LMFDB - Embedded newform 2736.2.s.i.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, degree $2$, 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:	$21.8470699930$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.577
Dual form			2736.2.s.i.1873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{7} +O(q^{10})$	$q-3.00000 q^{7} +2.00000 q^{11} +(-0.500000 + 0.866025i) q^{13} +(2.00000 + 3.46410i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} -7.00000 q^{31} -1.00000 q^{37} +(2.00000 + 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} +(1.00000 - 1.73205i) q^{47} +2.00000 q^{49} +(6.00000 - 10.3923i) q^{53} +(1.00000 + 1.73205i) q^{59} +(0.500000 - 0.866025i) q^{61} +(6.50000 - 11.2583i) q^{67} +(-5.00000 - 8.66025i) q^{71} +(1.50000 + 2.59808i) q^{73} -6.00000 q^{77} +(-6.50000 - 11.2583i) q^{79} -8.00000 q^{83} +(9.00000 - 15.5885i) q^{89} +(1.50000 - 2.59808i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7}+O(q^{10}) $ 2 * q - 6 * q^7	$ 2 q - 6 q^{7} + 4 q^{11} - q^{13} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 5 q^{25} - 14 q^{31} - 2 q^{37} + 4 q^{41} + q^{43} + 2 q^{47} + 4 q^{49} + 12 q^{53} + 2 q^{59} + q^{61} + 13 q^{67} - 10 q^{71} + 3 q^{73} - 12 q^{77} - 13 q^{79} - 16 q^{83} + 18 q^{89} + 3 q^{91} - 2 q^{97}+O(q^{100}) $ 2 * q - 6 * q^7 + 4 * q^11 - q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 5 * q^25 - 14 * q^31 - 2 * q^37 + 4 * q^41 + q^43 + 2 * q^47 + 4 * q^49 + 12 * q^53 + 2 * q^59 + q^61 + 13 * q^67 - 10 * q^71 + 3 * q^73 - 12 * q^77 - 13 * q^79 - 16 * q^83 + 18 * q^89 + 3 * q^91 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i	0.788320	+	0.615265i	$0.210951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$19$	−4.00000	−	1.73205i	−0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000	+	3.46410i	0.312348	+	0.541002i	0.978870	−	0.204483i	$-0.0655513\pi$
$41$	2.00000	+	3.46410i	0.312348	+	0.541002i	−0.666523	+	0.745485i	$0.732218\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	−0.0783936	−	0.996922i	$-0.524979\pi$
$53$	6.00000	−	10.3923i	0.824163	−	1.42749i	0.902557	−	0.430570i	$-0.141688\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.00000	+	1.73205i	0.130189	+	0.225494i	0.923749	−	0.382998i	$-0.125108\pi$
$59$	1.00000	+	1.73205i	0.130189	+	0.225494i	−0.793560	+	0.608492i	$0.791775\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	0.923408	−	0.383819i	$-0.125391\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.00000	−	8.66025i	−0.593391	−	1.02778i	−0.993772	−	0.111434i	$-0.964456\pi$
$71$	−5.00000	−	8.66025i	−0.593391	−	1.02778i	0.400381	−	0.916349i	$-0.368878\pi$
$72$		0			0
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	0.940356	−	0.340193i	$-0.110493\pi$
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	−0.764794	+	0.644275i	$0.777159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.217354	−	0.976093i	$-0.430258\pi$
$89$	9.00000	−	15.5885i	0.953998	−	1.65237i	0.736644	−	0.676280i	$-0.236409\pi$
$90$		0			0
$91$	1.50000	−	2.59808i	0.157243	−	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	0.811976	+	0.583691i	$0.198392\pi$
$102$		0			0
$103$	−15.0000			−1.47799			−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000			−1.47799			−0.738997	+	0.673709i	$0.764700\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		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
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.00000	−	10.3923i	−0.550019	−	0.952661i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	12.0000	+	5.19615i	1.04053	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.00000	−	8.66025i	0.427179	−	0.739895i	−0.569442	−	0.822031i	$-0.692841\pi$
$137$	5.00000	−	8.66025i	0.427179	−	0.739895i	0.996621	+	0.0821359i	$0.0261741\pi$
$138$		0			0
$139$	1.50000	−	2.59808i	0.127228	−	0.220366i	−0.795373	−	0.606120i	$-0.792725\pi$
$139$	1.50000	−	2.59808i	0.127228	−	0.220366i	0.922602	+	0.385754i	$0.126059\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000	+	1.73205i	−0.0836242	+	0.144841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	0.422744	−	0.906249i	$-0.361067\pi$
