LMFDB - Embedded newform 1512.2.s.b.865.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(865,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1512.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.s (of order $3$, degree $2$, 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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1512.865
Dual form			1512.2.s.b.1297.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} +(-2.00000 - 1.73205i) q^{7} +3.00000 q^{13} +(-2.00000 + 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-1.00000 - 1.73205i) q^{23} +(0.500000 - 0.866025i) q^{25} -4.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-1.00000 + 5.19615i) q^{35} +(1.50000 + 2.59808i) q^{37} -8.00000 q^{41} -1.00000 q^{43} +(-4.00000 - 6.92820i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-7.00000 + 12.1244i) q^{53} +(-6.00000 + 10.3923i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-1.50000 + 2.59808i) q^{67} -6.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{79} -6.00000 q^{83} +8.00000 q^{85} +(1.00000 + 1.73205i) q^{89} +(-6.00000 - 5.19615i) q^{91} +(-4.00000 + 6.92820i) q^{95} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} + 6 q^{13} - 4 q^{17} - 4 q^{19} - 2 q^{23} + q^{25} - 8 q^{29} - q^{31} - 2 q^{35} + 3 q^{37} - 16 q^{41} - 2 q^{43} - 8 q^{47} + 2 q^{49} - 14 q^{53} - 12 q^{59} - q^{61} - 6 q^{65} - 3 q^{67} - 12 q^{71} - 10 q^{73} - q^{79} - 12 q^{83} + 16 q^{85} + 2 q^{89} - 12 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 + 6 * q^13 - 4 * q^17 - 4 * q^19 - 2 * q^23 + q^25 - 8 * q^29 - q^31 - 2 * q^35 + 3 * q^37 - 16 * q^41 - 2 * q^43 - 8 * q^47 + 2 * q^49 - 14 * q^53 - 12 * q^59 - q^61 - 6 * q^65 - 3 * q^67 - 12 * q^71 - 10 * q^73 - q^79 - 12 * q^83 + 16 * q^85 + 2 * q^89 - 12 * q^91 - 8 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	3.00000			0.832050			0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000			0.832050			0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	−0.999853	−	0.0171533i	$-0.994540\pi$
$17$	−2.00000	+	3.46410i	−0.485071	+	0.840168i	0.514782	+	0.857321i	$0.327873\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.00000	+	5.19615i	−0.169031	+	0.878310i
$36$		0			0
$37$	1.50000	+	2.59808i	0.246598	+	0.427121i	0.962580	−	0.270998i	$-0.0873538\pi$
$37$	1.50000	+	2.59808i	0.246598	+	0.427121i	−0.715981	+	0.698119i	$0.754020\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	−0.995066	−	0.0992202i	$-0.968365\pi$
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	0.411606	−	0.911362i	$-0.364968\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.00000	+	12.1244i	−0.961524	+	1.66541i	−0.242846	+	0.970065i	$0.578081\pi$
$53$	−7.00000	+	12.1244i	−0.961524	+	1.66541i	−0.718677	+	0.695344i	$0.755252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	$-0.891996\pi$
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	0.759733	+	0.650236i	$0.225330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	0.409644	+	0.912245i	$0.365653\pi$
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	0.836527	−	0.547926i	$-0.184582\pi$
$79$	−0.500000	−	0.866025i	−0.0562544	−	0.0974355i	−0.892781	+	0.450490i	$0.851249\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.00000	+	1.73205i	0.106000	+	0.183597i	0.914146	−	0.405385i	$-0.132862\pi$
$89$	1.00000	+	1.73205i	0.106000	+	0.183597i	−0.808146	+	0.588982i	$0.799529\pi$
$90$		0			0
$91$	−6.00000	−	5.19615i	−0.628971	−	0.544705i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	1.00000			0.101535			0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000			0.101535			0.0507673	+	0.998711i	$0.483833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	−7.50000	−	12.9904i	−0.738997	−	1.27998i	−0.952947	−	0.303136i	$-0.901966\pi$
$103$	−7.50000	−	12.9904i	−0.738997	−	1.27998i	0.213950	−	0.976845i	$-0.431367\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$		0			0
$109$	2.50000	−	4.33013i	0.239457	−	0.414751i	−0.721102	−	0.692829i	$-0.756364\pi$
$109$	2.50000	−	4.33013i	0.239457	−	0.414751i	0.960558	+	0.278078i	$0.0896974\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	−2.00000	+	3.46410i	−0.186501	+	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	10.0000	−	3.46410i	0.916698	−	0.317554i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	0.636676	−	0.771132i	$-0.280309\pi$
