LMFDB - Embedded newform 2736.2.s.z.1873.1

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:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.1
Root			$-1.62241 + 0.606458i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.z.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.33641 + 2.31473i) q^{5} +3.67282 q^{7} +O(q^{10})$	$q+(-1.33641 + 2.31473i) q^{5} +3.67282 q^{7} -3.81681 q^{11} +(-0.0719933 - 0.124696i) q^{13} +(-4.24482 - 0.990721i) q^{19} +(-3.76442 - 6.52016i) q^{23} +(-1.07199 - 1.85675i) q^{25} +(-2.67282 - 4.62947i) q^{29} -8.81681 q^{31} +(-4.90841 + 8.50161i) q^{35} -1.00000 q^{37} +(2.67282 - 4.62947i) q^{41} +(1.40841 - 2.43943i) q^{43} +(3.00000 + 5.19615i) q^{47} +6.48963 q^{49} +(4.00924 + 6.94420i) q^{53} +(5.10083 - 8.83490i) q^{55} +(-1.90841 + 3.30545i) q^{59} +(-5.74482 - 9.95031i) q^{61} +0.384851 q^{65} +(2.69243 + 4.66342i) q^{67} +(6.81681 - 11.8071i) q^{71} +(0.172824 - 0.299339i) q^{73} -14.0185 q^{77} +(3.26442 - 5.65414i) q^{79} -2.28797 q^{83} +(-4.33641 - 7.51089i) q^{89} +(-0.264419 - 0.457986i) q^{91} +(7.96608 - 8.50161i) q^{95} +(2.95684 - 5.12140i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 + 2 * q^7	$ 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	−1.33641	+	2.31473i	−0.597662	+	1.03518i	0.395504	+	0.918464i	$0.370570\pi$
$5$	−1.33641	+	2.31473i	−0.597662	+	1.03518i	−0.993165	+	0.116716i	$0.962763\pi$
$6$		0			0
$7$	3.67282			1.38820			0.694098	−	0.719880i	$-0.255803\pi$
$7$	3.67282			1.38820			0.694098	+	0.719880i	$0.255803\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.81681			−1.15081			−0.575406	−	0.817868i	$-0.695156\pi$
$11$	−3.81681			−1.15081			−0.575406	+	0.817868i	$0.695156\pi$
$12$		0			0
$13$	−0.0719933	−	0.124696i	−0.0199673	−	0.0345844i	0.855869	−	0.517193i	$-0.173023\pi$
$13$	−0.0719933	−	0.124696i	−0.0199673	−	0.0345844i	−0.875836	+	0.482608i	$0.839690\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−4.24482	−	0.990721i	−0.973828	−	0.227287i
$19$	−4.24482	−	0.990721i	−0.973828	−	0.227287i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.76442	−	6.52016i	−0.784936	−	1.35955i	−0.929038	−	0.369985i	$-0.879363\pi$
$23$	−3.76442	−	6.52016i	−0.784936	−	1.35955i	0.144102	−	0.989563i	$-0.453971\pi$
$24$		0			0
$25$	−1.07199	−	1.85675i	−0.214399	−	0.371349i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.67282	−	4.62947i	−0.496331	−	0.859670i	0.503660	−	0.863902i	$-0.331986\pi$
$29$	−2.67282	−	4.62947i	−0.496331	−	0.859670i	−0.999991	+	0.00423154i	$0.998653\pi$
$30$		0			0
$31$	−8.81681			−1.58355			−0.791773	−	0.610816i	$-0.790842\pi$
$31$	−8.81681			−1.58355			−0.791773	+	0.610816i	$0.790842\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.90841	+	8.50161i	−0.829672	+	1.43703i
$36$		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
$41$	2.67282	−	4.62947i	0.417425	−	0.723001i	−0.578255	−	0.815856i	$-0.696266\pi$
$41$	2.67282	−	4.62947i	0.417425	−	0.723001i	0.995680	+	0.0928551i	$0.0295993\pi$
$42$		0			0
$43$	1.40841	−	2.43943i	0.214780	−	0.372009i	−0.738425	−	0.674336i	$-0.764430\pi$
$43$	1.40841	−	2.43943i	0.214780	−	0.372009i	0.953204	+	0.302327i	$0.0977633\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	6.48963			0.927091
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.00924	+	6.94420i	0.550711	+	0.953859i	0.998223	+	0.0595815i	$0.0189766\pi$
$53$	4.00924	+	6.94420i	0.550711	+	0.953859i	−0.447513	+	0.894278i	$0.647690\pi$
$54$		0			0
$55$	5.10083	−	8.83490i	0.687796	−	1.19130i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.90841	+	3.30545i	−0.248453	+	0.430334i	−0.963097	−	0.269155i	$-0.913256\pi$
$59$	−1.90841	+	3.30545i	−0.248453	+	0.430334i	0.714644	+	0.699489i	$0.246589\pi$
$60$		0			0
$61$	−5.74482	−	9.95031i	−0.735548	−	1.27401i	−0.954482	−	0.298268i	$-0.903591\pi$
$61$	−5.74482	−	9.95031i	−0.735548	−	1.27401i	0.218934	−	0.975740i	$-0.429742\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.384851			0.0477348
$66$		0			0
$67$	2.69243	+	4.66342i	0.328932	+	0.569727i	0.982300	−	0.187313i	$-0.0599779\pi$
$67$	2.69243	+	4.66342i	0.328932	+	0.569727i	−0.653368	+	0.757040i	$0.726645\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.81681	−	11.8071i	0.809007	−	1.40124i	−0.104546	−	0.994520i	$-0.533339\pi$
$71$	6.81681	−	11.8071i	0.809007	−	1.40124i	0.913553	−	0.406720i	$-0.133328\pi$
$72$		0			0
$73$	0.172824	−	0.299339i	0.0202275	−	0.0350350i	−0.855734	−	0.517415i	$-0.826894\pi$
$73$	0.172824	−	0.299339i	0.0202275	−	0.0350350i	0.875962	+	0.482380i	$0.160228\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.0185			−1.59755
$78$		0			0
$79$	3.26442	−	5.65414i	0.367276	−	0.636140i	−0.621863	−	0.783126i	$-0.713624\pi$
$79$	3.26442	−	5.65414i	0.367276	−	0.636140i	0.989139	+	0.146986i	$0.0469572\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.28797			−0.251138			−0.125569	−	0.992085i	$-0.540076\pi$
$83$	−2.28797			−0.251138			−0.125569	+	0.992085i	$0.540076\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.33641	−	7.51089i	−0.459659	−	0.796152i	0.539284	−	0.842124i	$-0.318695\pi$
$89$	−4.33641	−	7.51089i	−0.459659	−	0.796152i	−0.998943	+	0.0459717i	$0.985362\pi$
$90$		0			0
$91$	−0.264419	−	0.457986i	−0.0277186	−	0.0480100i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.96608	−	8.50161i	0.817303	−	0.872246i
$96$		0			0
$97$	2.95684	−	5.12140i	0.300222	−	0.520000i	−0.675964	−	0.736935i	$-0.736273\pi$