$150$		0			0
$151$	20.0000			1.62758			0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000			1.62758			0.813788	+	0.581161i	$0.197401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.5000	−	19.9186i	−0.917800	−	1.58968i	−0.802749	−	0.596316i	$-0.796630\pi$
$157$	−11.5000	−	19.9186i	−0.917800	−	1.58968i	−0.115050	−	0.993360i	$-0.536703\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00000	+	10.3923i	−0.472866	+	0.819028i
$162$		0			0
$163$	19.0000			1.48819			0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000			1.48819			0.744097	+	0.668071i	$0.232880\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.00000	−	12.1244i	0.541676	−	0.938211i	−0.457132	−	0.889399i	$-0.651123\pi$
$167$	7.00000	−	12.1244i	0.541676	−	0.938211i	0.998808	−	0.0488118i	$-0.0155435\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		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
$175$	−7.50000	+	12.9904i	−0.566947	+	0.981981i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	−0.826465	−	0.562988i	$-0.809652\pi$
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	0.900794	+	0.434246i	$0.142985\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.00000	+	6.92820i	0.292509	+	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	0.806976	−	0.590584i	$-0.201102\pi$
$193$	−1.50000	−	2.59808i	−0.107972	−	0.187014i	−0.914949	+	0.403570i	$0.867769\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.0000			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$	−4.50000	+	7.79423i	−0.318997	+	0.552518i	−0.980279	−	0.197619i	$-0.936679\pi$
$199$	−4.50000	+	7.79423i	−0.318997	+	0.552518i	0.661282	+	0.750137i	$0.270013\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000	−	3.46410i	−0.553372	−	0.239617i
$210$		0			0
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	0.668524	−	0.743690i	$-0.266926\pi$
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	−0.978317	+	0.207114i	$0.933593\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	21.0000			1.42557
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	0.986267	+	0.165161i	$0.0528144\pi$
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	−0.350100	+	0.936713i	$0.613852\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$	−7.00000			−0.462573			−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000			−0.462573			−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000	+	1.73205i	0.0655122	+	0.113470i	0.896921	−	0.442191i	$-0.145799\pi$
$233$	1.00000	+	1.73205i	0.0655122	+	0.113470i	−0.831409	+	0.555661i	$0.812465\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−4.50000	+	7.79423i	−0.289870	+	0.502070i	−0.973779	−	0.227498i	$-0.926946\pi$
$241$	−4.50000	+	7.79423i	−0.289870	+	0.502070i	0.683908	+	0.729568i	$0.260279\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.50000	−	2.59808i	0.222700	−	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.00000	+	15.5885i	−0.568075	+	0.983935i	0.428681	+	0.903456i	$0.358978\pi$
$251$	−9.00000	+	15.5885i	−0.568075	+	0.983935i	−0.996756	+	0.0804789i	$0.974355\pi$
$252$		0			0
$253$	4.00000	−	6.92820i	0.251478	−	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.0000	+	24.2487i	−0.873296	+	1.51259i	−0.0147291	+	0.999892i	$0.504689\pi$
$257$	−14.0000	+	24.2487i	−0.873296	+	1.51259i	−0.858567	+	0.512702i	$0.828645\pi$
$258$		0			0
$259$	3.00000			0.186411
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.00000	+	8.66025i	0.308313	+	0.534014i	0.977993	−	0.208635i	$-0.0669022\pi$
$263$	5.00000	+	8.66025i	0.308313	+	0.534014i	−0.669680	+	0.742650i	$0.733569\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$	4.00000	+	6.92820i	0.242983	+	0.420858i	0.961563	−	0.274586i	$-0.0885408\pi$
$271$	4.00000	+	6.92820i	0.242983	+	0.420858i	−0.718580	+	0.695444i	$0.755208\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.00000	−	8.66025i	0.301511	−	0.522233i
$276$		0			0
$277$	−18.0000			−1.08152			−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000			−1.08152			−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.0000	−	20.7846i	0.715860	−	1.23991i	−0.246767	−	0.969075i	$-0.579368\pi$
