LMFDB - Embedded newform 312.2.bp.a.89.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,2,Mod(41,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 6, 1]))

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

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

chi := DirichletCharacter("312.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	312.bp (of order $12$, degree $4$, 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:	$2.49133254306$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.3
Character	$\chi$	$=$	312.89
Dual form			312.2.bp.a.305.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.43567 - 0.968940i) q^{3} +(2.19967 - 2.19967i) q^{5} +(1.17763 + 4.39498i) q^{7} +(1.12231 + 2.78216i) q^{9} +O(q^{10})$	\(q+(-1.43567 - 0.968940i) q^{3} +(2.19967 - 2.19967i) q^{5} +(1.17763 + 4.39498i) q^{7} +(1.12231 + 2.78216i) q^{9} +(0.917595 - 3.42451i) q^{11} +(0.225112 - 3.59852i) q^{13} +(-5.28937 + 1.02666i) q^{15} +(2.20507 - 3.81929i) q^{17} +(1.06243 - 0.284677i) q^{19} +(2.56778 - 7.45081i) q^{21} +(-0.812870 - 1.40793i) q^{23} -4.67714i q^{25} +(1.08448 - 5.08172i) q^{27} +(-4.61533 + 2.66466i) q^{29} +(3.28177 + 3.28177i) q^{31} +(-4.63551 + 4.02738i) q^{33} +(12.2579 + 7.07712i) q^{35} +(2.75348 + 0.737792i) q^{37} +(-3.80993 + 4.94817i) q^{39} +(9.76774 + 2.61726i) q^{41} +(-6.07183 - 3.50557i) q^{43} +(8.58857 + 3.65113i) q^{45} +(-4.14895 - 4.14895i) q^{47} +(-11.8669 + 6.85134i) q^{49} +(-6.86643 + 3.34667i) q^{51} +8.33491i q^{53} +(-5.51440 - 9.55122i) q^{55} +(-1.80113 - 0.620727i) q^{57} +(-1.97653 + 0.529610i) q^{59} +(-4.77501 + 8.27056i) q^{61} +(-10.9059 + 8.20890i) q^{63} +(-7.42039 - 8.41074i) q^{65} +(-1.42966 + 5.33556i) q^{67} +(-0.197186 + 2.80895i) q^{69} +(-1.43147 - 5.34234i) q^{71} +(2.99445 - 2.99445i) q^{73} +(-4.53187 + 6.71484i) q^{75} +16.1312 q^{77} +15.4490 q^{79} +(-6.48084 + 6.24490i) q^{81} +(-6.38106 + 6.38106i) q^{83} +(-3.55077 - 13.2516i) q^{85} +(9.20800 + 0.646395i) q^{87} +(-1.98271 + 7.39957i) q^{89} +(16.0805 - 3.24837i) q^{91} +(-1.53171 - 7.89138i) q^{93} +(1.71080 - 2.96319i) q^{95} +(-8.75030 + 2.34464i) q^{97} +(10.5574 - 1.29047i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 4 q^{7}+O(q^{10}) $ 56 * q + 4 * q^7	$ 56 q + 4 q^{7} + 8 q^{13} - 8 q^{15} + 4 q^{19} + 16 q^{21} + 24 q^{27} - 36 q^{31} + 28 q^{33} + 20 q^{37} + 16 q^{39} - 84 q^{43} + 12 q^{45} - 12 q^{49} - 24 q^{55} - 36 q^{57} - 24 q^{61} - 12 q^{63} - 32 q^{67} - 36 q^{69} - 20 q^{73} - 60 q^{75} - 32 q^{79} - 88 q^{85} - 16 q^{87} + 28 q^{91} - 88 q^{93} - 36 q^{97} + 44 q^{99}+O(q^{100}) $ 56 * q + 4 * q^7 + 8 * q^13 - 8 * q^15 + 4 * q^19 + 16 * q^21 + 24 * q^27 - 36 * q^31 + 28 * q^33 + 20 * q^37 + 16 * q^39 - 84 * q^43 + 12 * q^45 - 12 * q^49 - 24 * q^55 - 36 * q^57 - 24 * q^61 - 12 * q^63 - 32 * q^67 - 36 * q^69 - 20 * q^73 - 60 * q^75 - 32 * q^79 - 88 * q^85 - 16 * q^87 + 28 * q^91 - 88 * q^93 - 36 * q^97 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$e\left(\frac{7}{12}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.43567	−	0.968940i	−0.828886	−	0.559418i
$3$	−1.43567	−	0.968940i	−0.828886	−	0.559418i
$4$		0			0
$5$	2.19967	−	2.19967i	0.983725	−	0.983725i	−0.0161451	−	0.999870i	$-0.505139\pi$
$5$	2.19967	−	2.19967i	0.983725	−	0.983725i	0.999870	+	0.0161451i	$0.00513938\pi$
$6$		0			0
$7$	1.17763	+	4.39498i	0.445103	+	1.66115i	0.715665	+	0.698444i	$0.246124\pi$
$7$	1.17763	+	4.39498i	0.445103	+	1.66115i	−0.270562	+	0.962703i	$0.587209\pi$
$8$		0			0
$9$	1.12231	+	2.78216i	0.374104	+	0.927387i
$10$		0			0
$11$	0.917595	−	3.42451i	0.276665	−	1.03253i	−0.678052	−	0.735014i	$-0.737176\pi$
$11$	0.917595	−	3.42451i	0.276665	−	1.03253i	0.954717	−	0.297515i	$-0.0961577\pi$
$12$		0			0
$13$	0.225112	−	3.59852i	0.0624348	−	0.998049i
$13$	0.225112	−	3.59852i	0.0624348	−	0.998049i
$14$		0			0
$15$	−5.28937	+	1.02666i	−1.36571	+	0.265082i
$16$		0			0
$17$	2.20507	−	3.81929i	0.534808	−	0.926315i	−0.464364	−	0.885644i	$-0.653717\pi$
$17$	2.20507	−	3.81929i	0.534808	−	0.926315i	0.999173	−	0.0406707i	$-0.0129495\pi$
$18$		0			0
$19$	1.06243	−	0.284677i	0.243738	−	0.0653094i	−0.134882	−	0.990862i	$-0.543065\pi$
$19$	1.06243	−	0.284677i	0.243738	−	0.0653094i	0.378620	+	0.925552i	$0.376399\pi$
$20$		0			0
$21$	2.56778	−	7.45081i	0.560335	−	1.62590i
$22$		0			0
$23$	−0.812870	−	1.40793i	−0.169495	−	0.293574i	0.768747	−	0.639553i	$-0.220880\pi$
$23$	−0.812870	−	1.40793i	−0.169495	−	0.293574i	−0.938242	+	0.345978i	$0.887547\pi$
$24$		0			0
$25$		−	4.67714i		−	0.935428i
$26$		0			0
$27$	1.08448	−	5.08172i	0.208707	−	0.977978i
$28$		0			0
$29$	−4.61533	+	2.66466i	−0.857045	+	0.494815i	−0.863022	−	0.505167i	$-0.831431\pi$
$29$	−4.61533	+	2.66466i	−0.857045	+	0.494815i	0.00597654	+	0.999982i	$0.498098\pi$
$30$		0			0
$31$	3.28177	+	3.28177i	0.589423	+	0.589423i	0.937475	−	0.348052i	$-0.113157\pi$
$31$	3.28177	+	3.28177i	0.589423	+	0.589423i	−0.348052	+	0.937475i	$0.613157\pi$
$32$		0			0
$33$	−4.63551	+	4.02738i	−0.806939	+	0.701077i
$34$		0			0
$35$	12.2579	+	7.07712i	2.07197	+	1.19625i
$36$		0			0
$37$	2.75348	+	0.737792i	0.452669	+	0.121292i	0.477948	−	0.878388i	$-0.341381\pi$
$37$	2.75348	+	0.737792i	0.452669	+	0.121292i	−0.0252790	+	0.999680i	$0.508047\pi$
$38$		0			0
$39$	−3.80993	+	4.94817i	−0.610078	+	0.792342i
$40$		0			0
$41$	9.76774	+	2.61726i	1.52546	+	0.408747i	0.921536	−	0.388292i	$-0.126935\pi$
$41$	9.76774	+	2.61726i	1.52546	+	0.408747i	0.603928	+	0.797039i	$0.293602\pi$
$42$		0			0
$43$	−6.07183	−	3.50557i	−0.925946	−	0.534595i	−0.0404187	−	0.999183i	$-0.512869\pi$
$43$	−6.07183	−	3.50557i	−0.925946	−	0.534595i	−0.885527	+	0.464588i	$0.846203\pi$
$44$		0			0
$45$	8.58857	+	3.65113i	1.28031	+	0.544278i
$46$		0			0
$47$	−4.14895	−	4.14895i	−0.605186	−	0.605186i	0.336498	−	0.941684i	$-0.390757\pi$
$47$	−4.14895	−	4.14895i	−0.605186	−	0.605186i	−0.941684	+	0.336498i	$0.890757\pi$
$48$		0			0
$49$	−11.8669	+	6.85134i	−1.69527	+	0.978762i
$50$		0			0
$51$	−6.86643	+	3.34667i	−0.961492	+	0.468628i
$52$		0			0
$53$			8.33491i			1.14489i	0.819944	+	0.572444i	$0.194005\pi$
$53$			8.33491i			1.14489i	−0.819944	+	0.572444i	$0.805995\pi$
$54$		0			0
$55$	−5.51440	−	9.55122i	−0.743561	−	1.28789i
$56$		0			0
$57$	−1.80113	−	0.620727i	−0.238566	−	0.0822173i
$58$		0			0
$59$	−1.97653	+	0.529610i	−0.257322	+	0.0689493i	−0.385174	−	0.922844i	$-0.625859\pi$
$59$	−1.97653	+	0.529610i	−0.257322	+	0.0689493i	0.127852	+	0.991793i	$0.459192\pi$
$60$		0			0
$61$	−4.77501	+	8.27056i	−0.611378	+	1.05894i	0.379631	+	0.925138i	$0.376051\pi$
$61$	−4.77501	+	8.27056i	−0.611378	+	1.05894i	−0.991008	+	0.133799i	$0.957282\pi$
$62$		0			0
$63$	−10.9059	+	8.20890i	−1.37401	+	1.03422i
$64$		0			0
$65$	−7.42039	−	8.41074i	−0.920387	−	1.04322i
$66$		0			0
$67$	−1.42966	+	5.33556i	−0.174661	+	0.651842i	0.821949	+	0.569562i	$0.192887\pi$
