LMFDB - Embedded newform 624.2.cn.b.305.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(305,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 6, 5]))

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

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

chi := DirichletCharacter("624.305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.cn (of order $12$, degree $4$, 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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			305.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	624.305
Dual form			624.2.cn.b.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 1.50000i) q^{3} +(0.767949 - 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(-0.866025 + 1.50000i) q^{3} +(0.767949 - 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(-2.59808 - 2.50000i) q^{13} +(-7.83013 - 2.09808i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(7.83013 - 7.83013i) q^{31} +(2.09808 - 0.562178i) q^{37} +(6.00000 - 1.73205i) q^{39} +(1.50000 - 0.866025i) q^{43} +(-1.56218 - 0.901924i) q^{49} +(9.92820 - 9.92820i) q^{57} +(4.33013 + 7.50000i) q^{61} +(-8.59808 + 2.30385i) q^{63} +(-0.205771 - 0.767949i) q^{67} +(-9.36603 - 9.36603i) q^{73} +(7.50000 + 4.33013i) q^{75} -12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-9.16025 + 5.52628i) q^{91} +(4.96410 + 18.5263i) q^{93} +(-16.4282 - 4.40192i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 10 * q^7 - 6 * q^9	$ 4 q + 10 q^{7} - 6 q^{9} - 14 q^{19} + 18 q^{21} + 14 q^{31} - 2 q^{37} + 24 q^{39} + 6 q^{43} + 18 q^{49} + 12 q^{57} - 24 q^{63} - 32 q^{67} - 34 q^{73} + 30 q^{75} - 18 q^{81} - 2 q^{91} + 6 q^{93} - 38 q^{97}+O(q^{100}) $ 4 * q + 10 * q^7 - 6 * q^9 - 14 * q^19 + 18 * q^21 + 14 * q^31 - 2 * q^37 + 24 * q^39 + 6 * q^43 + 18 * q^49 + 12 * q^57 - 24 * q^63 - 32 * q^67 - 34 * q^73 + 30 * q^75 - 18 * q^81 - 2 * q^91 + 6 * q^93 - 38 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{5}{12}\right)$	$-1$	$1$

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
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$3$	−0.866025	+	1.50000i	−0.500000	+	0.866025i
$4$		0			0
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$	0.767949	−	2.86603i	0.290258	−	1.08326i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	0.767949	−	2.86603i	0.290258	−	1.08326i	0.944911	−	0.327327i	$-0.106148\pi$
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$11$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$12$		0			0
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$13$	−2.59808	−	2.50000i	−0.720577	−	0.693375i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−7.83013	−	2.09808i	−1.79635	−	0.481332i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.83013	−	2.09808i	−1.79635	−	0.481332i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$	3.63397	+	3.63397i	0.792998	+	0.792998i
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$		−	5.00000i		−	1.00000i
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	7.83013	−	7.83013i	1.40633	−	1.40633i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.83013	−	7.83013i	1.40633	−	1.40633i	0.777714	−	0.628619i	$-0.216379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.09808	−	0.562178i	0.344922	−	0.0924215i	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	2.09808	−	0.562178i	0.344922	−	0.0924215i	0.427121	+	0.904194i	$0.359528\pi$
$38$		0			0
$39$	6.00000	−	1.73205i	0.960769	−	0.277350i
$40$		0			0
$41$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$41$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$42$		0			0
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	1.50000	−	0.866025i	0.228748	−	0.132068i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	−1.56218	−	0.901924i	−0.223168	−	0.128846i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.92820	−	9.92820i	1.31502	−	1.31502i
$58$		0			0
$59$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$59$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$60$		0			0