$281$	12.0000	−	20.7846i	0.715860	−	1.23991i	0.962627	−	0.270831i	$-0.0872985\pi$
$282$		0			0
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	0.630738	−	0.775996i	$-0.282752\pi$
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	−0.987401	+	0.158237i	$0.949419\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	−	10.3923i	−0.354169	−	0.613438i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−22.0000			−1.28525			−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000			−1.28525			−0.642627	+	0.766179i	$0.722155\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.00000	+	3.46410i	0.115663	+	0.200334i
$300$		0			0
$301$	−1.50000	−	2.59808i	−0.0864586	−	0.149751i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6.00000	+	10.3923i	0.342438	+	0.593120i	0.984885	−	0.173210i	$-0.0554140\pi$
$307$	6.00000	+	10.3923i	0.342438	+	0.593120i	−0.642447	+	0.766330i	$0.722081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	30.0000			1.70114			0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000			1.70114			0.850572	+	0.525859i	$0.176256\pi$
$312$		0			0
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	−0.220006	−	0.975499i	$-0.570608\pi$
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	0.954810	−	0.297218i	$-0.0960589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	−0.646872	−	0.762598i	$-0.723923\pi$
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	0.983866	+	0.178908i	$0.0572566\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.00000	−	17.3205i	−0.111283	−	0.963739i
$324$		0			0
$325$	2.50000	+	4.33013i	0.138675	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.00000	+	5.19615i	−0.165395	+	0.286473i
$330$		0			0
$331$	−17.0000			−0.934405			−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000			−0.934405			−0.467202	+	0.884150i	$0.654738\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−10.5000	−	18.1865i	−0.571971	−	0.990684i	−0.996363	−	0.0852050i	$-0.972845\pi$
$337$	−10.5000	−	18.1865i	−0.571971	−	0.990684i	0.424392	−	0.905479i	$-0.360488\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.0000			−0.758143
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.0000	−	27.7128i	−0.858925	−	1.48770i	−0.872955	−	0.487800i	$-0.837799\pi$
$347$	−16.0000	−	27.7128i	−0.858925	−	1.48770i	0.0140303	−	0.999902i	$-0.495534\pi$
$348$		0			0
$349$	−17.0000			−0.909989			−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000			−0.909989			−0.454995	+	0.890494i	$0.650359\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.0000			1.49029			0.745145	−	0.666903i	$-0.232380\pi$
$353$	28.0000			1.49029			0.745145	+	0.666903i	$0.232380\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0000	+	24.2487i	0.738892	+	1.27980i	0.952995	+	0.302987i	$0.0979839\pi$
$359$	14.0000	+	24.2487i	0.738892	+	1.27980i	−0.214103	+	0.976811i	$0.568683\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	−0.645003	−	0.764180i	$-0.723144\pi$
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	0.984301	+	0.176500i	$0.0564774\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000	+	31.1769i	−0.934513	+	1.61862i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	27.0000			1.38690			0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000			1.38690			0.693448	+	0.720506i	$0.256091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.00000	+	5.19615i	0.153293	+	0.265511i	0.932436	−	0.361335i	$-0.117679\pi$
$383$	3.00000	+	5.19615i	0.153293	+	0.265511i	−0.779143	+	0.626846i	$0.784346\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.00000	−	13.8564i	0.405616	−	0.702548i	−0.588777	−	0.808296i	$-0.700390\pi$
$389$	8.00000	−	13.8564i	0.405616	−	0.702548i	0.994393	+	0.105748i	$0.0337237\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.50000	−	9.52628i	−0.276037	−	0.478110i	0.694359	−	0.719629i	$-0.255688\pi$
$397$	−5.50000	−	9.52628i	−0.276037	−	0.478110i	−0.970396	+	0.241518i	$0.922355\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.00000	+	1.73205i	0.0499376	+	0.0864945i	0.889914	−	0.456129i	$-0.150764\pi$
$401$	1.00000	+	1.73205i	0.0499376	+	0.0864945i	−0.839976	+	0.542623i	$0.817431\pi$
