$658$		0			0
$659$			18.9737i			0.739109i	0.929209	+	0.369555i	$0.120490\pi$
$659$			18.9737i			0.739109i	−0.929209	+	0.369555i	$0.879510\pi$
$660$		0			0
$661$	5.80948	+	3.35410i	0.225962	+	0.130459i	0.608708	−	0.793394i	$-0.291688\pi$
$661$	5.80948	+	3.35410i	0.225962	+	0.130459i	−0.382746	+	0.923854i	$0.625021\pi$
$662$		0			0
$663$	−30.9839	+	17.8885i	−1.20331	+	0.694733i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	41.0792	−	23.7171i	1.59059	−	0.918329i
$668$		0			0
$669$	−22.5000	+	38.9711i	−0.869900	+	1.50671i
$670$		0			0
$671$	4.24264	−	13.4164i	0.163785	−	0.517935i
$672$		0			0
$673$			9.48683i			0.365691i	0.983142	+	0.182845i	$0.0585307\pi$
$673$			9.48683i			0.365691i	−0.983142	+	0.182845i	$0.941469\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.3137	−	19.5959i	−0.434821	−	0.753132i	0.562460	−	0.826825i	$-0.309855\pi$
$677$	−11.3137	−	19.5959i	−0.434821	−	0.753132i	−0.997281	+	0.0736923i	$0.976522\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−57.5109	+	33.2039i	−2.20382	+	1.27238i
$682$		0			0
$683$	−15.0000	+	25.9808i	−0.573959	+	0.994126i	0.422195	+	0.906505i	$0.361260\pi$
$683$	−15.0000	+	25.9808i	−0.573959	+	0.994126i	−0.996154	+	0.0876211i	$0.972074\pi$
$684$		0			0
$685$			20.1246i			0.768922i
$686$		0			0
$687$	25.0000			0.953809
$688$		0			0
$689$	−11.3137	+	19.5959i	−0.431018	+	0.746545i
$690$		0			0
$691$	5.80948	−	3.35410i	0.221003	−	0.127596i	−0.385412	−	0.922745i	$-0.625941\pi$
$691$	5.80948	−	3.35410i	0.221003	−	0.127596i	0.606415	+	0.795149i	$0.292607\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−43.8178	+	25.2982i	−1.66210	+	0.959616i
$696$		0			0
$697$	28.0000	−	48.4974i	1.06058	−	1.83697i
$698$		0			0
$699$	21.2132			0.802357
$700$		0			0
$701$			9.48683i			0.358313i	0.983821	+	0.179156i	$0.0573368\pi$
$701$			9.48683i			0.358313i	−0.983821	+	0.179156i	$0.942663\pi$
$702$		0			0
$703$	7.77817	−	13.4722i	0.293359	−	0.508113i
$704$		0			0
$705$	−19.3649	+	11.1803i	−0.729325	+	0.421076i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.50000	−	11.2583i	−0.244113	−	0.422815i	0.717769	−	0.696281i	$-0.245163\pi$
$709$	−6.50000	−	11.2583i	−0.244113	−	0.422815i	−0.961882	+	0.273466i	$0.911830\pi$
$710$		0			0
$711$	10.9545	+	6.32456i	0.410824	+	0.237189i
$712$		0			0
$713$		−	11.1803i		−	0.418707i
$714$		0			0
$715$	−20.0000	−	6.32456i	−0.747958	−	0.236525i
$716$		0			0
$717$	17.6777	−	30.6186i	0.660185	−	1.14347i
$718$		0			0
$719$	−9.68246	+	5.59017i	−0.361095	+	0.208478i	−0.669561	−	0.742757i	$-0.733518\pi$
$719$	−9.68246	+	5.59017i	−0.361095	+	0.208478i	0.308466	+	0.951235i	$0.400184\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−32.8634	+	18.9737i	−1.22220	+	0.705638i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	6.70820i		−	0.248794i	−0.992233	−	0.124397i	$-0.960300\pi$