$61$	4.33013	+	7.50000i	0.554416	+	0.960277i	0.997949	+	0.0640184i	$0.0203916\pi$
$61$	4.33013	+	7.50000i	0.554416	+	0.960277i	−0.443533	+	0.896258i	$0.646275\pi$
$62$		0			0
$63$	−8.59808	+	2.30385i	−1.08326	+	0.290258i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.205771	−	0.767949i	−0.0251390	−	0.0938199i	0.952217	−	0.305424i	$-0.0987981\pi$
$67$	−0.205771	−	0.767949i	−0.0251390	−	0.0938199i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$71$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$72$		0			0
$73$	−9.36603	−	9.36603i	−1.09621	−	1.09621i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−9.36603	−	9.36603i	−1.09621	−	1.09621i	−0.101361	−	0.994850i	$-0.532320\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−12.1244			−1.36410			−0.682048	−	0.731307i	$-0.738911\pi$
$79$	−12.1244			−1.36410			−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$89$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$	−9.16025	+	5.52628i	−0.960256	+	0.579311i
$92$		0			0
$93$	4.96410	+	18.5263i	0.514753	+	1.92109i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.4282	−	4.40192i	−1.66803	−	0.446948i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−16.4282	−	4.40192i	−1.66803	−	0.446948i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$			15.5885i			1.53598i	0.640464	+	0.767988i	$0.278742\pi$
$103$			15.5885i			1.53598i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	5.16987	−	5.16987i	0.495184	−	0.495184i	−0.414751	−	0.909935i	$-0.636131\pi$
$109$	5.16987	−	5.16987i	0.495184	−	0.495184i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	−0.973721	+	3.63397i	−0.0924215	+	0.344922i
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.59808	+	10.5000i	−0.240192	+	0.970725i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.52628	−	5.50000i	0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.866025	+	0.500000i	0.0768473	+	0.0443678i	0.537931	−	0.842989i	$-0.319206\pi$
$127$	0.866025	+	0.500000i	0.0768473	+	0.0443678i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$			3.00000i			0.264135i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−12.0263	+	20.8301i	−1.04281	+	1.80620i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$137$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$138$		0			0
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.70577	−	1.56218i	0.223168	−	0.128846i
$148$		0			0
$149$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$149$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$150$		0			0
$151$	14.1244	+	14.1244i	1.14942	+	1.14942i	0.986666	+	0.162758i	$0.0520389\pi$
$151$	14.1244	+	14.1244i	1.14942	+	1.14942i	0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.0000			−0.877896			−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000			−0.877896			−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.52628	−	20.6244i	0.432852	−	1.61542i	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	5.52628	−	20.6244i	0.432852	−	1.61542i	0.746156	−	0.665771i	$-0.231897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$167$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	6.29423	+	23.4904i	0.481332	+	1.79635i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−14.3301	−	3.83975i	−1.08326	−	0.290258i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	−15.0000			−1.10883
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3.99038	−	14.8923i	0.290258	−	1.08326i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	26.3564	−	7.06218i	1.89718	−	0.508347i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	26.3564	−	7.06218i	1.89718	−	0.508347i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$197$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$198$		0			0
$199$	14.7224	−	8.50000i	1.04365	−	0.602549i	0.122782	−	0.992434i	$-0.460818\pi$
$199$	14.7224	−	8.50000i	1.04365	−	0.602549i	0.920864	+	0.389885i	$0.127485\pi$
$200$		0			0