$402$		0			0
$403$	3.50000	−	6.06218i	0.174347	−	0.301979i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.00000			−0.0991363
$408$		0			0
$409$	−17.0000	+	29.4449i	−0.840596	+	1.45595i	0.0487958	+	0.998809i	$0.484462\pi$
$409$	−17.0000	+	29.4449i	−0.840596	+	1.45595i	−0.889392	+	0.457146i	$0.848872\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.00000	−	5.19615i	−0.147620	−	0.255686i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	20.0000			0.970143
$426$		0			0
$427$	−1.50000	+	2.59808i	−0.0725901	+	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	−0.997173	−	0.0751385i	$-0.976060\pi$
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	0.563658	+	0.826008i	$0.309393\pi$
$432$		0			0
$433$	0.500000	−	0.866025i	0.0240285	−	0.0416185i	−0.853761	−	0.520665i	$-0.825684\pi$
$433$	0.500000	−	0.866025i	0.0240285	−	0.0416185i	0.877790	+	0.479046i	$0.159017\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.0000	+	10.3923i	−0.669711	+	0.497131i
$438$		0			0
$439$	−3.50000	−	6.06218i	−0.167046	−	0.289332i	0.770334	−	0.637641i	$-0.220089\pi$
$439$	−3.50000	−	6.06218i	−0.167046	−	0.289332i	−0.937380	+	0.348309i	$0.886756\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	0.426414	+	0.904528i	$0.359777\pi$
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	−0.996551	+	0.0829786i	$0.973557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.0000			0.943858			0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000			0.943858			0.471929	+	0.881636i	$0.343558\pi$
$450$		0			0
$451$	4.00000	+	6.92820i	0.188353	+	0.326236i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−35.0000			−1.63723			−0.818615	−	0.574342i	$-0.805258\pi$
$457$	−35.0000			−1.63723			−0.818615	+	0.574342i	$0.805258\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.00000	+	12.1244i	0.326023	+	0.564688i	0.981719	−	0.190337i	$-0.0609581\pi$
$461$	7.00000	+	12.1244i	0.326023	+	0.564688i	−0.655696	+	0.755025i	$0.727625\pi$
$462$		0			0
$463$	−37.0000			−1.71954			−0.859768	−	0.510685i	$-0.829392\pi$
$463$	−37.0000			−1.71954			−0.859768	+	0.510685i	$0.829392\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$	−19.5000	+	33.7750i	−0.900426	+	1.55958i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.00000	+	1.73205i	0.0459800	+	0.0796398i
$474$		0			0
$475$	−17.5000	+	12.9904i	−0.802955	+	0.596040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	−0.995023	−	0.0996406i	$-0.968231\pi$
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	0.583803	+	0.811895i	$0.301564\pi$
$480$		0			0
$481$	0.500000	−	0.866025i	0.0227980	−	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	0.975898	+	0.218229i	$0.0700279\pi$
$491$	15.0000	+	25.9808i	0.676941	+	1.17250i	−0.298957	+	0.954267i	$0.596639\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.0000	+	25.9808i	0.672842	+	1.16540i
$498$		0			0
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	0.804627	−	0.593780i	$-0.202365\pi$
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	−0.916542	+	0.399937i	$0.869032\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	−0.978421	−	0.206623i	$-0.933753\pi$
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	0.668151	+	0.744026i	$0.267086\pi$
$510$		0			0
$511$	−4.50000	−	7.79423i	−0.199068	−	0.344796i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.00000	−	3.46410i	0.0879599	−	0.152351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	9.50000	−	16.4545i	0.415406	−	0.719504i	−0.580065	−	0.814570i	$-0.696973\pi$
$523$	9.50000	−	16.4545i	0.415406	−	0.719504i	0.995471	+	0.0950659i	$0.0303062\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.0000	−	24.2487i	−0.609850	−	1.05629i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.00000			−0.173259
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.00000			0.172292
$540$		0			0
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.131765	−	0.991281i	$-0.457935\pi$
$541$	21.5000	−	37.2391i	0.924357	−	1.60103i	0.792592	−	0.609753i	$-0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−17.5000	+	30.3109i	−0.748246	+	1.29600i	0.200417	+	0.979711i	$0.435770\pi$