$727$		−	6.70820i		−	0.248794i	0.992233	−	0.124397i	$-0.0396996\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$	−15.4919	−	8.94427i	−0.572990	−	0.330816i
$732$		0			0
$733$	−25.4558	−	44.0908i	−0.940233	−	1.62853i	−0.765026	−	0.644000i	$-0.777274\pi$
$733$	−25.4558	−	44.0908i	−0.940233	−	1.62853i	−0.175207	−	0.984532i	$-0.556060\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.1475	−	6.43602i	1.07366	−	0.237074i
$738$		0			0
$739$	−32.8634	−	18.9737i	−1.20890	−	0.697958i	−0.246379	−	0.969173i	$-0.579241\pi$
$739$	−32.8634	−	18.9737i	−1.20890	−	0.697958i	−0.962519	+	0.271216i	$0.912574\pi$
$740$		0			0
$741$		−	8.94427i		−	0.328576i
$742$		0			0
$743$			18.9737i			0.696076i	0.937480	+	0.348038i	$0.113152\pi$
$743$			18.9737i			0.696076i	−0.937480	+	0.348038i	$0.886848\pi$
$744$		0			0
$745$	−7.07107	+	12.2474i	−0.259064	+	0.448712i
$746$		0			0
$747$	−1.41421	−	2.44949i	−0.0517434	−	0.0896221i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	−0.996993	−	0.0774878i	$-0.975310\pi$
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	0.431390	−	0.902165i	$-0.358023\pi$
$752$		0			0
$753$	−27.5000	+	47.6314i	−1.00216	+	1.73578i
$754$		0			0
$755$	−28.2843			−1.02937
$756$		0			0
$757$	20.0000			0.726912			0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000			0.726912			0.363456	+	0.931611i	$0.381597\pi$
$758$		0			0
$759$	−27.3601	+	25.0285i	−0.993110	+	0.908475i
$760$		0			0
$761$	22.6274	+	39.1918i	0.820243	+	1.42070i	0.905501	+	0.424344i	$0.139495\pi$
$761$	22.6274	+	39.1918i	0.820243	+	1.42070i	−0.0852578	+	0.996359i	$0.527171\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	21.9089	−	12.6491i	0.792118	−	0.457330i
$766$		0			0
$767$	5.47723	+	3.16228i	0.197771	+	0.114183i
$768$		0			0
$769$	−42.4264			−1.52994			−0.764968	−	0.644069i	$-0.777245\pi$
$769$	−42.4264			−1.52994			−0.764968	+	0.644069i	$0.777245\pi$
$770$		0			0
$771$	30.0000			1.08042
$772$		0			0
$773$	38.7298	+	22.3607i	1.39302	+	0.804258i	0.993648	−	0.112533i	$-0.0358965\pi$
$773$	38.7298	+	22.3607i	1.39302	+	0.804258i	0.399367	+	0.916791i	$0.369230\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.00000	+	12.1244i	0.250801	+	0.434400i
$780$		0			0
$781$	2.23861	+	2.44716i	0.0801039	+	0.0875664i
$782$		0			0
$783$	21.2132			0.758098
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	7.77817	−	13.4722i	0.277262	−	0.480232i	−0.693441	−	0.720513i	$-0.743906\pi$
$787$	7.77817	−	13.4722i	0.277262	−	0.480232i	0.970703	+	0.240281i	$0.0772397\pi$
$788$		0			0
$789$	14.1421	+	24.4949i	0.503473	+	0.872041i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	+	10.3923i	0.213066	+	0.369042i
$794$		0			0
$795$	20.0000	−	34.6410i	0.709327	−	1.22859i
$796$		0			0
$797$			33.5410i			1.18808i	0.804434	+	0.594042i	$0.202469\pi$