$97$	2.95684	−	5.12140i	0.300222	−	0.520000i	0.976186	+	0.216935i	$0.0696059\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.14399	+	7.17760i	0.412342	+	0.714197i	0.995145	−	0.0984158i	$-0.0313775\pi$
$101$	4.14399	+	7.17760i	0.412342	+	0.714197i	−0.582803	+	0.812613i	$0.698044\pi$
$102$		0			0
$103$	−14.6521			−1.44371			−0.721857	−	0.692043i	$-0.756711\pi$
$103$	−14.6521			−1.44371			−0.721857	+	0.692043i	$0.756711\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.6913			−1.61361			−0.806804	−	0.590819i	$-0.798805\pi$
$107$	−16.6913			−1.61361			−0.806804	+	0.590819i	$0.798805\pi$
$108$		0			0
$109$	−3.91764	+	6.78555i	−0.375242	+	0.649938i	−0.990363	−	0.138494i	$-0.955774\pi$
$109$	−3.91764	+	6.78555i	−0.375242	+	0.649938i	0.615121	+	0.788432i	$0.289107\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.38485			0.600636			0.300318	−	0.953839i	$-0.402907\pi$
$113$	6.38485			0.600636			0.300318	+	0.953839i	$0.402907\pi$
$114$		0			0
$115$	20.1233			1.87650
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.56804			0.324367
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.63362			−0.682772
$126$		0			0
$127$	1.32718	+	2.29874i	0.117768	+	0.203980i	0.918883	−	0.394531i	$-0.129093\pi$
$127$	1.32718	+	2.29874i	0.117768	+	0.203980i	−0.801115	+	0.598511i	$0.795759\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.67282	−	9.82562i	0.495637	−	0.858468i	−0.504350	−	0.863499i	$-0.668268\pi$
$131$	5.67282	−	9.82562i	0.495637	−	0.858468i	0.999987	+	0.00503076i	$0.00160135\pi$
$132$		0			0
$133$	−15.5905	−	3.63875i	−1.35186	−	0.315519i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.816810	−	1.41476i	−0.0697848	−	0.120871i	0.829022	−	0.559216i	$-0.188898\pi$
$137$	−0.816810	−	1.41476i	−0.0697848	−	0.120871i	−0.898806	+	0.438346i	$0.855565\pi$
$138$		0			0
$139$	3.75405	+	6.50221i	0.318415	+	0.551510i	0.980157	−	0.198221i	$-0.0635163\pi$
$139$	3.75405	+	6.50221i	0.318415	+	0.551510i	−0.661743	+	0.749731i	$0.730183\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.274785	+	0.475941i	0.0229786	+	0.0398002i
$144$		0			0
$145$	14.2880			1.18655
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.00924	−	12.1404i	0.574219	−	0.994576i	−0.421907	−	0.906639i	$-0.638639\pi$
$149$	7.00924	−	12.1404i	0.574219	−	0.994576i	0.996126	−	0.0879373i	$-0.0280275\pi$
$150$		0			0
$151$	−11.0577			−0.899861			−0.449930	−	0.893064i	$-0.648551\pi$
$151$	−11.0577			−0.899861			−0.449930	+	0.893064i	$0.648551\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.7829	−	20.4086i	0.946424	−	1.63926i
$156$		0			0
$157$	−7.02884	+	12.1743i	−0.560962	+	0.971615i	0.436451	+	0.899728i	$0.356235\pi$
$157$	−7.02884	+	12.1743i	−0.560962	+	0.971615i	−0.997413	+	0.0718869i	$0.977098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.8260	−	23.9474i	−1.08965	−	1.88732i
$162$		0			0
$163$	−4.61515			−0.361486			−0.180743	−	0.983530i	$-0.557850\pi$
$163$	−4.61515			−0.361486			−0.180743	+	0.983530i	$0.557850\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.11007	+	10.5829i	0.472811	+	0.818933i	0.999516	−	0.0311155i	$-0.00990596\pi$
$167$	6.11007	+	10.5829i	0.472811	+	0.818933i	−0.526705	+	0.850048i	$0.676573\pi$
$168$		0			0
$169$	6.48963	−	11.2404i	0.499203	−	0.864644i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.52884	−	13.0403i	0.572407	−	0.991438i	−0.423911	−	0.905704i	$-0.639343\pi$
$173$	7.52884	−	13.0403i	0.572407	−	0.991438i	0.996318	−	0.0857340i	$-0.0273235\pi$
$174$		0			0
$175$	−3.93724	−	6.81950i	−0.297628	−	0.515506i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.1625			−1.13330			−0.566648	−	0.823960i	$-0.691760\pi$
$179$	−15.1625			−1.13330			−0.566648	+	0.823960i	$0.691760\pi$
$180$		0			0
$181$	−6.10083	−	10.5669i	−0.453471	−	0.785435i	0.545128	−	0.838353i	$-0.316481\pi$
$181$	−6.10083	−	10.5669i	−0.453471	−	0.785435i	−0.998599	+	0.0529179i	$0.983148\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.33641	−	2.31473i	0.0982550	−	0.170183i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.45043			0.394379			0.197190	−	0.980365i	$-0.436819\pi$
$191$	5.45043			0.394379			0.197190	+	0.980365i	$0.436819\pi$
$192$		0			0
$193$	0.255183	−	0.441990i	0.0183685	−	0.0318151i	−0.856695	−	0.515823i	$-0.827486\pi$
$193$	0.255183	−	0.441990i	0.0183685	−	0.0318151i	0.875064	+	0.484008i	$0.160819\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.9608			−1.63589			−0.817945	−	0.575297i	$-0.804886\pi$
$197$	−22.9608			−1.63589			−0.817945	+	0.575297i	$0.804886\pi$
$198$		0			0
$199$	−0.0627577	−	0.108700i	−0.00444878	−	0.00770551i	0.863792	−	0.503848i	$-0.168083\pi$
$199$	−0.0627577	−	0.108700i	−0.00444878	−	0.00770551i	−0.868241	+	0.496142i	$0.834749\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.81681	−	17.0032i	−0.689005	−	1.19339i
$204$		0			0
$205$	7.14399	+	12.3737i	0.498958	+	0.864220i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.2017	+	3.78140i	1.12069	+	0.261565i
$210$		0			0
$211$	12.4269	−	21.5240i	0.855501	−	1.48177i	−0.0206776	−	0.999786i	$-0.506582\pi$
$211$	12.4269	−	21.5240i	0.855501	−	1.48177i	0.876179	−	0.481986i	$-0.160084\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.76442	+	6.52016i	0.256731	+	0.444672i
$216$		0			0
$217$	−32.3826			−2.19827
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−11.8980	+	20.6080i	−0.796752	+	1.38001i	0.124969	+	0.992161i	$0.460117\pi$