$131$	−4.00000	−	6.92820i	−0.349482	−	0.605320i	−0.986157	+	0.165812i	$0.946976\pi$
$132$		0			0
$133$	−2.00000	+	10.3923i	−0.173422	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	−15.0000			−1.27228			−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000			−1.27228			−0.636142	+	0.771572i	$0.719471\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	4.00000	+	6.92820i	0.332182	+	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.00000	+	12.1244i	0.573462	+	0.993266i	0.996207	+	0.0870170i	$0.0277334\pi$
$149$	7.00000	+	12.1244i	0.573462	+	0.993266i	−0.422744	+	0.906249i	$0.638933\pi$
$150$		0			0
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	8.50000	−	14.7224i	0.691720	−	1.19809i	0.971274	−	0.237964i	$-0.0764802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000			0.160644
$156$		0			0
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	−0.243403	−	0.969925i	$-0.578264\pi$
$157$	9.00000	−	15.5885i	0.718278	−	1.24409i	0.961681	−	0.274169i	$-0.0884028\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.00000	+	5.19615i	−0.0788110	+	0.409514i
$162$		0			0
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	−0.950615	−	0.310372i	$-0.899546\pi$
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	0.206518	−	0.978443i	$-0.433787\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.00000	−	6.92820i	−0.304114	−	0.526742i	0.672949	−	0.739689i	$-0.265027\pi$
$173$	−4.00000	−	6.92820i	−0.304114	−	0.526742i	−0.977064	+	0.212947i	$0.931694\pi$
$174$		0			0
$175$	−2.50000	+	0.866025i	−0.188982	+	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.0000	+	22.5167i	−0.971666	+	1.68297i	−0.281139	+	0.959667i	$0.590712\pi$
$179$	−13.0000	+	22.5167i	−0.971666	+	1.68297i	−0.690526	+	0.723307i	$0.742621\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	0.626468	−	0.779447i	$-0.284500\pi$
$191$	−5.00000	−	8.66025i	−0.361787	−	0.626634i	−0.988255	+	0.152813i	$0.951167\pi$
$192$		0			0
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	−0.657376	−	0.753563i	$-0.728333\pi$
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	0.981293	+	0.192522i	$0.0616668\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−2.50000	+	4.33013i	−0.177220	+	0.306955i	−0.940927	−	0.338608i	$-0.890044\pi$
$199$	−2.50000	+	4.33013i	−0.177220	+	0.306955i	0.763707	+	0.645563i	$0.223377\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.00000	+	6.92820i	0.561490	+	0.486265i
$204$		0			0
$205$	8.00000	+	13.8564i	0.558744	+	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−25.0000			−1.72107			−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000			−1.72107			−0.860535	+	0.509390i	$0.829871\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.00000	+	1.73205i	0.0681994	+	0.118125i
$216$		0			0
$217$	2.50000	−	0.866025i	0.169711	−	0.0587896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	+	10.3923i	−0.403604	+	0.699062i
$222$		0			0
$223$	12.0000			0.803579			0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000			0.803579			0.401790	+	0.915732i	$0.368388\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0000	−	19.0526i	0.730096	−	1.26456i	−0.226746	−	0.973954i	$-0.572809\pi$
$227$	11.0000	−	19.0526i	0.730096	−	1.26456i	0.956842	−	0.290609i	$-0.0938578\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	0.882073	−	0.471113i	$-0.156147\pi$
$229$	0.500000	+	0.866025i	0.0330409	+	0.0572286i	−0.849032	+	0.528341i	$0.822814\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	8.00000	+	13.8564i	0.524097	+	0.907763i	0.999606	+	0.0280525i	$0.00893057\pi$
$233$	8.00000	+	13.8564i	0.524097	+	0.907763i	−0.475509	+	0.879711i	$0.657736\pi$
$234$		0			0
$235$	−8.00000	+	13.8564i	−0.521862	+	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	26.0000			1.68180			0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000			1.68180			0.840900	+	0.541190i	$0.182026\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.0000	−	8.66025i	0.702764	−	0.553283i
$246$		0			0
$247$	−6.00000	−	10.3923i	−0.381771	−	0.661247i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.0000			0.631194			0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000			0.631194			0.315597	+	0.948893i	$0.397795\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.0000	−	25.9808i	−0.935674	−	1.62064i	−0.773427	−	0.633885i	$-0.781459\pi$