$797$			33.5410i			1.18808i	−0.804434	+	0.594042i	$0.797531\pi$
$798$		0			0
$799$		−	25.2982i		−	0.894987i
$800$		0			0
$801$	−11.6190	−	6.70820i	−0.410535	−	0.237023i
$802$		0			0
$803$	−13.7403	+	3.03397i	−0.484884	+	0.107066i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−20.0000	−	34.6410i	−0.704033	−	1.21942i
$808$		0			0
$809$	24.6475	+	14.2302i	0.866560	+	0.500309i	0.866204	−	0.499691i	$-0.166553\pi$
$809$	24.6475	+	14.2302i	0.866560	+	0.500309i	0.000356756		1.00000i	$0.499886\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		−	37.9473i		−	1.33087i
$814$		0			0
$815$	−19.3649	−	11.1803i	−0.678323	−	0.391630i
$816$		0			0
$817$	3.87298	−	2.23607i	0.135499	−	0.0782301i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.47723	−	3.16228i	0.191156	−	0.110364i	−0.401367	−	0.915917i	$-0.631465\pi$
$821$	5.47723	−	3.16228i	0.191156	−	0.110364i	0.592524	+	0.805553i	$0.298132\pi$
$822$		0			0
$823$	−17.5000	+	30.3109i	−0.610012	+	1.05657i	0.381226	+	0.924482i	$0.375502\pi$
$823$	−17.5000	+	30.3109i	−0.610012	+	1.05657i	−0.991238	+	0.132089i	$0.957831\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.9473i		−	1.31956i	−0.751460	−	0.659779i	$-0.770650\pi$
$827$		−	37.9473i		−	1.31956i	0.751460	−	0.659779i	$-0.229350\pi$
$828$		0			0
$829$	−13.5554	−	7.82624i	−0.470800	−	0.271816i	0.245775	−	0.969327i	$-0.420958\pi$
$829$	−13.5554	−	7.82624i	−0.470800	−	0.271816i	−0.716575	+	0.697511i	$0.754291\pi$
$830$		0			0
$831$	−24.7487	−	42.8661i	−0.858524	−	1.48701i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	2.73861	−	1.58114i	0.0947736	−	0.0547176i
$836$		0			0
$837$	2.50000	−	4.33013i	0.0864126	−	0.149671i
$838$		0			0
$839$			29.0689i			1.00357i	0.864993	+	0.501785i	$0.167323\pi$
$839$			29.0689i			1.00357i	−0.864993	+	0.501785i	$0.832677\pi$
$840$		0			0
$841$	−61.0000			−2.10345
$842$		0			0
$843$	−3.53553	+	6.12372i	−0.121770	+	0.210912i
$844$		0			0
$845$	−9.68246	+	5.59017i	−0.333087	+	0.192308i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−16.4317	+	9.48683i	−0.563934	+	0.325587i
$850$		0			0
$851$	27.5000	−	47.6314i	0.942688	−	1.63278i
$852$		0			0
$853$	45.2548			1.54950			0.774748	−	0.632270i	$-0.217877\pi$
$853$	45.2548			1.54950			0.774748	+	0.632270i	$0.217877\pi$
$854$		0			0
$855$			6.32456i			0.216295i
$856$		0			0
$857$	−9.89949	+	17.1464i	−0.338160	+	0.585711i	−0.984087	−	0.177689i	$-0.943138\pi$
$857$	−9.89949	+	17.1464i	−0.338160	+	0.585711i	0.645926	+	0.763400i	$0.276471\pi$
$858$		0			0
$859$	40.6663	−	23.4787i	1.38752	−	0.801083i	0.394482	−	0.918904i	$-0.370924\pi$
$859$	40.6663	−	23.4787i	1.38752	−	0.801083i	0.993035	+	0.117820i	$0.0375907\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	0.977564	−	0.210639i	$-0.0675543\pi$
$863$	9.00000	+	15.5885i	0.306364	+	0.530637i	−0.671200	+	0.741276i	$0.734221\pi$
$864$		0			0