$223$	−11.8980	+	20.6080i	−0.796752	+	1.38001i	−0.921721	+	0.387854i	$0.873217\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.81681			0.651565			0.325782	−	0.945445i	$-0.394372\pi$
$227$	9.81681			0.651565			0.325782	+	0.945445i	$0.394372\pi$
$228$		0			0
$229$	0.143987			0.00951490			0.00475745	−	0.999989i	$-0.498486\pi$
$229$	0.143987			0.00951490			0.00475745	+	0.999989i	$0.498486\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.28797	−	9.15904i	0.346427	−	0.600029i	−0.639185	−	0.769053i	$-0.720728\pi$
$233$	5.28797	−	9.15904i	0.346427	−	0.600029i	0.985612	+	0.169024i	$0.0540616\pi$
$234$		0			0
$235$	−16.0369			−1.04613
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.81681			−0.634997			−0.317498	−	0.948259i	$-0.602843\pi$
$239$	−9.81681			−0.634997			−0.317498	+	0.948259i	$0.602843\pi$
$240$		0			0
$241$	−0.971163	−	1.68210i	−0.0625581	−	0.108354i	0.833050	−	0.553198i	$-0.186593\pi$
$241$	−0.971163	−	1.68210i	−0.0625581	−	0.108354i	−0.895608	+	0.444844i	$0.853259\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.67282	+	15.0218i	−0.554086	+	0.959706i
$246$		0			0
$247$	0.182059	+	0.600637i	0.0115842	+	0.0382176i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.81681	−	11.8071i	−0.430273	−	0.745255i	0.566623	−	0.823977i	$-0.308249\pi$
$251$	−6.81681	−	11.8071i	−0.430273	−	0.745255i	−0.996897	+	0.0787218i	$0.974916\pi$
$252$		0			0
$253$	14.3681	+	24.8862i	0.903313	+	1.56458i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.29721	−	10.9071i	−0.392809	−	0.680365i	0.600010	−	0.799993i	$-0.295163\pi$
$257$	−6.29721	−	10.9071i	−0.392809	−	0.680365i	−0.992819	+	0.119627i	$0.961830\pi$
$258$		0			0
$259$	−3.67282			−0.228218
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.816810	−	1.41476i	0.0503667	−	0.0872376i	−0.839743	−	0.542984i	$-0.817294\pi$
$263$	0.816810	−	1.41476i	0.0503667	−	0.0872376i	0.890110	+	0.455747i	$0.150628\pi$
$264$		0			0
$265$	−21.4320			−1.31655
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.8260	+	18.7513i	−0.660076	+	1.14328i	0.320520	+	0.947242i	$0.396142\pi$
$269$	−10.8260	+	18.7513i	−0.660076	+	1.14328i	−0.980595	+	0.196043i	$0.937191\pi$
$270$		0			0
$271$	−10.7776	+	18.6674i	−0.654693	+	1.13396i	0.327278	+	0.944928i	$0.393869\pi$
$271$	−10.7776	+	18.6674i	−0.654693	+	1.13396i	−0.981971	+	0.189033i	$0.939465\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.09159	+	7.08685i	0.246732	+	0.427353i
$276$		0			0
$277$	7.62571			0.458185			0.229092	−	0.973405i	$-0.426424\pi$
$277$	7.62571			0.458185			0.229092	+	0.973405i	$0.426424\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.846778	−	1.46666i	−0.0505145	−	0.0874937i	0.839663	−	0.543109i	$-0.182753\pi$
$281$	−0.846778	−	1.46666i	−0.0505145	−	0.0874937i	−0.890177	+	0.455615i	$0.849419\pi$
$282$		0			0
$283$	5.14399	−	8.90965i	0.305778	−	0.529623i	−0.671656	−	0.740863i	$-0.734417\pi$
$283$	5.14399	−	8.90965i	0.305778	−	0.529623i	0.977434	+	0.211240i	$0.0677500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.81681	−	17.0032i	0.579468	−	1.00367i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.1153			−0.707786			−0.353893	−	0.935286i	$-0.615142\pi$
$293$	−12.1153			−0.707786			−0.353893	+	0.935286i	$0.615142\pi$
$294$		0			0
$295$	−5.10083	−	8.83490i	−0.296982	−	0.514388i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.542026	+	0.938816i	−0.0313461	+	0.0542931i
$300$		0			0
$301$	5.17282	−	8.95959i	0.298157	−	0.516422i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	30.7098			1.75844
$306$		0			0
$307$	−2.24482	+	3.88814i	−0.128118	+	0.221908i	−0.922948	−	0.384926i	$-0.874227\pi$
$307$	−2.24482	+	3.88814i	−0.128118	+	0.221908i	0.794829	+	0.606833i	$0.207560\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.759136			0.0430466			0.0215233	−	0.999768i	$-0.493148\pi$
$311$	0.759136			0.0430466			0.0215233	+	0.999768i	$0.493148\pi$
$312$		0			0
$313$	−5.71598	−	9.90037i	−0.323086	−	0.559602i	0.658037	−	0.752986i	$-0.271387\pi$
$313$	−5.71598	−	9.90037i	−0.323086	−	0.559602i	−0.981123	+	0.193384i	$0.938054\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.80757	−	10.0590i	−0.326186	−	0.564971i	0.655566	−	0.755138i	$-0.272430\pi$
$317$	−5.80757	−	10.0590i	−0.326186	−	0.564971i	−0.981752	+	0.190168i	$0.939097\pi$
$318$		0			0
$319$	10.2017	+	17.6698i	0.571183	+	0.989319i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.154353	+	0.267347i	−0.00856194	+	0.0148297i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.0185	+	19.0846i	0.607468	+	1.05217i
$330$		0			0
$331$	11.4712			0.630512			0.315256	−	0.949007i	$-0.397910\pi$
$331$	11.4712			0.630512			0.315256	+	0.949007i	$0.397910\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.3928			−0.786360
$336$		0			0
$337$	−3.78402	+	6.55412i	−0.206129	+	0.357025i	−0.950492	−	0.310750i	$-0.899420\pi$
$337$	−3.78402	+	6.55412i	−0.206129	+	0.357025i	0.744363	+	0.667775i	$0.232753\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.6521			1.82236
$342$		0			0
$343$	−1.87448			−0.101213
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.9084	+	18.8939i	−0.585594	+	1.01428i	0.409207	+	0.912441i	$0.365805\pi$
$347$	−10.9084	+	18.8939i	−0.585594	+	1.01428i	−0.994801	+	0.101837i	$0.967528\pi$
$348$		0			0
$349$	−4.54731			−0.243412			−0.121706	−	0.992566i	$-0.538836\pi$
$349$	−4.54731			−0.243412			−0.121706	+	0.992566i	$0.538836\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.99774			0.478901			0.239451	−	0.970909i	$-0.423033\pi$