$201$	1.33013	+	0.356406i	0.0938199	+	0.0251390i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.9904	+	22.5000i	−0.894295	+	1.54896i	−0.0596196	+	0.998221i	$0.518989\pi$
$211$	−12.9904	+	22.5000i	−0.894295	+	1.54896i	−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.4282	−	28.4545i	−1.11522	−	1.93162i
$218$		0			0
$219$	22.1603	−	5.93782i	1.49745	−	0.401241i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.22243	−	12.0263i	−0.215790	−	0.805339i	−0.985887	−	0.167412i	$-0.946459\pi$
$223$	−3.22243	−	12.0263i	−0.215790	−	0.805339i	0.770097	−	0.637927i	$-0.220208\pi$
$224$		0			0
$225$	−12.9904	+	7.50000i	−0.866025	+	0.500000i
$226$		0			0
$227$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$227$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$228$		0			0
$229$	21.3923	+	21.3923i	1.41364	+	1.41364i	0.726900	+	0.686743i	$0.240960\pi$
$229$	21.3923	+	21.3923i	1.41364	+	1.41364i	0.686743	+	0.726900i	$0.259040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	10.5000	−	18.1865i	0.682048	−	1.18134i
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−2.50962	+	9.36603i	−0.161659	+	0.603319i	0.836784	+	0.547533i	$0.184433\pi$
$241$	−2.50962	+	9.36603i	−0.161659	+	0.603319i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	−7.79423	−	13.5000i	−0.500000	−	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	15.0981	+	25.0263i	0.960668	+	1.59238i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		−	6.44486i		−	0.400464i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	9.16025	−	2.45448i	0.556446	−	0.149099i	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	9.16025	−	2.45448i	0.556446	−	0.149099i	0.526073	+	0.850439i	$0.323664\pi$
$272$		0			0
$273$	−0.356406	−	18.5263i	−0.0215707	−	1.12126i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	18.0000	−	10.3923i	1.08152	−	0.624413i	0.150210	−	0.988654i	$-0.452005\pi$
$277$	18.0000	−	10.3923i	1.08152	−	0.624413i	0.931305	+	0.364241i	$0.118672\pi$
$278$		0			0
$279$	−32.0885	−	8.59808i	−1.92109	−	0.514753i
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$	−21.6506	−	12.5000i	−1.28700	−	0.743048i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	−21.6506	−	12.5000i	−1.28700	−	0.743048i	−0.978117	+	0.208053i	$0.933287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$	20.8301	−	20.8301i	1.22108	−	1.22108i
$292$		0			0
$293$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$293$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.33013	−	4.96410i	−0.0766672	−	0.286126i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−16.6340	−	16.6340i	−0.949351	−	0.949351i	0.0494267	−	0.998778i	$-0.484261\pi$
$307$	−16.6340	−	16.6340i	−0.949351	−	0.949351i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	−23.3827	−	13.5000i	−1.33019	−	0.767988i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−32.9090			−1.86012			−0.930062	−	0.367402i	$-0.880247\pi$
$313$	−32.9090			−1.86012			−0.930062	+	0.367402i	$0.880247\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.5000	+	12.9904i	−0.693375	+	0.720577i
$326$		0			0
$327$	3.27757	+	12.2321i	0.181250	+	0.676434i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	34.1865	+	9.16025i	1.87906	+	0.503493i	0.999622	+	0.0274825i	$0.00874905\pi$
$331$	34.1865	+	9.16025i	1.87906	+	0.503493i	0.879440	+	0.476011i	$0.157918\pi$
$332$		0			0
$333$	−4.60770	−	4.60770i	−0.252500	−	0.252500i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			29.0000i			1.57973i	0.613280	+	0.789865i	$0.289850\pi$
$337$			29.0000i			1.57973i	−0.613280	+	0.789865i	$0.710150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.9019	−	10.9019i	0.588649	−	0.588649i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−4.40192	+	1.17949i	−0.235630	+	0.0631368i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−4.40192	+	1.17949i	−0.235630	+	0.0631368i	0.139072	+	0.990282i	$0.455588\pi$
$350$		0			0
$351$	−13.5000	−	12.9904i	−0.720577	−	0.693375i
$352$		0			0
