LMFDB - Embedded newform 2736.2.s.g.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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
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.g.1873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} -4.00000 q^{11} +(0.500000 - 0.866025i) q^{13} +(1.50000 + 2.59808i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-2.50000 + 4.33013i) q^{23} +(2.00000 - 3.46410i) q^{25} +(3.50000 - 6.06218i) q^{29} -4.00000 q^{31} +10.0000 q^{37} +(-2.50000 - 4.33013i) q^{41} +(-2.50000 - 4.33013i) q^{43} +(3.50000 - 6.06218i) q^{47} -7.00000 q^{49} +(5.50000 - 9.52628i) q^{53} +(2.00000 + 3.46410i) q^{55} +(-1.50000 - 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} -1.00000 q^{65} +(-1.50000 + 2.59808i) q^{67} +(-5.50000 - 9.52628i) q^{71} +(-7.50000 - 12.9904i) q^{73} +(-6.50000 - 11.2583i) q^{79} +(1.50000 - 2.59808i) q^{85} +(1.50000 - 2.59808i) q^{89} +(-0.500000 - 4.33013i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 8 q^{11} + q^{13} + 3 q^{17} + 8 q^{19} - 5 q^{23} + 4 q^{25} + 7 q^{29} - 8 q^{31} + 20 q^{37} - 5 q^{41} - 5 q^{43} + 7 q^{47} - 14 q^{49} + 11 q^{53} + 4 q^{55} - 3 q^{59} - 11 q^{61}+ \cdots + 5 q^{97}+O(q^{100}) $ 2 * q - q^5 - 8 * q^11 + q^13 + 3 * q^17 + 8 * q^19 - 5 * q^23 + 4 * q^25 + 7 * q^29 - 8 * q^31 + 20 * q^37 - 5 * q^41 - 5 * q^43 + 7 * q^47 - 14 * q^49 + 11 * q^53 + 4 * q^55 - 3 * q^59 - 11 * q^61 - 2 * q^65 - 3 * q^67 - 11 * q^71 - 15 * q^73 - 13 * q^79 + 3 * q^85 + 3 * q^89 - q^95 + 5 * q^97

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	0.732294	−	0.680989i	$-0.238450\pi$
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i	−0.955901	+	0.293691i	$0.905116\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		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
$23$	−2.50000	+	4.33013i	−0.521286	+	0.902894i	0.478407	+	0.878138i	$0.341214\pi$
$23$	−2.50000	+	4.33013i	−0.521286	+	0.902894i	−0.999694	+	0.0247559i	$0.992119\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	−0.333205	−	0.942855i	$-0.608130\pi$
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	0.983138	−	0.182864i	$-0.0585367\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	0.602072	−	0.798441i	$-0.294342\pi$
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	−0.992507	+	0.122189i	$0.961009\pi$
$42$		0			0
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.50000	−	6.06218i	0.510527	−	0.884260i	−0.489398	−	0.872060i	$-0.662783\pi$
$47$	3.50000	−	6.06218i	0.510527	−	0.884260i	0.999926	−	0.0121990i	$-0.00388317\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.50000	−	9.52628i	0.755483	−	1.30854i	−0.189651	−	0.981852i	$-0.560736\pi$
$53$	5.50000	−	9.52628i	0.755483	−	1.30854i	0.945134	−	0.326683i	$-0.105931\pi$
$54$		0			0
$55$	2.00000	+	3.46410i	0.269680	+	0.467099i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.00000			−0.124035
$66$		0			0
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	−0.942987	−	0.332830i	$-0.891996\pi$
$67$	−1.50000	+	2.59808i	−0.183254	+	0.317406i	0.759733	+	0.650236i	$0.225330\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5.50000	−	9.52628i	−0.652730	−	1.13056i	−0.982458	−	0.186485i	$-0.940290\pi$
$71$	−5.50000	−	9.52628i	−0.652730	−	1.13056i	0.329728	−	0.944076i	$-0.393043\pi$
$72$		0			0
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.853740	−	0.520699i	$-0.825671\pi$
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.0240681	−	0.999710i	$-0.507662\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		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
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	1.50000	−	2.59808i	0.162698	−	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	−0.775509	−	0.631337i	$-0.782506\pi$
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	0.934508	+	0.355942i	$0.115840\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.500000	−	4.33013i	−0.0512989	−	0.444262i