$353$	8.99774			0.478901			0.239451	+	0.970909i	$0.423033\pi$
$354$		0			0
$355$	18.2201	+	31.5582i	0.967024	+	1.67494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.48963	+	11.2404i	−0.342510	+	0.593244i	−0.984898	−	0.173135i	$-0.944610\pi$
$359$	−6.48963	+	11.2404i	−0.342510	+	0.593244i	0.642388	+	0.766379i	$0.277944\pi$
$360$		0			0
$361$	17.0369	+	8.41086i	0.896681	+	0.442677i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.461927	+	0.800082i	0.0241784	+	0.0418782i
$366$		0			0
$367$	−2.02355	−	3.50490i	−0.105629	−	0.182954i	0.808366	−	0.588680i	$-0.200352\pi$
$367$	−2.02355	−	3.50490i	−0.105629	−	0.182954i	−0.913995	+	0.405726i	$0.867019\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	14.7252	+	25.5048i	0.764495	+	1.32414i
$372$		0			0
$373$	3.79834			0.196671			0.0983353	−	0.995153i	$-0.468648\pi$
$373$	3.79834			0.196671			0.0983353	+	0.995153i	$0.468648\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.384851	+	0.666581i	−0.0198208	+	0.0343307i
$378$		0			0
$379$	−4.89522			−0.251450			−0.125725	−	0.992065i	$-0.540126\pi$
$379$	−4.89522			−0.251450			−0.125725	+	0.992065i	$0.540126\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.0709	+	29.5676i	−0.872280	+	1.51083i	−0.0126484	+	0.999920i	$0.504026\pi$
$383$	−17.0709	+	29.5676i	−0.872280	+	1.51083i	−0.859632	+	0.510914i	$0.829307\pi$
$384$		0			0
$385$	18.7345	−	32.4490i	0.954796	−	1.65376i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.4412	+	23.2808i	0.681496	+	1.18039i	0.974524	+	0.224281i	$0.0720035\pi$
$389$	13.4412	+	23.2808i	0.681496	+	1.18039i	−0.293029	+	0.956104i	$0.594663\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.72522	+	15.1125i	0.439013	+	0.760393i
$396$		0			0
$397$	−12.6193	+	21.8573i	−0.633345	+	1.09699i	0.353519	+	0.935427i	$0.384985\pi$
$397$	−12.6193	+	21.8573i	−0.633345	+	1.09699i	−0.986863	+	0.161558i	$0.948348\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.66359	+	2.88142i	−0.0830756	+	0.143891i	−0.904570	−	0.426326i	$-0.859808\pi$
$401$	−1.66359	+	2.88142i	−0.0830756	+	0.143891i	0.821494	+	0.570217i	$0.193141\pi$
$402$		0			0
$403$	0.634751	+	1.09942i	0.0316192	+	0.0547661i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.81681			0.189192
$408$		0			0
$409$	−17.0616	−	29.5516i	−0.843643	−	1.46123i	−0.886794	−	0.462164i	$-0.847073\pi$
$409$	−17.0616	−	29.5516i	−0.843643	−	1.46123i	0.0431512	−	0.999069i	$-0.486260\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.00924	+	12.1404i	−0.344902	+	0.597388i
$414$		0			0
$415$	3.05767	−	5.29605i	0.150095	−	0.259973i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−9.16246			−0.447615			−0.223808	−	0.974633i	$-0.571849\pi$
$419$	−9.16246			−0.447615			−0.223808	+	0.974633i	$0.571849\pi$
$420$		0			0
$421$	7.91764	−	13.7138i	0.385882	−	0.668368i	−0.606009	−	0.795458i	$-0.707230\pi$
$421$	7.91764	−	13.7138i	0.385882	−	0.668368i	0.991891	+	0.127090i	$0.0405638\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−21.0997	−	36.5458i	−1.02109	−	1.76857i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.63362	+	8.02567i	0.223194	+	0.386583i	0.955776	−	0.294096i	$-0.0950184\pi$
$431$	4.63362	+	8.02567i	0.223194	+	0.386583i	−0.732582	+	0.680678i	$0.761685\pi$
$432$		0			0
$433$	9.90727	+	17.1599i	0.476113	+	0.824652i	0.999625	−	0.0273658i	$-0.00871190\pi$
$433$	9.90727	+	17.1599i	0.476113	+	0.824652i	−0.523512	+	0.852018i	$0.675379\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.51960	+	31.4064i	0.455384	+	1.50237i
$438$		0			0
$439$	3.45156	−	5.97828i	0.164734	−	0.285328i	−0.771827	−	0.635833i	$-0.780657\pi$
$439$	3.45156	−	5.97828i	0.164734	−	0.285328i	0.936561	+	0.350505i	$0.113990\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9.81681	+	17.0032i	0.466411	+	0.807847i	0.999264	−	0.0383606i	$-0.0122136\pi$
$443$	9.81681	+	17.0032i	0.466411	+	0.807847i	−0.532853	+	0.846208i	$0.678880\pi$
$444$		0			0
$445$	23.1809			1.09888
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−29.0162			−1.36936			−0.684680	−	0.728844i	$-0.740058\pi$
$449$	−29.0162			−1.36936			−0.684680	+	0.728844i	$0.740058\pi$
$450$		0			0
$451$	−10.2017	+	17.6698i	−0.480377	+	0.832038i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.41349			0.0662654
$456$		0			0
$457$	32.5473			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$457$	32.5473			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.82605	+	8.35896i	−0.224771	+	0.389315i	−0.956251	−	0.292548i	$-0.905497\pi$
$461$	−4.82605	+	8.35896i	−0.224771	+	0.389315i	0.731479	+	0.681863i	$0.238830\pi$
$462$		0			0
$463$	31.0554			1.44327			0.721634	−	0.692275i	$-0.243392\pi$
$463$	31.0554			1.44327			0.721634	+	0.692275i	$0.243392\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.61289			0.398557			0.199278	−	0.979943i	$-0.436140\pi$
$467$	8.61289			0.398557			0.199278	+	0.979943i	$0.436140\pi$
$468$		0			0
$469$	9.88880	+	17.1279i	0.456623	+	0.790893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.37562	+	9.31084i	−0.247171	+	0.428113i
$474$		0			0
$475$	2.71090	+	8.94360i	0.124384	+	0.410360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.73050	−	15.1217i	−0.398907	−	0.690927i	0.594685	−	0.803959i	$-0.297277\pi$
$479$	−8.73050	−	15.1217i	−0.398907	−	0.690927i	−0.993591	+	0.113032i	$0.963944\pi$
$480$		0			0
$481$	0.0719933	+	0.124696i	0.00328261	+	0.00568565i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.90312	+	13.6886i	0.358862	+	0.621568i