$547$	−17.5000	+	30.3109i	−0.748246	+	1.29600i	−0.948663	+	0.316289i	$0.897563\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	19.5000	+	33.7750i	0.829224	+	1.43626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	−0.922557	−	0.385860i	$-0.873905\pi$
$557$	−3.00000	+	5.19615i	−0.127114	+	0.220168i	0.795443	+	0.606028i	$0.207238\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.0000			0.927189			0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000			0.927189			0.463595	+	0.886047i	$0.346559\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.0000	−	17.3205i	−0.417029	−	0.722315i
$576$		0			0
$577$	−14.0000			−0.582828			−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000			−0.582828			−0.291414	+	0.956597i	$0.594126\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$	12.0000	−	20.7846i	0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.00000	+	12.1244i	0.288921	+	0.500426i	0.973552	−	0.228464i	$-0.0733702\pi$
$587$	7.00000	+	12.1244i	0.288921	+	0.500426i	−0.684632	+	0.728889i	$0.740037\pi$
$588$		0			0
$589$	28.0000	+	12.1244i	1.15372	+	0.499575i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−23.0000	+	39.8372i	−0.944497	+	1.63592i	−0.187741	+	0.982219i	$0.560117\pi$
$593$	−23.0000	+	39.8372i	−0.944497	+	1.63592i	−0.756756	+	0.653698i	$0.773217\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	0.157731	+	0.987482i	$0.449582\pi$
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	−0.934050	+	0.357142i	$0.883751\pi$
$600$		0			0
$601$	35.0000			1.42768			0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000			1.42768			0.713840	+	0.700309i	$0.246954\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	9.00000			0.365299			0.182649	−	0.983178i	$-0.441533\pi$
$607$	9.00000			0.365299			0.182649	+	0.983178i	$0.441533\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.00000	+	1.73205i	0.0404557	+	0.0700713i
$612$		0			0
$613$	23.0000	+	39.8372i	0.928961	+	1.60901i	0.785063	+	0.619416i	$0.212630\pi$
$613$	23.0000	+	39.8372i	0.928961	+	1.60901i	0.143898	+	0.989593i	$0.454036\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	0.0398207	+	0.999207i	$0.487321\pi$
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	−0.885249	+	0.465118i	$0.846012\pi$
$618$		0			0
$619$	−31.0000			−1.24600			−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000			−1.24600			−0.622998	+	0.782224i	$0.714085\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−27.0000	+	46.7654i	−1.08173	+	1.87362i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.00000	−	3.46410i	−0.0797452	−	0.138123i
$630$		0			0
$631$	−14.5000	+	25.1147i	−0.577236	+	0.999802i	0.418559	+	0.908190i	$0.362535\pi$
$631$	−14.5000	+	25.1147i	−0.577236	+	0.999802i	−0.995795	+	0.0916122i	$0.970798\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.00000	+	1.73205i	−0.0396214	+	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	0.987200	−	0.159489i	$-0.0509845\pi$
$641$	9.00000	+	15.5885i	0.355479	+	0.615707i	−0.631721	+	0.775196i	$0.717651\pi$
$642$		0			0
$643$	17.5000	+	30.3109i	0.690133	+	1.19534i	0.971794	+	0.235831i	$0.0757812\pi$
$643$	17.5000	+	30.3109i	0.690133	+	1.19534i	−0.281661	+	0.959514i	$0.590886\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	2.00000	+	3.46410i	0.0785069	+	0.135978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.0000			−1.17399			−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000			−1.17399			−0.586995	+	0.809590i	$0.699689\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	25.0000	−	43.3013i	0.973862	−	1.68678i	0.290220	−	0.956960i	$-0.406271\pi$
$659$	25.0000	−	43.3013i	0.973862	−	1.68678i	0.683641	−	0.729818i	$-0.260395\pi$
$660$		0			0
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	19.0000	−	32.9090i	0.739014	−	1.28001i	0.952940	−	0.303160i	$-0.0980418\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.00000	−	1.73205i	0.0386046	−	0.0668651i
$672$		0			0
$673$	39.0000			1.50334			0.751670	−	0.659540i	$-0.229249\pi$
