LMFDB - Embedded newform 624.2.cn.a.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 156)
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.a.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.50000i) q^{3} +(1.23205 - 4.59808i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(0.866025 - 1.50000i) q^{3} +(1.23205 - 4.59808i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(0.866025 + 3.50000i) q^{13} +(-3.09808 - 0.830127i) q^{19} +(-5.83013 - 5.83013i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(-6.36603 + 6.36603i) q^{31} +(11.5622 - 3.09808i) q^{37} +(6.00000 + 1.73205i) q^{39} +(10.5000 - 6.06218i) q^{43} +(-13.5622 - 7.83013i) q^{49} +(-3.92820 + 3.92820i) q^{57} +(7.79423 + 13.5000i) q^{61} +(-13.7942 + 3.69615i) q^{63} +(-1.93782 - 7.23205i) q^{67} +(-4.29423 - 4.29423i) q^{73} +(-7.50000 - 4.33013i) q^{75} +5.19615 q^{79} +(-4.50000 + 7.79423i) q^{81} +(17.1603 + 0.330127i) q^{91} +(4.03590 + 15.0622i) q^{93} +(9.42820 + 2.52628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^7 - 6 * q^9	$ 4 q - 2 q^{7} - 6 q^{9} - 2 q^{19} - 6 q^{21} - 22 q^{31} + 22 q^{37} + 24 q^{39} + 42 q^{43} - 30 q^{49} + 12 q^{57} - 24 q^{63} - 32 q^{67} + 14 q^{73} - 30 q^{75} - 18 q^{81} + 34 q^{91} + 30 q^{93} + 10 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 - 6 * q^9 - 2 * q^19 - 6 * q^21 - 22 * q^31 + 22 * q^37 + 24 * q^39 + 42 * q^43 - 30 * q^49 + 12 * q^57 - 24 * q^63 - 32 * q^67 + 14 * q^73 - 30 * q^75 - 18 * q^81 + 34 * q^91 + 30 * q^93 + 10 * 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
$7$	1.23205	−	4.59808i	0.465671	−	1.73791i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	1.23205	−	4.59808i	0.465671	−	1.73791i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		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
$13$	0.866025	+	3.50000i	0.240192	+	0.970725i
$13$	0.866025	+	3.50000i	0.240192	+	0.970725i
$14$		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
$19$	−3.09808	−	0.830127i	−0.710747	−	0.190444i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−3.09808	−	0.830127i	−0.710747	−	0.190444i	−0.596040	+	0.802955i	$0.703260\pi$
$20$		0			0
$21$	−5.83013	−	5.83013i	−1.27224	−	1.27224i
$22$		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
$31$	−6.36603	+	6.36603i	−1.14337	+	1.14337i	−0.155543	+	0.987829i	$0.549713\pi$
$31$	−6.36603	+	6.36603i	−1.14337	+	1.14337i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	11.5622	−	3.09808i	1.90081	−	0.509321i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.5622	−	3.09808i	1.90081	−	0.509321i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$		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
$43$	10.5000	−	6.06218i	1.60123	−	0.924473i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	10.5000	−	6.06218i	1.60123	−	0.924473i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		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
$49$	−13.5622	−	7.83013i	−1.93745	−	1.11859i
$50$		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
$57$	−3.92820	+	3.92820i	−0.520303	+	0.520303i
$58$		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
$61$	7.79423	+	13.5000i	0.997949	+	1.72850i	0.554416	+	0.832240i	$0.312942\pi$
$61$	7.79423	+	13.5000i	0.997949	+	1.72850i	0.443533	+	0.896258i	$0.353725\pi$
$62$		0			0
$63$	−13.7942	+	3.69615i	−1.73791	+	0.465671i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.93782	−	7.23205i	−0.236743	−	0.883536i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−1.93782	−	7.23205i	−0.236743	−	0.883536i	0.740613	−	0.671932i	$-0.234535\pi$
$68$		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
$73$	−4.29423	−	4.29423i	−0.502601	−	0.502601i	0.409644	−	0.912245i	$-0.365653\pi$
$73$	−4.29423	−	4.29423i	−0.502601	−	0.502601i	−0.912245	+	0.409644i	$0.865653\pi$
$74$		0			0
$75$	−7.50000	−	4.33013i	−0.866025	−	0.500000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.19615			0.584613			0.292306	−	0.956325i	$-0.405577\pi$