$67$	−1.42966	+	5.33556i	−0.174661	+	0.651842i	−0.996609	+	0.0822804i	$0.973780\pi$
$68$		0			0
$69$	−0.197186	+	2.80895i	−0.0237385	+	0.338158i
$70$		0			0
$71$	−1.43147	−	5.34234i	−0.169885	−	0.634019i	−0.997367	−	0.0725259i	$-0.976894\pi$
$71$	−1.43147	−	5.34234i	−0.169885	−	0.634019i	0.827482	−	0.561493i	$-0.189773\pi$
$72$		0			0
$73$	2.99445	−	2.99445i	0.350473	−	0.350473i	−0.509812	−	0.860286i	$-0.670285\pi$
$73$	2.99445	−	2.99445i	0.350473	−	0.350473i	0.860286	+	0.509812i	$0.170285\pi$
$74$		0			0
$75$	−4.53187	+	6.71484i	−0.523295	+	0.775363i
$76$		0			0
$77$	16.1312			1.83833
$78$		0			0
$79$	15.4490			1.73815			0.869077	−	0.494677i	$-0.164714\pi$
$79$	15.4490			1.73815			0.869077	+	0.494677i	$0.164714\pi$
$80$		0			0
$81$	−6.48084	+	6.24490i	−0.720093	+	0.693878i
$82$		0			0
$83$	−6.38106	+	6.38106i	−0.700413	+	0.700413i	−0.964499	−	0.264086i	$-0.914930\pi$
$83$	−6.38106	+	6.38106i	−0.700413	+	0.700413i	0.264086	+	0.964499i	$0.414930\pi$
$84$		0			0
$85$	−3.55077	−	13.2516i	−0.385135	−	1.43734i
$86$		0			0
$87$	9.20800	+	0.646395i	0.987201	+	0.0693008i
$88$		0			0
$89$	−1.98271	+	7.39957i	−0.210167	+	0.784353i	0.777646	+	0.628703i	$0.216414\pi$
$89$	−1.98271	+	7.39957i	−0.210167	+	0.784353i	−0.987812	+	0.155650i	$0.950253\pi$
$90$		0			0
$91$	16.0805	−	3.24837i	1.68570	−	0.340521i
$92$		0			0
$93$	−1.53171	−	7.89138i	−0.158831	−	0.818298i
$94$		0			0
$95$	1.71080	−	2.96319i	0.175525	−	0.304017i
$96$		0			0
$97$	−8.75030	+	2.34464i	−0.888458	+	0.238062i	−0.674053	−	0.738683i	$-0.735448\pi$
$97$	−8.75030	+	2.34464i	−0.888458	+	0.238062i	−0.214405	+	0.976745i	$0.568781\pi$
$98$		0			0
$99$	10.5574	−	1.29047i	1.06106	−	0.129697i
$100$		0			0
$101$	3.52149	+	6.09939i	0.350401	+	0.606912i	0.986320	−	0.164844i	$-0.0527120\pi$
$101$	3.52149	+	6.09939i	0.350401	+	0.606912i	−0.635919	+	0.771756i	$0.719379\pi$
$102$		0			0
$103$			7.10076i			0.699658i	0.936813	+	0.349829i	$0.113760\pi$
$103$			7.10076i			0.699658i	−0.936813	+	0.349829i	$0.886240\pi$
$104$		0			0
$105$	−10.7411	−	22.0376i	−1.04822	−	2.15065i
$106$		0			0
$107$	−10.7690	+	6.21749i	−1.04108	+	0.601068i	−0.920139	−	0.391592i	$-0.871924\pi$
$107$	−10.7690	+	6.21749i	−1.04108	+	0.601068i	−0.120941	+	0.992660i	$0.538591\pi$
$108$		0			0
$109$	5.09377	+	5.09377i	0.487895	+	0.487895i	0.907641	−	0.419746i	$-0.137881\pi$
$109$	5.09377	+	5.09377i	0.487895	+	0.487895i	−0.419746	+	0.907641i	$0.637881\pi$
$110$		0			0
$111$	−3.23822	−	3.72718i	−0.307358	−	0.353769i
$112$		0			0
$113$	−10.4784	−	6.04969i	−0.985722	−	0.569107i	−0.0817293	−	0.996655i	$-0.526044\pi$
$113$	−10.4784	−	6.04969i	−0.985722	−	0.569107i	−0.903993	+	0.427548i	$0.859378\pi$
$114$		0			0
$115$	−4.88504	−	1.30894i	−0.455533	−	0.122060i
$116$		0			0
$117$	10.2643	−	3.41236i	0.948935	−	0.315473i
$118$		0			0
$119$	19.3825	+	5.19352i	1.77679	+	0.476089i
$120$		0			0
$121$	−1.35901	−	0.784625i	−0.123546	−	0.0713296i
$122$		0			0
$123$	−11.4873	−	13.2219i	−1.03578	−	1.19218i
$124$		0			0
$125$	0.710189	+	0.710189i	0.0635212	+	0.0635212i
$126$		0			0
$127$	11.6158	−	6.70637i	1.03073	−	0.595094i	0.113539	−	0.993534i	$-0.463781\pi$
$127$	11.6158	−	6.70637i	1.03073	−	0.595094i	0.917195	+	0.398440i	$0.130448\pi$
$128$		0			0
$129$	5.32047	+	10.9161i	0.468441	+	0.961109i
$130$		0			0
$131$		−	10.7616i		−	0.940244i	−0.882601	−	0.470122i	$-0.844210\pi$
$131$		−	10.7616i		−	0.940244i	0.882601	−	0.470122i	$-0.155790\pi$
$132$		0			0
$133$	2.50230	+	4.33411i	0.216977	+	0.375815i
$134$		0			0
$135$	−8.79265	−	13.5636i	−0.756750	−	1.16737i
$136$		0			0
$137$	−18.9042	+	5.06536i	−1.61509	+	0.432763i	−0.949555	−	0.313600i	$-0.898465\pi$
$137$	−18.9042	+	5.06536i	−1.61509	+	0.432763i	−0.665539	+	0.746363i	$0.731798\pi$
$138$		0			0
$139$	1.42992	−	2.47669i	0.121284	−	0.210070i	−0.798990	−	0.601344i	$-0.794632\pi$
$139$	1.42992	−	2.47669i	0.121284	−	0.210070i	0.920274	+	0.391274i	$0.127966\pi$
$140$		0			0
$141$	1.93645	+	9.97661i	0.163078	+	0.840182i
$142$		0			0
$143$	−12.1166	−	4.07288i	−1.01324	−	0.340591i
$144$		0			0
$145$	−4.29083	+	16.0136i	−0.356334	+	1.32986i
$146$		0			0
$147$	23.6755	+	1.66200i	1.95272	+	0.137080i
$148$		0			0
$149$	−5.07006	−	18.9217i	−0.415356	−	1.55013i	−0.784122	−	0.620607i	$-0.786886\pi$
$149$	−5.07006	−	18.9217i	−0.415356	−	1.55013i	0.368766	−	0.929522i	$-0.379780\pi$
$150$		0			0
$151$	−15.8267	+	15.8267i	−1.28796	+	1.28796i	−0.351940	+	0.936023i	$0.614478\pi$
$151$	−15.8267	+	15.8267i	−1.28796	+	1.28796i	−0.936023	+	0.351940i	$0.885522\pi$
$152$		0			0
$153$	13.1007	+	1.84842i	1.05913	+	0.149436i
$154$		0			0
$155$	14.4376			1.15966
$156$		0			0
$157$	1.15713			0.0923494			0.0461747	−	0.998933i	$-0.485297\pi$
$157$	1.15713			0.0923494			0.0461747	+	0.998933i	$0.485297\pi$
$158$		0			0
$159$	8.07603	−	11.9662i	0.640471	−	0.948982i
$160$		0			0
$161$	5.23057	−	5.23057i	0.412227	−	0.412227i
$162$		0			0
$163$	−3.58002	−	13.3608i	−0.280409	−	1.04650i	−0.952129	−	0.305696i	$-0.901111\pi$
$163$	−3.58002	−	13.3608i	−0.280409	−	1.04650i	0.671720	−	0.740805i	$-0.265556\pi$
$164$		0			0
$165$	−1.33769	+	19.0555i	−0.104139	+	1.48347i
$166$		0			0
$167$	−2.53415	+	9.45758i	−0.196098	+	0.731849i	0.795881	+	0.605453i	$0.207008\pi$
$167$	−2.53415	+	9.45758i	−0.196098	+	0.731849i	−0.991980	+	0.126397i	$0.959659\pi$
$168$		0			0
$169$	−12.8986	−	1.62014i	−0.992204	−	0.124626i
$170$		0			0
$171$	1.98439	+	2.63635i	0.151750	+	0.201607i
$172$		0			0
$173$	2.36780	−	4.10115i	0.180020	−	0.311804i	−0.761867	−	0.647734i	$-0.775717\pi$
$173$	2.36780	−	4.10115i	0.180020	−	0.311804i	0.941887	+	0.335929i	$0.109050\pi$
$174$		0			0
$175$	20.5559	−	5.50795i	1.55388	−	0.416362i
$176$		0			0
$177$	3.35081	+	1.15479i	0.251862	+	0.0867995i
$178$		0			0
$179$	4.29215	+	7.43422i	0.320810	+	0.555660i	0.980655	−	0.195742i	$-0.0627115\pi$
$179$	4.29215	+	7.43422i	0.320810	+	0.555660i	−0.659845	+	0.751402i	$0.729378\pi$
$180$		0			0
$181$		−	6.64164i		−	0.493670i	−0.969058	−	0.246835i	$-0.920610\pi$
$181$		−	6.64164i		−	0.493670i	0.969058	−	0.246835i	$-0.0793905\pi$
$182$		0			0
$183$	14.8690	−	7.24712i	1.09915	−	0.535723i
$184$		0			0
$185$	7.67966	−	4.43385i	0.564620	−	0.325983i
$186$		0			0
$187$	−11.0559	−	11.0559i	−0.808484	−	0.808484i
$188$		0			0
$189$	23.6112	−	1.21815i	1.71746	−	0.0886073i
$190$		0			0
$191$	6.75249	+	3.89855i	0.488593	+	0.282089i	0.723991	−	0.689810i	$-0.242306\pi$
$191$	6.75249	+	3.89855i	0.488593	+	0.282089i	−0.235398	+	0.971899i	$0.575639\pi$
$192$		0			0
$193$	−2.57872	−	0.690967i	−0.185621	−	0.0497369i	0.164811	−	0.986325i	$-0.447299\pi$
$193$	−2.57872	−	0.690967i	−0.185621	−	0.0497369i	−0.350432	+	0.936588i	$0.613965\pi$
$194$		0			0
$195$	2.50375	+	19.2650i	0.179298	+	1.37959i
$196$		0			0
$197$	−18.6823	−	5.00590i	−1.33106	−	0.356655i	−0.477947	−	0.878388i	$-0.658619\pi$
$197$	−18.6823	−	5.00590i	−1.33106	−	0.356655i	−0.853109	+	0.521733i	$0.825286\pi$
$198$		0			0
