LMFDB - Embedded newform 1300.2.bs.b.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(193,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 9, 7]))

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

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

chi := DirichletCharacter("1300.193");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.bs (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:	$10.3805522628$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			193.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1300.193
Dual form			1300.2.bs.b.357.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 1.86603i) q^{3} +(3.86603 + 2.23205i) q^{7} +(-0.633975 - 0.366025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 1.86603i) q^{3} +(3.86603 + 2.23205i) q^{7} +(-0.633975 - 0.366025i) q^{9} +(-2.86603 - 0.767949i) q^{11} +(2.00000 - 3.00000i) q^{13} +(-3.23205 + 0.866025i) q^{17} +(-1.13397 - 4.23205i) q^{19} +(6.09808 - 6.09808i) q^{21} +(6.96410 + 1.86603i) q^{23} +(3.09808 - 3.09808i) q^{27} +(-1.50000 + 0.866025i) q^{29} +(5.19615 + 5.19615i) q^{31} +(-2.86603 + 4.96410i) q^{33} +(7.33013 - 4.23205i) q^{37} +(-4.59808 - 5.23205i) q^{39} +(0.669873 - 2.50000i) q^{41} +(-0.964102 - 3.59808i) q^{43} +2.92820i q^{47} +(6.46410 + 11.1962i) q^{49} +6.46410i q^{51} +(4.46410 + 4.46410i) q^{53} -8.46410 q^{57} +(-6.33013 + 1.69615i) q^{59} +(4.50000 - 7.79423i) q^{61} +(-1.63397 - 2.83013i) q^{63} +(-7.86603 - 13.6244i) q^{67} +(6.96410 - 12.0622i) q^{69} +(-4.59808 + 1.23205i) q^{71} +1.07180 q^{73} +(-9.36603 - 9.36603i) q^{77} -7.46410i q^{79} +(-5.33013 - 9.23205i) q^{81} +10.9282i q^{83} +(0.866025 + 3.23205i) q^{87} +(0.794229 - 2.96410i) q^{89} +(14.4282 - 7.13397i) q^{91} +(12.2942 - 7.09808i) q^{93} +(-2.13397 + 3.69615i) q^{97} +(1.53590 + 1.53590i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9	$ 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 8 q^{19} + 14 q^{21} + 14 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 20 q^{41} + 10 q^{43} + 12 q^{49} + 4 q^{53} - 20 q^{57} - 8 q^{59} + 18 q^{61} - 10 q^{63} - 28 q^{67} + 14 q^{69} - 8 q^{71} + 32 q^{73} - 34 q^{77} - 4 q^{81} - 28 q^{89} + 30 q^{91} + 18 q^{93} - 12 q^{97} + 20 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 8 * q^19 + 14 * q^21 + 14 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 20 * q^41 + 10 * q^43 + 12 * q^49 + 4 * q^53 - 20 * q^57 - 8 * q^59 + 18 * q^61 - 10 * q^63 - 28 * q^67 + 14 * q^69 - 8 * q^71 + 32 * q^73 - 34 * q^77 - 4 * q^81 - 28 * q^89 + 30 * q^91 + 18 * q^93 - 12 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$e\left(\frac{7}{12}\right)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.500000	−	1.86603i	0.288675	−	1.07735i	−0.657437	−	0.753510i	$-0.728359\pi$
$3$	0.500000	−	1.86603i	0.288675	−	1.07735i	0.946112	−	0.323840i	$-0.104974\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.86603	+	2.23205i	1.46122	+	0.843636i	0.999068	−	0.0431647i	$-0.0137440\pi$
$7$	3.86603	+	2.23205i	1.46122	+	0.843636i	0.462152	+	0.886801i	$0.347077\pi$
$8$		0			0
$9$	−0.633975	−	0.366025i	−0.211325	−	0.122008i
$10$		0			0
$11$	−2.86603	−	0.767949i	−0.864139	−	0.231545i	−0.200587	−	0.979676i	$-0.564285\pi$
$11$	−2.86603	−	0.767949i	−0.864139	−	0.231545i	−0.663552	+	0.748130i	$0.730952\pi$
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.23205	+	0.866025i	−0.783887	+	0.210042i	−0.628498	−	0.777811i	$-0.716330\pi$
$17$	−3.23205	+	0.866025i	−0.783887	+	0.210042i	−0.155390	+	0.987853i	$0.549663\pi$
$18$		0			0
$19$	−1.13397	−	4.23205i	−0.260152	−	0.970899i	−0.965152	−	0.261692i	$-0.915720\pi$
$19$	−1.13397	−	4.23205i	−0.260152	−	0.970899i	0.705000	−	0.709207i	$-0.250947\pi$
$20$		0			0
$21$	6.09808	−	6.09808i	1.33071	−	1.33071i
$22$		0			0
$23$	6.96410	+	1.86603i	1.45212	+	0.389093i	0.896759	−	0.442519i	$-0.145915\pi$
$23$	6.96410	+	1.86603i	1.45212	+	0.389093i	0.555357	+	0.831612i	$0.312582\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.09808	−	3.09808i	0.596225	−	0.596225i
$28$		0			0
$29$	−1.50000	+	0.866025i	−0.278543	+	0.160817i	−0.632764	−	0.774345i	$-0.718080\pi$
$29$	−1.50000	+	0.866025i	−0.278543	+	0.160817i	0.354221	+	0.935162i	$0.384746\pi$
$30$		0			0
$31$	5.19615	+	5.19615i	0.933257	+	0.933257i	0.997908	−	0.0646514i	$-0.0205935\pi$
$31$	5.19615	+	5.19615i	0.933257	+	0.933257i	−0.0646514	+	0.997908i	$0.520594\pi$
$32$		0			0
$33$	−2.86603	+	4.96410i	−0.498911	+	0.864139i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.33013	−	4.23205i	1.20507	−	0.695745i	0.243388	−	0.969929i	$-0.421741\pi$
$37$	7.33013	−	4.23205i	1.20507	−	0.695745i	0.961677	+	0.274184i	$0.0884078\pi$
$38$		0			0
$39$	−4.59808	−	5.23205i	−0.736281	−	0.837799i
$40$		0			0
$41$	0.669873	−	2.50000i	0.104617	−	0.390434i	−0.893685	−	0.448695i	$-0.851889\pi$
$41$	0.669873	−	2.50000i	0.104617	−	0.390434i	0.998301	+	0.0582609i	$0.0185555\pi$
$42$		0			0
$43$	−0.964102	−	3.59808i	−0.147024	−	0.548701i	−0.999657	−	0.0261910i	$-0.991662\pi$
$43$	−0.964102	−	3.59808i	−0.147024	−	0.548701i	0.852633	−	0.522511i	$-0.175004\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.92820i			0.427122i	0.976930	+	0.213561i	$0.0685063\pi$