$79$	5.19615			0.584613			0.292306	+	0.956325i	$0.405577\pi$
$80$		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
$91$	17.1603	+	0.330127i	1.79888	+	0.0346067i
$92$		0			0
$93$	4.03590	+	15.0622i	0.418503	+	1.56188i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.42820	+	2.52628i	0.957289	+	0.256505i	0.703452	−	0.710742i	$-0.251641\pi$
$97$	9.42820	+	2.52628i	0.957289	+	0.256505i	0.253837	+	0.967247i	$0.418307\pi$
$98$		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
$103$			19.0526i			1.87730i	0.344865	+	0.938652i	$0.387925\pi$
$103$			19.0526i			1.87730i	−0.344865	+	0.938652i	$0.612075\pi$
$104$		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
$109$	−2.43782	+	2.43782i	−0.233501	+	0.233501i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−2.43782	+	2.43782i	−0.233501	+	0.233501i	0.580651	+	0.814152i	$0.302798\pi$
$110$		0			0
$111$	5.36603	−	20.0263i	0.509321	−	1.90081i
$112$		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
$117$	7.79423	−	7.50000i	0.720577	−	0.693375i
$118$		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
$127$	16.4545	+	9.50000i	1.46010	+	0.842989i	0.999015	−	0.0443678i	$-0.0141274\pi$
$127$	16.4545	+	9.50000i	1.46010	+	0.842989i	0.461084	+	0.887357i	$0.347461\pi$
$128$		0			0
$129$		−	21.0000i		−	1.84895i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−7.63397	+	13.2224i	−0.661950	+	1.14653i
$134$		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
$139$	11.5000	+	19.9186i	0.975417	+	1.68947i	0.678551	+	0.734553i	$0.262608\pi$
$139$	11.5000	+	19.9186i	0.975417	+	1.68947i	0.296866	+	0.954919i	$0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−23.4904	+	13.5622i	−1.93745	+	1.11859i
$148$		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
$151$	−10.1244	−	10.1244i	−0.823908	−	0.823908i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	−10.1244	−	10.1244i	−0.823908	−	0.823908i	−0.986666	+	0.162758i	$0.947961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	25.0000			1.99522			0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	25.0000			1.99522			0.997609	+	0.0691164i	$0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.59808	+	24.6244i	−0.516801	+	1.92873i	−0.203497	+	0.979076i	$0.565231\pi$
$163$	−6.59808	+	24.6244i	−0.516801	+	1.92873i	−0.313304	+	0.949653i	$0.601436\pi$
$164$		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
$169$	−11.5000	+	6.06218i	−0.884615	+	0.466321i
$170$		0			0
$171$	2.49038	+	9.29423i	0.190444	+	0.710747i
$172$		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
$175$	−22.9904	−	6.16025i	−1.73791	−	0.465671i
$176$		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	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	27.0000			1.99590
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−6.40192	+	23.8923i	−0.465671	+	1.73791i
$190$		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
$193$	−25.3564	+	6.79423i	−1.82519	+	0.489059i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	−25.3564	+	6.79423i	−1.82519	+	0.489059i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		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
$199$	9.52628	−	5.50000i	0.675300	−	0.389885i	−0.122782	−	0.992434i	$-0.539182\pi$
$199$	9.52628	−	5.50000i	0.675300	−	0.389885i	0.798082	+	0.602549i	$0.205848\pi$
$200$		0			0
$201$	−12.5263	−	3.35641i	−0.883536	−	0.236743i
$202$		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
$211$	−0.866025	+	1.50000i	−0.0596196	+	0.103264i	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	−0.866025	+	1.50000i	−0.0596196	+	0.103264i	0.834675	+	0.550743i	$0.185655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	21.4282	+	37.1147i	1.45464	+	2.51951i
$218$		0			0
$219$	−10.1603	+	2.72243i	−0.686566	+	0.183965i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−7.02628	−	26.2224i	−0.470514	−	1.75598i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	−7.02628	−	26.2224i	−0.470514	−	1.75598i	0.167412	−	0.985887i	$-0.446459\pi$