$486$		0			0
$487$	11.6336			0.527170			0.263585	−	0.964636i	$-0.415095\pi$
$487$	11.6336			0.527170			0.263585	+	0.964636i	$0.415095\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.1625	−	21.0660i	0.548884	−	0.950695i	−0.449467	−	0.893297i	$-0.648386\pi$
$491$	12.1625	−	21.0660i	0.548884	−	0.950695i	0.998351	−	0.0573983i	$-0.0182805\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.0369	−	43.3653i	1.12306	−	1.94520i
$498$		0			0
$499$	19.0565	−	33.0069i	0.853088	−	1.47759i	−0.0253194	−	0.999679i	$-0.508060\pi$
$499$	19.0565	−	33.0069i	0.853088	−	1.47759i	0.878407	−	0.477912i	$-0.158606\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.18319	−	3.78140i	−0.0973436	−	0.168604i	0.813241	−	0.581927i	$-0.197701\pi$
$503$	−2.18319	−	3.78140i	−0.0973436	−	0.168604i	−0.910584	+	0.413323i	$0.864368\pi$
$504$		0			0
$505$	−22.1523			−0.985764
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.85601	−	13.6070i	−0.348212	−	0.603120i	0.637720	−	0.770268i	$-0.279878\pi$
$509$	−7.85601	−	13.6070i	−0.348212	−	0.603120i	−0.985932	+	0.167148i	$0.946544\pi$
$510$		0			0
$511$	0.634751	−	1.09942i	0.0280797	−	0.0486355i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.5812	−	33.9157i	0.862852	−	1.49450i
$516$		0			0
$517$	−11.4504	−	19.8327i	−0.503589	−	0.872242i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.4425			−1.46514			−0.732572	−	0.680690i	$-0.761680\pi$
$521$	−33.4425			−1.46514			−0.732572	+	0.680690i	$0.761680\pi$
$522$		0			0
$523$	−14.7109	−	25.4800i	−0.643263	−	1.11416i	−0.984700	−	0.174259i	$-0.944247\pi$
$523$	−14.7109	−	25.4800i	−0.643263	−	1.11416i	0.341437	−	0.939905i	$-0.389086\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−16.8417	+	29.1707i	−0.732248	+	1.26829i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.769701			−0.0333395
$534$		0			0
$535$	22.3064	−	38.6359i	0.964392	−	1.67038i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.7697			−1.06691
$540$		0			0
$541$	−13.0865	−	22.6665i	−0.562633	−	0.974509i	−0.997266	−	0.0739012i	$-0.976455\pi$
$541$	−13.0865	−	22.6665i	−0.562633	−	0.974509i	0.434632	−	0.900608i	$-0.356878\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10.4712	−	18.1366i	−0.448535	−	0.776886i
$546$		0			0
$547$	10.9557	+	18.9759i	0.468432	+	0.811349i	0.999349	−	0.0360752i	$-0.0114856\pi$
$547$	10.9557	+	18.9759i	0.468432	+	0.811349i	−0.530917	+	0.847424i	$0.678152\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.75914	+	22.2993i	0.287949	+	0.949980i
$552$		0			0
$553$	11.9896	−	20.7667i	0.509851	−	0.883088i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.0185	+	29.4769i	0.721096	+	1.24897i	0.960561	+	0.278070i	$0.0896946\pi$
$557$	17.0185	+	29.4769i	0.721096	+	1.24897i	−0.239465	+	0.970905i	$0.576972\pi$
$558$		0			0
$559$	−0.405583			−0.0171543
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.5552			0.486994			0.243497	−	0.969902i	$-0.421705\pi$
$563$	11.5552			0.486994			0.243497	+	0.969902i	$0.421705\pi$
$564$		0			0
$565$	−8.53279	+	14.7792i	−0.358977	+	0.621767i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−39.7075			−1.66463			−0.832313	−	0.554307i	$-0.812984\pi$
$569$	−39.7075			−1.66463			−0.832313	+	0.554307i	$0.812984\pi$
$570$		0			0
$571$	−14.6521			−0.613171			−0.306585	−	0.951843i	$-0.599187\pi$
$571$	−14.6521			−0.613171			−0.306585	+	0.951843i	$0.599187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.07086	+	13.9791i	−0.336578	+	0.582971i
$576$		0			0
$577$	20.8145			0.866521			0.433261	−	0.901269i	$-0.357363\pi$
$577$	20.8145			0.866521			0.433261	+	0.901269i	$0.357363\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.40332			−0.348629
$582$		0			0
$583$	−15.3025	−	26.5047i	−0.633764	−	1.09771i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.82209	+	17.0124i	−0.405401	+	0.702175i	−0.994368	−	0.105982i	$-0.966201\pi$
$587$	−9.82209	+	17.0124i	−0.405401	+	0.702175i	0.588967	+	0.808157i	$0.299535\pi$
$588$		0			0
$589$	37.4257	+	8.73500i	1.54210	+	0.359920i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	10.8260	+	18.7513i	0.444572	+	0.770022i	0.998022	−	0.0628605i	$-0.0200223\pi$
$593$	10.8260	+	18.7513i	0.444572	+	0.770022i	−0.553450	+	0.832882i	$0.686689\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.88993	+	10.2017i	0.240656	+	0.416829i	0.960901	−	0.276891i	$-0.0893040\pi$
$599$	5.88993	+	10.2017i	0.240656	+	0.416829i	−0.720245	+	0.693720i	$0.755971\pi$
$600$		0			0
$601$	−11.4112			−0.465474			−0.232737	−	0.972540i	$-0.574768\pi$
$601$	−11.4112			−0.465474			−0.232737	+	0.972540i	$0.574768\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.76837	+	8.25906i	−0.193862	+	0.335779i
$606$		0			0
$607$	1.10478			0.0448418			0.0224209	−	0.999749i	$-0.492863\pi$
$607$	1.10478			0.0448418			0.0224209	+	0.999749i	$0.492863\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.431960	−	0.748176i	0.0174752	−	0.0302680i
$612$		0			0
$613$	−2.38880	+	4.13753i	−0.0964829	+	0.167113i	−0.910227	−	0.414111i	$-0.864093\pi$
$613$	−2.38880	+	4.13753i	−0.0964829	+	0.167113i	0.813744	+	0.581224i	$0.197426\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.7397	+	32.4582i	0.754433	+	1.30672i	0.945656	+	0.325170i	$0.105422\pi$
$617$	18.7397	+	32.4582i	0.754433	+	1.30672i	−0.191222	+	0.981547i	$0.561245\pi$
$618$		0			0
$619$	0.615149			0.0247249			0.0123625	−	0.999924i	$-0.496065\pi$