$47$			2.92820i			0.427122i	−0.976930	+	0.213561i	$0.931494\pi$
$48$		0			0
$49$	6.46410	+	11.1962i	0.923443	+	1.59945i
$50$		0			0
$51$			6.46410i			0.905155i
$52$		0			0
$53$	4.46410	+	4.46410i	0.613192	+	0.613192i	0.943776	−	0.330585i	$-0.107246\pi$
$53$	4.46410	+	4.46410i	0.613192	+	0.613192i	−0.330585	+	0.943776i	$0.607246\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.46410			−1.12110
$58$		0			0
$59$	−6.33013	+	1.69615i	−0.824112	+	0.220820i	−0.646144	−	0.763216i	$-0.723619\pi$
$59$	−6.33013	+	1.69615i	−0.824112	+	0.220820i	−0.177969	+	0.984036i	$0.556953\pi$
$60$		0			0
$61$	4.50000	−	7.79423i	0.576166	−	0.997949i	−0.419748	−	0.907641i	$-0.637882\pi$
$61$	4.50000	−	7.79423i	0.576166	−	0.997949i	0.995914	−	0.0903080i	$-0.0287851\pi$
$62$		0			0
$63$	−1.63397	−	2.83013i	−0.205861	−	0.356562i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.86603	−	13.6244i	−0.960988	−	1.66448i	−0.720028	−	0.693945i	$-0.755871\pi$
$67$	−7.86603	−	13.6244i	−0.960988	−	1.66448i	−0.240960	−	0.970535i	$-0.577462\pi$
$68$		0			0
$69$	6.96410	−	12.0622i	0.838379	−	1.45212i
$70$		0			0
$71$	−4.59808	+	1.23205i	−0.545691	+	0.146218i	−0.521125	−	0.853481i	$-0.674487\pi$
$71$	−4.59808	+	1.23205i	−0.545691	+	0.146218i	−0.0245667	+	0.999698i	$0.507821\pi$
$72$		0			0
$73$	1.07180			0.125444			0.0627222	−	0.998031i	$-0.480022\pi$
$73$	1.07180			0.125444			0.0627222	+	0.998031i	$0.480022\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.36603	−	9.36603i	−1.06736	−	1.06736i
$78$		0			0
$79$		−	7.46410i		−	0.839777i	−0.907576	−	0.419889i	$-0.862069\pi$
$79$		−	7.46410i		−	0.839777i	0.907576	−	0.419889i	$-0.137931\pi$
$80$		0			0
$81$	−5.33013	−	9.23205i	−0.592236	−	1.02578i
$82$		0			0
$83$			10.9282i			1.19953i	0.800178	+	0.599763i	$0.204739\pi$
$83$			10.9282i			1.19953i	−0.800178	+	0.599763i	$0.795261\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.866025	+	3.23205i	0.0928477	+	0.346512i
$88$		0			0
$89$	0.794229	−	2.96410i	0.0841881	−	0.314194i	−0.910971	−	0.412470i	$-0.864666\pi$
$89$	0.794229	−	2.96410i	0.0841881	−	0.314194i	0.995159	+	0.0982760i	$0.0313328\pi$
$90$		0			0
$91$	14.4282	−	7.13397i	1.51249	−	0.747844i
$92$		0			0
$93$	12.2942	−	7.09808i	1.27485	−	0.736036i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.13397	+	3.69615i	−0.216672	+	0.375287i	−0.953789	−	0.300478i	$-0.902854\pi$
$97$	−2.13397	+	3.69615i	−0.216672	+	0.375287i	0.737116	+	0.675766i	$0.236187\pi$
$98$		0			0
$99$	1.53590	+	1.53590i	0.154364	+	0.154364i
$100$		0			0
$101$	5.89230	−	3.40192i	0.586306	−	0.338504i	−0.177329	−	0.984152i	$-0.556746\pi$
$101$	5.89230	−	3.40192i	0.586306	−	0.338504i	0.763636	+	0.645647i	$0.223412\pi$
$102$		0			0
$103$	0.803848	−	0.803848i	0.0792055	−	0.0792055i	−0.666394	−	0.745600i	$-0.732163\pi$
$103$	0.803848	−	0.803848i	0.0792055	−	0.0792055i	0.745600	+	0.666394i	$0.232163\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.96410	−	1.33013i	−0.479898	−	0.128588i	0.0107572	−	0.999942i	$-0.496576\pi$
$107$	−4.96410	−	1.33013i	−0.479898	−	0.128588i	−0.490655	+	0.871354i	$0.663242\pi$
$108$		0			0
$109$	−10.4641	+	10.4641i	−1.00228	+	1.00228i	−0.00228176	+	0.999997i	$0.500726\pi$
$109$	−10.4641	+	10.4641i	−1.00228	+	1.00228i	−0.999997	+	0.00228176i	$0.999274\pi$
$110$		0			0
$111$	−4.23205	−	15.7942i	−0.401688	−	1.49912i
$112$		0			0
$113$	−4.23205	+	1.13397i	−0.398118	+	0.106675i	−0.452322	−	0.891854i	$-0.649404\pi$
$113$	−4.23205	+	1.13397i	−0.398118	+	0.106675i	0.0542046	+	0.998530i	$0.482738\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.36603	+	1.16987i	−0.218739	+	0.108155i
$118$		0			0
$119$	−14.4282	−	3.86603i	−1.32263	−	0.354398i
$120$		0			0
$121$	−1.90192	−	1.09808i	−0.172902	−	0.0998251i
$122$		0			0
$123$	−4.33013	−	2.50000i	−0.390434	−	0.225417i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−0.892305	+	3.33013i	−0.0791793	+	0.295501i	−0.994148	−	0.108023i	$-0.965548\pi$
$127$	−0.892305	+	3.33013i	−0.0791793	+	0.295501i	0.914969	+	0.403524i	$0.132215\pi$
$128$		0			0
$129$	−7.19615			−0.633586
$130$		0			0
$131$	−21.8564			−1.90960			−0.954802	−	0.297244i	$-0.903933\pi$
$131$	−21.8564			−1.90960			−0.954802	+	0.297244i	$0.903933\pi$
$132$		0			0
$133$	5.06218	−	18.8923i	0.438946	−	1.63817i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.5981	+	10.1603i	1.50351	+	0.868049i	0.999992	+	0.00406165i	$0.00129287\pi$
$137$	17.5981	+	10.1603i	1.50351	+	0.868049i	0.503513	+	0.863987i	$0.332040\pi$
$138$		0			0
$139$	−4.50000	−	2.59808i	−0.381685	−	0.220366i	0.296866	−	0.954919i	$-0.404058\pi$
$139$	−4.50000	−	2.59808i	−0.381685	−	0.220366i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$	5.46410	+	1.46410i	0.460160	+	0.123300i
$142$		0			0
$143$	−8.03590	+	7.06218i	−0.671996	+	0.590569i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	24.1244	−	6.46410i	1.98974	−	0.533150i
$148$		0			0
$149$	5.33013	+	19.8923i	0.436661	+	1.62964i	0.737060	+	0.675827i	$0.236214\pi$
$149$	5.33013	+	19.8923i	0.436661	+	1.62964i	−0.300399	+	0.953814i	$0.597120\pi$
$150$		0			0
$151$	−3.73205	+	3.73205i	−0.303710	+	0.303710i	−0.842463	−	0.538753i	$-0.818895\pi$