$353$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$353$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$		0			0
$361$	40.4545	+	23.3564i	2.12918	+	1.22928i
$362$		0			0
$363$			19.0526i			1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.5000	−	26.8468i	0.809093	−	1.40139i	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	15.5000	−	26.8468i	0.809093	−	1.40139i	0.913493	−	0.406855i	$-0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−18.1865	−	31.5000i	−0.941663	−	1.63101i	−0.762299	−	0.647225i	$-0.775929\pi$
$373$	−18.1865	−	31.5000i	−0.941663	−	1.63101i	−0.179364	−	0.983783i	$-0.557404\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	4.55256	+	16.9904i	0.233849	+	0.872737i	0.978664	+	0.205466i	$0.0658711\pi$
$379$	4.55256	+	16.9904i	0.233849	+	0.872737i	−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$	−1.50000	+	0.866025i	−0.0768473	+	0.0443678i
$382$		0			0
$383$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$383$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.50000	−	2.59808i	−0.228748	−	0.132068i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	7.47372	−	27.8923i	0.375095	−	1.39987i	−0.478110	−	0.878300i	$-0.658678\pi$
$397$	7.47372	−	27.8923i	0.375095	−	1.39987i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$	−20.8301	−	36.0788i	−1.04281	−	1.80620i
$400$		0			0
$401$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$401$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$	−39.9186	+	0.767949i	−1.98849	+	0.0382543i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.42820	−	0.918584i	−0.169514	−	0.0454211i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	−3.42820	−	0.918584i	−0.169514	−	0.0454211i	−0.342578	+	0.939490i	$0.611300\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	12.1244			0.593732
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.463002	−	0.886357i	$-0.346772\pi$
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.886357	−	0.463002i	$-0.153228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	24.8205	−	6.65064i	1.20115	−	0.321847i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$431$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$432$		0			0
$433$	30.3109	−	17.5000i	1.45665	−	0.840996i	0.457804	−	0.889053i	$-0.348636\pi$
$433$	30.3109	−	17.5000i	1.45665	−	0.840996i	0.998845	+	0.0480569i	$0.0153029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$			5.41154i			0.257693i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$449$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−33.4186	+	8.95448i	−1.57014	+	0.420718i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.72243	+	36.2846i	0.454796	+	1.69732i	0.688686	+	0.725059i	$0.258188\pi$
$457$	9.72243	+	36.2846i	0.454796	+	1.69732i	−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$461$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$462$		0			0
$463$	22.3660	+	22.3660i	1.03944	+	1.03944i	0.999190	+	0.0402476i	$0.0128147\pi$
$463$	22.3660	+	22.3660i	1.03944	+	1.03944i	0.0402476	+	0.999190i	$0.487185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−2.35898			−0.108928
$470$		0			0
$471$	9.52628	−	16.5000i	0.438948	−	0.760280i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−10.4904	+	39.1506i	−0.481332	+	1.79635i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$479$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$480$		0			0
$481$	−6.85641	−	3.78461i	−0.312625	−	0.172563i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	27.6865	+	7.41858i	1.25460	+	0.336168i	0.824110	−	0.566429i	$-0.191675\pi$
$487$	27.6865	+	7.41858i	1.25460	+	0.336168i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$	26.1506	+	26.1506i	1.18257	+	1.18257i
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.411543	+	0.411543i	−0.0184232	+	0.0184232i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−0.411543	+	0.411543i	−0.0184232	+	0.0184232i	0.697835	+	0.716258i	$0.254147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19.9186	−	10.5000i	−0.884615	−	0.466321i