$257$	−15.0000	−	25.9808i	−0.935674	−	1.62064i	−0.162247	−	0.986750i	$-0.551874\pi$
$258$		0			0
$259$	1.50000	−	7.79423i	0.0932055	−	0.484310i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	28.0000			1.72003
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	6.50000	+	11.2583i	0.394847	+	0.683895i	0.993082	−	0.117426i	$-0.0374643\pi$
$271$	6.50000	+	11.2583i	0.394847	+	0.683895i	−0.598235	+	0.801321i	$0.704131\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.150210	+	0.988654i	$0.547995\pi$
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	−11.5000	+	19.9186i	−0.683604	+	1.18404i	0.290269	+	0.956945i	$0.406255\pi$
$283$	−11.5000	+	19.9186i	−0.683604	+	1.18404i	−0.973873	+	0.227092i	$0.927078\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.0000	+	13.8564i	0.944450	+	0.817918i
$288$		0			0
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	2.00000	+	1.73205i	0.115278	+	0.0998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.00000	+	1.73205i	−0.0572598	+	0.0991769i
$306$		0			0
$307$	−23.0000			−1.31268			−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000			−1.31268			−0.656340	+	0.754466i	$0.727896\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	−0.768345	−	0.640036i	$-0.778920\pi$
$311$	3.00000	−	5.19615i	0.170114	−	0.294647i	0.938460	+	0.345389i	$0.112253\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	−0.976764	−	0.214318i	$-0.931247\pi$
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	0.302777	−	0.953062i	$-0.402086\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000			0.890264
$324$		0			0
$325$	1.50000	−	2.59808i	0.0832050	−	0.144115i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.00000	+	20.7846i	−0.220527	+	1.14589i
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.00000			0.327815
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	−0.347024	−	0.937856i	$-0.612808\pi$
$353$	12.0000	−	20.7846i	0.638696	−	1.10625i	0.985719	−	0.168397i	$-0.0538590\pi$
$354$		0			0
$355$	6.00000	+	10.3923i	0.318447	+	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3.00000	+	5.19615i	0.158334	+	0.274242i	0.934268	−	0.356572i	$-0.116054\pi$
$359$	3.00000	+	5.19615i	0.158334	+	0.274242i	−0.775934	+	0.630814i	$0.782721\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	20.0000			1.04685
$366$		0			0
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	−0.578101	−	0.815966i	$-0.696206\pi$
$367$	8.00000	−	13.8564i	0.417597	−	0.723299i	0.995697	+	0.0926670i	$0.0295392\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	35.0000	−	12.1244i	1.81711	−	0.629465i
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	15.0000			0.770498			0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000			0.770498			0.385249	+	0.922813i	$0.374116\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.0000	−	25.9808i	−0.766464	−	1.32755i	−0.939469	−	0.342634i	$-0.888681\pi$
$383$	−15.0000	−	25.9808i	−0.766464	−	1.32755i	0.173005	−	0.984921i	$-0.444652\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0000	+	24.2487i	−0.709828	+	1.22946i	0.255092	+	0.966917i	$0.417894\pi$
$389$	−14.0000	+	24.2487i	−0.709828	+	1.22946i	−0.964921	+	0.262542i	$0.915439\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.00000	+	1.73205i	−0.0503155	+	0.0871489i
$396$		0			0
$397$	−6.50000	−	11.2583i	−0.326226	−	0.565039i	0.655534	−	0.755166i	$-0.272444\pi$
$397$	−6.50000	−	11.2583i	−0.326226	−	0.565039i	−0.981760	+	0.190126i	$0.939110\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−1.50000	+	2.59808i	−0.0747203	+	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	19.5000	−	33.7750i	0.964213	−	1.67007i	0.252498	−	0.967597i	$-0.418748\pi$
$409$	19.5000	−	33.7750i	0.964213	−	1.67007i	0.711715	−	0.702468i	$-0.247919\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	30.0000	−	10.3923i	1.47620	−	0.511372i
$414$		0			0
$415$	6.00000	+	10.3923i	0.294528	+	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.0000			0.683945			0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000			0.683945			0.341972	+	0.939710i	$0.388905\pi$
$420$		0			0