$151$	−3.73205	+	3.73205i	−0.303710	+	0.303710i	0.538753	+	0.842463i	$0.318895\pi$
$152$		0			0
$153$	2.36603	+	0.633975i	0.191282	+	0.0512538i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.464102	−	0.464102i	0.0370393	−	0.0370393i	−0.688345	−	0.725384i	$-0.741662\pi$
$157$	0.464102	−	0.464102i	0.0370393	−	0.0370393i	0.725384	+	0.688345i	$0.241662\pi$
$158$		0			0
$159$	10.5622	−	6.09808i	0.837635	−	0.483609i
$160$		0			0
$161$	22.7583	+	22.7583i	1.79361	+	1.79361i
$162$		0			0
$163$	−3.59808	+	6.23205i	−0.281823	+	0.488132i	−0.971834	−	0.235667i	$-0.924272\pi$
$163$	−3.59808	+	6.23205i	−0.281823	+	0.488132i	0.690011	+	0.723799i	$0.257606\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.99038	+	5.76795i	−0.773079	+	0.446337i	−0.833972	−	0.551807i	$-0.813939\pi$
$167$	−9.99038	+	5.76795i	−0.773079	+	0.446337i	0.0608930	+	0.998144i	$0.480605\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$	−0.830127	+	3.09808i	−0.0634814	+	0.236916i
$172$		0			0
$173$	6.23205	+	23.2583i	0.473814	+	1.76830i	0.625871	+	0.779926i	$0.284744\pi$
$173$	6.23205	+	23.2583i	0.473814	+	1.76830i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			12.6603i			0.951603i
$178$		0			0
$179$	0.0358984	+	0.0621778i	0.00268317	+	0.00464739i	0.867364	−	0.497675i	$-0.165813\pi$
$179$	0.0358984	+	0.0621778i	0.00268317	+	0.00464739i	−0.864681	+	0.502322i	$0.832479\pi$
$180$		0			0
$181$			9.07180i			0.674301i	0.941451	+	0.337151i	$0.109463\pi$
$181$			9.07180i			0.674301i	−0.941451	+	0.337151i	$0.890537\pi$
$182$		0			0
$183$	−12.2942	−	12.2942i	−0.908816	−	0.908816i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	9.92820			0.726022
$188$		0			0
$189$	18.8923	−	5.06218i	1.37421	−	0.368219i
$190$		0			0
$191$	7.50000	−	12.9904i	0.542681	−	0.939951i	−0.456068	−	0.889945i	$-0.650743\pi$
$191$	7.50000	−	12.9904i	0.542681	−	0.939951i	0.998749	−	0.0500060i	$-0.0159241\pi$
$192$		0			0
$193$	2.79423	+	4.83975i	0.201133	+	0.348373i	0.948894	−	0.315596i	$-0.102204\pi$
$193$	2.79423	+	4.83975i	0.201133	+	0.348373i	−0.747761	+	0.663968i	$0.768871\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.86603	−	6.69615i	−0.275443	−	0.477081i	0.694804	−	0.719199i	$-0.255491\pi$
$197$	−3.86603	−	6.69615i	−0.275443	−	0.477081i	−0.970247	+	0.242118i	$0.922158\pi$
$198$		0			0
$199$	−4.42820	+	7.66987i	−0.313907	+	0.543703i	−0.979205	−	0.202875i	$-0.934971\pi$
$199$	−4.42820	+	7.66987i	−0.313907	+	0.543703i	0.665298	+	0.746578i	$0.268305\pi$
$200$		0			0
$201$	−29.3564	+	7.86603i	−2.07064	+	0.554827i
$202$		0			0
$203$	−7.73205			−0.542684
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.73205	−	3.73205i	−0.259395	−	0.259395i
$208$		0			0
$209$			13.0000i			0.899229i
$210$		0			0
$211$	3.96410	+	6.86603i	0.272900	+	0.472677i	0.969603	−	0.244683i	$-0.0786838\pi$
$211$	3.96410	+	6.86603i	0.272900	+	0.472677i	−0.696703	+	0.717360i	$0.745351\pi$
$212$		0			0
$213$			9.19615i			0.630110i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	8.49038	+	31.6865i	0.576365	+	2.15102i
$218$		0			0
$219$	0.535898	−	2.00000i	0.0362127	−	0.135147i
$220$		0			0
$221$	−3.86603	+	11.4282i	−0.260057	+	0.768744i
$222$		0			0
$223$	−7.33013	+	4.23205i	−0.490862	+	0.283399i	−0.724932	−	0.688821i	$-0.758129\pi$
$223$	−7.33013	+	4.23205i	−0.490862	+	0.283399i	0.234070	+	0.972220i	$0.424795\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.59808	−	6.23205i	0.238813	−	0.413636i	−0.721561	−	0.692351i	$-0.756575\pi$
$227$	3.59808	−	6.23205i	0.238813	−	0.413636i	0.960374	+	0.278715i	$0.0899085\pi$
$228$		0			0
$229$	13.9282	+	13.9282i	0.920402	+	0.920402i	0.997058	−	0.0766560i	$-0.0244243\pi$
$229$	13.9282	+	13.9282i	0.920402	+	0.920402i	−0.0766560	+	0.997058i	$0.524424\pi$
$230$		0			0
$231$	−22.1603	+	12.7942i	−1.45804	+	0.841798i
$232$		0			0
$233$	14.8564	−	14.8564i	0.973276	−	0.973276i	−0.0263765	−	0.999652i	$-0.508397\pi$
$233$	14.8564	−	14.8564i	0.973276	−	0.973276i	0.999652	+	0.0263765i	$0.00839688\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−13.9282	−	3.73205i	−0.904734	−	0.242423i
$238$		0			0
$239$	−0.660254	+	0.660254i	−0.0427083	+	0.0427083i	−0.728138	−	0.685430i	$-0.759614\pi$
$239$	−0.660254	+	0.660254i	−0.0427083	+	0.0427083i	0.685430	+	0.728138i	$0.259614\pi$
$240$		0			0
$241$	2.66987	+	9.96410i	0.171982	+	0.641844i	0.997046	+	0.0768056i	$0.0244721\pi$
$241$	2.66987	+	9.96410i	0.171982	+	0.641844i	−0.825064	+	0.565039i	$0.808861\pi$
$242$		0			0
$243$	−7.19615	+	1.92820i	−0.461633	+	0.123694i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−14.9641	−	5.06218i	−0.952143	−	0.322099i
$248$		0			0
$249$	20.3923	+	5.46410i	1.29231	+	0.346273i
$250$		0			0
$251$	12.3564	+	7.13397i	0.779929	+	0.450292i	0.836405	−	0.548111i	$-0.184653\pi$
$251$	12.3564	+	7.13397i	0.779929	+	0.450292i	−0.0564758	+	0.998404i	$0.517986\pi$
$252$		0			0
$253$	−18.5263	−	10.6962i	−1.16474	−	0.672461i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.62436	−	9.79423i	0.163703	−	0.610947i	−0.834499	−	0.551009i	$-0.814243\pi$
$257$	2.62436	−	9.79423i	0.163703	−	0.610947i	0.998202	−	0.0599382i	$-0.0190904\pi$
$258$		0			0
$259$	37.7846			2.34782
$260$		0			0