$865$	−24.6475	−	14.2302i	−0.838041	−	0.483843i
$866$		0			0
$867$			33.5410i			1.13911i
$868$		0			0
$869$	−20.0000	−	6.32456i	−0.678454	−	0.214546i
$870$		0			0
$871$	−12.7279	+	22.0454i	−0.431269	+	0.746980i
$872$		0			0
$873$	27.1109	−	15.6525i	0.917564	−	0.529756i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	30.1247	−	17.3925i	1.01724	−	0.587304i	0.103937	−	0.994584i	$-0.466856\pi$
$877$	30.1247	−	17.3925i	1.01724	−	0.587304i	0.913303	+	0.407280i	$0.133523\pi$
$878$		0			0
$879$	−30.1247	−	17.3925i	−1.01608	−	0.586635i
$880$		0			0
$881$		−	42.4853i		−	1.43137i	−0.698425	−	0.715683i	$-0.746116\pi$
$881$		−	42.4853i		−	1.43137i	0.698425	−	0.715683i	$-0.253884\pi$
$882$		0			0
$883$	−32.0000			−1.07689			−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000			−1.07689			−0.538443	+	0.842662i	$0.680987\pi$
$884$		0			0
$885$	−9.68246	−	5.59017i	−0.325472	−	0.187912i
$886$		0			0
$887$	18.3848	+	31.8434i	0.617300	+	1.06920i	0.989976	+	0.141234i	$0.0451070\pi$
$887$	18.3848	+	31.8434i	0.617300	+	1.06920i	−0.372676	+	0.927962i	$0.621560\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−35.6247	+	7.86625i	−1.19347	+	0.263529i
$892$		0			0
$893$	5.47723	+	3.16228i	0.183288	+	0.105822i
$894$		0			0
$895$		−	29.0689i		−	0.971666i
$896$		0			0
$897$		−	31.6228i		−	1.05585i
$898$		0			0
$899$	−10.6066	+	18.3712i	−0.353750	+	0.612713i
$900$		0			0
$901$	22.6274	+	39.1918i	0.753829	+	1.30567i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.5000	+	38.9711i	0.747925	+	1.29544i
$906$		0			0
$907$	−5.00000	+	8.66025i	−0.166022	+	0.287559i	−0.937018	−	0.349281i	$-0.886426\pi$
$907$	−5.00000	+	8.66025i	−0.166022	+	0.287559i	0.770996	+	0.636841i	$0.219759\pi$
$908$		0			0
$909$	8.48528			0.281439
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$	3.16588	+	3.46081i	0.104775	+	0.114536i
$914$		0			0
$915$	−10.6066	−	18.3712i	−0.350643	−	0.607332i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2.73861	−	1.58114i	0.0903385	−	0.0521570i	−0.454150	−	0.890925i	$-0.650057\pi$
$919$	2.73861	−	1.58114i	0.0903385	−	0.0521570i	0.544489	+	0.838768i	$0.316724\pi$
$920$		0			0
$921$	−38.3406	−	22.1359i	−1.26337	−	0.729404i
$922$		0			0
$923$	−2.82843			−0.0930988
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−7.74597	−	4.47214i	−0.254411	−	0.146884i
$928$		0			0
$929$	−15.4919	+	8.94427i	−0.508274	+	0.293452i	−0.732124	−	0.681172i	$-0.761471\pi$
$929$	−15.4919	+	8.94427i	−0.508274	+	0.293452i	0.223850	+	0.974624i	$0.428137\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	25.0000	+	43.3013i	0.818463	+	1.41762i
$934$		0			0
$935$	−30.9545	+	28.3165i	−1.01232	+	0.926047i
$936$		0			0
$937$	5.65685			0.184801			0.0924007	−	0.995722i	$-0.470546\pi$
$937$	5.65685			0.184801			0.0924007	+	0.995722i	$0.470546\pi$
$938$		0			0
$939$	−45.0000			−1.46852