$224$		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
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	0.607695	+	0.607695i	0.0401576	+	0.0401576i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		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
$237$	4.50000	−	7.79423i	0.292306	−	0.506290i
$238$		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
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	0.547533	−	0.836784i	$-0.315567\pi$
$242$		0			0
$243$	7.79423	+	13.5000i	0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.222432	−	11.5622i	0.0141530	−	0.735684i
$248$		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
$259$		−	56.9808i		−	3.54061i
$260$		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
$271$	−23.1603	+	6.20577i	−1.40689	+	0.376974i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−23.1603	+	6.20577i	−1.40689	+	0.376974i	−0.526073	+	0.850439i	$0.676336\pi$
$272$		0			0
$273$	15.3564	−	25.4545i	0.929412	−	1.54058i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−18.0000	+	10.3923i	−1.08152	+	0.624413i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−18.0000	+	10.3923i	−1.08152	+	0.624413i	−0.150210	+	0.988654i	$0.547995\pi$
$278$		0			0
$279$	26.0885	+	6.99038i	1.56188	+	0.418503i
$280$		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
$283$	−6.06218	−	3.50000i	−0.360359	−	0.208053i	0.308879	−	0.951101i	$-0.400046\pi$
$283$	−6.06218	−	3.50000i	−0.360359	−	0.208053i	−0.669238	+	0.743048i	$0.733379\pi$
$284$		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
$291$	11.9545	−	11.9545i	0.700784	−	0.700784i
$292$		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
$301$	−14.9378	−	55.7487i	−0.861002	−	3.21330i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−5.22243	−	5.22243i	−0.298060	−	0.298060i	0.542194	−	0.840254i	$-0.317594\pi$
$307$	−5.22243	−	5.22243i	−0.298060	−	0.298060i	−0.840254	+	0.542194i	$0.817594\pi$
$308$		0			0
$309$	28.5788	+	16.5000i	1.62579	+	0.938652i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−5.19615			−0.293704			−0.146852	−	0.989158i	$-0.546914\pi$
$313$	−5.19615			−0.293704			−0.146852	+	0.989158i	$0.546914\pi$
$314$		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
$325$	17.5000	−	4.33013i	0.970725	−	0.240192i
$326$		0			0
$327$	1.54552	+	5.76795i	0.0854673	+	0.318968i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	25.5263	+	6.83975i	1.40305	+	0.375946i	0.879440	−	0.476011i	$-0.157918\pi$
$331$	25.5263	+	6.83975i	1.40305	+	0.375946i	0.523612	+	0.851957i	$0.324584\pi$
$332$		0			0
$333$	−25.3923	−	25.3923i	−1.39149	−	1.39149i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			5.00000i			0.272367i	0.990684	+	0.136184i	$0.0434837\pi$
$337$			5.00000i			0.272367i	−0.990684	+	0.136184i	$0.956516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−29.1506	+	29.1506i	−1.57399	+	1.57399i
$344$		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
$349$	−21.7224	+	5.82051i	−1.16278	+	0.311565i	−0.788074	−	0.615581i	$-0.788921\pi$
$349$	−21.7224	+	5.82051i	−1.16278	+	0.311565i	−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−4.50000	−	18.1865i	−0.240192	−	0.970725i
$352$		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
$361$	−7.54552	−	4.35641i	−0.397132	−	0.229285i
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−17.5000	+	30.3109i	−0.913493	+	1.58222i	−0.104399	+	0.994535i	$0.533292\pi$
$367$	−17.5000	+	30.3109i	−0.913493	+	1.58222i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−14.7224	−	25.5000i	−0.762299	−	1.32034i	−0.941663	−	0.336557i	$-0.890737\pi$
$373$	−14.7224	−	25.5000i	−0.762299	−	1.32034i	0.179364	−	0.983783i	$-0.442596\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.552559	−	2.06218i	−0.0283830	−	0.105927i	0.950281	−	0.311393i	$-0.100796\pi$