$261$	1.26795			0.0784841
$262$		0			0
$263$	6.89230	−	25.7224i	0.424998	−	1.58611i	−0.338930	−	0.940812i	$-0.610065\pi$
$263$	6.89230	−	25.7224i	0.424998	−	1.58611i	0.763928	−	0.645302i	$-0.223268\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.13397	−	2.96410i	−0.314194	−	0.181400i
$268$		0			0
$269$	−18.8205	−	10.8660i	−1.14751	−	0.662513i	−0.199228	−	0.979953i	$-0.563844\pi$
$269$	−18.8205	−	10.8660i	−1.14751	−	0.662513i	−0.948278	+	0.317440i	$0.897177\pi$
$270$		0			0
$271$	−4.33013	−	1.16025i	−0.263036	−	0.0704804i	0.124890	−	0.992171i	$-0.460142\pi$
$271$	−4.33013	−	1.16025i	−0.263036	−	0.0704804i	−0.387927	+	0.921690i	$0.626809\pi$
$272$		0			0
$273$	−6.09808	−	30.4904i	−0.369072	−	1.84536i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	6.23205	−	1.66987i	0.374448	−	0.100333i	−0.0666868	−	0.997774i	$-0.521243\pi$
$277$	6.23205	−	1.66987i	0.374448	−	0.100333i	0.441134	+	0.897441i	$0.354576\pi$
$278$		0			0
$279$	−1.39230	−	5.19615i	−0.0833551	−	0.311086i
$280$		0			0
$281$	−9.53590	+	9.53590i	−0.568864	+	0.568864i	−0.931810	−	0.362946i	$-0.881771\pi$
$281$	−9.53590	+	9.53590i	−0.568864	+	0.568864i	0.362946	+	0.931810i	$0.381771\pi$
$282$		0			0
$283$	−1.03590	−	0.277568i	−0.0615778	−	0.0164997i	0.227899	−	0.973685i	$-0.426815\pi$
$283$	−1.03590	−	0.277568i	−0.0615778	−	0.0164997i	−0.289476	+	0.957185i	$0.593481\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.16987	−	8.16987i	0.482252	−	0.482252i
$288$		0			0
$289$	−5.02628	+	2.90192i	−0.295663	+	0.170701i
$290$		0			0
$291$	5.83013	+	5.83013i	0.341768	+	0.341768i
$292$		0			0
$293$	−12.2583	+	21.2321i	−0.716139	+	1.24039i	0.246379	+	0.969174i	$0.420759\pi$
$293$	−12.2583	+	21.2321i	−0.716139	+	1.24039i	−0.962518	+	0.271216i	$0.912574\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−11.2583	+	6.50000i	−0.653275	+	0.377168i
$298$		0			0
$299$	19.5263	−	17.1603i	1.12923	−	0.992403i
$300$		0			0
$301$	4.30385	−	16.0622i	0.248070	−	0.925809i
$302$		0			0
$303$	−3.40192	−	12.6962i	−0.195435	−	0.729375i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		0			0
$309$	−1.09808	−	1.90192i	−0.0624674	−	0.108197i
$310$		0			0
$311$			3.46410i			0.196431i	0.995165	+	0.0982156i	$0.0313135\pi$
$311$			3.46410i			0.196431i	−0.995165	+	0.0982156i	$0.968687\pi$
$312$		0			0
$313$	12.4641	+	12.4641i	0.704513	+	0.704513i	0.965376	−	0.260863i	$-0.0840071\pi$
$313$	12.4641	+	12.4641i	0.704513	+	0.704513i	−0.260863	+	0.965376i	$0.584007\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.07180			0.284860			0.142430	−	0.989805i	$-0.454508\pi$
$317$	5.07180			0.284860			0.142430	+	0.989805i	$0.454508\pi$
$318$		0			0
$319$	4.96410	−	1.33013i	0.277936	−	0.0744728i
$320$		0			0
$321$	−4.96410	+	8.59808i	−0.277069	+	0.479898i
$322$		0			0
$323$	7.33013	+	12.6962i	0.407859	+	0.706433i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	14.2942	+	24.7583i	0.790473	+	1.36914i
$328$		0			0
$329$	−6.53590	+	11.3205i	−0.360336	+	0.624120i
$330$		0			0
$331$	−28.4545	+	7.62436i	−1.56400	+	0.419072i	−0.933927	−	0.357464i	$-0.883642\pi$
$331$	−28.4545	+	7.62436i	−1.56400	+	0.419072i	−0.630073	+	0.776536i	$0.716975\pi$
$332$		0			0
$333$	−6.19615			−0.339547
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.07180	+	2.07180i	0.112858	+	0.112858i	0.761281	−	0.648423i	$-0.224571\pi$
$337$	2.07180	+	2.07180i	0.112858	+	0.112858i	−0.648423	+	0.761281i	$0.724571\pi$
$338$		0			0
$339$			8.46410i			0.459707i
$340$		0			0
$341$	−10.9019	−	18.8827i	−0.590372	−	1.02255i
$342$		0			0
$343$			26.4641i			1.42893i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.42820	−	12.7942i	−0.184036	−	0.686830i	−0.994835	−	0.101506i	$-0.967634\pi$
$347$	−3.42820	−	12.7942i	−0.184036	−	0.686830i	0.810799	−	0.585324i	$-0.199033\pi$
$348$		0			0
$349$	−6.13397	+	22.8923i	−0.328344	+	1.22540i	0.582563	+	0.812786i	$0.302050\pi$
$349$	−6.13397	+	22.8923i	−0.328344	+	1.22540i	−0.910907	+	0.412611i	$0.864617\pi$
$350$		0			0
$351$	−3.09808	−	15.4904i	−0.165363	−	0.826815i
$352$		0			0
$353$	3.06218	−	1.76795i	0.162983	−	0.0940984i	−0.416290	−	0.909232i	$-0.636670\pi$
$353$	3.06218	−	1.76795i	0.162983	−	0.0940984i	0.579273	+	0.815134i	$0.303336\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−14.4282	+	24.9904i	−0.763621	+	1.32263i
$358$		0			0
$359$	2.80385	+	2.80385i	0.147981	+	0.147981i	0.777216	−	0.629234i	$-0.216631\pi$
$359$	2.80385	+	2.80385i	0.147981	+	0.147981i	−0.629234	+	0.777216i	$0.716631\pi$
$360$		0			0
$361$	−0.169873	+	0.0980762i	−0.00894068	+	0.00516191i
$362$		0			0
$363$	−3.00000	+	3.00000i	−0.157459	+	0.157459i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.96410	−	0.794229i	−0.154725	−	0.0414584i	0.180625	−	0.983552i	$-0.442188\pi$
$367$	−2.96410	−	0.794229i	−0.154725	−	0.0414584i	−0.335350	+	0.942094i	$0.608855\pi$
$368$		0			0
$369$	−1.33975	+	1.33975i	−0.0697444	+	0.0697444i
$370$		0			0
$371$	7.29423	+	27.2224i	0.378697	+	1.41332i
$372$		0			0
$373$	−15.6962	+	4.20577i	−0.812716	+	0.217767i	−0.641159	−	0.767408i	$-0.721546\pi$
$373$	−15.6962	+	4.20577i	−0.812716	+	0.217767i	−0.171557	+	0.985174i	$0.554880\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.401924	+	6.23205i	−0.0207001	+	0.320967i