$96$		0			0
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	2.50000	+	4.33013i	0.253837	+	0.439658i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.500000	+	0.866025i	−0.0497519	+	0.0861727i	−0.889829	−	0.456294i	$-0.849176\pi$
$101$	−0.500000	+	0.866025i	−0.0497519	+	0.0861727i	0.840077	+	0.542467i	$0.182510\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$	−1.50000	−	2.59808i	−0.143674	−	0.248851i	0.785203	−	0.619238i	$-0.212558\pi$
$109$	−1.50000	−	2.59808i	−0.143674	−	0.248851i	−0.928877	+	0.370387i	$0.879225\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	5.00000			0.466252
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	−1.50000	+	2.59808i	−0.133103	+	0.230542i	−0.924871	−	0.380280i	$-0.875828\pi$
$127$	−1.50000	+	2.59808i	−0.133103	+	0.230542i	0.791768	+	0.610822i	$0.209161\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	−0.981824	−	0.189794i	$-0.939218\pi$
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	0.326546	−	0.945181i	$-0.394115\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.50000	+	4.33013i	−0.213589	+	0.369948i	−0.952835	−	0.303488i	$-0.901849\pi$
$137$	−2.50000	+	4.33013i	−0.213589	+	0.369948i	0.739246	+	0.673436i	$0.235182\pi$
$138$		0			0
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	0.991303	+	0.131597i	$0.0420106\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.00000	+	3.46410i	−0.167248	+	0.289683i
$144$		0			0
$145$	−7.00000			−0.581318
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	0.920904	−	0.389789i	$-0.127452\pi$
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	−0.798019	+	0.602632i	$0.794119\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	+	3.46410i	0.160644	+	0.278243i
$156$		0			0
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	0.691888	−	0.722005i	$-0.256779\pi$
$157$	−3.50000	−	6.06218i	−0.279330	−	0.483814i	−0.971219	+	0.238190i	$0.923446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.50000	−	12.9904i	0.580367	−	1.00523i	−0.415068	−	0.909790i	$-0.636242\pi$
$167$	7.50000	−	12.9904i	0.580367	−	1.00523i	0.995436	−	0.0954356i	$-0.0304244\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.50000	+	12.9904i	0.570214	+	0.987640i	0.996544	+	0.0830722i	$0.0264732\pi$
$173$	7.50000	+	12.9904i	0.570214	+	0.987640i	−0.426329	+	0.904568i	$0.640193\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.50000	−	4.33013i	0.185824	−	0.321856i	−0.758030	−	0.652219i	$-0.773838\pi$
$181$	2.50000	−	4.33013i	0.185824	−	0.321856i	0.943854	+	0.330364i	$0.107171\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.00000	−	8.66025i	−0.367607	−	0.636715i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$	−7.50000	−	12.9904i	−0.539862	−	0.935068i	−0.998911	−	0.0466572i	$-0.985143\pi$
$193$	−7.50000	−	12.9904i	−0.539862	−	0.935068i	0.459049	−	0.888411i	$-0.348190\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	−3.50000	+	6.06218i	−0.248108	+	0.429736i	−0.963001	−	0.269498i	$-0.913142\pi$
$199$	−3.50000	+	6.06218i	−0.248108	+	0.429736i	0.714893	+	0.699234i	$0.246476\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.50000	+	4.33013i	−0.174608	+	0.302429i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.0000	−	6.92820i	−1.10674	−	0.479234i
$210$		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
$215$	−2.50000	+	4.33013i	−0.170499	+	0.295312i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	−12.5000	−	21.6506i	−0.837062	−	1.44983i	−0.892341	−	0.451363i	$-0.850938\pi$
$223$	−12.5000	−	21.6506i	−0.837062	−	1.44983i	0.0552786	−	0.998471i	$-0.482395\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.0000			−1.32745			−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000			−1.32745			−0.663723	+	0.747978i	$0.731025\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	−0.972523	−	0.232806i	$-0.925209\pi$
$233$	−10.5000	−	18.1865i	−0.687878	−	1.19144i	0.284645	−	0.958633i	$-0.408124\pi$
$234$		0			0
$235$	−7.00000			−0.456630