$379$	−0.552559	−	2.06218i	−0.0283830	−	0.105927i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$	28.5000	−	16.4545i	1.46010	−	0.842989i
$382$		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
$387$	−31.5000	−	18.1865i	−1.60123	−	0.924473i
$388$		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
$397$	−2.91858	+	10.8923i	−0.146480	+	0.546669i	0.853206	+	0.521575i	$0.174655\pi$
$397$	−2.91858	+	10.8923i	−0.146480	+	0.546669i	−0.999685	+	0.0250943i	$0.992011\pi$
$398$		0			0
$399$	13.2224	+	22.9019i	0.661950	+	1.14653i
$400$		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
$403$	−27.7942	−	16.7679i	−1.38453	−	0.835271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.4282	+	6.00962i	1.10900	+	0.297157i	0.766426	−	0.642333i	$-0.222033\pi$
$409$	22.4282	+	6.00962i	1.10900	+	0.297157i	0.342578	+	0.939490i	$0.388700\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	39.8372			1.95083
$418$		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
$421$	−19.6340	+	19.6340i	−0.956901	+	0.956901i	−0.999109	−	0.0422075i	$-0.986561\pi$
$421$	−19.6340	+	19.6340i	−0.956901	+	0.956901i	0.0422075	+	0.999109i	$0.486561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	71.6769	−	19.2058i	3.46869	−	0.929432i
$428$		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
$433$	−32.0429	+	18.5000i	−1.53989	+	0.889053i	−0.541041	+	0.840996i	$0.681970\pi$
$433$	−32.0429	+	18.5000i	−1.53989	+	0.889053i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	7.50000	+	4.33013i	0.357955	+	0.206666i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	7.50000	+	4.33013i	0.357955	+	0.206666i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$			46.9808i			2.23718i
$442$		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
$453$	−23.9545	+	6.41858i	−1.12548	+	0.301571i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0622	−	41.2846i	−0.517467	−	1.93121i	−0.283577	−	0.958950i	$-0.591521\pi$
$457$	−11.0622	−	41.2846i	−0.517467	−	1.93121i	−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
$463$	−29.6865	−	29.6865i	−1.37965	−	1.37965i	−0.845200	−	0.534450i	$-0.820519\pi$
$463$	−29.6865	−	29.6865i	−1.37965	−	1.37965i	−0.534450	−	0.845200i	$-0.679481\pi$
$464$		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
$469$	−35.6410			−1.64575
$470$		0			0
$471$	21.6506	−	37.5000i	0.997609	−	1.72791i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.15064	+	15.4904i	−0.190444	+	0.710747i
$476$		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
$481$	20.8564	+	37.7846i	0.950970	+	1.72283i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	32.4186	+	8.68653i	1.46903	+	0.393624i	0.902597	−	0.430486i	$-0.141658\pi$
$487$	32.4186	+	8.68653i	1.46903	+	0.393624i	0.566429	+	0.824110i	$0.308325\pi$
$488$		0			0
$489$	31.2224	+	31.2224i	1.41193	+	1.41193i
$490$		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
$499$	−31.5885	+	31.5885i	−1.41409	+	1.41409i	−0.697835	+	0.716258i	$0.745853\pi$
$499$	−31.5885	+	31.5885i	−1.41409	+	1.41409i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		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
$507$	−0.866025	+	22.5000i	−0.0384615	+	0.999260i
$508$		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
$511$	−25.0359	+	14.4545i	−1.10752	+	0.639429i
$512$		0			0
$513$	16.0981	+	4.31347i	0.710747	+	0.190444i
$514$		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
$525$	−29.1506	+	29.1506i	−1.27224	+	1.27224i
$526$		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
$541$	3.68653	+	3.68653i	0.158496	+	0.158496i	0.781900	−	0.623404i	$-0.214251\pi$
$541$	3.68653	+	3.68653i	0.158496	+	0.158496i	−0.623404	+	0.781900i	$0.714251\pi$
$542$		0			0
$543$	−10.3923	−	6.00000i	−0.445976	−	0.257485i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00000			0.0427569			0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000			0.0427569			0.0213785	+	0.999771i	$0.493195\pi$
$548$		0			0