$199$	6.95427	+	4.01505i	0.492975	+	0.284619i	0.725808	−	0.687898i	$-0.241466\pi$
$199$	6.95427	+	4.01505i	0.492975	+	0.284619i	−0.232833	+	0.972517i	$0.574800\pi$
$200$		0			0
$201$	7.22235	−	6.27486i	0.509426	−	0.442594i
$202$		0			0
$203$	−17.1463	−	17.1463i	−1.20343	−	1.20343i
$204$		0			0
$205$	27.2430	−	15.7287i	1.90273	−	1.09854i
$206$		0			0
$207$	3.00480	−	3.84167i	0.208848	−	0.267015i
$208$		0			0
$209$		−	3.89952i		−	0.269735i
$210$		0			0
$211$	2.69364	+	4.66552i	0.185438	+	0.321187i	0.943724	−	0.330734i	$-0.107296\pi$
$211$	2.69364	+	4.66552i	0.185438	+	0.321187i	−0.758286	+	0.651922i	$0.773963\pi$
$212$		0			0
$213$	−3.12127	+	9.05686i	−0.213866	+	0.620566i
$214$		0			0
$215$	−21.0672	+	5.64493i	−1.43677	+	0.384981i
$216$		0			0
$217$	−10.5586	+	18.2880i	−0.716764	+	1.24147i
$218$		0			0
$219$	−7.20048	+	1.39761i	−0.486564	+	0.0944415i
$220$		0			0
$221$	−13.2474	−	8.79475i	−0.891117	−	0.591599i
$222$		0			0
$223$	2.18490	−	8.15417i	0.146312	−	0.546044i	−0.853381	−	0.521287i	$-0.825452\pi$
$223$	2.18490	−	8.15417i	0.146312	−	0.546044i	0.999693	−	0.0247569i	$-0.00788119\pi$
$224$		0			0
$225$	13.0126	−	5.24920i	0.867504	−	0.349947i
$226$		0			0
$227$	−0.237467	−	0.886239i	−0.0157612	−	0.0588217i	0.957597	−	0.288110i	$-0.0930268\pi$
$227$	−0.237467	−	0.886239i	−0.0157612	−	0.0588217i	−0.973359	+	0.229288i	$0.926360\pi$
$228$		0			0
$229$	−2.28432	+	2.28432i	−0.150952	+	0.150952i	−0.778543	−	0.627591i	$-0.784041\pi$
$229$	−2.28432	+	2.28432i	−0.150952	+	0.150952i	0.627591	+	0.778543i	$0.284041\pi$
$230$		0			0
$231$	−23.1592	−	15.6302i	−1.52376	−	1.02839i
$232$		0			0
$233$	−9.30420			−0.609538			−0.304769	−	0.952426i	$-0.598579\pi$
$233$	−9.30420			−0.609538			−0.304769	+	0.952426i	$0.598579\pi$
$234$		0			0
$235$	−18.2527			−1.19067
$236$		0			0
$237$	−22.1798	−	14.9692i	−1.44073	−	0.972354i
$238$		0			0
$239$	18.5336	−	18.5336i	1.19884	−	1.19884i	0.224323	−	0.974515i	$-0.427983\pi$
$239$	18.5336	−	18.5336i	1.19884	−	1.19884i	0.974515	−	0.224323i	$-0.0720171\pi$
$240$		0			0
$241$	−3.15760	−	11.7843i	−0.203399	−	0.759094i	−0.989932	−	0.141546i	$-0.954793\pi$
$241$	−3.15760	−	11.7843i	−0.203399	−	0.759094i	0.786533	−	0.617548i	$-0.211874\pi$
$242$		0			0
$243$	15.3553	−	2.68609i	0.985042	−	0.172313i
$244$		0			0
$245$	−11.0325	+	41.1740i	−0.704842	+	2.63051i
$246$		0			0
$247$	−0.785250	−	3.88725i	−0.0499642	−	0.247340i
$248$		0			0
$249$	15.3440	−	2.97825i	0.972386	−	0.188739i
$250$		0			0
$251$	−4.56257	+	7.90260i	−0.287987	+	0.498808i	−0.973329	−	0.229413i	$-0.926319\pi$
$251$	−4.56257	+	7.90260i	−0.287987	+	0.498808i	0.685342	+	0.728221i	$0.259653\pi$
$252$		0			0
$253$	−5.56736	+	1.49177i	−0.350017	+	0.0937868i
$254$		0			0
$255$	−7.74231	+	22.4655i	−0.484842	+	1.40684i
$256$		0			0
$257$	0.0791658	+	0.137119i	0.00493823	+	0.00855326i	0.868484	−	0.495717i	$-0.165095\pi$
$257$	0.0791658	+	0.137119i	0.00493823	+	0.00855326i	−0.863546	+	0.504271i	$0.831761\pi$
$258$		0			0
$259$			12.9703i			0.805937i
$260$		0			0
$261$	−12.5934	−	9.85001i	−0.779509	−	0.609700i
$262$		0			0
$263$	3.05457	−	1.76356i	0.188353	−	0.108746i	−0.402858	−	0.915262i	$-0.631983\pi$
$263$	3.05457	−	1.76356i	0.188353	−	0.108746i	0.591211	+	0.806517i	$0.298650\pi$
$264$		0			0
$265$	18.3341	+	18.3341i	1.12625	+	1.12625i
$266$		0			0
$267$	10.0163	−	8.70223i	0.612985	−	0.532568i
$268$		0			0
$269$	16.2959	+	9.40842i	0.993576	+	0.573641i	0.906341	−	0.422546i	$-0.138864\pi$
$269$	16.2959	+	9.40842i	0.993576	+	0.573641i	0.0872346	+	0.996188i	$0.472197\pi$
$270$		0			0
$271$	1.95323	+	0.523366i	0.118650	+	0.0317922i	0.317656	−	0.948206i	$-0.397104\pi$
$271$	1.95323	+	0.523366i	0.118650	+	0.0317922i	−0.199005	+	0.979998i	$0.563771\pi$
$272$		0			0
$273$	−26.2338	−	10.9175i	−1.58774	−	0.660755i
$274$		0			0
$275$	−16.0169	−	4.29172i	−0.965856	−	0.258800i
$276$		0			0
$277$	15.1882	+	8.76890i	0.912569	+	0.526872i	0.881257	−	0.472638i	$-0.156698\pi$
$277$	15.1882	+	8.76890i	0.912569	+	0.526872i	0.0313123	+	0.999510i	$0.490031\pi$
$278$		0			0
$279$	−5.44724	+	12.8136i	−0.326118	+	0.767128i
$280$		0			0
$281$	13.8319	+	13.8319i	0.825142	+	0.825142i	0.986840	−	0.161698i	$-0.0516972\pi$
$281$	13.8319	+	13.8319i	0.825142	+	0.825142i	−0.161698	+	0.986840i	$0.551697\pi$
$282$		0			0
$283$	−15.2851	+	8.82488i	−0.908607	+	0.524585i	−0.879983	−	0.475006i	$-0.842446\pi$
$283$	−15.2851	+	8.82488i	−0.908607	+	0.524585i	−0.0286245	+	0.999590i	$0.509113\pi$
$284$		0			0
$285$	−5.32731	+	2.59651i	−0.315562	+	0.153804i
$286$		0			0
$287$			46.0112i			2.71595i
$288$		0			0
$289$	−1.22467	−	2.12120i	−0.0720396	−	0.124776i
$290$		0			0
$291$	14.8344	+	5.11239i	0.869607	+	0.299693i
$292$		0			0
$293$	−12.2454	+	3.28114i	−0.715383	+	0.191686i	−0.598111	−	0.801414i	$-0.704082\pi$
$293$	−12.2454	+	3.28114i	−0.715383	+	0.191686i	−0.117272	+	0.993100i	$0.537415\pi$
$294$		0			0
$295$	−3.18275	+	5.51269i	−0.185307	+	0.320961i
$296$		0			0
$297$	−16.4073	−	8.37676i	−0.952048	−	0.486069i
$298$		0			0
$299$	−5.24945	+	2.60818i	−0.303584	+	0.150835i
$300$		0			0
$301$	8.25655	−	30.8139i	0.475900	−	1.77608i
$302$		0			0
$303$	0.854244	−	12.1688i	0.0490751	−	0.699082i
$304$		0			0
$305$	7.68908	+	28.6960i	0.440275	+	1.64313i
$306$		0			0
$307$	−6.09233	+	6.09233i	−0.347707	+	0.347707i	−0.859255	−	0.511548i	$-0.829072\pi$
$307$	−6.09233	+	6.09233i	−0.347707	+	0.347707i	0.511548	+	0.859255i	$0.329072\pi$
$308$		0			0
$309$	6.88021	−	10.1944i	0.391401	−	0.579937i
$310$		0			0
$311$	−9.87859			−0.560164			−0.280082	−	0.959976i	$-0.590362\pi$
$311$	−9.87859			−0.560164			−0.280082	+	0.959976i	$0.590362\pi$
$312$		0			0
$313$	17.2222			0.973456			0.486728	−	0.873554i	$-0.338190\pi$
$313$	17.2222			0.973456			0.486728	+	0.873554i	$0.338190\pi$
$314$		0			0
$315$	−5.93248	+	42.0463i	−0.334257	+	2.36904i
$316$		0			0
$317$	0.00884596	−	0.00884596i	0.000496839	−	0.000496839i	−0.706858	−	0.707355i	$-0.749888\pi$
$317$	0.00884596	−	0.00884596i	0.000496839	−	0.000496839i	0.707355	+	0.706858i	$0.249888\pi$
$318$		0			0
$319$	4.89016	+	18.2503i	0.273796	+	1.02182i
$320$		0			0
$321$	21.4852	+	1.50824i	1.19918	+	0.0841819i
$322$		0			0
$323$	1.25547	−	4.68546i	0.0698560	−	0.260706i
$324$		0			0
$325$	−16.8308	−	1.05288i	−0.933603	−	0.0584032i
$326$		0			0
$327$	−2.37743	−	12.2486i	−0.131472	−	0.677346i
$328$		0			0
$329$	13.3486	−	23.1205i	0.735933	−	1.27467i
$330$		0			0
$331$	0.360355	−	0.0965567i	0.0198069	−	0.00530724i	−0.248902	−	0.968529i	$-0.580070\pi$
$331$	0.360355	−	0.0965567i	0.0198069	−	0.00530724i	0.268709	+	0.963221i	$0.413403\pi$
$332$		0			0
$333$	1.03760	+	8.48865i	0.0568602	+	0.465175i
$334$		0			0
$335$	8.59171	+	14.8813i	0.469415	+	0.813051i
$336$		0			0
$337$		−	22.3015i		−	1.21484i	−0.794381	−	0.607420i	$-0.792204\pi$
$337$		−	22.3015i		−	1.21484i	0.794381	−	0.607420i	$-0.207796\pi$
$338$		0			0
$339$	9.18172	+	18.8383i	0.498683	+	1.02316i