$378$		0			0
$379$	−2.59808	−	0.696152i	−0.133454	−	0.0357589i	0.191474	−	0.981498i	$-0.438673\pi$
$379$	−2.59808	−	0.696152i	−0.133454	−	0.0357589i	−0.324928	+	0.945739i	$0.605340\pi$
$380$		0			0
$381$	5.76795	+	3.33013i	0.295501	+	0.170608i
$382$		0			0
$383$	−28.1147	−	16.2321i	−1.43660	−	0.829419i	−0.438985	−	0.898495i	$-0.644662\pi$
$383$	−28.1147	−	16.2321i	−1.43660	−	0.829419i	−0.997611	+	0.0690756i	$0.977995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.705771	+	2.63397i	−0.0358764	+	0.133892i
$388$		0			0
$389$	−28.9282			−1.46672			−0.733359	−	0.679842i	$-0.762049\pi$
$389$	−28.9282			−1.46672			−0.733359	+	0.679842i	$0.762049\pi$
$390$		0			0
$391$	−24.1244			−1.22002
$392$		0			0
$393$	−10.9282	+	40.7846i	−0.551255	+	2.05731i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.13397	+	4.69615i	0.408232	+	0.235693i	0.690030	−	0.723781i	$-0.257597\pi$
$397$	8.13397	+	4.69615i	0.408232	+	0.235693i	−0.281798	+	0.959474i	$0.590931\pi$
$398$		0			0
$399$	−32.7224	−	18.8923i	−1.63817	−	0.945798i
$400$		0			0
$401$	11.3301	+	3.03590i	0.565800	+	0.151606i	0.530369	−	0.847767i	$-0.322053\pi$
$401$	11.3301	+	3.03590i	0.565800	+	0.151606i	0.0354301	+	0.999372i	$0.488720\pi$
$402$		0			0
$403$	25.9808	−	5.19615i	1.29419	−	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.2583	+	6.50000i	−1.20244	+	0.322193i
$408$		0			0
$409$	−0.526279	−	1.96410i	−0.0260228	−	0.0971186i	0.951693	−	0.307051i	$-0.0993422\pi$
$409$	−0.526279	−	1.96410i	−0.0260228	−	0.0971186i	−0.977716	+	0.209932i	$0.932676\pi$
$410$		0			0
$411$	27.7583	−	27.7583i	1.36922	−	1.36922i
$412$		0			0
$413$	−28.2583	−	7.57180i	−1.39050	−	0.372584i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−7.09808	+	7.09808i	−0.347594	+	0.347594i
$418$		0			0
$419$	−31.5000	+	18.1865i	−1.53888	+	0.888470i	−0.539971	+	0.841684i	$0.681565\pi$
$419$	−31.5000	+	18.1865i	−1.53888	+	0.888470i	−0.998905	+	0.0467865i	$0.985102\pi$
$420$		0			0
$421$	−20.8564	−	20.8564i	−1.01648	−	1.01648i	−0.999862	−	0.0166171i	$-0.994710\pi$
$421$	−20.8564	−	20.8564i	−1.01648	−	1.01648i	−0.0166171	−	0.999862i	$-0.505290\pi$
$422$		0			0
$423$	1.07180	−	1.85641i	0.0521125	−	0.0902616i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	34.7942	−	20.0885i	1.68381	−	0.972149i
$428$		0			0
$429$	9.16025	+	18.5263i	0.442261	+	0.894457i
$430$		0			0
$431$	−0.741670	+	2.76795i	−0.0357250	+	0.133327i	−0.981485	−	0.191538i	$-0.938652\pi$
$431$	−0.741670	+	2.76795i	−0.0357250	+	0.133327i	0.945760	+	0.324866i	$0.105319\pi$
$432$		0			0
$433$	5.55256	+	20.7224i	0.266839	+	0.995857i	0.961115	+	0.276148i	$0.0890580\pi$
$433$	5.55256	+	20.7224i	0.266839	+	0.995857i	−0.694276	+	0.719709i	$0.744275\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	31.5885i		−	1.51108i
$438$		0			0
$439$	6.96410	+	12.0622i	0.332378	+	0.575696i	0.982978	−	0.183725i	$-0.0588156\pi$
$439$	6.96410	+	12.0622i	0.332378	+	0.575696i	−0.650599	+	0.759421i	$0.725482\pi$
$440$		0			0
$441$		−	9.46410i		−	0.450672i
$442$		0			0
$443$	−15.5885	−	15.5885i	−0.740630	−	0.740630i	0.232069	−	0.972699i	$-0.425450\pi$
$443$	−15.5885	−	15.5885i	−0.740630	−	0.740630i	−0.972699	+	0.232069i	$0.925450\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	39.7846			1.88175
$448$		0			0
$449$	−20.7942	+	5.57180i	−0.981340	+	0.262949i	−0.713609	−	0.700544i	$-0.752941\pi$
$449$	−20.7942	+	5.57180i	−0.981340	+	0.262949i	−0.267731	+	0.963494i	$0.586274\pi$
$450$		0			0
$451$	−3.83975	+	6.65064i	−0.180807	+	0.313166i
$452$		0			0
$453$	5.09808	+	8.83013i	0.239529	+	0.414876i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.79423	−	15.2321i	−0.411377	−	0.712525i	0.583664	−	0.811995i	$-0.301619\pi$
$457$	−8.79423	−	15.2321i	−0.411377	−	0.712525i	−0.995041	+	0.0994701i	$0.968285\pi$
$458$		0			0
$459$	−7.33013	+	12.6962i	−0.342141	+	0.592606i
$460$		0			0
$461$	37.1865	−	9.96410i	1.73195	−	0.464074i	0.751319	−	0.659940i	$-0.229418\pi$
$461$	37.1865	−	9.96410i	1.73195	−	0.464074i	0.980631	+	0.195865i	$0.0627515\pi$
$462$		0			0
$463$	42.3923			1.97014			0.985069	−	0.172161i	$-0.0550751\pi$
$463$	42.3923			1.97014			0.985069	+	0.172161i	$0.0550751\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.660254	+	0.660254i	0.0305529	+	0.0305529i	0.722218	−	0.691665i	$-0.243123\pi$
$467$	0.660254	+	0.660254i	0.0305529	+	0.0305529i	−0.691665	+	0.722218i	$0.743123\pi$
$468$		0			0
$469$		−	70.2295i		−	3.24290i
$470$		0			0
$471$	−0.633975	−	1.09808i	−0.0292120	−	0.0505967i
$472$		0			0
$473$			11.0526i			0.508197i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.19615	−	4.46410i	−0.0547681	−	0.204397i
$478$		0			0
$479$	−4.86603	+	18.1603i	−0.222334	+	0.829763i	0.761121	+	0.648610i	$0.224650\pi$
$479$	−4.86603	+	18.1603i	−0.222334	+	0.829763i	−0.983455	+	0.181153i	$0.942017\pi$
$480$		0			0
$481$	1.96410	−	30.4545i	0.0895553	−	1.38860i
$482$		0			0
$483$	53.8468	−	31.0885i	2.45011	−	1.41457i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−14.7942	+	25.6244i	−0.670390	+	1.16115i	0.307403	+	0.951579i	$0.400540\pi$
$487$	−14.7942	+	25.6244i	−0.670390	+	1.16115i	−0.977793	+	0.209571i	$0.932793\pi$