$940$		0			0
$941$	−14.1421	+	24.4949i	−0.461020	+	0.798511i	−0.999012	−	0.0444393i	$-0.985850\pi$
$941$	−14.1421	+	24.4949i	−0.461020	+	0.798511i	0.537992	+	0.842950i	$0.319183\pi$
$942$		0			0
$943$	24.7487	+	42.8661i	0.805930	+	1.39591i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.5000	+	52.8275i	0.991117	+	1.71666i	0.610735	+	0.791835i	$0.290874\pi$
$947$	30.5000	+	52.8275i	0.991117	+	1.71666i	0.380382	+	0.924829i	$0.375792\pi$
$948$		0			0
$949$	6.00000	−	10.3923i	0.194768	−	0.337348i
$950$		0			0
$951$		−	33.5410i		−	1.08764i
$952$		0			0
$953$			9.48683i			0.307309i	0.988125	+	0.153654i	$0.0491042\pi$
$953$			9.48683i			0.307309i	−0.988125	+	0.153654i	$0.950896\pi$
$954$		0			0
$955$	5.80948	+	3.35410i	0.187990	+	0.108536i
$956$		0			0
$957$	68.7014	−	15.1698i	2.22080	−	0.490371i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−13.0000	−	22.5167i	−0.419355	−	0.726344i
$962$		0			0
$963$	−10.9545	−	6.32456i	−0.353002	−	0.203806i
$964$		0			0
$965$	35.3553			1.13813
$966$		0			0
$967$		−	47.4342i		−	1.52538i	−0.646764	−	0.762690i	$-0.723878\pi$
$967$		−	47.4342i		−	1.52538i	0.646764	−	0.762690i	$-0.276122\pi$
$968$		0			0
$969$	−15.4919	−	8.94427i	−0.497673	−	0.287331i
$970$		0			0
$971$	−25.1744	+	14.5344i	−0.807885	+	0.466432i	−0.846221	−	0.532832i	$-0.821128\pi$
$971$	−25.1744	+	14.5344i	−0.807885	+	0.466432i	0.0383361	+	0.999265i	$0.487794\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.5000	−	18.1865i	0.335925	−	0.581839i	−0.647737	−	0.761864i	$-0.724285\pi$
$977$	10.5000	−	18.1865i	0.335925	−	0.581839i	0.983662	+	0.180025i	$0.0576179\pi$
$978$		0			0
$979$	21.2132	+	6.70820i	0.677977	+	0.214395i
$980$		0			0
$981$		−	31.6228i		−	1.00964i
$982$		0			0
$983$	1.93649	+	1.11803i	0.0617645	+	0.0356597i	0.530564	−	0.847645i	$-0.321980\pi$
$983$	1.93649	+	1.11803i	0.0617645	+	0.0356597i	−0.468800	+	0.883304i	$0.655313\pi$
$984$		0			0
$985$	14.1421	+	24.4949i	0.450606	+	0.780472i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.6931	−	7.90569i	0.435414	−	0.251386i
$990$		0			0
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	−0.849705	−	0.527258i	$-0.823220\pi$
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	0.881471	+	0.472237i	$0.156554\pi$
$992$		0			0
$993$		−	2.23607i		−	0.0709595i
$994$		0			0
$995$	40.0000			1.26809
$996$		0			0
$997$	6.36396	−	11.0227i	0.201549	−	0.349093i	−0.747479	−	0.664286i	$-0.768736\pi$
$997$	6.36396	−	11.0227i	0.201549	−	0.349093i	0.949028	+	0.315193i	$0.102069\pi$
$998$		0			0
$999$	21.3014	−	12.2984i	0.673947	−	0.389103i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.q.b.2089.1		8
7.2	even	3	inner	2156.2.q.b.901.4		8
7.3	odd	6		308.2.c.a.153.1	✓	4
7.4	even	3		308.2.c.a.153.4	yes	4
7.5	odd	6	inner	2156.2.q.b.901.2		8
7.6	odd	2	inner	2156.2.q.b.2089.3		8