$340$		0			0
$341$	14.2498	−	8.22711i	0.771669	−	0.445523i
$342$		0			0
$343$	−21.5648	−	21.5648i	−1.16439	−	1.16439i
$344$		0			0
$345$	5.74503	+	6.61253i	0.309302	+	0.356006i
$346$		0			0
$347$	0.801179	+	0.462561i	0.0430095	+	0.0248316i	0.521351	−	0.853343i	$-0.325428\pi$
$347$	0.801179	+	0.462561i	0.0430095	+	0.0248316i	−0.478341	+	0.878174i	$0.658762\pi$
$348$		0			0
$349$	−16.4270	−	4.40159i	−0.879315	−	0.235612i	−0.209204	−	0.977872i	$-0.567087\pi$
$349$	−16.4270	−	4.40159i	−0.879315	−	0.235612i	−0.670112	+	0.742260i	$0.733754\pi$
$350$		0			0
$351$	−18.0425	−	5.04646i	−0.963040	−	0.269360i
$352$		0			0
$353$	1.31686	+	0.352851i	0.0700893	+	0.0187804i	0.293693	−	0.955900i	$-0.405116\pi$
$353$	1.31686	+	0.352851i	0.0700893	+	0.0187804i	−0.223604	+	0.974680i	$0.571782\pi$
$354$		0			0
$355$	−14.9002	−	8.60262i	−0.790819	−	0.456580i
$356$		0			0
$357$	−22.7947	−	26.2367i	−1.20642	−	1.38859i
$358$		0			0
$359$	−23.2669	−	23.2669i	−1.22798	−	1.22798i	−0.964726	−	0.263256i	$-0.915204\pi$
$359$	−23.2669	−	23.2669i	−1.22798	−	1.22798i	−0.263256	−	0.964726i	$-0.584796\pi$
$360$		0			0
$361$	−15.4068	+	8.89510i	−0.810883	+	0.468163i
$362$		0			0
$363$	1.19084	+	2.44326i	0.0625029	+	0.128238i
$364$		0			0
$365$		−	13.1736i		−	0.689539i
$366$		0			0
$367$	8.66640	+	15.0106i	0.452382	+	0.783549i	0.998533	−	0.0541374i	$-0.0172409\pi$
$367$	8.66640	+	15.0106i	0.452382	+	0.783549i	−0.546151	+	0.837687i	$0.683908\pi$
$368$		0			0
$369$	3.68081	+	30.1128i	0.191615	+	1.56761i
$370$		0			0
$371$	−36.6318	+	9.81545i	−1.90183	+	0.509593i
$372$		0			0
$373$	1.40911	−	2.44066i	0.0729611	−	0.126372i	−0.827237	−	0.561854i	$-0.810088\pi$
$373$	1.40911	−	2.44066i	0.0729611	−	0.126372i	0.900198	+	0.435481i	$0.143422\pi$
$374$		0			0
$375$	−0.331468	−	1.70773i	−0.0171169	−	0.0881867i
$376$		0			0
$377$	8.54986	+	17.2082i	0.440340	+	0.886267i
$378$		0			0
$379$	−7.23342	+	26.9955i	−0.371556	+	1.38667i	0.486756	+	0.873538i	$0.338180\pi$
$379$	−7.23342	+	26.9955i	−0.371556	+	1.38667i	−0.858312	+	0.513128i	$0.828487\pi$
$380$		0			0
$381$	−23.1745	−	1.62684i	−1.18727	−	0.0833453i
$382$		0			0
$383$	1.58764	+	5.92516i	0.0811248	+	0.302762i	0.994552	−	0.104240i	$-0.0332410\pi$
$383$	1.58764	+	5.92516i	0.0811248	+	0.302762i	−0.913427	+	0.407002i	$0.866574\pi$
$384$		0			0
$385$	35.4835	−	35.4835i	1.80841	−	1.80841i
$386$		0			0
$387$	2.93859	−	20.8272i	0.149377	−	1.05870i
$388$		0			0
$389$	34.3070			1.73943			0.869717	−	0.493550i	$-0.164301\pi$
$389$	34.3070			1.73943			0.869717	+	0.493550i	$0.164301\pi$
$390$		0			0
$391$	−7.16974			−0.362589
$392$		0			0
$393$	−10.4273	+	15.4501i	−0.525989	+	0.779355i
$394$		0			0
$395$	33.9829	−	33.9829i	1.70986	−	1.70986i
$396$		0			0
$397$	2.44015	+	9.10677i	0.122468	+	0.457056i	0.999737	−	0.0229433i	$-0.00730371\pi$
$397$	2.44015	+	9.10677i	0.122468	+	0.457056i	−0.877269	+	0.479999i	$0.840637\pi$
$398$		0			0
$399$	0.607009	−	8.64694i	0.0303885	−	0.432888i
$400$		0			0
$401$	7.09363	−	26.4738i	0.354239	−	1.32204i	−0.527201	−	0.849741i	$-0.676758\pi$
$401$	7.09363	−	26.4738i	0.354239	−	1.32204i	0.881440	−	0.472297i	$-0.156575\pi$
$402$		0			0
$403$	12.5483	−	11.0707i	0.625073	−	0.551472i
$404$		0			0
$405$	−0.518985	+	27.9925i	−0.0257886	+	1.39096i
$406$		0			0
$407$	5.05315	−	8.75232i	0.250476	−	0.433836i
$408$		0			0
$409$	13.8641	−	3.71487i	0.685535	−	0.183689i	0.100793	−	0.994907i	$-0.467862\pi$
$409$	13.8641	−	3.71487i	0.685535	−	0.183689i	0.584743	+	0.811219i	$0.301195\pi$
$410$		0			0
$411$	32.0483	+	11.0448i	1.58082	+	0.544801i
$412$		0			0
$413$	−4.65525	−	8.06313i	−0.229070	−	0.396761i
$414$		0			0
$415$			28.0725i			1.37803i
$416$		0			0
$417$	−4.45265	+	2.17021i	−0.218047	+	0.106276i
$418$		0			0
$419$	27.2181	−	15.7144i	1.32969	−	0.767697i	0.344438	−	0.938809i	$-0.388069\pi$
$419$	27.2181	−	15.7144i	1.32969	−	0.767697i	0.985252	+	0.171112i	$0.0547361\pi$
$420$		0			0
$421$	15.8765	+	15.8765i	0.773773	+	0.773773i	0.978764	−	0.204991i	$-0.0657165\pi$
$421$	15.8765	+	15.8765i	0.773773	+	0.773773i	−0.204991	+	0.978764i	$0.565717\pi$
$422$		0			0
$423$	6.88663	−	16.1994i	0.334839	−	0.787644i
$424$		0			0
$425$	−17.8634	−	10.3134i	−0.866501	−	0.500274i
$426$		0			0
$427$	−41.9722	−	11.2464i	−2.03118	−	0.544252i
$428$		0			0
$429$	13.4491	+	17.5876i	0.649328	+	0.849136i
$430$		0			0
$431$	−12.3582	−	3.31137i	−0.595274	−	0.159503i	−0.0514116	−	0.998678i	$-0.516372\pi$
$431$	−12.3582	−	3.31137i	−0.595274	−	0.159503i	−0.543862	+	0.839174i	$0.683039\pi$
$432$		0			0
$433$	14.1339	+	8.16020i	0.679231	+	0.392154i	0.799565	−	0.600579i	$-0.205063\pi$
$433$	14.1339	+	8.16020i	0.679231	+	0.392154i	−0.120335	+	0.992733i	$0.538397\pi$
$434$		0			0
$435$	21.6765	−	18.8327i	1.03931	−	0.902961i
$436$		0			0
$437$	−1.26442	−	1.26442i	−0.0604855	−	0.0604855i
$438$		0			0
$439$	−20.2300	+	11.6798i	−0.965525	+	0.557446i	−0.897869	−	0.440262i	$-0.854885\pi$
$439$	−20.2300	+	11.6798i	−0.965525	+	0.557446i	−0.0676563	+	0.997709i	$0.521552\pi$
$440$		0			0
$441$	−32.3798	−	25.3262i	−1.54190	−	1.20601i
$442$		0			0
$443$		−	19.6001i		−	0.931228i	−0.884988	−	0.465614i	$-0.845834\pi$
$443$		−	19.6001i		−	0.931228i	0.884988	−	0.465614i	$-0.154166\pi$
$444$		0			0
$445$	11.9153	+	20.6380i	0.564841	+	0.978333i
$446$		0			0
$447$	−11.0551	+	32.0780i	−0.522887	+	1.51724i
$448$		0			0
$449$	−11.4364	+	3.06438i	−0.539717	+	0.144617i	−0.518372	−	0.855155i	$-0.673462\pi$
$449$	−11.4364	+	3.06438i	−0.539717	+	0.144617i	−0.0213456	+	0.999772i	$0.506795\pi$
$450$		0			0
$451$	17.9256	−	31.0481i	0.844086	−	1.46200i
$452$		0			0
$453$	38.0572	−	7.38686i	1.78808	−	0.347065i
$454$		0			0
$455$	28.2266	−	42.5172i	1.32328	−	1.99324i
$456$		0			0
$457$	8.92124	−	33.2945i	0.417318	−	1.55745i	−0.362830	−	0.931856i	$-0.618189\pi$
$457$	8.92124	−	33.2945i	0.417318	−	1.55745i	0.780147	−	0.625596i	$-0.215144\pi$
$458$		0			0
$459$	−17.0173	−	15.3475i	−0.794297	−	0.716359i
$460$		0			0
$461$	−4.84992	−	18.1002i	−0.225883	−	0.843008i	−0.982049	−	0.188628i	$-0.939596\pi$
$461$	−4.84992	−	18.1002i	−0.225883	−	0.843008i	0.756165	−	0.654381i	$-0.227071\pi$
$462$		0			0
$463$	16.7806	−	16.7806i	0.779860	−	0.779860i	−0.199947	−	0.979807i	$-0.564077\pi$
$463$	16.7806	−	16.7806i	0.779860	−	0.779860i	0.979807	+	0.199947i	$0.0640769\pi$
$464$		0			0
$465$	−20.7277	−	13.9892i	−0.961225	−	0.648734i
$466$		0			0
$467$	30.7628			1.42353			0.711767	−	0.702416i	$-0.247895\pi$
$467$	30.7628			1.42353			0.711767	+	0.702416i	$0.247895\pi$
$468$		0			0
$469$	−25.1333			−1.16055
$470$		0			0
$471$	−1.66127	−	1.12119i	−0.0765471	−	0.0516619i
$472$		0			0
$473$	−17.5764	+	17.5764i	−0.808162	+	0.808162i
$474$		0			0
$475$	−1.33147	−	4.96913i	−0.0610922	−	0.227999i
$476$		0			0
$477$	−23.1891	+	9.35436i	−1.06175	+	0.428307i
$478$		0			0
$479$	−4.38660	+	16.3710i	−0.200429	+	0.748010i	0.790366	+	0.612635i	$0.209891\pi$