$508$		0			0
$509$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$509$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$510$		0			0
$511$	−34.0359	+	19.6506i	−1.50566	+	0.869293i
$512$		0			0
$513$	−40.6865	−	10.9019i	−1.79635	−	0.481332i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−4.00000	+	6.92820i	−0.174908	+	0.302949i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−4.00000	+	6.92820i	−0.174908	+	0.302949i	0.765222	+	0.643767i	$0.222629\pi$
$524$		0			0
$525$	18.1699	−	18.1699i	0.792998	−	0.792998i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	13.1506	+	13.1506i	0.565390	+	0.565390i	0.930834	−	0.365444i	$-0.119083\pi$
$541$	13.1506	+	13.1506i	0.565390	+	0.565390i	−0.365444	+	0.930834i	$0.619083\pi$
$542$		0			0
$543$	−10.3923	−	6.00000i	−0.445976	−	0.257485i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−41.0000			−1.75303			−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−41.0000			−1.75303			−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$	12.9904	−	22.5000i	0.554416	−	0.960277i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−9.31089	+	34.7487i	−0.395939	+	1.47767i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$557$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$558$		0			0
$559$	−6.06218	−	1.50000i	−0.256403	−	0.0634432i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.8827	+	18.8827i	0.792998	+	0.792998i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		−	16.0000i		−	0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−16.0718	+	16.0718i	−0.669078	+	0.669078i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−16.0718	+	16.0718i	−0.669078	+	0.669078i	0.288425	+	0.957503i	$0.406868\pi$
$578$		0			0
$579$	−12.2321	+	45.6506i	−0.508347	+	1.89718i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$587$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$588$		0			0
$589$	−77.7391	+	44.8827i	−3.20318	+	1.84936i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			29.4449i			1.20510i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−20.7846	+	36.0000i	−0.847822	+	1.46847i	0.0353259	+	0.999376i	$0.488753\pi$
$601$	−20.7846	+	36.0000i	−0.847822	+	1.46847i	−0.883148	+	0.469095i	$0.844580\pi$
$602$		0			0
$603$	−1.68653	+	1.68653i	−0.0686810	+	0.0686810i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−10.0000	−	17.3205i	−0.405887	−	0.703018i	0.588537	−	0.808470i	$-0.299704\pi$
$607$	−10.0000	−	17.3205i	−0.405887	−	0.703018i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−12.7942	−	47.7487i	−0.516754	−	1.92855i	−0.314806	−	0.949156i	$-0.601939\pi$
$613$	−12.7942	−	47.7487i	−0.516754	−	1.92855i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$617$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$618$		0			0
$619$	31.8827	+	31.8827i	1.28147	+	1.28147i	0.939829	+	0.341644i	$0.110984\pi$
$619$	31.8827	+	31.8827i	1.28147	+	1.28147i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	9.00962	−	33.6244i	0.358667	−	1.33856i	−0.517139	−	0.855901i	$-0.673003\pi$
$631$	9.00962	−	33.6244i	0.358667	−	1.33856i	0.875806	−	0.482663i	$-0.160330\pi$
$632$		0			0
$633$	−22.5000	−	38.9711i	−0.894295	−	1.54896i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.80385	+	6.24871i	0.0714710	+	0.247583i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−19.0885	−	5.11474i	−0.752775	−	0.201706i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−19.0885	−	5.11474i	−0.752775	−	0.201706i	−0.614749	+	0.788723i	$0.710743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	56.9090			2.23044
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−10.2846	+	38.3827i	−0.401241	+	1.49745i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	44.1147	−	11.8205i	1.71586	−	0.459764i	0.739014	−	0.673690i	$-0.235292\pi$
$661$	44.1147	−	11.8205i	1.71586	−	0.459764i	0.976850	+	0.213925i	$0.0686249\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	20.8301	+	5.58142i	0.805339	+	0.215790i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	43.5000	+	25.1147i	1.67680	+	0.968102i	0.963679	+	0.267063i	$0.0860531\pi$