$488$		0			0
$489$	9.83013	+	9.83013i	0.444534	+	0.444534i
$490$		0			0
$491$	−14.8923	+	8.59808i	−0.672080	+	0.388026i	−0.796864	−	0.604158i	$-0.793510\pi$
$491$	−14.8923	+	8.59808i	−0.672080	+	0.388026i	0.124784	+	0.992184i	$0.460176\pi$
$492$		0			0
$493$	4.09808	−	4.09808i	0.184568	−	0.184568i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−20.5263	−	5.50000i	−0.920729	−	0.246709i
$498$		0			0
$499$	23.7321	−	23.7321i	1.06239	−	1.06239i	0.0644731	−	0.997919i	$-0.479463\pi$
$499$	23.7321	−	23.7321i	1.06239	−	1.06239i	0.997919	−	0.0644731i	$-0.0205367\pi$
$500$		0			0
$501$	5.76795	+	21.5263i	0.257693	+	0.961723i
$502$		0			0
$503$	−24.3564	+	6.52628i	−1.08600	+	0.290992i	−0.757051	−	0.653356i	$-0.773361\pi$
$503$	−24.3564	+	6.52628i	−1.08600	+	0.290992i	−0.328947	+	0.944348i	$0.606694\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−24.8923	+	3.33013i	−1.10551	+	0.147896i
$508$		0			0
$509$	25.4545	+	6.82051i	1.12825	+	0.302314i	0.774218	−	0.632919i	$-0.218144\pi$
$509$	25.4545	+	6.82051i	1.12825	+	0.302314i	0.354032	+	0.935233i	$0.384810\pi$
$510$		0			0
$511$	4.14359	+	2.39230i	0.183302	+	0.105829i
$512$		0			0
$513$	−16.6244	−	9.59808i	−0.733983	−	0.423765i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.24871	−	8.39230i	0.0988982	−	0.369093i
$518$		0			0
$519$	46.5167			2.04185
$520$		0			0
$521$	19.8564			0.869925			0.434962	−	0.900449i	$-0.356762\pi$
$521$	19.8564			0.869925			0.434962	+	0.900449i	$0.356762\pi$
$522$		0			0
$523$	−4.03590	+	15.0622i	−0.176478	+	0.658623i	0.819818	+	0.572625i	$0.194075\pi$
$523$	−4.03590	+	15.0622i	−0.176478	+	0.658623i	−0.996295	+	0.0859985i	$0.972592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.2942	−	12.2942i	−0.927591	−	0.535545i
$528$		0			0
$529$	25.0981	+	14.4904i	1.09122	+	0.630017i
$530$		0			0
$531$	4.63397	+	1.24167i	0.201097	+	0.0538839i
$532$		0			0
$533$	−6.16025	−	7.00962i	−0.266830	−	0.303620i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.133975	−	0.0358984i	0.00578143	−	0.00154913i
$538$		0			0
$539$	−9.92820	−	37.0526i	−0.427638	−	1.59597i
$540$		0			0
$541$	−19.7846	+	19.7846i	−0.850607	+	0.850607i	−0.990208	−	0.139601i	$-0.955418\pi$
$541$	−19.7846	+	19.7846i	−0.850607	+	0.850607i	0.139601	+	0.990208i	$0.455418\pi$
$542$		0			0
$543$	16.9282	+	4.53590i	0.726459	+	0.194654i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−24.1244	+	24.1244i	−1.03148	+	1.03148i	−0.0319949	+	0.999488i	$0.510186\pi$
$547$	−24.1244	+	24.1244i	−1.03148	+	1.03148i	−0.999488	+	0.0319949i	$0.989814\pi$
$548$		0			0
$549$	−5.70577	+	3.29423i	−0.243516	+	0.140594i
$550$		0			0
$551$	5.36603	+	5.36603i	0.228600	+	0.228600i
$552$		0			0
$553$	16.6603	−	28.8564i	0.708466	−	1.22710i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.66987	+	2.69615i	−0.197869	+	0.114240i	−0.595661	−	0.803236i	$-0.703110\pi$
$557$	−4.66987	+	2.69615i	−0.197869	+	0.114240i	0.397792	+	0.917476i	$0.369777\pi$
$558$		0			0
$559$	−12.7224	−	4.30385i	−0.538102	−	0.182033i
$560$		0			0
$561$	4.96410	−	18.5263i	0.209585	−	0.782180i
$562$		0			0
$563$	4.10770	+	15.3301i	0.173119	+	0.646088i	0.996864	+	0.0791284i	$0.0252137\pi$
$563$	4.10770	+	15.3301i	0.173119	+	0.646088i	−0.823746	+	0.566959i	$0.808120\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	47.5885i		−	1.99853i
$568$		0			0
$569$	−15.8205	−	27.4019i	−0.663230	−	1.14875i	−0.979762	−	0.200166i	$-0.935852\pi$
$569$	−15.8205	−	27.4019i	−0.663230	−	1.14875i	0.316532	−	0.948582i	$-0.397482\pi$
$570$		0			0
$571$		−	42.3923i		−	1.77406i	−0.461709	−	0.887031i	$-0.652764\pi$
$571$		−	42.3923i		−	1.77406i	0.461709	−	0.887031i	$-0.347236\pi$
$572$		0			0
$573$	−20.4904	−	20.4904i	−0.855998	−	0.855998i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	20.7846			0.865275			0.432637	−	0.901568i	$-0.357583\pi$
$577$	20.7846			0.865275			0.432637	+	0.901568i	$0.357583\pi$
$578$		0			0
$579$	10.4282	−	2.79423i	0.433381	−	0.116124i
$580$		0			0
$581$	−24.3923	+	42.2487i	−1.01196	+	1.75277i
$582$		0			0
$583$	−9.36603	−	16.2224i	−0.387901	−	0.671864i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.72243	−	9.91154i	−0.236190	−	0.409093i	0.723428	−	0.690400i	$-0.242565\pi$
$587$	−5.72243	−	9.91154i	−0.236190	−	0.409093i	−0.959618	+	0.281307i	$0.909232\pi$
$588$		0			0
$589$	16.0981	−	27.8827i	0.663310	−	1.14889i
$590$		0			0
$591$	−14.4282	+	3.86603i	−0.593497	+	0.159027i
$592$		0			0
$593$	17.0718			0.701055			0.350527	−	0.936553i	$-0.386002\pi$
$593$	17.0718			0.701055			0.350527	+	0.936553i	$0.386002\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12.0981	+	12.0981i	0.495141	+	0.495141i
$598$		0			0
$599$			12.2487i			0.500469i	0.968185	+	0.250234i	$0.0805077\pi$
$599$			12.2487i			0.500469i	−0.968185	+	0.250234i	$0.919492\pi$
$600$		0			0
$601$	−6.42820	−	11.1340i	−0.262212	−	0.454164i	0.704618	−	0.709587i	$-0.251119\pi$
$601$	−6.42820	−	11.1340i	−0.262212	−	0.454164i	−0.966829	+	0.255423i	$0.917785\pi$
$602$		0			0
$603$			11.5167i			0.468995i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−9.96410	−	37.1865i	−0.404430	−	1.50935i	−0.805104	−	0.593134i	$-0.797890\pi$