$479$	−4.38660	+	16.3710i	−0.200429	+	0.748010i	−0.990794	+	0.135375i	$0.956776\pi$
$480$		0			0
$481$	3.27480	−	9.74235i	0.149318	−	0.444213i
$482$		0			0
$483$	−12.5775	+	2.44128i	−0.572296	+	0.111082i
$484$		0			0
$485$	−14.0904	+	24.4053i	−0.639811	+	1.10819i
$486$		0			0
$487$	27.0620	−	7.25125i	1.22630	−	0.328585i	0.413161	−	0.910658i	$-0.364425\pi$
$487$	27.0620	−	7.25125i	1.22630	−	0.328585i	0.813137	+	0.582073i	$0.197758\pi$
$488$		0			0
$489$	−7.80610	+	22.6506i	−0.353004	+	1.02430i
$490$		0			0
$491$	6.45173	+	11.1747i	0.291162	+	0.504308i	0.974085	−	0.226183i	$-0.0726247\pi$
$491$	6.45173	+	11.1747i	0.291162	+	0.504308i	−0.682922	+	0.730491i	$0.739291\pi$
$492$		0			0
$493$			23.5031i			1.05852i
$494$		0			0
$495$	20.3842	−	26.0614i	0.916200	−	1.17137i
$496$		0			0
$497$	21.7937	−	12.5826i	0.977582	−	0.564407i
$498$		0			0
$499$	−2.22600	−	2.22600i	−0.0996495	−	0.0996495i	0.655524	−	0.755174i	$-0.272448\pi$
$499$	−2.22600	−	2.22600i	−0.0996495	−	0.0996495i	−0.755174	+	0.655524i	$0.772448\pi$
$500$		0			0
$501$	12.8020	−	11.1225i	0.571953	−	0.496919i
$502$		0			0
$503$	3.89079	+	2.24635i	0.173482	+	0.100160i	0.584226	−	0.811591i	$-0.301398\pi$
$503$	3.89079	+	2.24635i	0.173482	+	0.100160i	−0.410745	+	0.911750i	$0.634731\pi$
$504$		0			0
$505$	21.1628	+	5.67056i	0.941733	+	0.252336i
$506$		0			0
$507$	16.9484	+	14.8240i	0.752706	+	0.658357i
$508$		0			0
$509$	22.9448	+	6.14805i	1.01701	+	0.272508i	0.728556	−	0.684986i	$-0.240192\pi$
$509$	22.9448	+	6.14805i	1.01701	+	0.272508i	0.288456	+	0.957493i	$0.406858\pi$
$510$		0			0
$511$	16.6869	+	9.63418i	0.738185	+	0.426191i
$512$		0			0
$513$	−0.294472	−	5.70769i	−0.0130012	−	0.252001i
$514$		0			0
$515$	15.6194	+	15.6194i	0.688271	+	0.688271i
$516$		0			0
$517$	−18.0152	+	10.4011i	−0.792306	+	0.457438i
$518$		0			0
$519$	−7.37314	+	3.59365i	−0.323645	+	0.157744i
$520$		0			0
$521$			9.05417i			0.396670i	0.980134	+	0.198335i	$0.0635534\pi$
$521$			9.05417i			0.396670i	−0.980134	+	0.198335i	$0.936447\pi$
$522$		0			0
$523$	−14.6521	−	25.3781i	−0.640691	−	1.10971i	−0.985279	−	0.170955i	$-0.945315\pi$
$523$	−14.6521	−	25.3781i	−0.640691	−	1.10971i	0.344588	−	0.938754i	$-0.388019\pi$
$524$		0			0
$525$	−34.8485	−	12.0099i	−1.52091	−	0.524153i
$526$		0			0
$527$	19.7706	−	5.29751i	0.861219	−	0.230763i
$528$		0			0
$529$	10.1785	−	17.6297i	0.442543	−	0.766507i
$530$		0			0
$531$	−3.69174	−	4.90464i	−0.160208	−	0.212843i
$532$		0			0
$533$	11.6171	−	34.5602i	0.503191	−	1.49697i
$534$		0			0
$535$	−10.0119	+	37.3648i	−0.432851	+	1.61542i
$536$		0			0
$537$	1.04119	−	14.8319i	0.0449308	−	0.640045i
$538$		0			0
$539$	12.5735	+	46.9249i	0.541579	+	2.02120i
$540$		0			0
$541$	−29.7524	+	29.7524i	−1.27916	+	1.27916i	−0.338015	+	0.941141i	$0.609756\pi$
$541$	−29.7524	+	29.7524i	−1.27916	+	1.27916i	−0.941141	+	0.338015i	$0.890244\pi$
$542$		0			0
$543$	−6.43535	+	9.53523i	−0.276167	+	0.409196i
$544$		0			0
$545$	22.4093			0.959909
$546$		0			0
$547$	−9.72797			−0.415938			−0.207969	−	0.978135i	$-0.566685\pi$
$547$	−9.72797			−0.415938			−0.207969	+	0.978135i	$0.566685\pi$
$548$		0			0
$549$	−28.3691	−	4.00271i	−1.21076	−	0.170831i
$550$		0			0
$551$	−4.14489	+	4.14489i	−0.176578	+	0.176578i
$552$		0			0
$553$	18.1933	+	67.8983i	0.773657	+	2.88733i
$554$		0			0
$555$	−15.3216	−	1.07557i	−0.650366	−	0.0456553i
$556$		0			0
$557$	2.30023	−	8.58457i	0.0974638	−	0.363740i	−0.899918	−	0.436060i	$-0.856374\pi$
$557$	2.30023	−	8.58457i	0.0974638	−	0.363740i	0.997381	+	0.0723199i	$0.0230402\pi$
$558$		0			0
$559$	−13.9817	+	21.0604i	−0.591363	+	0.890762i
$560$		0			0
$561$	5.16013	+	26.5850i	0.217861	+	1.12242i
$562$		0			0
$563$	−0.952635	+	1.65001i	−0.0401488	+	0.0695398i	−0.885402	−	0.464827i	$-0.846117\pi$
$563$	−0.952635	+	1.65001i	−0.0401488	+	0.0695398i	0.845253	+	0.534367i	$0.179450\pi$
$564$		0			0
$565$	−36.3564	+	9.74166i	−1.52952	+	0.409835i
$566$		0			0
$567$	−35.0782	−	21.1290i	−1.47315	−	0.887333i
$568$		0			0
$569$	5.02931	+	8.71103i	0.210840	+	0.365185i	0.951978	−	0.306168i	$-0.0990468\pi$
$569$	5.02931	+	8.71103i	0.210840	+	0.365185i	−0.741138	+	0.671353i	$0.765713\pi$
$570$		0			0
$571$		−	20.2231i		−	0.846309i	−0.906057	−	0.423155i	$-0.860923\pi$
$571$		−	20.2231i		−	0.846309i	0.906057	−	0.423155i	$-0.139077\pi$
$572$		0			0
$573$	−5.91690	−	12.1398i	−0.247182	−	0.507147i
$574$		0			0
$575$	−6.58509	+	3.80191i	−0.274617	+	0.158550i
$576$		0			0
$577$	−7.71564	−	7.71564i	−0.321206	−	0.321206i	0.528024	−	0.849230i	$-0.322933\pi$
$577$	−7.71564	−	7.71564i	−0.321206	−	0.321206i	−0.849230	+	0.528024i	$0.822933\pi$
$578$		0			0
$579$	3.03270	+	3.49063i	0.126035	+	0.145066i
$580$		0			0
$581$	−35.5592	−	20.5301i	−1.47524	−	0.851733i
$582$		0			0
$583$	28.5430	+	7.64807i	1.18213	+	0.316751i
$584$		0			0
$585$	15.0720	−	30.0842i	0.623152	−	1.24383i
$586$		0			0
$587$	1.87678	+	0.502881i	0.0774630	+	0.0207561i	0.297342	−	0.954771i	$-0.403900\pi$
$587$	1.87678	+	0.502881i	0.0774630	+	0.0207561i	−0.219879	+	0.975527i	$0.570566\pi$
$588$		0			0
$589$	4.42089	+	2.55240i	0.182159	+	0.105170i
$590$		0			0
$591$	21.9712	+	25.2888i	0.903774	+	1.04024i
$592$		0			0
$593$	−23.1037	−	23.1037i	−0.948756	−	0.948756i	0.0499940	−	0.998750i	$-0.484080\pi$
$593$	−23.1037	−	23.1037i	−0.948756	−	0.948756i	−0.998750	+	0.0499940i	$0.984080\pi$
$594$		0			0
$595$	54.0592	−	31.2111i	2.21621	−	1.27953i
$596$		0			0
$597$	−6.09371	−	12.5026i	−0.249399	−	0.511696i
$598$		0			0
$599$			8.98472i			0.367106i	0.983010	+	0.183553i	$0.0587598\pi$
$599$			8.98472i			0.367106i	−0.983010	+	0.183553i	$0.941240\pi$
$600$		0			0
$601$	−16.6602	−	28.8563i	−0.679583	−	1.17707i	−0.975107	−	0.221737i	$-0.928827\pi$
$601$	−16.6602	−	28.8563i	−0.679583	−	1.17707i	0.295523	−	0.955336i	$-0.404506\pi$
$602$		0			0
$603$	−16.4489	+	2.01062i	−0.669851	+	0.0818786i
$604$		0			0
$605$	−4.71530	+	1.26346i	−0.191704	+	0.0513670i
$606$		0			0
$607$	22.1672	−	38.3948i	0.899740	−	1.55840i	0.0719135	−	0.997411i	$-0.477089\pi$
$607$	22.1672	−	38.3948i	0.899740	−	1.55840i	0.827826	−	0.560984i	$-0.189577\pi$
$608$		0			0
$609$	8.00273	+	41.2302i	0.324287	+	1.67073i
$610$		0			0
$611$	−15.8640	+	13.9961i	−0.641790	+	0.566221i
$612$		0			0
$613$	11.5021	−	42.9265i	0.464566	−	1.73379i	−0.193756	−	0.981050i	$-0.562067\pi$
$613$	11.5021	−	42.9265i	0.464566	−	1.73379i	0.658323	−	0.752736i	$-0.271266\pi$
$614$		0			0
$615$	−54.3522	−	3.81549i	−2.19169	−	0.153855i
$616$		0			0
$617$	−2.01395	−	7.51616i	−0.0810785	−	0.302589i	0.913464	−	0.406919i	$-0.133397\pi$
$617$	−2.01395	−	7.51616i	−0.0810785	−	0.302589i	−0.994543	+	0.104330i	$0.966730\pi$
$618$		0			0
$619$	−3.82416	+	3.82416i	−0.153706	+	0.153706i	−0.779771	−	0.626065i	$-0.784664\pi$
$619$	−3.82416	+	3.82416i	−0.153706	+	0.153706i	0.626065	+	0.779771i	$0.284664\pi$
$620$		0			0
$621$	−8.03626	+	2.60391i	−0.322484	+	0.104491i
$622$		0			0
$623$	−34.8559			−1.39647