$673$	43.5000	+	25.1147i	1.67680	+	0.968102i	0.713123	+	0.701039i	$0.247280\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−25.2321	+	43.7032i	−0.968317	+	1.67717i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$683$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−50.6147	+	13.5622i	−1.93107	+	0.517429i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.48076	−	5.52628i	−0.0563308	−	0.210230i	0.932024	−	0.362397i	$-0.118041\pi$
$691$	−1.48076	−	5.52628i	−0.0563308	−	0.210230i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−17.6077			−0.664087
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	8.74871	−	32.6506i	0.328565	−	1.22622i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	8.74871	−	32.6506i	0.328565	−	1.22622i	0.910679	−	0.413114i	$-0.135559\pi$
$710$		0			0
$711$	18.1865	+	31.5000i	0.682048	+	1.18134i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	44.6769	+	11.9711i	1.66386	+	0.445829i
$722$		0			0
$723$	−11.8756	−	11.8756i	−0.441660	−	0.441660i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			49.0000i			1.81731i	0.417548	+	0.908655i	$0.362889\pi$
$727$			49.0000i			1.81731i	−0.417548	+	0.908655i	$0.637111\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−30.3468	+	30.3468i	−1.12088	+	1.12088i	−0.129275	+	0.991609i	$0.541265\pi$
$733$	−30.3468	+	30.3468i	−1.12088	+	1.12088i	−0.991609	+	0.129275i	$0.958735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	46.4186	−	12.4378i	1.70754	−	0.457533i	0.732717	−	0.680534i	$-0.238252\pi$
$739$	46.4186	−	12.4378i	1.70754	−	0.457533i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$	−50.6147	+	0.973721i	−1.85938	+	0.0357705i
$742$		0			0
$743$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$743$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	15.0000	+	8.66025i	0.547358	+	0.316017i	0.748056	−	0.663636i	$-0.230988\pi$
$751$	15.0000	+	8.66025i	0.547358	+	0.316017i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.2487	−	42.0000i	0.881334	−	1.52652i	0.0314762	−	0.999505i	$-0.489979\pi$
$757$	24.2487	−	42.0000i	0.881334	−	1.52652i	0.849858	−	0.527011i	$-0.176688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$761$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$762$		0			0
$763$	−10.8468	−	18.7872i	−0.392680	−	0.680142i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−9.77757	−	36.4904i	−0.352588	−	1.31588i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−9.77757	−	36.4904i	−0.352588	−	1.31588i	0.530904	−	0.847432i	$-0.321852\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$	−39.1506	−	39.1506i	−1.40633	−	1.40633i
$776$		0			0
$777$	9.66730	+	5.58142i	0.346812	+	0.200232i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	13.7679	−	51.3827i	0.490774	−	1.83159i	−0.0617409	−	0.998092i	$-0.519665\pi$
$787$	13.7679	−	51.3827i	0.490774	−	1.83159i	0.552515	−	0.833503i	$-0.313668\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	7.50000	−	30.3109i	0.266333	−	1.07637i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	17.3468	−	17.3468i	0.609128	−	0.609128i	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	17.3468	−	17.3468i	0.609128	−	0.609128i	0.942718	+	0.333590i	$0.108260\pi$
$812$		0			0
$813$	−4.25129	+	15.8660i	−0.149099	+	0.556446i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−13.5622	+	3.63397i	−0.474481	+	0.127137i
$818$		0			0
$819$	28.0981	+	15.5096i	0.981826	+	0.541950i
$820$		0			0
$821$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$821$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$822$		0			0
$823$	21.0000	−	12.1244i	0.732014	−	0.422628i	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	21.0000	−	12.1244i	0.732014	−	0.422628i	0.819159	+	0.573567i	$0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−45.8993	−	26.5000i	−1.59415	−	0.920383i	−0.992584	−	0.121560i	$-0.961210\pi$