$607$	−9.96410	−	37.1865i	−0.404430	−	1.50935i	0.400673	−	0.916221i	$-0.368776\pi$
$608$		0			0
$609$	−3.86603	+	14.4282i	−0.156659	+	0.584660i
$610$		0			0
$611$	8.78461	+	5.85641i	0.355387	+	0.236925i
$612$		0			0
$613$	−15.1865	+	8.76795i	−0.613378	+	0.354134i	−0.774286	−	0.632835i	$-0.781891\pi$
$613$	−15.1865	+	8.76795i	−0.613378	+	0.354134i	0.160908	+	0.986969i	$0.448558\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.79423	−	11.7679i	0.273525	−	0.473760i	−0.696237	−	0.717812i	$-0.745144\pi$
$617$	6.79423	−	11.7679i	0.273525	−	0.473760i	0.969762	+	0.244053i	$0.0784769\pi$
$618$		0			0
$619$	−2.66025	−	2.66025i	−0.106925	−	0.106925i	0.651620	−	0.758545i	$-0.274089\pi$
$619$	−2.66025	−	2.66025i	−0.106925	−	0.106925i	−0.758545	+	0.651620i	$0.774089\pi$
$620$		0			0
$621$	27.3564	−	15.7942i	1.09777	−	0.633801i
$622$		0			0
$623$	9.68653	−	9.68653i	0.388083	−	0.388083i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	24.2583	+	6.50000i	0.968784	+	0.259585i
$628$		0			0
$629$	−20.0263	+	20.0263i	−0.798500	+	0.798500i
$630$		0			0
$631$	−1.65064	−	6.16025i	−0.0657107	−	0.245236i	0.925256	−	0.379343i	$-0.123850\pi$
$631$	−1.65064	−	6.16025i	−0.0657107	−	0.245236i	−0.990967	+	0.134107i	$0.957183\pi$
$632$		0			0
$633$	14.7942	−	3.96410i	0.588018	−	0.157559i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	46.5167	+	3.00000i	1.84306	+	0.118864i
$638$		0			0
$639$	3.36603	+	0.901924i	0.133158	+	0.0356796i
$640$		0			0
$641$	33.3564	+	19.2583i	1.31750	+	0.760658i	0.983326	−	0.181853i	$-0.0582096\pi$
$641$	33.3564	+	19.2583i	1.31750	+	0.760658i	0.334173	+	0.942512i	$0.391543\pi$
$642$		0			0
$643$	22.6699	+	13.0885i	0.894013	+	0.516158i	0.875253	−	0.483666i	$-0.160695\pi$
$643$	22.6699	+	13.0885i	0.894013	+	0.516158i	0.0187597	+	0.999824i	$0.494028\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.50000	+	9.33013i	−0.0982851	+	0.366805i	−0.997497	−	0.0707082i	$-0.977474\pi$
$647$	−2.50000	+	9.33013i	−0.0982851	+	0.366805i	0.899212	+	0.437513i	$0.144141\pi$
$648$		0			0
$649$	19.4449			0.763278
$650$		0			0
$651$	63.3731			2.48379
$652$		0			0
$653$	10.9449	−	40.8468i	0.428306	−	1.59846i	−0.328291	−	0.944577i	$-0.606473\pi$
$653$	10.9449	−	40.8468i	0.428306	−	1.59846i	0.756597	−	0.653882i	$-0.226861\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.679492	−	0.392305i	−0.0265095	−	0.0153053i
$658$		0			0
$659$	−21.5718	−	12.4545i	−0.840318	−	0.485158i	0.0170544	−	0.999855i	$-0.494571\pi$
$659$	−21.5718	−	12.4545i	−0.840318	−	0.485158i	−0.857372	+	0.514697i	$0.827904\pi$
$660$		0			0
$661$	14.7942	+	3.96410i	0.575429	+	0.154186i	0.534785	−	0.844988i	$-0.320393\pi$
$661$	14.7942	+	3.96410i	0.575429	+	0.154186i	0.0406436	+	0.999174i	$0.487059\pi$
$662$		0			0
$663$	19.3923	+	12.9282i	0.753135	+	0.502090i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−12.0622	+	3.23205i	−0.467049	+	0.125146i
$668$		0			0
$669$	4.23205	+	15.7942i	0.163621	+	0.610640i
$670$		0			0
$671$	−18.8827	+	18.8827i	−0.728958	+	0.728958i
$672$		0			0
$673$	14.6244	+	3.91858i	0.563727	+	0.151050i	0.529418	−	0.848361i	$-0.322410\pi$
$673$	14.6244	+	3.91858i	0.563727	+	0.151050i	0.0343092	+	0.999411i	$0.489077\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.1769	+	34.1769i	−1.31353	+	1.31353i	−0.394727	+	0.918798i	$0.629161\pi$
$677$	−34.1769	+	34.1769i	−1.31353	+	1.31353i	−0.918798	+	0.394727i	$0.870839\pi$
$678$		0			0
$679$	−16.5000	+	9.52628i	−0.633212	+	0.365585i
$680$		0			0
$681$	−9.83013	−	9.83013i	−0.376691	−	0.376691i
$682$		0			0
$683$	1.33013	−	2.30385i	0.0508959	−	0.0881543i	−0.839455	−	0.543429i	$-0.817126\pi$
$683$	1.33013	−	2.30385i	0.0508959	−	0.0881543i	0.890351	+	0.455275i	$0.150459\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	32.9545	−	19.0263i	1.25729	−	0.725898i
$688$		0			0
$689$	22.3205	−	4.46410i	0.850344	−	0.170069i
$690$		0			0
$691$	5.65064	−	21.0885i	0.214960	−	0.802243i	−0.771220	−	0.636568i	$-0.780353\pi$
$691$	5.65064	−	21.0885i	0.214960	−	0.802243i	0.986181	−	0.165674i	$-0.0529800\pi$
$692$		0			0
$693$	2.50962	+	9.36603i	0.0953325	+	0.355786i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			8.66025i			0.328031i
$698$		0			0
$699$	−20.2942	−	35.1506i	−0.767598	−	1.32952i
$700$		0			0
$701$		−	34.9282i		−	1.31922i	−0.751608	−	0.659610i	$-0.770721\pi$
$701$		−	34.9282i		−	1.31922i	0.751608	−	0.659610i	$-0.229279\pi$
$702$		0			0
$703$	−26.2224	−	26.2224i	−0.988998	−	0.988998i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.3731			1.14230
$708$		0			0
$709$	−33.7224	+	9.03590i	−1.26647	+	0.339350i	−0.828678	−	0.559725i	$-0.810907\pi$
$709$	−33.7224	+	9.03590i	−1.26647	+	0.339350i	−0.437794	+	0.899075i	$0.644240\pi$
$710$		0			0
$711$	−2.73205	+	4.73205i	−0.102460	+	0.177466i
$712$		0			0
$713$	26.4904	+	45.8827i	0.992073	+	1.71832i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.901924	+	1.56218i	0.0336830	+	0.0583406i
$718$		0			0
$719$	−14.0359	+	24.3109i	−0.523451	+	0.906643i	0.476177	+	0.879350i	$0.342022\pi$
$719$	−14.0359	+	24.3109i	−0.523451	+	0.906643i	−0.999627	+	0.0272936i	$0.991311\pi$
$720$		0			0
$721$	4.90192	−	1.31347i	0.182557	−	0.0489160i
$722$		0			0
$723$	19.9282			0.741138