$421$	−38.0000			−1.85201			−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000			−1.85201			−0.926003	+	0.377515i	$0.876779\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	+	3.46410i	0.0970143	+	0.168034i
$426$		0			0
$427$	−0.500000	+	2.59808i	−0.0241967	+	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	0.417687	+	0.908591i	$0.362841\pi$
$431$	−12.0000	+	20.7846i	−0.578020	+	1.00116i	−0.995706	+	0.0925683i	$0.970492\pi$
$432$		0			0
$433$	7.00000			0.336399			0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000			0.336399			0.168199	+	0.985753i	$0.446205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.00000	+	6.92820i	−0.191346	+	0.331421i
$438$		0			0
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	0.996284	+	0.0861252i	$0.0274485\pi$
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	−0.423556	+	0.905870i	$0.639218\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.00000	−	6.92820i	−0.190046	−	0.329169i	0.755219	−	0.655472i	$-0.227530\pi$
$443$	−4.00000	−	6.92820i	−0.190046	−	0.329169i	−0.945265	+	0.326303i	$0.894197\pi$
$444$		0			0
$445$	2.00000	−	3.46410i	0.0948091	−	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.00000	+	15.5885i	−0.140642	+	0.730798i
$456$		0			0
$457$	−12.5000	−	21.6506i	−0.584725	−	1.01277i	−0.994910	−	0.100771i	$-0.967869\pi$
$457$	−12.5000	−	21.6506i	−0.584725	−	1.01277i	0.410184	−	0.912003i	$-0.365464\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.00000	−	5.19615i	−0.138823	−	0.240449i	0.788228	−	0.615383i	$-0.210999\pi$
$467$	−3.00000	−	5.19615i	−0.138823	−	0.240449i	−0.927052	+	0.374934i	$0.877665\pi$
$468$		0			0
$469$	7.50000	−	2.59808i	0.346318	−	0.119968i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.00000	+	8.66025i	−0.228456	+	0.395697i	−0.957351	−	0.288929i	$-0.906701\pi$
$479$	−5.00000	+	8.66025i	−0.228456	+	0.395697i	0.728895	+	0.684626i	$0.240034\pi$
$480$		0			0
$481$	4.50000	+	7.79423i	0.205182	+	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00000	−	1.73205i	−0.0454077	−	0.0786484i
$486$		0			0
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	−0.761052	−	0.648691i	$-0.775317\pi$
$487$	4.00000	−	6.92820i	0.181257	−	0.313947i	0.942309	+	0.334744i	$0.108650\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.00000			0.270776			0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000			0.270776			0.135388	+	0.990793i	$0.456772\pi$
$492$		0			0
$493$	8.00000	−	13.8564i	0.360302	−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.0000	+	10.3923i	0.538274	+	0.466159i
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.0000			−0.624229			−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000			−0.624229			−0.312115	+	0.950044i	$0.601037\pi$
$504$		0			0
$505$	−36.0000			−1.60198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	0.791849	−	0.610718i	$-0.209119\pi$
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	−0.924821	+	0.380402i	$0.875786\pi$
$510$		0			0
$511$	25.0000	−	8.66025i	1.10593	−	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−15.0000	+	25.9808i	−0.660979	+	1.14485i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000	−	5.19615i	0.131432	−	0.227648i	−0.792797	−	0.609486i	$-0.791376\pi$
$521$	3.00000	−	5.19615i	0.131432	−	0.227648i	0.924229	+	0.381839i	$0.124709\pi$
$522$		0			0
$523$	12.5000	+	21.6506i	0.546587	+	0.946716i	0.998505	+	0.0546569i	$0.0174065\pi$
$523$	12.5000	+	21.6506i	0.546587	+	0.946716i	−0.451918	+	0.892059i	$0.649260\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.00000	−	3.46410i	−0.0871214	−	0.150899i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−24.0000			−1.03956
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.0000	−	29.4449i	−0.730887	−	1.26593i	−0.956504	−	0.291718i	$-0.905773\pi$
$541$	−17.0000	−	29.4449i	−0.730887	−	1.26593i	0.225617	−	0.974216i	$-0.427560\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$	7.00000			0.299298			0.149649	−	0.988739i	$-0.452186\pi$
$547$	7.00000			0.299298			0.149649	+	0.988739i	$0.452186\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.00000	+	13.8564i	0.340811	+	0.590303i
$552$		0			0
$553$	−0.500000	+	2.59808i	−0.0212622	+	0.110481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	−0.844062	−	0.536246i	$-0.819842\pi$
$557$	1.00000	−	1.73205i	0.0423714	−	0.0733893i	0.886433	+	0.462856i	$0.153175\pi$