$829$	−45.8993	−	26.5000i	−1.59415	−	0.920383i	−0.601566	−	0.798823i	$-0.705456\pi$
$830$		0			0
$831$			36.0000i			1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	40.6865	−	40.6865i	1.40633	−	1.40633i
$838$		0			0
$839$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$839$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−8.44744	−	31.5263i	−0.290258	−	1.08326i
$848$		0			0
$849$	37.5000	−	21.6506i	1.28700	−	0.743048i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	−0.800608	+	0.599189i	$0.795490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−57.1577			−1.95019			−0.975097	−	0.221777i	$-0.928814\pi$
$859$	−57.1577			−1.95019			−0.975097	+	0.221777i	$0.928814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$863$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	14.7224	+	25.5000i	0.500000	+	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1.38526	+	2.50962i	−0.0469379	+	0.0850352i
$872$		0			0
$873$	13.2058	+	49.2846i	0.446948	+	1.66803i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	56.3468	+	15.0981i	1.90270	+	0.509826i	0.996144	+	0.0877308i	$0.0279615\pi$
$877$	56.3468	+	15.0981i	1.90270	+	0.509826i	0.906552	+	0.422095i	$0.138705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$		−	55.0000i		−	1.85090i	−0.378873	−	0.925449i	$-0.623688\pi$
$883$		−	55.0000i		−	1.85090i	0.378873	−	0.925449i	$-0.376312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	2.09808	−	2.09808i	0.0703672	−	0.0703672i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	8.59808	+	2.30385i	0.286126	+	0.0766672i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	34.6410	+	20.0000i	1.15024	+	0.664089i	0.948945	−	0.315442i	$-0.102153\pi$
$907$	34.6410	+	20.0000i	1.15024	+	0.664089i	0.201291	+	0.979531i	$0.435486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−15.5885	−	27.0000i	−0.514216	−	0.890648i	−0.999864	−	0.0164935i	$-0.994750\pi$
$919$	−15.5885	−	27.0000i	−0.514216	−	0.890648i	0.485648	−	0.874154i	$-0.338584\pi$
$920$		0			0
$921$	39.3564	−	10.5455i	1.29684	−	0.347487i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.81089	−	10.4904i	−0.0924215	−	0.344922i
$926$		0			0
$927$	40.5000	−	23.3827i	1.33019	−	0.767988i
$928$		0			0
$929$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$929$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$930$		0			0
$931$	10.3397	+	10.3397i	0.338871	+	0.338871i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55.4256			−1.81068			−0.905338	−	0.424691i	$-0.860383\pi$
$937$	−55.4256			−1.81068			−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$	28.5000	−	49.3634i	0.930062	−	1.61092i
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$947$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$948$		0			0
$949$	0.918584	+	47.7487i	0.0298185	+	1.54999i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	91.6218i		−	2.95554i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−19.4449	+	19.4449i	−0.625305	+	0.625305i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	−19.4449	+	19.4449i	−0.625305	+	0.625305i	0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	−20.0622	+	5.37564i	−0.643164	+	0.172335i
$974$		0			0
$975$	−8.66025	−	30.0000i	−0.277350	−	0.960769i
$976$		0			0
$977$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$977$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−21.1865	−	5.67691i	−0.676434	−	0.181250i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	0.968864	−	0.247592i	$-0.0796392\pi$
$992$		0			0
$993$	−43.3468	+	43.3468i	−1.37557	+	1.37557i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.775923	+	0.630828i	$0.217285\pi$
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	10.9019	−	2.92116i	0.344922	−	0.0924215i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.cn.b.305.1		4
3.2	odd	2	CM	624.2.cn.b.305.1		4
4.3	odd	2		39.2.k.a.32.1	yes	4
12.11	even	2		39.2.k.a.32.1	yes	4
13.11	odd	12	inner	624.2.cn.b.401.1		4
20.3	even	4		975.2.bp.d.149.1		4