$673$	39.0000			1.50334			0.751670	+	0.659540i	$0.229249\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−42.0000			−1.61419			−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000			−1.61419			−0.807096	+	0.590421i	$0.798962\pi$
$678$		0			0
$679$	3.00000	+	5.19615i	0.115129	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−8.00000	+	13.8564i	−0.303022	+	0.524849i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.0000	+	17.3205i	0.377695	+	0.654187i	0.990726	−	0.135872i	$-0.0433835\pi$
$701$	10.0000	+	17.3205i	0.377695	+	0.654187i	−0.613032	+	0.790058i	$0.710050\pi$
$702$		0			0
$703$	4.00000	+	1.73205i	0.150863	+	0.0653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.00000	−	5.19615i	0.112827	−	0.195421i
$708$		0			0
$709$	−18.5000	+	32.0429i	−0.694782	+	1.20340i	0.275472	+	0.961309i	$0.411166\pi$
$709$	−18.5000	+	32.0429i	−0.694782	+	1.20340i	−0.970254	+	0.242089i	$0.922167\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−14.0000	+	24.2487i	−0.524304	+	0.908121i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.0000	+	20.7846i	0.447524	+	0.775135i	0.998224	−	0.0595683i	$-0.0189724\pi$
$719$	12.0000	+	20.7846i	0.447524	+	0.775135i	−0.550700	+	0.834703i	$0.685639\pi$
$720$		0			0
$721$	45.0000			1.67589
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−14.5000	−	25.1147i	−0.537775	−	0.931454i	−0.999023	−	0.0441829i	$-0.985932\pi$
$727$	−14.5000	−	25.1147i	−0.537775	−	0.931454i	0.461248	−	0.887271i	$-0.347402\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.00000	+	3.46410i	−0.0739727	+	0.128124i
$732$		0			0
$733$	−18.0000			−0.664845			−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000			−0.664845			−0.332423	+	0.943131i	$0.607866\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.0000	−	22.5167i	0.478861	−	0.829412i
$738$		0			0
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	0.746955	−	0.664875i	$-0.231515\pi$
$739$	−5.50000	−	9.52628i	−0.202321	−	0.350430i	−0.949276	+	0.314445i	$0.898182\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.0000	−	22.5167i	−0.476924	−	0.826056i	0.522727	−	0.852500i	$-0.324915\pi$
$743$	−13.0000	−	22.5167i	−0.476924	−	0.826056i	−0.999650	+	0.0264443i	$0.991582\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000			0.657706
$750$		0			0
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	−0.795075	−	0.606511i	$-0.792568\pi$
$751$	3.50000	−	6.06218i	0.127717	−	0.221212i	0.922792	+	0.385299i	$0.125902\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	15.5000	+	26.8468i	0.563357	+	0.975763i	0.997200	+	0.0747748i	$0.0238238\pi$
$757$	15.5000	+	26.8468i	0.563357	+	0.975763i	−0.433843	+	0.900988i	$0.642843\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14.0000			−0.507500			−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000			−0.507500			−0.253750	+	0.967270i	$0.581664\pi$
$762$		0			0
$763$	−3.00000	−	5.19615i	−0.108607	−	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000			−0.0722158
$768$		0			0
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	−0.311577	−	0.950221i	$-0.600857\pi$
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	0.978704	−	0.205277i	$-0.0658095\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	−0.290560	−	0.956857i	$-0.593841\pi$
$773$	19.0000	−	32.9090i	0.683383	−	1.18365i	0.973942	−	0.226796i	$-0.0728252\pi$
$774$		0			0
$775$	−17.5000	+	30.3109i	−0.628619	+	1.08880i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.00000	−	17.3205i	−0.0716574	−	0.620572i
$780$		0			0
$781$	−10.0000	−	17.3205i	−0.357828	−	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−23.0000			−0.819861			−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000			−0.819861			−0.409931	+	0.912117i	$0.634447\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−18.0000			−0.640006
$792$		0			0
$793$	0.500000	+	0.866025i	0.0177555	+	0.0307535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	+	5.19615i	0.105868	+	0.183368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	6.00000	−	10.3923i	0.210688	−	0.364923i	−0.741242	−	0.671238i	$-0.765763\pi$