$619$	0.615149			0.0247249			0.0123625	+	0.999924i	$0.496065\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.9269	−	27.5862i	−0.638097	−	1.10522i
$624$		0			0
$625$	15.5616	−	26.9535i	0.622465	−	1.07814i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	10.6532	+	18.4519i	0.424098	+	0.734559i	0.996336	−	0.0855284i	$-0.0272578\pi$
$631$	10.6532	+	18.4519i	0.424098	+	0.734559i	−0.572238	+	0.820088i	$0.693925\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.09462			−0.281541
$636$		0			0
$637$	−0.467210	−	0.809231i	−0.0185115	−	0.0320629i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.9608	−	34.5731i	0.788404	−	1.36556i	−0.138540	−	0.990357i	$-0.544241\pi$
$641$	19.9608	−	34.5731i	0.788404	−	1.36556i	0.926944	−	0.375199i	$-0.122426\pi$
$642$		0			0
$643$	−18.6924	+	32.3762i	−0.737157	+	1.27679i	0.216613	+	0.976258i	$0.430499\pi$
$643$	−18.6924	+	32.3762i	−0.737157	+	1.27679i	−0.953770	+	0.300536i	$0.902834\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.0369			0.630477			0.315239	−	0.949012i	$-0.397915\pi$
$647$	16.0369			0.630477			0.315239	+	0.949012i	$0.397915\pi$
$648$		0			0
$649$	7.28402	−	12.6163i	0.285923	−	0.495233i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13.4241			−0.525324			−0.262662	−	0.964888i	$-0.584600\pi$
$653$	−13.4241			−0.525324			−0.262662	+	0.964888i	$0.584600\pi$
$654$		0			0
$655$	15.1625	+	26.2621i	0.592446	+	1.02615i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.27252	+	16.0605i	0.361206	+	0.625628i	0.988160	−	0.153429i	$-0.0490317\pi$
$659$	9.27252	+	16.0605i	0.361206	+	0.625628i	−0.626953	+	0.779057i	$0.715698\pi$
$660$		0			0
$661$	−3.28797	−	5.69494i	−0.127887	−	0.221507i	0.794971	−	0.606648i	$-0.207486\pi$
$661$	−3.28797	−	5.69494i	−0.127887	−	0.221507i	−0.922858	+	0.385141i	$0.874153\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	29.2580	−	31.2249i	1.13458	−	1.21085i
$666$		0			0
$667$	−20.1233	+	34.8545i	−0.779176	+	1.34957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.9269	+	37.9785i	0.846478	+	1.46614i
$672$		0			0
$673$	−39.2386			−1.51254			−0.756268	−	0.654261i	$-0.772980\pi$
$673$	−39.2386			−1.51254			−0.756268	+	0.654261i	$0.772980\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.8432			0.993234			0.496617	−	0.867970i	$-0.334575\pi$
$677$	25.8432			0.993234			0.496617	+	0.867970i	$0.334575\pi$
$678$		0			0
$679$	10.8600	−	18.8100i	0.416767	−	0.721862i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	42.9898			1.64496			0.822480	−	0.568794i	$-0.192590\pi$
$683$	42.9898			1.64496			0.822480	+	0.568794i	$0.192590\pi$
$684$		0			0
$685$	4.36638			0.166831
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.577276	−	0.999871i	0.0219925	−	0.0380921i
$690$		0			0
$691$	48.4033			1.84135			0.920675	−	0.390331i	$-0.127639\pi$
$691$	48.4033			1.84135			0.920675	+	0.390331i	$0.127639\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.0678			−0.761217
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.60591	+	2.78152i	−0.0606545	+	0.105057i	−0.894758	−	0.446551i	$-0.852652\pi$
$701$	−1.60591	+	2.78152i	−0.0606545	+	0.105057i	0.834104	+	0.551608i	$0.185985\pi$
$702$		0			0
$703$	4.24482	+	0.990721i	0.160096	+	0.0373658i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.2201	+	26.3620i	0.572412	+	0.991447i
$708$		0			0
$709$	−11.0328	−	19.1094i	−0.414345	−	0.717667i	0.581014	−	0.813893i	$-0.302656\pi$
$709$	−11.0328	−	19.1094i	−0.414345	−	0.717667i	−0.995359	+	0.0962265i	$0.969323\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	33.1902	+	57.4871i	1.24298	+	2.15291i
$714$		0			0
$715$	−1.46890			−0.0549338
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.83000	+	15.2940i	−0.329303	+	0.570370i	−0.982374	−	0.186927i	$-0.940147\pi$
$719$	−8.83000	+	15.2940i	−0.329303	+	0.570370i	0.653070	+	0.757297i	$0.273481\pi$
$720$		0			0
$721$	−53.8145			−2.00416
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.73050	+	9.92551i	−0.212825	+	0.368624i
$726$		0			0
$727$	−20.1966	+	34.9815i	−0.749050	+	1.29739i	0.199229	+	0.979953i	$0.436156\pi$
$727$	−20.1966	+	34.9815i	−0.749050	+	1.29739i	−0.948279	+	0.317439i	$0.897177\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−17.9137			−0.661657			−0.330829	−	0.943691i	$-0.607328\pi$
$733$	−17.9137			−0.661657			−0.330829	+	0.943691i	$0.607328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.2765	−	17.7994i	−0.378539	−	0.655649i
$738$		0			0
$739$	−8.48850	+	14.7025i	−0.312255	+	0.540841i	−0.978850	−	0.204579i	$-0.934418\pi$
$739$	−8.48850	+	14.7025i	−0.312255	+	0.540841i	0.666595	+	0.745420i	$0.267751\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.3588	−	43.9228i	0.930325	−	1.61137i	0.147561	−	0.989053i	$-0.452858\pi$
$743$	25.3588	−	43.9228i	0.930325	−	1.61137i	0.782765	−	0.622318i	$-0.213809\pi$
$744$		0			0
$745$	18.7345	+	32.4490i	0.686377	+	1.18884i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−61.3042			−2.24001
$750$		0			0
$751$	−12.7972	−	22.1654i	−0.466977	−	0.808828i	0.532312	−	0.846549i	$-0.321324\pi$
$751$	−12.7972	−	22.1654i	−0.466977	−	0.808828i	−0.999288	+	0.0377210i	$0.987990\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.7776	−	25.5956i	0.537812	−	0.931518i
$756$		0			0
$757$	−22.8681	+	39.6087i	−0.831154	+	1.43960i	0.0659694	+	0.997822i	$0.478986\pi$
$757$	−22.8681	+	39.6087i	−0.831154	+	1.43960i	−0.897124	+	0.441780i	$0.854347\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	44.1338			1.59985			0.799925	−	0.600100i	$-0.204873\pi$