20.7	even	4		975.2.bp.a.149.1		4
20.19	odd	2		975.2.bo.c.851.1		4
39.11	even	12	inner	624.2.cn.b.401.1		4
52.3	odd	6		507.2.k.b.80.1		4
52.7	even	12		507.2.f.c.437.1		4
52.11	even	12		39.2.k.a.11.1	✓	4
52.15	even	12		507.2.k.c.89.1		4
52.19	even	12		507.2.f.b.437.1		4
52.23	odd	6		507.2.k.a.80.1		4
52.31	even	4		507.2.k.a.488.1		4
52.35	odd	6		507.2.f.c.239.1		4
52.43	odd	6		507.2.f.b.239.1		4
52.47	even	4		507.2.k.b.488.1		4
52.51	odd	2		507.2.k.c.188.1		4
60.23	odd	4		975.2.bp.d.149.1		4
60.47	odd	4		975.2.bp.a.149.1		4
60.59	even	2		975.2.bo.c.851.1		4
156.11	odd	12		39.2.k.a.11.1	✓	4
156.23	even	6		507.2.k.a.80.1		4
156.35	even	6		507.2.f.c.239.1		4
156.47	odd	4		507.2.k.b.488.1		4
156.59	odd	12		507.2.f.c.437.1		4
156.71	odd	12		507.2.f.b.437.1		4
156.83	odd	4		507.2.k.a.488.1		4
156.95	even	6		507.2.f.b.239.1		4
156.107	even	6		507.2.k.b.80.1		4
156.119	odd	12		507.2.k.c.89.1		4
156.155	even	2		507.2.k.c.188.1		4
260.63	odd	12		975.2.bp.a.674.1		4
260.167	odd	12		975.2.bp.d.674.1		4
260.219	even	12		975.2.bo.c.401.1		4
780.167	even	12		975.2.bp.d.674.1		4
780.323	even	12		975.2.bp.a.674.1		4
780.479	odd	12		975.2.bo.c.401.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.k.a.11.1	✓	4	52.11	even	12
39.2.k.a.11.1	✓	4	156.11	odd	12
39.2.k.a.32.1	yes	4	4.3	odd	2
39.2.k.a.32.1	yes	4	12.11	even	2
507.2.f.b.239.1		4	52.43	odd	6
507.2.f.b.239.1		4	156.95	even	6
507.2.f.b.437.1		4	52.19	even	12
507.2.f.b.437.1		4	156.71	odd	12
507.2.f.c.239.1		4	52.35	odd	6
507.2.f.c.239.1		4	156.35	even	6
507.2.f.c.437.1		4	52.7	even	12
507.2.f.c.437.1		4	156.59	odd	12
507.2.k.a.80.1		4	52.23	odd	6
507.2.k.a.80.1		4	156.23	even	6
507.2.k.a.488.1		4	52.31	even	4
507.2.k.a.488.1		4	156.83	odd	4
507.2.k.b.80.1		4	52.3	odd	6
507.2.k.b.80.1		4	156.107	even	6
507.2.k.b.488.1		4	52.47	even	4
507.2.k.b.488.1		4	156.47	odd	4
507.2.k.c.89.1		4	52.15	even	12
507.2.k.c.89.1		4	156.119	odd	12
507.2.k.c.188.1		4	52.51	odd	2
507.2.k.c.188.1		4	156.155	even	2
624.2.cn.b.305.1		4	1.1	even	1	trivial
624.2.cn.b.305.1		4	3.2	odd	2	CM
624.2.cn.b.401.1		4	13.11	odd	12	inner
624.2.cn.b.401.1		4	39.11	even	12	inner
975.2.bo.c.401.1		4	260.219	even	12
975.2.bo.c.401.1		4	780.479	odd	12
975.2.bo.c.851.1		4	20.19	odd	2
975.2.bo.c.851.1		4	60.59	even	2
975.2.bp.a.149.1		4	20.7	even	4
975.2.bp.a.149.1		4	60.47	odd	4
975.2.bp.a.674.1		4	260.63	odd	12
975.2.bp.a.674.1		4	780.323	even	12
975.2.bp.d.149.1		4	20.3	even	4
975.2.bp.d.149.1		4	60.23	odd	4
975.2.bp.d.674.1		4	260.167	odd	12
975.2.bp.d.674.1		4	780.167	even	12

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.cn (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 1.50000i) q^{3} +(0.767949 - 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.866025 + 1.50000i) q^{3} +(0.767949 - 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(-2.59808 - 2.50000i) q^{13} +(-7.83013 - 2.09808i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} +5.19615 q^{27} +(7.83013 - 7.83013i) q^{31} +(2.09808 - 0.562178i) q^{37} +(6.00000 - 1.73205i) q^{39} +(1.50000 - 0.866025i) q^{43} +(-1.56218 - 0.901924i) q^{49} +(9.92820 - 9.92820i) q^{57} +(4.33013 + 7.50000i) q^{61} +(-8.59808 + 2.30385i) q^{63} +(-0.205771 - 0.767949i) q^{67} +(-9.36603 - 9.36603i) q^{73} +(7.50000 + 4.33013i) q^{75} -12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-9.16025 + 5.52628i) q^{91} +(4.96410 + 18.5263i) q^{93} +(-16.4282 - 4.40192i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 10 * q^7 - 6 * q^9	\( 4 q + 10 q^{7} - 6 q^{9} - 14 q^{19} + 18 q^{21} + 14 q^{31} - 2 q^{37} + 24 q^{39} + 6 q^{43} + 18 q^{49} + 12 q^{57} - 24 q^{63} - 32 q^{67} - 34 q^{73} + 30 q^{75} - 18 q^{81} - 2 q^{91} + 6 q^{93} - 38 q^{97}+O(q^{100}) \) 4 * q + 10 * q^7 - 6 * q^9 - 14 * q^19 + 18 * q^21 + 14 * q^31 - 2 * q^37 + 24 * q^39 + 6 * q^43 + 18 * q^49 + 12 * q^57 - 24 * q^63 - 32 * q^67 - 34 * q^73 + 30 * q^75 - 18 * q^81 - 2 * q^91 + 6 * q^93 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more