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.41154	+	2.41154i	0.0894392	+	0.0894392i	0.750411	−	0.660972i	$-0.229856\pi$
$727$	2.41154	+	2.41154i	0.0894392	+	0.0894392i	−0.660972	+	0.750411i	$0.729856\pi$
$728$		0			0
$729$		−	17.5885i		−	0.651424i
$730$		0			0
$731$	6.23205	+	10.7942i	0.230501	+	0.399239i
$732$		0			0
$733$			14.7846i			0.546082i	0.962002	+	0.273041i	$0.0880295\pi$
$733$			14.7846i			0.546082i	−0.962002	+	0.273041i	$0.911971\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0814	+	45.0885i	0.445025	+	1.66085i
$738$		0			0
$739$	1.91858	−	7.16025i	0.0705763	−	0.263394i	−0.921618	−	0.388099i	$-0.873132\pi$
$739$	1.91858	−	7.16025i	0.0705763	−	0.263394i	0.992194	+	0.124705i	$0.0397985\pi$
$740$		0			0
$741$	−16.9282	+	25.3923i	−0.621873	+	0.932810i
$742$		0			0
$743$	10.6699	−	6.16025i	0.391440	−	0.225998i	−0.291344	−	0.956618i	$-0.594102\pi$
$743$	10.6699	−	6.16025i	0.391440	−	0.225998i	0.682784	+	0.730621i	$0.260769\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.00000	−	6.92820i	0.146352	−	0.253490i
$748$		0			0
$749$	−16.2224	−	16.2224i	−0.592755	−	0.592755i
$750$		0			0
$751$	45.3564	−	26.1865i	1.65508	−	0.955560i	0.680141	−	0.733081i	$-0.261918\pi$
$751$	45.3564	−	26.1865i	1.65508	−	0.955560i	0.974937	−	0.222479i	$-0.0714149\pi$
$752$		0			0
$753$	19.4904	−	19.4904i	0.710269	−	0.710269i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.1603	+	5.93782i	0.805428	+	0.215814i	0.637966	−	0.770065i	$-0.279776\pi$
$757$	22.1603	+	5.93782i	0.805428	+	0.215814i	0.167462	+	0.985878i	$0.446443\pi$
$758$		0			0
$759$	−29.2224	+	29.2224i	−1.06071	+	1.06071i
$760$		0			0
$761$	−9.33013	−	34.8205i	−0.338217	−	1.26224i	−0.900339	−	0.435188i	$-0.856682\pi$
$761$	−9.33013	−	34.8205i	−0.338217	−	1.26224i	0.562123	−	0.827054i	$-0.309985\pi$
$762$		0			0
$763$	−63.8109	+	17.0981i	−2.31011	+	0.618992i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.57180	+	22.3827i	−0.273402	+	0.808192i
$768$		0			0
$769$	24.5263	+	6.57180i	0.884440	+	0.236985i	0.672322	−	0.740259i	$-0.265297\pi$
$769$	24.5263	+	6.57180i	0.884440	+	0.236985i	0.212118	+	0.977244i	$0.431964\pi$
$770$		0			0
$771$	−16.9641	−	9.79423i	−0.610947	−	0.352731i
$772$		0			0
$773$	47.0429	+	27.1603i	1.69202	+	0.976886i	0.952888	+	0.303323i	$0.0980960\pi$
$773$	47.0429	+	27.1603i	1.69202	+	0.976886i	0.739129	+	0.673564i	$0.235237\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	18.8923	−	70.5070i	0.677758	−	2.52943i
$778$		0			0
$779$	−11.3397			−0.406289
$780$		0			0
$781$	14.1244			0.505409
$782$		0			0
$783$	−1.96410	+	7.33013i	−0.0701913	+	0.261957i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	4.54552	+	2.62436i	0.162030	+	0.0935482i	0.578822	−	0.815454i	$-0.303512\pi$
$787$	4.54552	+	2.62436i	0.162030	+	0.0935482i	−0.416792	+	0.909002i	$0.636846\pi$
$788$		0			0
$789$	−44.5526	−	25.7224i	−1.58611	−	0.915743i
$790$		0			0
$791$	−18.8923	−	5.06218i	−0.671733	−	0.179990i
$792$		0			0

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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.bs (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 1.86603i) q^{3} +(3.86603 + 2.23205i) q^{7} +(-0.633975 - 0.366025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 1.86603i) q^{3} +(3.86603 + 2.23205i) q^{7} +(-0.633975 - 0.366025i) q^{9} +(-2.86603 - 0.767949i) q^{11} +(2.00000 - 3.00000i) q^{13} +(-3.23205 + 0.866025i) q^{17} +(-1.13397 - 4.23205i) q^{19} +(6.09808 - 6.09808i) q^{21} +(6.96410 + 1.86603i) q^{23} +(3.09808 - 3.09808i) q^{27} +(-1.50000 + 0.866025i) q^{29} +(5.19615 + 5.19615i) q^{31} +(-2.86603 + 4.96410i) q^{33} +(7.33013 - 4.23205i) q^{37} +(-4.59808 - 5.23205i) q^{39} +(0.669873 - 2.50000i) q^{41} +(-0.964102 - 3.59808i) q^{43} +2.92820i q^{47} +(6.46410 + 11.1962i) q^{49} +6.46410i q^{51} +(4.46410 + 4.46410i) q^{53} -8.46410 q^{57} +(-6.33013 + 1.69615i) q^{59} +(4.50000 - 7.79423i) q^{61} +(-1.63397 - 2.83013i) q^{63} +(-7.86603 - 13.6244i) q^{67} +(6.96410 - 12.0622i) q^{69} +(-4.59808 + 1.23205i) q^{71} +1.07180 q^{73} +(-9.36603 - 9.36603i) q^{77} -7.46410i q^{79} +(-5.33013 - 9.23205i) q^{81} +10.9282i q^{83} +(0.866025 + 3.23205i) q^{87} +(0.794229 - 2.96410i) q^{89} +(14.4282 - 7.13397i) q^{91} +(12.2942 - 7.09808i) q^{93} +(-2.13397 + 3.69615i) q^{97} +(1.53590 + 1.53590i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9	\( 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 8 q^{19} + 14 q^{21} + 14 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 20 q^{41} + 10 q^{43} + 12 q^{49} + 4 q^{53} - 20 q^{57} - 8 q^{59} + 18 q^{61} - 10 q^{63} - 28 q^{67} + 14 q^{69} - 8 q^{71} + 32 q^{73} - 34 q^{77} - 4 q^{81} - 28 q^{89} + 30 q^{91} + 18 q^{93} - 12 q^{97} + 20 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 8 * q^19 + 14 * q^21 + 14 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 20 * q^41 + 10 * q^43 + 12 * q^49 + 4 * q^53 - 20 * q^57 - 8 * q^59 + 18 * q^61 - 10 * q^63 - 28 * q^67 + 14 * q^69 - 8 * q^71 + 32 * q^73 - 34 * q^77 - 4 * q^81 - 28 * q^89 + 30 * q^91 + 18 * q^93 - 12 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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