$761$	44.1338			1.59985			0.799925	+	0.600100i	$0.204873\pi$
$762$		0			0
$763$	−14.3888	+	24.9221i	−0.520910	+	0.902242i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.549569			0.0198438
$768$		0			0
$769$	22.0473	+	38.1871i	0.795046	+	1.37706i	0.922810	+	0.385256i	$0.125887\pi$
$769$	22.0473	+	38.1871i	0.795046	+	1.37706i	−0.127764	+	0.991805i	$0.540780\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.45043	+	4.24427i	0.0881359	+	0.152656i	0.906723	−	0.421726i	$-0.138576\pi$
$773$	2.45043	+	4.24427i	0.0881359	+	0.152656i	−0.818587	+	0.574382i	$0.805242\pi$
$774$		0			0
$775$	9.45156	+	16.3706i	0.339510	+	0.588049i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.9322	+	17.0032i	−0.570829	+	0.609203i
$780$		0			0
$781$	−26.0185	+	45.0653i	−0.931014	+	1.61256i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−18.7868	−	32.5398i	−0.670531	−	1.16139i
$786$		0			0
$787$	−35.0554			−1.24959			−0.624795	−	0.780789i	$-0.714818\pi$
$787$	−35.0554			−1.24959			−0.624795	+	0.780789i	$0.714818\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	23.4504			0.833801
$792$		0			0
$793$	−0.827176	+	1.43271i	−0.0293739	+	0.0508771i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.67056			0.200862			0.100431	−	0.994944i	$-0.467978\pi$
$797$	5.67056			0.200862			0.100431	+	0.994944i	$0.467978\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.659635	+	1.14252i	−0.0232780	+	0.0403187i
$804$		0			0
$805$	73.9092			2.60496
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.6359			−0.373938			−0.186969	−	0.982366i	$-0.559866\pi$
$809$	−10.6359			−0.373938			−0.186969	+	0.982366i	$0.559866\pi$
$810$		0			0
$811$	13.6521	+	23.6461i	0.479390	+	0.830327i	0.999721	−	0.0236373i	$-0.00752470\pi$
$811$	13.6521	+	23.6461i	0.479390	+	0.830327i	−0.520331	+	0.853965i	$0.674191\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.16774	−	10.6828i	0.216047	−	0.374204i
$816$		0			0
$817$	−8.39522	+	8.95959i	−0.293711	+	0.313456i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.5865	+	18.3364i	0.369472	+	0.639944i	0.989483	−	0.144649i	$-0.0462053\pi$
$821$	10.5865	+	18.3364i	0.369472	+	0.639944i	−0.620011	+	0.784593i	$0.712872\pi$
$822$		0			0
$823$	23.3641	+	40.4678i	0.814422	+	1.41062i	0.909742	+	0.415174i	$0.136279\pi$
$823$	23.3641	+	40.4678i	0.814422	+	1.41062i	−0.0953202	+	0.995447i	$0.530388\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.83754	+	4.91477i	0.0986710	+	0.170903i	0.911135	−	0.412108i	$-0.135208\pi$
$827$	2.83754	+	4.91477i	0.0986710	+	0.170903i	−0.812464	+	0.583012i	$0.801874\pi$
$828$		0			0
$829$	−22.3377			−0.775822			−0.387911	−	0.921697i	$-0.626803\pi$
$829$	−22.3377			−0.775822			−0.387911	+	0.921697i	$0.626803\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−32.6623			−1.13032
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.54203	−	11.3311i	0.225856	−	0.391194i	−0.730720	−	0.682677i	$-0.760816\pi$
$839$	6.54203	−	11.3311i	0.225856	−	0.391194i	0.956576	+	0.291484i	$0.0941488\pi$
$840$		0			0
$841$	0.212027	−	0.367241i	0.00731127	−	0.0126635i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.3456	+	30.0435i	0.596708	+	1.03353i
$846$		0			0
$847$	13.1048			0.450286
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.76442	+	6.52016i	0.129043	+	0.223508i
$852$		0			0
$853$	−20.1129	+	34.8365i	−0.688652	+	1.19278i	0.283622	+	0.958936i	$0.408464\pi$
$853$	−20.1129	+	34.8365i	−0.688652	+	1.19278i	−0.972274	+	0.233844i	$0.924869\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	−10.9412	−	18.9507i	−0.373309	−	0.646590i	0.616764	−	0.787148i	$-0.288443\pi$
$859$	−10.9412	−	18.9507i	−0.373309	−	0.646590i	−0.990072	+	0.140559i	$0.955110\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.5759			−0.360009			−0.180005	−	0.983666i	$-0.557611\pi$
$863$	−10.5759			−0.360009			−0.180005	+	0.983666i	$0.557611\pi$
$864$		0			0
$865$	20.1233	+	34.8545i	0.684211	+	1.18509i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.4597	+	21.5808i	−0.422665	+	0.732078i
$870$		0			0
$871$	0.387673	−	0.671469i	0.0131358	−	0.0227519i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.0369			−0.947822
$876$		0			0
$877$	21.7672	−	37.7020i	0.735028	−	1.27310i	−0.219684	−	0.975571i	$-0.570503\pi$
$877$	21.7672	−	37.7020i	0.735028	−	1.27310i	0.954711	−	0.297534i	$-0.0961641\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.96080			−0.167133			−0.0835667	−	0.996502i	$-0.526631\pi$
$881$	−4.96080			−0.167133			−0.0835667	+	0.996502i	$0.526631\pi$
$882$		0			0
$883$	−6.50528	−	11.2675i	−0.218920	−	0.379181i	0.735558	−	0.677462i	$-0.236920\pi$
$883$	−6.50528	−	11.2675i	−0.218920	−	0.379181i	−0.954478	+	0.298281i	$0.903587\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.0709	−	29.5676i	−0.573183	−	0.992783i	−0.996236	−	0.0866779i	$-0.972375\pi$
$887$	−17.0709	−	29.5676i	−0.573183	−	0.992783i	0.423053	−	0.906105i	$-0.360958\pi$
$888$		0			0
$889$	4.87448	+	8.44285i	0.163485	+	0.283164i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.58651	−	25.0289i	−0.253873	−	0.837560i
$894$		0			0
$895$	20.2633	−	35.0970i	0.677327	−	1.17316i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23.5658	+	40.8171i	0.785963	+	1.36133i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	32.6129			1.08409
$906$		0			0
$907$	23.7490	−	41.1344i	0.788572	−	1.36585i	−0.138270	−	0.990395i	$-0.544154\pi$