$549$	23.3827	−	40.5000i	0.997949	−	1.72850i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	6.40192	−	23.8923i	0.272237	−	1.01600i
$554$		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
$559$	30.3109	+	31.5000i	1.28201	+	1.33231i
$560$		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
$567$	30.2942	+	30.2942i	1.27224	+	1.27224i
$568$		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
$577$	−29.9282	+	29.9282i	−1.24593	+	1.24593i	−0.288425	+	0.957503i	$0.593132\pi$
$577$	−29.9282	+	29.9282i	−1.24593	+	1.24593i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−11.7679	+	43.9186i	−0.489059	+	1.82519i
$580$		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
$589$	25.0070	−	14.4378i	1.03040	−	0.594900i
$590$		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
$597$		−	19.0526i		−	0.779769i
$598$		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	$-0.511247\pi$
$601$	20.7846	−	36.0000i	0.847822	−	1.46847i	0.883148	−	0.469095i	$-0.155420\pi$
$602$		0			0
$603$	−15.8827	+	15.8827i	−0.646793	+	0.646793i
$604$		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
$613$	11.4545	+	42.7487i	0.462642	+	1.72660i	0.664590	+	0.747208i	$0.268606\pi$
$613$	11.4545	+	42.7487i	0.462642	+	1.72660i	−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
$619$	−20.1699	−	20.1699i	−0.810696	−	0.810696i	0.174042	−	0.984738i	$-0.444317\pi$
$619$	−20.1699	−	20.1699i	−0.810696	−	0.810696i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		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
$631$	−3.11474	+	11.6244i	−0.123996	+	0.462758i	−0.999802	−	0.0199047i	$-0.993664\pi$
$631$	−3.11474	+	11.6244i	−0.123996	+	0.462758i	0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$	1.50000	+	2.59808i	0.0596196	+	0.103264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.6603	−	54.2487i	0.620482	−	2.14941i
$638$		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
$643$	39.0885	+	10.4737i	1.54150	+	0.413043i	0.926750	−	0.375680i	$-0.122591\pi$
$643$	39.0885	+	10.4737i	1.54150	+	0.413043i	0.614749	+	0.788723i	$0.289257\pi$
$644$		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
$651$	74.2295			2.90928
$652$		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
$657$	−4.71539	+	17.5981i	−0.183965	+	0.686566i
$658$		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
$661$	26.7942	−	7.17949i	1.04217	−	0.279250i	0.303160	−	0.952940i	$-0.401958\pi$
$661$	26.7942	−	7.17949i	1.04217	−	0.279250i	0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−45.4186	−	12.1699i	−1.75598	−	0.470514i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	31.5000	+	18.1865i	1.21424	+	0.701039i	0.963679	−	0.267063i	$-0.0860531\pi$
$673$	31.5000	+	18.1865i	1.21424	+	0.701039i	0.250557	+	0.968102i	$0.419386\pi$
$674$		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
$679$	23.2321	−	40.2391i	0.891564	−	1.54423i
$680$		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
$687$	1.43782	−	0.385263i	0.0548563	−	0.0146987i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.48076	+	20.4545i	0.208498	+	0.778125i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	5.48076	+	20.4545i	0.208498	+	0.778125i	−0.779857	+	0.625958i	$0.784708\pi$
$692$		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
$703$	−38.3923			−1.44799
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.25129	−	8.40192i	0.0845489	−	0.315541i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	2.25129	−	8.40192i	0.0845489	−	0.315541i	0.995228	+	0.0975728i	$0.0311079\pi$
$710$		0			0
$711$	−7.79423	−	13.5000i	−0.292306	−	0.506290i
$712$		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
$721$	87.6051	+	23.4737i	3.26259	+	0.874207i
$722$		0			0
$723$	−36.1244	−	36.1244i	−1.34348	−	1.34348i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	5.00000i		−	0.185440i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$		−	5.00000i		−	0.185440i	0.995692	−	0.0927199i	$-0.0295561\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−37.9545	+	37.9545i	−1.40188	+	1.40188i	−0.607760	+	0.794121i	$0.707932\pi$