$811$	6.00000	−	10.3923i	0.210688	−	0.364923i	0.951930	+	0.306315i	$0.0990961\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.500000	−	4.33013i	−0.0174928	−	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16.0000	−	27.7128i	0.558404	−	0.967184i	−0.439226	−	0.898377i	$-0.644747\pi$
$821$	16.0000	−	27.7128i	0.558404	−	0.967184i	0.997630	−	0.0688073i	$-0.0219194\pi$
$822$		0			0
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.819159	−	0.573567i	$-0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.00000	+	10.3923i	−0.208640	+	0.361376i	−0.951286	−	0.308308i	$-0.900237\pi$
$827$	−6.00000	+	10.3923i	−0.208640	+	0.361376i	0.742646	+	0.669684i	$0.233571\pi$
$828$		0			0
$829$	−27.0000			−0.937749			−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000			−0.937749			−0.468874	+	0.883265i	$0.655340\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.00000	+	6.92820i	0.138592	+	0.240048i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.0000	−	24.2487i	−0.483334	−	0.837158i	0.516483	−	0.856297i	$-0.327241\pi$
$839$	−14.0000	−	24.2487i	−0.483334	−	0.837158i	−0.999817	+	0.0191389i	$0.993908\pi$
$840$		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
$847$	21.0000			0.721569
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.00000	+	3.46410i	−0.0685591	+	0.118748i
$852$		0			0
$853$	−7.50000	−	12.9904i	−0.256795	−	0.444782i	0.708586	−	0.705624i	$-0.249333\pi$
$853$	−7.50000	−	12.9904i	−0.256795	−	0.444782i	−0.965382	+	0.260842i	$0.916000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.0000	−	32.9090i	−0.649028	−	1.12415i	−0.983355	−	0.181692i	$-0.941843\pi$
$857$	−19.0000	−	32.9090i	−0.649028	−	1.12415i	0.334328	−	0.942457i	$-0.391491\pi$
$858$		0			0
$859$	−15.5000	+	26.8468i	−0.528853	+	0.916001i	0.470581	+	0.882357i	$0.344044\pi$
$859$	−15.5000	+	26.8468i	−0.528853	+	0.916001i	−0.999434	+	0.0336436i	$0.989289\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.0000			−0.612727			−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000			−0.612727			−0.306364	+	0.951915i	$0.599112\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13.0000	−	22.5167i	−0.440995	−	0.763825i
$870$		0			0
$871$	6.50000	+	11.2583i	0.220244	+	0.381474i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	28.0000			0.943344			0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000			0.943344			0.471672	+	0.881774i	$0.343651\pi$
$882$		0			0
$883$	0.500000	−	0.866025i	0.0168263	−	0.0291441i	−0.857490	−	0.514501i	$-0.827977\pi$
$883$	0.500000	−	0.866025i	0.0168263	−	0.0291441i	0.874316	+	0.485357i	$0.161310\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	−0.261540	−	0.965193i	$-0.584230\pi$
$887$	21.0000	−	36.3731i	0.705111	−	1.22129i	0.966651	−	0.256096i	$-0.0824362\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.00000	+	5.19615i	−0.234246	+	0.173883i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	48.0000			1.59911
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	0.999195	−	0.0401089i	$-0.0127705\pi$
$907$	14.0000	+	24.2487i	0.464862	+	0.805165i	−0.534333	+	0.845274i	$0.679437\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−20.0000			−0.662630			−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000			−0.662630			−0.331315	+	0.943520i	$0.607492\pi$
$912$		0			0
$913$	−16.0000			−0.529523
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.00000	+	15.5885i	0.297206	+	0.514776i
$918$		0			0
$919$	19.0000			0.626752			0.313376	−	0.949629i	$-0.398540\pi$
$919$	19.0000			0.626752			0.313376	+	0.949629i	$0.398540\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.0000			0.329154
$924$		0			0
$925$	−2.50000	+	4.33013i	−0.0821995	+	0.142374i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.0000	−	17.3205i	−0.328089	−	0.568267i	0.654043	−	0.756457i	$-0.273071\pi$
$929$	−10.0000	−	17.3205i	−0.328089	−	0.568267i	−0.982133	+	0.188190i	$0.939738\pi$
$930$		0			0
$931$	−8.00000	−	3.46410i	−0.262189	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−3.50000	+	6.06218i	−0.114340	+	0.198043i	−0.917516	−	0.397699i	$-0.869809\pi$
$937$	−3.50000	+	6.06218i	−0.114340	+	0.198043i	0.803176	+	0.595742i	$0.203142\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−24.0000	+	41.5692i	−0.782378	+	1.35512i	0.148176	+	0.988961i	$0.452660\pi$