$907$	23.7490	−	41.1344i	0.788572	−	1.36585i	0.926842	−	0.375452i	$-0.122512\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−46.1523			−1.52909			−0.764547	−	0.644568i	$-0.777037\pi$
$911$	−46.1523			−1.52909			−0.764547	+	0.644568i	$0.777037\pi$
$912$		0			0
$913$	8.73276			0.289012
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.8353	−	36.0878i	0.688042	−	1.19172i
$918$		0			0
$919$	29.5160			0.973643			0.486822	−	0.873501i	$-0.338156\pi$
$919$	29.5160			0.973643			0.486822	+	0.873501i	$0.338156\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.96306			−0.0646148
$924$		0			0
$925$	1.07199	+	1.85675i	0.0352469	+	0.0610495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.7213	+	34.1582i	−0.647034	+	1.12070i	0.336794	+	0.941578i	$0.390657\pi$
$929$	−19.7213	+	34.1582i	−0.647034	+	1.12070i	−0.983828	+	0.179117i	$0.942676\pi$
$930$		0			0
$931$	−27.5473	−	6.42942i	−0.902827	−	0.210716i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−13.8681	−	24.0202i	−0.453050	−	0.784706i	0.545524	−	0.838095i	$-0.316331\pi$
$937$	−13.8681	−	24.0202i	−0.453050	−	0.784706i	−0.998574	+	0.0533896i	$0.982997\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.12325	−	14.0699i	−0.264811	−	0.458665i	0.702703	−	0.711483i	$-0.251976\pi$
$941$	−8.12325	−	14.0699i	−0.264811	−	0.458665i	−0.967514	+	0.252818i	$0.918643\pi$
$942$		0			0
$943$	−40.2465			−1.31061
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.96080	−	13.7885i	0.258691	−	0.448066i	−0.707200	−	0.707013i	$-0.750042\pi$
$947$	7.96080	−	13.7885i	0.258691	−	0.448066i	0.965892	+	0.258947i	$0.0833755\pi$
$948$		0			0
$949$	−0.0497686			−0.00161556
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17.9700	−	31.1250i	0.582106	−	1.00824i	−0.413123	−	0.910675i	$-0.635562\pi$
$953$	17.9700	−	31.1250i	0.582106	−	1.00824i	0.995229	−	0.0975627i	$-0.0311046\pi$
$954$		0			0
$955$	−7.28402	+	12.6163i	−0.235705	+	0.408254i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	46.7361			1.50762
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.682059	+	1.18136i	0.0219563	+	0.0380294i
$966$		0			0
$967$	1.30362	−	2.25794i	0.0419217	−	0.0726104i	−0.844303	−	0.535866i	$-0.819985\pi$
$967$	1.30362	−	2.25794i	0.0419217	−	0.0726104i	0.886225	+	0.463255i	$0.153319\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.06558	+	13.9700i	−0.258837	+	0.448318i	−0.965931	−	0.258801i	$-0.916673\pi$
$971$	−8.06558	+	13.9700i	−0.258837	+	0.448318i	0.707094	+	0.707120i	$0.250006\pi$
$972$		0			0
$973$	13.7880	+	23.8815i	0.442022	+	0.765605i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.17302			0.101514			0.0507570	−	0.998711i	$-0.483837\pi$
$977$	3.17302			0.101514			0.0507570	+	0.998711i	$0.483837\pi$
$978$		0			0
$979$	16.5513	+	28.6676i	0.528981	+	0.916221i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−10.4165	+	18.0419i	−0.332235	+	0.575448i	−0.982950	−	0.183874i	$-0.941136\pi$
$983$	−10.4165	+	18.0419i	−0.332235	+	0.575448i	0.650715	+	0.759322i	$0.274469\pi$
$984$		0			0
$985$	30.6851	−	53.1481i	0.977708	−	1.69344i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21.2073			−0.674353
$990$		0			0
$991$	−22.9165	+	39.6926i	−0.727967	+	1.26088i	0.229774	+	0.973244i	$0.426201\pi$
$991$	−22.9165	+	39.6926i	−0.727967	+	1.26088i	−0.957741	+	0.287632i	$0.907132\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.335481			0.0106355
$996$		0			0
$997$	−27.3168	−	47.3141i	−0.865132	−	1.49845i	−0.866916	−	0.498454i	$-0.833901\pi$
$997$	−27.3168	−	47.3141i	−0.865132	−	1.49845i	0.00178393	−	0.999998i	$-0.499432\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.z.1873.1		6
3.2	odd	2		912.2.q.l.49.3		6
4.3	odd	2		171.2.f.b.163.2		6
12.11	even	2		57.2.e.b.49.2	yes	6
19.7	even	3	inner	2736.2.s.z.577.1		6
57.26	odd	6		912.2.q.l.577.3		6
76.7	odd	6		171.2.f.b.64.2		6
76.11	odd	6		3249.2.a.y.1.2		3
76.27	even	6		3249.2.a.t.1.2		3
228.11	even	6		1083.2.a.l.1.2		3
228.83	even	6		57.2.e.b.7.2	✓	6
228.179	odd	6		1083.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.2	✓	6	228.83	even	6
57.2.e.b.49.2	yes	6	12.11	even	2
171.2.f.b.64.2		6	76.7	odd	6
171.2.f.b.163.2		6	4.3	odd	2
912.2.q.l.49.3		6	3.2	odd	2
912.2.q.l.577.3		6	57.26	odd	6
1083.2.a.l.1.2		3	228.11	even	6
1083.2.a.o.1.2		3	228.179	odd	6
2736.2.s.z.577.1		6	19.7	even	3	inner
2736.2.s.z.1873.1		6	1.1	even	1	trivial
3249.2.a.t.1.2		3	76.27	even	6
3249.2.a.y.1.2		3	76.11	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.33641 + 2.31473i) q^{5} +3.67282 q^{7} +O(q^{10})\)	\(q+(-1.33641 + 2.31473i) q^{5} +3.67282 q^{7} -3.81681 q^{11} +(-0.0719933 - 0.124696i) q^{13} +(-4.24482 - 0.990721i) q^{19} +(-3.76442 - 6.52016i) q^{23} +(-1.07199 - 1.85675i) q^{25} +(-2.67282 - 4.62947i) q^{29} -8.81681 q^{31} +(-4.90841 + 8.50161i) q^{35} -1.00000 q^{37} +(2.67282 - 4.62947i) q^{41} +(1.40841 - 2.43943i) q^{43} +(3.00000 + 5.19615i) q^{47} +6.48963 q^{49} +(4.00924 + 6.94420i) q^{53} +(5.10083 - 8.83490i) q^{55} +(-1.90841 + 3.30545i) q^{59} +(-5.74482 - 9.95031i) q^{61} +0.384851 q^{65} +(2.69243 + 4.66342i) q^{67} +(6.81681 - 11.8071i) q^{71} +(0.172824 - 0.299339i) q^{73} -14.0185 q^{77} +(3.26442 - 5.65414i) q^{79} -2.28797 q^{83} +(-4.33641 - 7.51089i) q^{89} +(-0.264419 - 0.457986i) q^{91} +(7.96608 - 8.50161i) q^{95} +(2.95684 - 5.12140i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 + 2 * q^7	\( 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

Introduction

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