$624$		0			0
$625$	26.5101			1.06040
$626$		0			0
$627$	−3.77840	+	5.59843i	−0.150895	+	0.223580i
$628$		0			0
$629$	8.88946	−	8.88946i	0.354446	−	0.354446i
$630$		0			0
$631$	−4.24796	−	15.8536i	−0.169109	−	0.631122i	−0.997480	−	0.0709434i	$-0.977399\pi$
$631$	−4.24796	−	15.8536i	−0.169109	−	0.631122i	0.828372	−	0.560179i	$-0.189268\pi$
$632$		0			0
$633$	0.653424	−	9.30812i	0.0259713	−	0.369965i
$634$		0			0
$635$	10.7991	−	40.3028i	0.428549	−	1.59937i
$636$		0			0
$637$	21.9833	+	44.2454i	0.871009	+	1.75307i
$638$		0			0
$639$	13.2567	−	9.97835i	0.524426	−	0.394738i
$640$		0			0
$641$	−24.3557	+	42.1853i	−0.961993	+	1.66622i	−0.244506	+	0.969648i	$0.578626\pi$
$641$	−24.3557	+	42.1853i	−0.961993	+	1.66622i	−0.717487	+	0.696572i	$0.754708\pi$
$642$		0			0
$643$	−16.3066	+	4.36934i	−0.643069	+	0.172310i	−0.565593	−	0.824684i	$-0.691353\pi$
$643$	−16.3066	+	4.36934i	−0.643069	+	0.172310i	−0.0774759	+	0.996994i	$0.524686\pi$
$644$		0			0
$645$	35.7152	+	12.3086i	1.40628	+	0.484649i
$646$		0			0
$647$	10.9410	+	18.9504i	0.430137	+	0.745018i	0.996885	−	0.0788727i	$-0.0251321\pi$
$647$	10.9410	+	18.9504i	0.430137	+	0.745018i	−0.566748	+	0.823891i	$0.691799\pi$
$648$		0			0
$649$			7.25461i			0.284769i
$650$		0			0
$651$	32.8787	−	16.0250i	1.28862	−	0.628068i
$652$		0			0
$653$	19.2265	−	11.1004i	0.752392	−	0.434393i	−0.0741658	−	0.997246i	$-0.523629\pi$
$653$	19.2265	−	11.1004i	0.752392	−	0.434393i	0.826557	+	0.562852i	$0.190296\pi$
$654$		0			0
$655$	−23.6720	−	23.6720i	−0.924941	−	0.924941i
$656$		0			0
$657$	11.6917	+	4.97033i	0.456138	+	0.193911i
$658$		0			0
$659$	−37.6857	−	21.7578i	−1.46803	−	0.847565i	−0.468667	−	0.883375i	$-0.655266\pi$
$659$	−37.6857	−	21.7578i	−1.46803	−	0.847565i	−0.999359	+	0.0358100i	$0.988599\pi$
$660$		0			0
$661$	−5.55235	−	1.48775i	−0.215961	−	0.0578667i	0.149216	−	0.988805i	$-0.452325\pi$
$661$	−5.55235	−	1.48775i	−0.215961	−	0.0578667i	−0.365177	+	0.930938i	$0.618992\pi$
$662$		0			0
$663$	10.4974	+	25.4623i	0.407683	+	0.988875i
$664$		0			0
$665$	15.0379	+	4.02939i	0.583144	+	0.156253i
$666$		0			0
$667$	7.50332	+	4.33205i	0.290530	+	0.167738i
$668$		0			0
$669$	−11.0377	+	9.58968i	−0.426743	+	0.370759i
$670$		0			0
$671$	23.9411	+	23.9411i	0.924236	+	0.924236i
$672$		0			0
$673$	−19.1318	+	11.0458i	−0.737478	+	0.425783i	−0.821152	−	0.570710i	$-0.806668\pi$
$673$	−19.1318	+	11.0458i	−0.737478	+	0.425783i	0.0836738	+	0.996493i	$0.473335\pi$
$674$		0			0
$675$	−23.7679	−	5.07224i	−0.914828	−	0.195231i
$676$		0			0
$677$			29.1088i			1.11874i	0.828917	+	0.559371i	$0.188957\pi$
$677$			29.1088i			1.11874i	−0.828917	+	0.559371i	$0.811043\pi$
$678$		0			0
$679$	−20.6093	−	35.6963i	−0.790911	−	1.36990i
$680$		0			0
$681$	−0.517787	+	1.50244i	−0.0198416	+	0.0575736i
$682$		0			0
$683$	19.8514	−	5.31917i	0.759593	−	0.203532i	0.141824	−	0.989892i	$-0.454703\pi$
$683$	19.8514	−	5.31917i	0.759593	−	0.203532i	0.617769	+	0.786360i	$0.288037\pi$
$684$		0			0
$685$	−30.4409	+	52.7252i	−1.16309	+	2.01453i
$686$		0			0
$687$	5.49290	−	1.06617i	0.209567	−	0.0406768i
$688$		0			0
$689$	29.9933	+	1.87629i	1.14265	+	0.0714808i
$690$		0			0
$691$	−7.53183	+	28.1092i	−0.286524	+	1.06932i	0.661194	+	0.750215i	$0.270050\pi$
$691$	−7.53183	+	28.1092i	−0.286524	+	1.06932i	−0.947718	+	0.319109i	$0.896617\pi$
$692$		0			0
$693$	18.1043	+	44.8797i	0.687724	+	1.70484i
$694$		0			0
$695$	−2.30256	−	8.59325i	−0.0873409	−	0.325961i
$696$		0			0
$697$	31.5346	−	31.5346i	1.19446	−	1.19446i
$698$		0			0
$699$	13.3578	+	9.01521i	0.505238	+	0.340986i
$700$		0			0
$701$	−47.9676			−1.81171			−0.905855	−	0.423588i	$-0.860770\pi$
$701$	−47.9676			−1.81171			−0.905855	+	0.423588i	$0.860770\pi$
$702$		0			0
$703$	3.13541			0.118254
$704$		0			0
$705$	26.2048	+	17.6857i	0.986932	+	0.666083i
$706$		0			0
$707$	−22.6597	+	22.6597i	−0.852206	+	0.852206i
$708$		0			0
$709$	−13.0177	−	48.5829i	−0.488891	−	1.82457i	−0.561858	−	0.827234i	$-0.689913\pi$
$709$	−13.0177	−	48.5829i	−0.488891	−	1.82457i	0.0729667	−	0.997334i	$-0.476753\pi$
$710$		0			0
$711$	17.3386	+	42.9817i	0.650250	+	1.61194i
$712$		0			0
$713$	1.95285	−	7.28815i	0.0731350	−	0.272943i
$714$		0			0
$715$	−35.6116	+	17.6936i	−1.33180	+	0.661702i
$716$		0			0
$717$	−44.5661	+	8.65023i	−1.66435	+	0.323049i
$718$		0			0
$719$	4.43706	−	7.68521i	0.165474	−	0.286610i	−0.771349	−	0.636412i	$-0.780418\pi$
$719$	4.43706	−	7.68521i	0.165474	−	0.286610i	0.936824	+	0.349802i	$0.113751\pi$
$720$		0			0
$721$	−31.2077	+	8.36208i	−1.16224	+	0.311420i
$722$		0			0
$723$	−6.88501	+	19.9779i	−0.256056	+	0.742987i
$724$		0			0
$725$	12.4630	+	21.5865i	0.462864	+	0.801704i
$726$		0			0
$727$			11.4138i			0.423315i	0.977344	+	0.211658i	$0.0678861\pi$
$727$			11.4138i			0.423315i	−0.977344	+	0.211658i	$0.932114\pi$
$728$		0			0
$729$	−24.6478	−	11.0220i	−0.912882	−	0.408223i
$730$		0			0
$731$	−26.7776	+	15.4601i	−0.990407	+	0.571812i
$732$		0			0
$733$	−14.4825	−	14.4825i	−0.534923	−	0.534923i	0.387110	−	0.922033i	$-0.373473\pi$
$733$	−14.4825	−	14.4825i	−0.534923	−	0.534923i	−0.922033	+	0.387110i	$0.873473\pi$
$734$		0			0
$735$	55.7342	−	48.4225i	2.05579	−	1.78609i
$736$		0			0
$737$	16.9598	+	9.79176i	0.624723	+	0.360684i
$738$		0			0
$739$	35.8764	+	9.61305i	1.31973	+	0.353622i	0.848879	−	0.528588i	$-0.177278\pi$
$739$	35.8764	+	9.61305i	1.31973	+	0.353622i	0.470856	+	0.882210i	$0.343945\pi$
$740$		0			0
$741$	−2.63915	+	6.34168i	−0.0969517	+	0.232967i
$742$		0			0
$743$	12.1935	+	3.26724i	0.447337	+	0.119864i	0.475453	−	0.879741i	$-0.342284\pi$
$743$	12.1935	+	3.26724i	0.447337	+	0.119864i	−0.0281161	+	0.999605i	$0.508951\pi$
$744$		0			0
$745$	−52.7741	−	30.4692i	−1.93350	−	1.11630i
$746$		0			0
$747$	−24.9147	−	10.5916i	−0.911581	−	0.387527i
$748$		0			0
$749$	−40.0077	−	40.0077i	−1.46185	−	1.46185i
$750$		0			0
$751$	−2.30561	+	1.33114i	−0.0841328	+	0.0485741i	−0.541476	−	0.840716i	$-0.682134\pi$
$751$	−2.30561	+	1.33114i	−0.0841328	+	0.0485741i	0.457343	+	0.889290i	$0.348801\pi$
$752$		0			0
$753$	14.2075	−	6.92469i	0.517750	−	0.252350i
$754$		0			0
$755$			69.6274i			2.53400i
$756$		0			0
$757$	−4.12709	−	7.14833i	−0.150002	−	0.259810i	0.781226	−	0.624248i	$-0.214595\pi$
$757$	−4.12709	−	7.14833i	−0.150002	−	0.259810i	−0.931228	+	0.364438i	$0.881261\pi$
$758$		0			0
$759$	9.43834	+	3.25275i	0.342590	+	0.118067i
$760$		0			0
$761$	−17.6088	+	4.71825i	−0.638317	+	0.171037i	−0.563441	−	0.826156i	$-0.690523\pi$
$761$	−17.6088	+	4.71825i	−0.638317	+	0.171037i	−0.0748761	+	0.997193i	$0.523856\pi$
$762$		0			0
$763$	−16.3884	+	28.3856i	−0.593302	+	1.02763i
$764$		0			0
$765$	32.8831	−	24.7513i	1.18889	−	0.894884i
$766$		0			0
$767$	1.46087	+	7.23180i	0.0527489	+	0.261125i
$768$		0			0
$769$	1.20203	−	4.48602i	0.0433462	−	0.161770i	−0.940860	−	0.338795i	$-0.889981\pi$
$769$	1.20203	−	4.48602i	0.0433462	−	0.161770i	0.984206	+	0.177025i	$0.0566473\pi$