11.10	odd	2	inner	2156.2.q.b.2089.2		8
21.11	odd	6		2772.2.i.c.1693.2		4
21.17	even	6		2772.2.i.c.1693.3		4
28.3	even	6		1232.2.e.b.769.4		4
28.11	odd	6		1232.2.e.b.769.1		4
77.10	even	6		308.2.c.a.153.2	yes	4
77.32	odd	6		308.2.c.a.153.3	yes	4
77.54	even	6	inner	2156.2.q.b.901.1		8
77.65	odd	6	inner	2156.2.q.b.901.3		8
77.76	even	2	inner	2156.2.q.b.2089.4		8
231.32	even	6		2772.2.i.c.1693.1		4
231.164	odd	6		2772.2.i.c.1693.4		4
308.87	odd	6		1232.2.e.b.769.3		4
308.263	even	6		1232.2.e.b.769.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.c.a.153.1	✓	4	7.3	odd	6
308.2.c.a.153.2	yes	4	77.10	even	6
308.2.c.a.153.3	yes	4	77.32	odd	6
308.2.c.a.153.4	yes	4	7.4	even	3
1232.2.e.b.769.1		4	28.11	odd	6
1232.2.e.b.769.2		4	308.263	even	6
1232.2.e.b.769.3		4	308.87	odd	6
1232.2.e.b.769.4		4	28.3	even	6
2156.2.q.b.901.1		8	77.54	even	6	inner
2156.2.q.b.901.2		8	7.5	odd	6	inner
2156.2.q.b.901.3		8	77.65	odd	6	inner
2156.2.q.b.901.4		8	7.2	even	3	inner
2156.2.q.b.2089.1		8	1.1	even	1	trivial
2156.2.q.b.2089.2		8	11.10	odd	2	inner
2156.2.q.b.2089.3		8	7.6	odd	2	inner
2156.2.q.b.2089.4		8	77.76	even	2	inner
2772.2.i.c.1693.1		4	231.32	even	6
2772.2.i.c.1693.2		4	21.11	odd	6
2772.2.i.c.1693.3		4	21.17	even	6
2772.2.i.c.1693.4		4	231.164	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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{3} +(1.93649 - 1.11803i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{3} +(1.93649 - 1.11803i) q^{5} +(1.00000 + 1.73205i) q^{9} +(-2.23861 - 2.44716i) q^{11} +2.82843 q^{13} -5.00000 q^{15} +(2.82843 - 4.89898i) q^{17} +(0.707107 + 1.22474i) q^{19} +(2.50000 + 4.33013i) q^{23} +2.23607i q^{27} -9.48683i q^{29} +(-1.93649 - 1.11803i) q^{31} +(1.59904 + 7.24176i) q^{33} +(-5.50000 - 9.52628i) q^{37} +(-5.47723 - 3.16228i) q^{39} +9.89949 q^{41} -3.16228i q^{43} +(3.87298 + 2.23607i) q^{45} +(3.87298 - 2.23607i) q^{47} +(-10.9545 + 6.32456i) q^{51} +(-4.00000 + 6.92820i) q^{53} +(-7.07107 - 2.23607i) q^{55} -3.16228i q^{57} +(1.93649 + 1.11803i) q^{59} +(2.12132 + 3.67423i) q^{61} +(5.47723 - 3.16228i) q^{65} +(-4.50000 + 7.79423i) q^{67} -11.1803i q^{69} -1.00000 q^{71} +(2.12132 - 3.67423i) q^{73} +(5.47723 - 3.16228i) q^{79} +(5.50000 - 9.52628i) q^{81} -1.41421 q^{83} -12.6491i q^{85} +(-10.6066 + 18.3712i) q^{87} +(-5.80948 + 3.35410i) q^{89} +(2.50000 + 4.33013i) q^{93} +(2.73861 + 1.58114i) q^{95} -15.6525i q^{97} +(2.00000 - 6.32456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 4 q^{11} - 40 q^{15} + 20 q^{23} - 44 q^{37} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 44 q^{81} + 20 q^{93} + 16 q^{99}+O(q^{100}) \) 8 * q + 8 * q^9 + 4 * q^11 - 40 * q^15 + 20 * q^23 - 44 * q^37 - 32 * q^53 - 36 * q^67 - 8 * q^71 + 44 * q^81 + 20 * q^93 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more