$558$		0			0
$559$	−3.00000			−0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.00000	−	12.1244i	0.295015	−	0.510981i	−0.679974	−	0.733237i	$-0.738009\pi$
$563$	7.00000	−	12.1244i	0.295015	−	0.510981i	0.974988	+	0.222256i	$0.0713421\pi$
$564$		0			0
$565$	−6.00000	−	10.3923i	−0.252422	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	−0.995788	−	0.0916910i	$-0.970773\pi$
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	0.577301	+	0.816532i	$0.304106\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.00000			−0.0834058
$576$		0			0
$577$	−13.5000	+	23.3827i	−0.562012	+	0.973434i	0.435308	+	0.900281i	$0.356639\pi$
$577$	−13.5000	+	23.3827i	−0.562012	+	0.973434i	−0.997321	+	0.0731526i	$0.976694\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000	+	10.3923i	0.497844	+	0.431145i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−38.0000			−1.56843			−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000			−1.56843			−0.784214	+	0.620491i	$0.786934\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.0000	−	36.3731i	−0.862367	−	1.49366i	−0.869638	−	0.493689i	$-0.835648\pi$
$593$	−21.0000	−	36.3731i	−0.862367	−	1.49366i	0.00727173	−	0.999974i	$-0.497685\pi$
$594$		0			0
$595$	−16.0000	−	13.8564i	−0.655936	−	0.568057i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.0000	+	34.6410i	−0.817178	+	1.41539i	0.0905757	+	0.995890i	$0.471129\pi$
$599$	−20.0000	+	34.6410i	−0.817178	+	1.41539i	−0.907754	+	0.419504i	$0.862204\pi$
$600$		0			0
$601$	−5.00000			−0.203954			−0.101977	−	0.994787i	$-0.532517\pi$
$601$	−5.00000			−0.203954			−0.101977	+	0.994787i	$0.532517\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	−	19.0526i	0.447214	−	0.774597i
$606$		0			0
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	0.983262	+	0.182199i	$0.0583216\pi$
$607$	16.0000	+	27.7128i	0.649420	+	1.12483i	−0.333842	+	0.942629i	$0.608345\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	20.7846i	−0.485468	−	0.840855i
$612$		0			0
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	−0.875946	−	0.482409i	$-0.839762\pi$
$613$	−0.500000	+	0.866025i	−0.0201948	+	0.0349784i	0.855751	+	0.517387i	$0.173095\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	36.0000			1.44931			0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000			1.44931			0.724653	+	0.689114i	$0.242000\pi$
$618$		0			0
$619$	−13.5000	+	23.3827i	−0.542611	+	0.939829i	0.456142	+	0.889907i	$0.349231\pi$
$619$	−13.5000	+	23.3827i	−0.542611	+	0.939829i	−0.998753	+	0.0499226i	$0.984103\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.00000	−	5.19615i	0.0400642	−	0.208179i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	47.0000			1.87104			0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000			1.87104			0.935520	+	0.353273i	$0.114931\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.0000	−	29.4449i	−0.674624	−	1.16848i
$636$		0			0
$637$	3.00000	+	20.7846i	0.118864	+	0.823516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0000	+	34.6410i	−0.789953	+	1.36824i	0.136043	+	0.990703i	$0.456562\pi$
$641$	−20.0000	+	34.6410i	−0.789953	+	1.36824i	−0.925995	+	0.377535i	$0.876772\pi$
$642$		0			0
$643$	−29.0000			−1.14365			−0.571824	−	0.820376i	$-0.693764\pi$
$643$	−29.0000			−1.14365			−0.571824	+	0.820376i	$0.693764\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	−0.527710	−	0.849425i	$-0.676949\pi$
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	0.999478	+	0.0322975i	$0.0102824\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	0.695934	−	0.718106i	$-0.254991\pi$
$653$	−7.00000	−	12.1244i	−0.273931	−	0.474463i	−0.969865	+	0.243643i	$0.921657\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10.0000			−0.389545			−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000			−0.389545			−0.194772	+	0.980848i	$0.562397\pi$
$660$		0			0
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	−0.568831	−	0.822454i	$-0.692604\pi$
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	0.996682	+	0.0813955i	$0.0259377\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.0000	−	6.92820i	0.775567	−	0.268664i
$666$		0			0
$667$	4.00000	+	6.92820i	0.154881	+	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000	+	31.1769i	0.691796	+	1.19823i	0.971249	+	0.238067i	$0.0765137\pi$