$770$		0			0
$771$	0.0192041	−	0.273565i	0.000691618	−	0.00985221i
$772$		0			0
$773$	4.06155	+	15.1579i	0.146084	+	0.545193i	0.999705	+	0.0242978i	$0.00773498\pi$
$773$	4.06155	+	15.1579i	0.146084	+	0.545193i	−0.853621	+	0.520895i	$0.825598\pi$
$774$		0			0
$775$	15.3493	−	15.3493i	0.551362	−	0.551362i
$776$		0			0
$777$	12.5675	−	18.6211i	0.450855	−	0.668030i
$778$		0			0
$779$	11.1226			0.398508
$780$		0			0
$781$	−19.6084			−0.701644
$782$		0			0
$783$	8.53586	+	26.3436i	0.305047	+	0.941443i
$784$		0			0
$785$	2.54532	−	2.54532i	0.0908464	−	0.0908464i
$786$		0			0
$787$	12.6603	+	47.2487i	0.451290	+	1.68424i	0.698773	+	0.715343i	$0.253730\pi$
$787$	12.6603	+	47.2487i	0.451290	+	1.68424i	−0.247483	+	0.968892i	$0.579604\pi$
$788$		0			0
$789$	−6.09415	−	0.427805i	−0.216957	−	0.0152303i
$790$		0			0
$791$	14.2486	−	53.1765i	0.506622	−	1.89074i
$792$		0			0
$793$	28.6869	+	19.0448i	1.01870	+	0.676299i
$794$		0			0
$795$	−8.55712	−	44.0864i	−0.303490	−	1.56358i
$796$		0			0
$797$	−22.4830	+	38.9417i	−0.796388	+	1.37938i	0.125566	+	0.992085i	$0.459925\pi$
$797$	−22.4830	+	38.9417i	−0.796388	+	1.37938i	−0.921954	+	0.387299i	$0.873408\pi$
$798$		0			0
$799$	−24.9948	+	6.69733i	−0.884251	+	0.236934i
$800$		0			0
$801$	−22.8120	+	2.78840i	−0.806022	+	0.0985234i
$802$		0			0
$803$	−7.50682	−	13.0022i	−0.264910	−	0.458838i
$804$		0			0
$805$		−	23.0111i		−	0.811035i
$806$		0			0
$807$	−14.2793	−	29.2971i	−0.502656	−	1.03131i
$808$		0			0
$809$	−23.6360	+	13.6462i	−0.830996	+	0.479776i	−0.854194	−	0.519955i	$-0.825949\pi$
$809$	−23.6360	+	13.6462i	−0.830996	+	0.479776i	0.0231976	+	0.999731i	$0.492615\pi$
$810$		0			0
$811$	−11.2312	−	11.2312i	−0.394382	−	0.394382i	0.481864	−	0.876246i	$-0.339960\pi$
$811$	−11.2312	−	11.2312i	−0.394382	−	0.394382i	−0.876246	+	0.481864i	$0.839960\pi$
$812$		0			0
$813$	−2.29709	−	2.64394i	−0.0805624	−	0.0927272i
$814$		0			0
$815$	−37.2644	−	21.5146i	−1.30531	−	0.753623i
$816$		0			0
$817$	−7.44885	−	1.99591i	−0.260602	−	0.0698281i
$818$		0			0
$819$	27.0848	+	41.0929i	0.946420	+	1.43590i
$820$		0			0
$821$	18.0916	+	4.84764i	0.631402	+	0.169184i	0.560306	−	0.828286i	$-0.310684\pi$
$821$	18.0916	+	4.84764i	0.631402	+	0.169184i	0.0710962	+	0.997469i	$0.477350\pi$
$822$		0			0
$823$	0.110358	+	0.0637153i	0.00384684	+	0.00222098i	0.501922	−	0.864913i	$-0.332626\pi$
$823$	0.110358	+	0.0637153i	0.00384684	+	0.00222098i	−0.498075	+	0.867134i	$0.665960\pi$
$824$		0			0
$825$	18.8366	+	21.6809i	0.655807	+	0.754833i
$826$		0			0
$827$	−26.2443	−	26.2443i	−0.912603	−	0.912603i	0.0838739	−	0.996476i	$-0.473271\pi$
$827$	−26.2443	−	26.2443i	−0.912603	−	0.912603i	−0.996476	+	0.0838739i	$0.973271\pi$
$828$		0			0
$829$	48.2799	−	27.8744i	1.67683	−	0.968119i	0.713170	−	0.700991i	$-0.247259\pi$
$829$	48.2799	−	27.8744i	1.67683	−	0.968119i	0.963661	−	0.267128i	$-0.0860747\pi$
$830$		0			0
$831$	−13.3087	−	27.3057i	−0.461674	−	0.947224i
$832$		0			0
$833$			60.4307i			2.09380i
$834$		0			0
$835$	15.2293	+	26.3779i	0.527031	+	0.912845i
$836$		0			0
$837$	20.2360	−	13.1180i	0.699459	−	0.453426i
$838$		0			0
$839$	11.9950	−	3.21404i	0.414112	−	0.110961i	−0.0457468	−	0.998953i	$-0.514567\pi$
$839$	11.9950	−	3.21404i	0.414112	−	0.110961i	0.459859	+	0.887992i	$0.347900\pi$
$840$		0			0
$841$	−0.299157	+	0.518156i	−0.0103158	+	0.0178674i
$842$		0			0
$843$	−6.45580	−	33.2603i	−0.222349	−	1.14555i
$844$		0			0
$845$	−31.9366	+	24.8091i	−1.09865	+	0.853458i
$846$		0			0
$847$	1.84800	−	6.89682i	0.0634980	−	0.236978i
$848$		0			0
$849$	30.4952	+	2.14074i	1.04659	+	0.0734702i
$850$		0			0
$851$	−1.19946	−	4.47644i	−0.0411169	−	0.153450i
$852$		0			0
$853$	−3.60751	+	3.60751i	−0.123519	+	0.123519i	−0.766164	−	0.642645i	$-0.777837\pi$
$853$	−3.60751	+	3.60751i	−0.123519	+	0.123519i	0.642645	+	0.766164i	$0.277837\pi$
$854$		0			0
$855$	10.1641	+	1.43410i	0.347606	+	0.0490451i
$856$		0			0
$857$	41.9079			1.43155			0.715773	−	0.698333i	$-0.246075\pi$
$857$	41.9079			1.43155			0.715773	+	0.698333i	$0.246075\pi$
$858$		0			0
$859$	27.5846			0.941174			0.470587	−	0.882354i	$-0.344042\pi$
$859$	27.5846			0.941174			0.470587	+	0.882354i	$0.344042\pi$
$860$		0			0
$861$	44.5821	−	66.0570i	1.51935	−	2.25122i
$862$		0			0
$863$	19.5622	−	19.5622i	0.665905	−	0.665905i	−0.290861	−	0.956765i	$-0.593942\pi$
$863$	19.5622	−	19.5622i	0.665905	−	0.665905i	0.956765	+	0.290861i	$0.0939416\pi$
$864$		0			0
$865$	−3.81280	−	14.2296i	−0.129639	−	0.483820i
$866$		0			0
$867$	−0.297082	+	4.23198i	−0.0100894	+	0.143725i
$868$		0			0
$869$	14.1760	−	52.9054i	0.480887	−	1.79469i
$870$		0			0
$871$	18.8783	+	6.34575i	0.639665	+	0.215017i
$872$		0			0
$873$	−16.3437	−	21.7133i	−0.553151	−	0.734885i
$874$		0			0
$875$	−2.28492	+	3.95761i	−0.0772446	+	0.133792i
$876$		0			0
$877$	9.29152	−	2.48965i	0.313752	−	0.0840697i	−0.0985066	−	0.995136i	$-0.531407\pi$
$877$	9.29152	−	2.48965i	0.313752	−	0.0840697i	0.412259	+	0.911067i	$0.364740\pi$
$878$		0			0
$879$	20.7596	+	7.15440i	0.700203	+	0.241312i
$880$		0			0
$881$	−2.63029	−	4.55580i	−0.0886168	−	0.153489i	0.818310	−	0.574777i	$-0.194911\pi$
$881$	−2.63029	−	4.55580i	−0.0886168	−	0.153489i	−0.906927	+	0.421288i	$0.861578\pi$
$882$		0			0
$883$		−	37.8287i		−	1.27304i	−0.771262	−	0.636518i	$-0.780374\pi$
$883$		−	37.8287i		−	1.27304i	0.771262	−	0.636518i	$-0.219626\pi$
$884$		0			0
$885$	9.91086	−	4.83052i	0.333150	−	0.162376i
$886$		0			0
$887$	−5.84110	+	3.37236i	−0.196125	+	0.113233i	−0.594847	−	0.803839i	$-0.702787\pi$
$887$	−5.84110	+	3.37236i	−0.196125	+	0.113233i	0.398722	+	0.917072i	$0.369454\pi$
$888$		0			0
$889$	43.1535	+	43.1535i	1.44732	+	1.44732i
$890$		0			0
$891$	15.4389	+	27.9240i	0.517224	+	0.935488i
$892$		0			0
$893$	−5.58907	−	3.22685i	−0.187031	−	0.107982i
$894$		0			0
$895$	25.7942	+	6.91154i	0.862205	+	0.231027i
$896$		0			0
$897$	10.0637	+	1.34191i	0.336016	+	0.0448050i
$898$		0			0
$899$	−23.8912	−	6.40164i	−0.796817	−	0.213507i
$900$		0			0
$901$	31.8335	+	18.3791i	1.06053	+	0.612296i
$902$		0			0
$903$	−41.7105	+	36.2385i	−1.38804	+	1.20594i
$904$		0			0
$905$	−14.6095	−	14.6095i	−0.485635	−	0.485635i
$906$		0			0
$907$	−14.8555	+	8.57683i	−0.493269	+	0.284789i	−0.725930	−	0.687769i	$-0.758590\pi$
$907$	−14.8555	+	8.57683i	−0.493269	+	0.284789i	0.232661	+	0.972558i	$0.425257\pi$
$908$		0			0
$909$	−13.0173	+	16.6428i	−0.431756	+	0.552005i
$910$		0			0
$911$		−	0.770931i		−	0.0255421i	−0.999918	−	0.0127710i	$-0.995935\pi$
$911$		−	0.770931i		−	0.0255421i	0.999918	−	0.0127710i	$-0.00406526\pi$
$912$		0			0
$913$	15.9968	+	27.7072i	0.529416	+	0.916976i
$914$		0			0
$915$	16.7657	−	48.6483i	0.554258	−	1.60826i
$916$		0			0
$917$	47.2970	−	12.6732i	1.56188	−	0.418505i
$918$		0			0
$919$	14.3203	−	24.8035i	0.472384	−	0.818192i	−0.527117	−	0.849793i	$-0.676727\pi$
$919$	14.3203	−	24.8035i	0.472384	−	0.818192i	0.999501	+	0.0316003i	$0.0100604\pi$