$733$	−37.9545	+	37.9545i	−1.40188	+	1.40188i	−0.794121	+	0.607760i	$0.792068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−24.5622	+	6.58142i	−0.903534	+	0.242101i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−24.5622	+	6.58142i	−0.903534	+	0.242101i	−0.223001	+	0.974818i	$0.571585\pi$
$740$		0			0
$741$	−17.1506	−	10.3468i	−0.630044	−	0.380099i
$742$		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	0.200698	−	0.979653i	$-0.435679\pi$
$751$	−15.0000	−	8.66025i	−0.547358	−	0.316017i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		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	$0.510021\pi$
$757$	−24.2487	+	42.0000i	−0.881334	+	1.52652i	−0.849858	+	0.527011i	$0.823312\pi$
$758$		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
$763$	8.20577	+	14.2128i	0.297069	+	0.514538i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	10.5096	+	39.2224i	0.378987	+	1.41440i	0.847432	+	0.530904i	$0.178148\pi$
$769$	10.5096	+	39.2224i	0.378987	+	1.41440i	−0.468445	+	0.883493i	$0.655186\pi$
$770$		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
$775$	31.8301	+	31.8301i	1.14337	+	1.14337i
$776$		0			0
$777$	−85.4711	−	49.3468i	−3.06626	−	1.77031i
$778$		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
$787$	14.2321	−	53.1147i	0.507318	−	1.89334i	0.0617409	−	0.998092i	$-0.480335\pi$
$787$	14.2321	−	53.1147i	0.507318	−	1.89334i	0.445577	−	0.895244i	$-0.352999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−40.5000	+	38.9711i	−1.43820	+	1.38391i
$794$		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
$811$	3.15064	−	3.15064i	0.110634	−	0.110634i	−0.649623	−	0.760257i	$-0.725073\pi$
$811$	3.15064	−	3.15064i	0.110634	−	0.110634i	0.760257	+	0.649623i	$0.225073\pi$
$812$		0			0
$813$	−10.7487	+	40.1147i	−0.376974	+	1.40689i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−37.5622	+	10.0648i	−1.31413	+	0.352121i
$818$		0			0
$819$	−24.8827	−	45.0788i	−0.869471	−	1.57518i
$820$		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	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−21.0000	+	12.1244i	−0.732014	+	0.422628i	0.0871445	+	0.996196i	$0.472226\pi$
$824$		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
$829$	6.06218	+	3.50000i	0.210548	+	0.121560i	0.601566	−	0.798823i	$-0.294544\pi$
$829$	6.06218	+	3.50000i	0.210548	+	0.121560i	−0.391018	+	0.920383i	$0.627877\pi$
$830$		0			0
$831$			36.0000i			1.24883i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	33.0788	−	33.0788i	1.14337	−	1.14337i
$838$		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
$847$	−13.5526	−	50.5788i	−0.465671	−	1.73791i
$848$		0			0
$849$	−10.5000	+	6.06218i	−0.360359	+	0.208053i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−15.3468	−	15.3468i	−0.525464	−	0.525464i	0.393753	−	0.919216i	$-0.371177\pi$
$853$	−15.3468	−	15.3468i	−0.525464	−	0.525464i	−0.919216	+	0.393753i	$0.871177\pi$
$854$		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
$859$	−39.8372			−1.35923			−0.679613	−	0.733571i	$-0.737852\pi$
$859$	−39.8372			−1.35923			−0.679613	+	0.733571i	$0.737852\pi$
$860$		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
$871$	23.6340	−	13.0455i	0.800807	−	0.442030i
$872$		0			0
$873$	−7.57884	−	28.2846i	−0.256505	−	0.957289i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−9.90192	−	2.65321i	−0.334364	−	0.0895926i	0.0877308	−	0.996144i	$-0.472038\pi$
$877$	−9.90192	−	2.65321i	−0.334364	−	0.0895926i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		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
$883$			47.0000i			1.58168i	0.612026	+	0.790838i	$0.290355\pi$
$883$			47.0000i			1.58168i	−0.612026	+	0.790838i	$0.709645\pi$
$884$		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
$889$	63.9545	−	63.9545i	2.14496	−	2.14496i
$890$		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