$677$	18.0000	+	31.1769i	0.691796	+	1.19823i	−0.279453	+	0.960159i	$0.590153\pi$
$678$		0			0
$679$	−2.00000	−	1.73205i	−0.0767530	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.00000	−	15.5885i	0.344375	−	0.596476i	−0.640865	−	0.767654i	$-0.721424\pi$
$683$	9.00000	−	15.5885i	0.344375	−	0.596476i	0.985240	+	0.171178i	$0.0547574\pi$
$684$		0			0
$685$	4.00000			0.152832
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.0000	+	36.3731i	−0.800036	+	1.38570i
$690$		0			0
$691$	3.50000	+	6.06218i	0.133146	+	0.230616i	0.924888	−	0.380240i	$-0.124159\pi$
$691$	3.50000	+	6.06218i	0.133146	+	0.230616i	−0.791742	+	0.610856i	$0.790825\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.0000	+	25.9808i	0.568982	+	0.985506i
$696$		0			0
$697$	16.0000	−	27.7128i	0.606043	−	1.04970i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−44.0000			−1.66186			−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000			−1.66186			−0.830929	+	0.556379i	$0.812190\pi$
$702$		0			0
$703$	6.00000	−	10.3923i	0.226294	−	0.391953i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−45.0000	+	15.5885i	−1.69240	+	0.586264i
$708$		0			0
$709$	8.50000	+	14.7224i	0.319224	+	0.552913i	0.980326	−	0.197383i	$-0.0632444\pi$
$709$	8.50000	+	14.7224i	0.319224	+	0.552913i	−0.661102	+	0.750296i	$0.729911\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.00000			0.0749006
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.00000	−	15.5885i	−0.335643	−	0.581351i	0.647965	−	0.761670i	$-0.275620\pi$
$719$	−9.00000	−	15.5885i	−0.335643	−	0.581351i	−0.983608	+	0.180319i	$0.942287\pi$
$720$		0			0
$721$	−7.50000	+	38.9711i	−0.279315	+	1.45136i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.00000	+	3.46410i	−0.0742781	+	0.128654i
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.00000	−	3.46410i	0.0739727	−	0.128124i
$732$		0			0
$733$	−15.5000	−	26.8468i	−0.572506	−	0.991609i	−0.996308	−	0.0858539i	$-0.972638\pi$
$733$	−15.5000	−	26.8468i	−0.572506	−	0.991609i	0.423802	−	0.905755i	$-0.360695\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	−0.986154	−	0.165831i	$-0.946969\pi$
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	0.636691	+	0.771119i	$0.280303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	44.0000			1.61420			0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000			1.61420			0.807102	+	0.590412i	$0.201035\pi$
$744$		0			0
$745$	14.0000	−	24.2487i	0.512920	−	0.888404i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.00000	+	15.5885i	−0.109618	+	0.569590i
$750$		0			0
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	0.623856	−	0.781540i	$-0.285565\pi$
$751$	−10.0000	−	17.3205i	−0.364905	−	0.632034i	−0.988761	+	0.149505i	$0.952232\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−34.0000			−1.23739
$756$		0			0
$757$	−17.0000			−0.617876			−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000			−0.617876			−0.308938	+	0.951082i	$0.599973\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−24.0000	−	41.5692i	−0.869999	−	1.50688i	−0.861996	−	0.506915i	$-0.830786\pi$
$761$	−24.0000	−	41.5692i	−0.869999	−	1.50688i	−0.00800331	−	0.999968i	$-0.502548\pi$
$762$		0			0
$763$	−12.5000	+	4.33013i	−0.452530	+	0.156761i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.0000	+	31.1769i	−0.649942	+	1.12573i
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.0000	−	17.3205i	0.359675	−	0.622975i	−0.628231	−	0.778027i	$-0.716221\pi$
$773$	10.0000	−	17.3205i	0.359675	−	0.622975i	0.987906	+	0.155051i	$0.0495542\pi$
$774$		0			0
$775$	0.500000	+	0.866025i	0.0179605	+	0.0311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	16.0000	+	27.7128i	0.573259	+	0.992915i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−36.0000			−1.28490
$786$		0			0
$787$	−18.5000	+	32.0429i	−0.659454	+	1.14221i	0.321303	+	0.946976i	$0.395879\pi$
$787$	−18.5000	+	32.0429i	−0.659454	+	1.14221i	−0.980757	+	0.195231i	$0.937454\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000	−	10.3923i	−0.426671	−	0.369508i
$792$		0			0
$793$	−1.50000	−	2.59808i	−0.0532666	−	0.0922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.0000			−0.850124			−0.425062	−	0.905164i	$-0.639748\pi$