$920$		0			0
$921$	14.6497	−	2.84349i	0.482723	−	0.0936961i
$922$		0			0
$923$	−19.5467	+	3.94856i	−0.643388	+	0.129969i
$924$		0			0
$925$	3.45076	−	12.8784i	0.113460	−	0.423439i
$926$		0			0
$927$	−19.7554	+	7.96926i	−0.648854	+	0.261745i
$928$		0			0
$929$	0.882669	+	3.29417i	0.0289594	+	0.108078i	0.978893	−	0.204375i	$-0.0655161\pi$
$929$	0.882669	+	3.29417i	0.0289594	+	0.108078i	−0.949933	+	0.312453i	$0.898849\pi$
$930$		0			0
$931$	−10.6573	+	10.6573i	−0.349278	+	0.349278i
$932$		0			0
$933$	14.1824	+	9.57176i	0.464312	+	0.313365i
$934$		0			0
$935$	−48.6386			−1.59065
$936$		0			0
$937$	−24.0973			−0.787225			−0.393613	−	0.919276i	$-0.628775\pi$
$937$	−24.0973			−0.787225			−0.393613	+	0.919276i	$0.628775\pi$
$938$		0			0
$939$	−24.7254	−	16.6873i	−0.806884	−	0.544568i
$940$		0			0
$941$	23.8741	−	23.8741i	0.778274	−	0.778274i	−0.201263	−	0.979537i	$-0.564505\pi$
$941$	23.8741	−	23.8741i	0.778274	−	0.778274i	0.979537	+	0.201263i	$0.0645045\pi$
$942$		0			0
$943$	−4.25498	−	15.8798i	−0.138561	−	0.517117i
$944$		0			0
$945$	49.2574	−	54.6165i	1.60234	−	1.77667i
$946$		0			0
$947$	−3.42796	+	12.7933i	−0.111394	+	0.415727i	−0.998992	−	0.0448919i	$-0.985706\pi$
$947$	−3.42796	+	12.7933i	−0.111394	+	0.415727i	0.887598	+	0.460618i	$0.152372\pi$
$948$		0			0
$949$	−10.1015	−	11.4497i	−0.327908	−	0.371671i
$950$		0			0
$951$	−0.0212711	+	0.00412870i	−0.000689763	+	0.000133882i
$952$		0			0
$953$	17.9590	−	31.1060i	0.581751	−	1.00762i	−0.413521	−	0.910494i	$-0.635701\pi$
$953$	17.9590	−	31.1060i	0.581751	−	1.00762i	0.995272	−	0.0971272i	$-0.0309654\pi$
$954$		0			0
$955$	23.4288	−	6.27773i	0.758139	−	0.203143i
$956$		0			0
$957$	10.6628	−	30.9398i	0.344679	−	1.00014i
$958$		0			0
$959$	−44.5243	−	77.1184i	−1.43777	−	2.49028i
$960$		0			0
$961$		−	9.46001i		−	0.305162i
$962$		0			0
$963$	−29.3842	−	22.9832i	−0.946894	−	0.740622i
$964$		0			0
$965$	−7.19226	+	4.15245i	−0.231527	+	0.133672i
$966$		0			0
$967$	32.3505	+	32.3505i	1.04032	+	1.04032i	0.999152	+	0.0411686i	$0.0131081\pi$
$967$	32.3505	+	32.3505i	1.04032	+	1.04032i	0.0411686	+	0.999152i	$0.486892\pi$
$968$		0			0
$969$	−6.34237	+	5.51032i	−0.203746	+	0.177017i
$970$		0			0
$971$	−14.1052	−	8.14364i	−0.452657	−	0.261342i	0.256294	−	0.966599i	$-0.417498\pi$
$971$	−14.1052	−	8.14364i	−0.452657	−	0.261342i	−0.708952	+	0.705257i	$0.750832\pi$
$972$		0			0
$973$	12.5689	+	3.36783i	0.402941	+	0.107968i
$974$		0			0
$975$	23.1433	+	17.8196i	0.741178	+	0.570684i
$976$		0			0
$977$	−38.6294	−	10.3507i	−1.23587	−	0.331149i	−0.419005	−	0.907984i	$-0.637621\pi$
$977$	−38.6294	−	10.3507i	−1.23587	−	0.331149i	−0.816861	+	0.576835i	$0.804288\pi$
$978$		0			0
$979$	23.5206	+	13.5796i	0.751721	+	0.434006i
$980$		0			0
$981$	−8.45490	+	19.8885i	−0.269944	+	0.634991i
$982$		0			0
$983$	7.22554	+	7.22554i	0.230459	+	0.230459i	0.812884	−	0.582425i	$-0.197896\pi$
$983$	7.22554	+	7.22554i	0.230459	+	0.230459i	−0.582425	+	0.812884i	$0.697896\pi$
$984$		0			0
$985$	−52.1063	+	30.0836i	−1.66024	+	0.958542i
$986$		0			0
$987$	−41.5666	+	20.2594i	−1.32308	+	0.644864i
$988$		0			0
$989$			11.3983i			0.362445i
$990$		0			0
$991$	13.7017	+	23.7321i	0.435250	+	0.753875i	0.997316	−	0.0732176i	$-0.0233268\pi$
$991$	13.7017	+	23.7321i	0.435250	+	0.753875i	−0.562066	+	0.827092i	$0.689993\pi$
$992$		0			0
$993$	−0.610909	−	0.210538i	−0.0193866	−	0.00668122i
$994$		0			0
$995$	24.1289	−	6.46532i	0.764938	−	0.204965i
$996$		0			0
$997$	18.8049	−	32.5710i	0.595556	−	1.03153i	−0.397912	−	0.917424i	$-0.630265\pi$
$997$	18.8049	−	32.5710i	0.595556	−	1.03153i	0.993468	−	0.114110i	$-0.0364016\pi$
$998$		0			0
$999$	6.73534	−	13.1923i	0.213097	−	0.417386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.2.bp.a.89.3	✓	56
3.2	odd	2	inner	312.2.bp.a.89.7	yes	56
4.3	odd	2		624.2.cn.f.401.12		56
12.11	even	2		624.2.cn.f.401.8		56
13.6	odd	12	inner	312.2.bp.a.305.7	yes	56
39.32	even	12	inner	312.2.bp.a.305.3	yes	56
52.19	even	12		624.2.cn.f.305.8		56
156.71	odd	12		624.2.cn.f.305.12		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bp.a.89.3	✓	56	1.1	even	1	trivial
312.2.bp.a.89.7	yes	56	3.2	odd	2	inner
312.2.bp.a.305.3	yes	56	39.32	even	12	inner
312.2.bp.a.305.7	yes	56	13.6	odd	12	inner
624.2.cn.f.305.8		56	52.19	even	12
624.2.cn.f.305.12		56	156.71	odd	12
624.2.cn.f.401.8		56	12.11	even	2
624.2.cn.f.401.12		56	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	312.bp (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.43567 - 0.968940i) q^{3} +(2.19967 - 2.19967i) q^{5} +(1.17763 + 4.39498i) q^{7} +(1.12231 + 2.78216i) q^{9} +O(q^{10})\)	\(q+(-1.43567 - 0.968940i) q^{3} +(2.19967 - 2.19967i) q^{5} +(1.17763 + 4.39498i) q^{7} +(1.12231 + 2.78216i) q^{9} +(0.917595 - 3.42451i) q^{11} +(0.225112 - 3.59852i) q^{13} +(-5.28937 + 1.02666i) q^{15} +(2.20507 - 3.81929i) q^{17} +(1.06243 - 0.284677i) q^{19} +(2.56778 - 7.45081i) q^{21} +(-0.812870 - 1.40793i) q^{23} -4.67714i q^{25} +(1.08448 - 5.08172i) q^{27} +(-4.61533 + 2.66466i) q^{29} +(3.28177 + 3.28177i) q^{31} +(-4.63551 + 4.02738i) q^{33} +(12.2579 + 7.07712i) q^{35} +(2.75348 + 0.737792i) q^{37} +(-3.80993 + 4.94817i) q^{39} +(9.76774 + 2.61726i) q^{41} +(-6.07183 - 3.50557i) q^{43} +(8.58857 + 3.65113i) q^{45} +(-4.14895 - 4.14895i) q^{47} +(-11.8669 + 6.85134i) q^{49} +(-6.86643 + 3.34667i) q^{51} +8.33491i q^{53} +(-5.51440 - 9.55122i) q^{55} +(-1.80113 - 0.620727i) q^{57} +(-1.97653 + 0.529610i) q^{59} +(-4.77501 + 8.27056i) q^{61} +(-10.9059 + 8.20890i) q^{63} +(-7.42039 - 8.41074i) q^{65} +(-1.42966 + 5.33556i) q^{67} +(-0.197186 + 2.80895i) q^{69} +(-1.43147 - 5.34234i) q^{71} +(2.99445 - 2.99445i) q^{73} +(-4.53187 + 6.71484i) q^{75} +16.1312 q^{77} +15.4490 q^{79} +(-6.48084 + 6.24490i) q^{81} +(-6.38106 + 6.38106i) q^{83} +(-3.55077 - 13.2516i) q^{85} +(9.20800 + 0.646395i) q^{87} +(-1.98271 + 7.39957i) q^{89} +(16.0805 - 3.24837i) q^{91} +(-1.53171 - 7.89138i) q^{93} +(1.71080 - 2.96319i) q^{95} +(-8.75030 + 2.34464i) q^{97} +(10.5574 - 1.29047i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 4 q^{7}+O(q^{10}) \) 56 * q + 4 * q^7	\( 56 q + 4 q^{7} + 8 q^{13} - 8 q^{15} + 4 q^{19} + 16 q^{21} + 24 q^{27} - 36 q^{31} + 28 q^{33} + 20 q^{37} + 16 q^{39} - 84 q^{43} + 12 q^{45} - 12 q^{49} - 24 q^{55} - 36 q^{57} - 24 q^{61} - 12 q^{63} - 32 q^{67} - 36 q^{69} - 20 q^{73} - 60 q^{75} - 32 q^{79} - 88 q^{85} - 16 q^{87} + 28 q^{91} - 88 q^{93} - 36 q^{97} + 44 q^{99}+O(q^{100}) \) 56 * q + 4 * q^7 + 8 * q^13 - 8 * q^15 + 4 * q^19 + 16 * q^21 + 24 * q^27 - 36 * q^31 + 28 * q^33 + 20 * q^37 + 16 * q^39 - 84 * q^43 + 12 * q^45 - 12 * q^49 - 24 * q^55 - 36 * q^57 - 24 * q^61 - 12 * q^63 - 32 * q^67 - 36 * q^69 - 20 * q^73 - 60 * q^75 - 32 * q^79 - 88 * q^85 - 16 * q^87 + 28 * q^91 - 88 * q^93 - 36 * q^97 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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