$941$	−24.0000	+	41.5692i	−0.782378	+	1.35512i	−0.930553	+	0.366157i	$0.880673\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.0000	−	19.0526i	−0.357452	−	0.619125i	0.630082	−	0.776528i	$-0.283021\pi$
$947$	−11.0000	−	19.0526i	−0.357452	−	0.619125i	−0.987534	+	0.157403i	$0.949688\pi$
$948$		0			0
$949$	−3.00000			−0.0973841
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15.0000	−	25.9808i	−0.485898	−	0.841599i	0.513971	−	0.857808i	$-0.328174\pi$
$953$	−15.0000	−	25.9808i	−0.485898	−	0.841599i	−0.999869	+	0.0162081i	$0.994841\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.0000	+	25.9808i	−0.484375	+	0.838963i
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	0.619721	−	0.784822i	$-0.287246\pi$
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	−0.989536	+	0.144283i	$0.953912\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.0000	−	36.3731i	−0.673922	−	1.16727i	−0.976783	−	0.214232i	$-0.931275\pi$
$971$	−21.0000	−	36.3731i	−0.673922	−	1.16727i	0.302861	−	0.953035i	$-0.402058\pi$
$972$		0			0
$973$	−4.50000	+	7.79423i	−0.144263	+	0.249871i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.0000			1.34370			0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000			1.34370			0.671850	+	0.740688i	$0.265500\pi$
$978$		0			0
$979$	18.0000	−	31.1769i	0.575282	−	0.996419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.00000	+	1.73205i	0.0318950	+	0.0552438i	0.881532	−	0.472124i	$-0.156512\pi$
$983$	1.00000	+	1.73205i	0.0318950	+	0.0552438i	−0.849637	+	0.527368i	$0.823179\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.00000			0.127193
$990$		0			0
$991$	2.50000	+	4.33013i	0.0794151	+	0.137551i	0.902998	−	0.429645i	$-0.141361\pi$
$991$	2.50000	+	4.33013i	0.0794151	+	0.137551i	−0.823583	+	0.567196i	$0.808028\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	0.443732	+	0.896159i	$0.353654\pi$
$997$	−17.5000	+	30.3109i	−0.554231	+	0.959955i	−0.997963	+	0.0637961i	$0.979679\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.i.577.1		2
3.2	odd	2		912.2.q.f.577.1		2
4.3	odd	2		1368.2.s.d.577.1		2
12.11	even	2		456.2.q.b.121.1	yes	2
19.11	even	3	inner	2736.2.s.i.1873.1		2
57.11	odd	6		912.2.q.f.49.1		2
76.11	odd	6		1368.2.s.d.505.1		2
228.11	even	6		456.2.q.b.49.1	✓	2
228.83	even	6		8664.2.a.k.1.1		1
228.107	odd	6		8664.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.b.49.1	✓	2	228.11	even	6
456.2.q.b.121.1	yes	2	12.11	even	2
912.2.q.f.49.1		2	57.11	odd	6
912.2.q.f.577.1		2	3.2	odd	2
1368.2.s.d.505.1		2	76.11	odd	6
1368.2.s.d.577.1		2	4.3	odd	2
2736.2.s.i.577.1		2	1.1	even	1	trivial
2736.2.s.i.1873.1		2	19.11	even	3	inner
8664.2.a.d.1.1		1	228.107	odd	6
8664.2.a.k.1.1		1	228.83	even	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{7} +O(q^{10})\)	\(q-3.00000 q^{7} +2.00000 q^{11} +(-0.500000 + 0.866025i) q^{13} +(2.00000 + 3.46410i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} -7.00000 q^{31} -1.00000 q^{37} +(2.00000 + 3.46410i) q^{41} +(0.500000 + 0.866025i) q^{43} +(1.00000 - 1.73205i) q^{47} +2.00000 q^{49} +(6.00000 - 10.3923i) q^{53} +(1.00000 + 1.73205i) q^{59} +(0.500000 - 0.866025i) q^{61} +(6.50000 - 11.2583i) q^{67} +(-5.00000 - 8.66025i) q^{71} +(1.50000 + 2.59808i) q^{73} -6.00000 q^{77} +(-6.50000 - 11.2583i) q^{79} -8.00000 q^{83} +(9.00000 - 15.5885i) q^{89} +(1.50000 - 2.59808i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7}+O(q^{10}) \) 2 * q - 6 * q^7	\( 2 q - 6 q^{7} + 4 q^{11} - q^{13} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 5 q^{25} - 14 q^{31} - 2 q^{37} + 4 q^{41} + q^{43} + 2 q^{47} + 4 q^{49} + 12 q^{53} + 2 q^{59} + q^{61} + 13 q^{67} - 10 q^{71} + 3 q^{73} - 12 q^{77} - 13 q^{79} - 16 q^{83} + 18 q^{89} + 3 q^{91} - 2 q^{97}+O(q^{100}) \) 2 * q - 6 * q^7 + 4 * q^11 - q^13 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 5 * q^25 - 14 * q^31 - 2 * q^37 + 4 * q^41 + q^43 + 2 * q^47 + 4 * q^49 + 12 * q^53 + 2 * q^59 + q^61 + 13 * q^67 - 10 * q^71 + 3 * q^73 - 12 * q^77 - 13 * q^79 - 16 * q^83 + 18 * q^89 + 3 * q^91 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more