$797$	−24.0000			−0.850124			−0.425062	+	0.905164i	$0.639748\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	10.0000	−	3.46410i	0.352454	−	0.122094i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.0000	+	25.9808i	−0.527372	+	0.913435i	0.472119	+	0.881535i	$0.343489\pi$
$809$	−15.0000	+	25.9808i	−0.527372	+	0.913435i	−0.999491	+	0.0319002i	$0.989844\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.0000	+	32.9090i	−0.665541	+	1.15275i
$816$		0			0
$817$	2.00000	+	3.46410i	0.0699711	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	0.665144	−	0.746715i	$-0.268370\pi$
$821$	−9.00000	−	15.5885i	−0.314102	−	0.544041i	−0.979246	+	0.202674i	$0.935037\pi$
$822$		0			0
$823$	0.500000	−	0.866025i	0.0174289	−	0.0301877i	−0.857179	−	0.515018i	$-0.827785\pi$
$823$	0.500000	−	0.866025i	0.0174289	−	0.0301877i	0.874608	+	0.484830i	$0.161119\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	23.0000	−	39.8372i	0.798823	−	1.38360i	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	23.0000	−	39.8372i	0.798823	−	1.38360i	0.920383	−	0.391018i	$-0.127877\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26.0000	−	10.3923i	−0.900847	−	0.360072i
$834$		0			0
$835$	6.00000	+	10.3923i	0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−54.0000			−1.86429			−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000			−1.86429			−0.932144	+	0.362089i	$0.882064\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.00000	+	6.92820i	0.137604	+	0.238337i
$846$		0			0
$847$	5.50000	−	28.5788i	0.188982	−	0.981981i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	−	5.19615i	0.102839	−	0.178122i
$852$		0			0
$853$	−2.00000			−0.0684787			−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000			−0.0684787			−0.0342393	+	0.999414i	$0.510901\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	0.244701	+	0.969599i	$0.421310\pi$
$857$	−21.0000	+	36.3731i	−0.717346	+	1.24248i	−0.962048	+	0.272882i	$0.912023\pi$
$858$		0			0
$859$	−15.5000	−	26.8468i	−0.528853	−	0.916001i	−0.999434	−	0.0336436i	$-0.989289\pi$
$859$	−15.5000	−	26.8468i	−0.528853	−	0.916001i	0.470581	−	0.882357i	$-0.344044\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.0000	−	17.3205i	−0.340404	−	0.589597i	0.644104	−	0.764938i	$-0.277230\pi$
$863$	−10.0000	−	17.3205i	−0.340404	−	0.589597i	−0.984508	+	0.175341i	$0.943897\pi$
$864$		0			0
$865$	−8.00000	+	13.8564i	−0.272008	+	0.471132i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−4.50000	+	7.79423i	−0.152477	+	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000	+	20.7846i	0.811348	+	0.702648i
$876$		0			0
$877$	−8.50000	−	14.7224i	−0.287025	−	0.497141i	0.686074	−	0.727532i	$-0.259333\pi$
$877$	−8.50000	−	14.7224i	−0.287025	−	0.497141i	−0.973098	+	0.230391i	$0.925999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.0000	−	25.9808i	−0.503651	−	0.872349i	−0.999991	−	0.00422062i	$-0.998657\pi$
$887$	−15.0000	−	25.9808i	−0.503651	−	0.872349i	0.496340	−	0.868128i	$-0.334677\pi$
$888$		0			0
$889$	−34.0000	−	29.4449i	−1.14032	−	0.987549i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0000	+	27.7128i	−0.535420	+	0.927374i
$894$		0			0
$895$	52.0000			1.73817
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.00000	−	3.46410i	0.0667037	−	0.115534i
$900$		0			0
$901$	−28.0000	−	48.4974i	−0.932815	−	1.61568i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.00000	−	3.46410i	−0.0664822	−	0.115151i
$906$		0			0
$907$	0.500000	−	0.866025i	0.0166022	−	0.0287559i	−0.857605	−	0.514309i	$-0.828048\pi$
$907$	0.500000	−	0.866025i	0.0166022	−	0.0287559i	0.874207	+	0.485553i	$0.161382\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−44.0000			−1.45779			−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000			−1.45779			−0.728893	+	0.684628i	$0.759965\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.00000	+	20.7846i	−0.132092	+	0.686368i
$918$		0			0
$919$	20.5000	+	35.5070i	0.676233	+	1.17127i	0.976107	+	0.217291i	$0.0697219\pi$
$919$	20.5000	+	35.5070i	0.676233	+	1.17127i	−0.299874	+	0.953979i	$0.596945\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.0000			−0.592477
$924$		0			0
$925$	3.00000			0.0986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	0.957708	−	0.287742i	$-0.0929044\pi$
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	−0.728046	+	0.685529i	$0.759571\pi$