$903$	−96.5596	−	25.8731i	−3.21330	−	0.861002i
$904$		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	$0.00525028\pi$
$919$	15.5885	+	27.0000i	0.514216	+	0.890648i	−0.485648	+	0.874154i	$0.661416\pi$
$920$		0			0
$921$	−12.3564	+	3.31089i	−0.407157	+	0.109098i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−15.4904	−	57.8109i	−0.509321	−	1.90081i
$926$		0			0
$927$	49.5000	−	28.5788i	1.62579	−	0.938652i
$928$		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
$931$	35.5167	+	35.5167i	1.16401	+	1.16401i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	55.4256			1.81068			0.905338	−	0.424691i	$-0.139617\pi$
$937$	55.4256			1.81068			0.905338	+	0.424691i	$0.139617\pi$
$938$		0			0
$939$	−4.50000	+	7.79423i	−0.146852	+	0.254355i
$940$		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
$949$	11.3109	−	18.7487i	0.367167	−	0.608609i
$950$		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
$961$		−	50.0526i		−	1.61460i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	39.4449	−	39.4449i	1.26846	−	1.26846i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	39.4449	−	39.4449i	1.26846	−	1.26846i	0.946883	−	0.321578i	$-0.104213\pi$
$968$		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
$973$	105.756	−	28.3372i	3.39037	−	0.908448i
$974$		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
$981$	9.99038	+	2.67691i	0.318968	+	0.0854673i
$982$		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
$993$	32.3660	−	32.3660i	1.02710	−	1.02710i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.5000	−	42.4352i	−0.775923	−	1.34394i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−24.5000	−	42.4352i	−0.775923	−	1.34394i	0.158352	−	0.987383i	$-0.449382\pi$
$998$		0			0
$999$	−60.0788	+	16.0981i	−1.90081	+	0.509321i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.cn.a.305.1		4
3.2	odd	2	CM	624.2.cn.a.305.1		4
4.3	odd	2		156.2.u.a.149.1	yes	4
12.11	even	2		156.2.u.a.149.1	yes	4
13.11	odd	12	inner	624.2.cn.a.401.1		4
39.11	even	12	inner	624.2.cn.a.401.1		4
52.11	even	12		156.2.u.a.89.1	✓	4
156.11	odd	12		156.2.u.a.89.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.u.a.89.1	✓	4	52.11	even	12
156.2.u.a.89.1	✓	4	156.11	odd	12
156.2.u.a.149.1	yes	4	4.3	odd	2
156.2.u.a.149.1	yes	4	12.11	even	2
624.2.cn.a.305.1		4	1.1	even	1	trivial
624.2.cn.a.305.1		4	3.2	odd	2	CM
624.2.cn.a.401.1		4	13.11	odd	12	inner
624.2.cn.a.401.1		4	39.11	even	12	inner

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)

\(f(q)\)	\(=\)	\(q+(0.866025 - 1.50000i) q^{3} +(1.23205 - 4.59808i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 1.50000i) q^{3} +(1.23205 - 4.59808i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(0.866025 + 3.50000i) q^{13} +(-3.09808 - 0.830127i) q^{19} +(-5.83013 - 5.83013i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(-6.36603 + 6.36603i) q^{31} +(11.5622 - 3.09808i) q^{37} +(6.00000 + 1.73205i) q^{39} +(10.5000 - 6.06218i) q^{43} +(-13.5622 - 7.83013i) q^{49} +(-3.92820 + 3.92820i) q^{57} +(7.79423 + 13.5000i) q^{61} +(-13.7942 + 3.69615i) q^{63} +(-1.93782 - 7.23205i) q^{67} +(-4.29423 - 4.29423i) q^{73} +(-7.50000 - 4.33013i) q^{75} +5.19615 q^{79} +(-4.50000 + 7.79423i) q^{81} +(17.1603 + 0.330127i) q^{91} +(4.03590 + 15.0622i) q^{93} +(9.42820 + 2.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^7 - 6 * q^9	\( 4 q - 2 q^{7} - 6 q^{9} - 2 q^{19} - 6 q^{21} - 22 q^{31} + 22 q^{37} + 24 q^{39} + 42 q^{43} - 30 q^{49} + 12 q^{57} - 24 q^{63} - 32 q^{67} + 14 q^{73} - 30 q^{75} - 18 q^{81} + 34 q^{91} + 30 q^{93} + 10 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 - 6 * q^9 - 2 * q^19 - 6 * q^21 - 22 * q^31 + 22 * q^37 + 24 * q^39 + 42 * q^43 - 30 * q^49 + 12 * q^57 - 24 * q^63 - 32 * q^67 + 14 * q^73 - 30 * q^75 - 18 * q^81 + 34 * q^91 + 30 * q^93 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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