$930$		0			0
$931$	22.0000	−	17.3205i	0.721021	−	0.567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	27.0000			0.882052			0.441026	−	0.897494i	$-0.354615\pi$
$937$	27.0000			0.882052			0.441026	+	0.897494i	$0.354615\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$	8.00000	+	13.8564i	0.260516	+	0.451227i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.0000	+	27.7128i	0.519930	+	0.900545i	0.999732	+	0.0231683i	$0.00737536\pi$
$947$	16.0000	+	27.7128i	0.519930	+	0.900545i	−0.479801	+	0.877377i	$0.659291\pi$
$948$		0			0
$949$	−15.0000	+	25.9808i	−0.486921	+	0.843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.0000			0.323932			0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000			0.323932			0.161966	+	0.986796i	$0.448217\pi$
$954$		0			0
$955$	−10.0000	+	17.3205i	−0.323592	+	0.560478i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.00000	−	1.73205i	0.161458	−	0.0559308i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.0000			−0.579441
$966$		0			0
$967$	33.0000			1.06121			0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000			1.06121			0.530604	+	0.847620i	$0.321965\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.00000	+	6.92820i	0.128366	+	0.222337i	0.923044	−	0.384695i	$-0.125693\pi$
$971$	4.00000	+	6.92820i	0.128366	+	0.222337i	−0.794678	+	0.607032i	$0.792360\pi$
$972$		0			0
$973$	30.0000	+	25.9808i	0.961756	+	0.832905i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	−0.519841	−	0.854263i	$-0.674009\pi$
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	0.999734	+	0.0230645i	$0.00734232\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.0000	−	36.3731i	0.669796	−	1.16012i	−0.308165	−	0.951333i	$-0.599715\pi$
$983$	21.0000	−	36.3731i	0.669796	−	1.16012i	0.977961	−	0.208788i	$-0.0669518\pi$
$984$		0			0
$985$	−12.0000	−	20.7846i	−0.382352	−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.00000	+	1.73205i	0.0317982	+	0.0550760i
$990$		0			0
$991$	10.5000	−	18.1865i	0.333543	−	0.577714i	−0.649660	−	0.760224i	$-0.725089\pi$
$991$	10.5000	−	18.1865i	0.333543	−	0.577714i	0.983204	+	0.182510i	$0.0584223\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.0000			0.317021
$996$		0			0
$997$	−18.5000	+	32.0429i	−0.585901	+	1.01481i	0.408862	+	0.912596i	$0.365926\pi$
$997$	−18.5000	+	32.0429i	−0.585901	+	1.01481i	−0.994762	+	0.102214i	$0.967407\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.s.b.865.1	✓	2
3.2	odd	2		1512.2.s.f.865.1	yes	2
7.2	even	3	inner	1512.2.s.b.1297.1	yes	2
21.2	odd	6		1512.2.s.f.1297.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.s.b.865.1	✓	2	1.1	even	1	trivial
1512.2.s.b.1297.1	yes	2	7.2	even	3	inner
1512.2.s.f.865.1	yes	2	3.2	odd	2
1512.2.s.f.1297.1	yes	2	21.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} +(-2.00000 - 1.73205i) q^{7} +3.00000 q^{13} +(-2.00000 + 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-1.00000 - 1.73205i) q^{23} +(0.500000 - 0.866025i) q^{25} -4.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(-1.00000 + 5.19615i) q^{35} +(1.50000 + 2.59808i) q^{37} -8.00000 q^{41} -1.00000 q^{43} +(-4.00000 - 6.92820i) q^{47} +(1.00000 + 6.92820i) q^{49} +(-7.00000 + 12.1244i) q^{53} +(-6.00000 + 10.3923i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-1.50000 + 2.59808i) q^{67} -6.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{79} -6.00000 q^{83} +8.00000 q^{85} +(1.00000 + 1.73205i) q^{89} +(-6.00000 - 5.19615i) q^{91} +(-4.00000 + 6.92820i) q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} + 6 q^{13} - 4 q^{17} - 4 q^{19} - 2 q^{23} + q^{25} - 8 q^{29} - q^{31} - 2 q^{35} + 3 q^{37} - 16 q^{41} - 2 q^{43} - 8 q^{47} + 2 q^{49} - 14 q^{53} - 12 q^{59} - q^{61} - 6 q^{65} - 3 q^{67} - 12 q^{71} - 10 q^{73} - q^{79} - 12 q^{83} + 16 q^{85} + 2 q^{89} - 12 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 + 6 * q^13 - 4 * q^17 - 4 * q^19 - 2 * q^23 + q^25 - 8 * q^29 - q^31 - 2 * q^35 + 3 * q^37 - 16 * q^41 - 2 * q^43 - 8 * q^47 + 2 * q^49 - 14 * q^53 - 12 * q^59 - q^61 - 6 * q^65 - 3 * q^67 - 12 * q^71 - 10 * q^73 - q^79 - 12 * q^83 + 16 * q^85 + 2 * q^89 - 12 * q^91 - 8 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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