$236$		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
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	0.378963	+	0.925412i	$0.376281\pi$
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	−0.990912	+	0.134515i	$0.957053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.50000	+	6.06218i	0.223607	+	0.387298i
$246$		0			0
$247$	3.50000	−	2.59808i	0.222700	−	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.5000	−	26.8468i	0.978351	−	1.69455i	0.309951	−	0.950753i	$-0.399687\pi$
$251$	15.5000	−	26.8468i	0.978351	−	1.69455i	0.668400	−	0.743802i	$-0.266979\pi$
$252$		0			0
$253$	10.0000	−	17.3205i	0.628695	−	1.08893i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.5000	−	19.9186i	0.717350	−	1.24249i	−0.244696	−	0.969600i	$-0.578688\pi$
$257$	11.5000	−	19.9186i	0.717350	−	1.24249i	0.962046	−	0.272887i	$-0.0879786\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	0.970758	−	0.240059i	$-0.0771668\pi$
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	−0.693276	+	0.720672i	$0.743833\pi$
$264$		0			0
$265$	−11.0000			−0.675725
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.5000	+	23.3827i	0.823110	+	1.42567i	0.903356	+	0.428892i	$0.141096\pi$
$269$	13.5000	+	23.3827i	0.823110	+	1.42567i	−0.0802460	+	0.996775i	$0.525571\pi$
$270$		0			0
$271$	15.5000	+	26.8468i	0.941558	+	1.63083i	0.762501	+	0.646988i	$0.223971\pi$
$271$	15.5000	+	26.8468i	0.941558	+	1.63083i	0.179057	+	0.983839i	$0.442695\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000	+	13.8564i	−0.482418	+	0.835573i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.50000	−	6.06218i	0.208792	−	0.361639i	−0.742542	−	0.669800i	$-0.766380\pi$
$281$	3.50000	−	6.06218i	0.208792	−	0.361639i	0.951334	+	0.308160i	$0.0997133\pi$
$282$		0			0
$283$	−4.50000	−	7.79423i	−0.267497	−	0.463319i	0.700718	−	0.713439i	$-0.252863\pi$
$283$	−4.50000	−	7.79423i	−0.267497	−	0.463319i	−0.968215	+	0.250120i	$0.919530\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	30.0000			1.75262			0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000			1.75262			0.876309	+	0.481749i	$0.159998\pi$
$294$		0			0
$295$	−1.50000	+	2.59808i	−0.0873334	+	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.50000	+	4.33013i	0.144579	+	0.250418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.0000			0.629858
$306$		0			0
$307$	13.5000	+	23.3827i	0.770486	+	1.33452i	0.937297	+	0.348532i	$0.113320\pi$
$307$	13.5000	+	23.3827i	0.770486	+	1.33452i	−0.166811	+	0.985989i	$0.553347\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	−0.978553	−	0.205996i	$-0.933957\pi$
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	0.667674	+	0.744453i	$0.267290\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.50000	−	12.9904i	0.421242	−	0.729612i	−0.574819	−	0.818280i	$-0.694928\pi$
$317$	7.50000	−	12.9904i	0.421242	−	0.729612i	0.996061	+	0.0886679i	$0.0282610\pi$
$318$		0			0
$319$	−14.0000	+	24.2487i	−0.783850	+	1.35767i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.50000	+	12.9904i	0.0834622	+	0.722804i
$324$		0			0
$325$	−2.00000	−	3.46410i	−0.110940	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.00000			0.163908
$336$		0			0
$337$	2.50000	+	4.33013i	0.136184	+	0.235877i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	2.50000	+	4.33013i	0.136184	+	0.235877i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.50000	+	4.33013i	0.134207	+	0.232453i	0.925294	−	0.379250i	$-0.123818\pi$
$347$	2.50000	+	4.33013i	0.134207	+	0.232453i	−0.791087	+	0.611703i	$0.790485\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$	−5.50000	+	9.52628i	−0.291910	+	0.505602i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.50000	−	12.9904i	−0.395835	−	0.685606i	0.597372	−	0.801964i	$-0.296211\pi$
$359$	−7.50000	−	12.9904i	−0.395835	−	0.685606i	−0.993207	+	0.116358i	$0.962878\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.50000	+	12.9904i	−0.392568	+	0.679948i
$366$		0			0
$367$	12.5000	−	21.6506i	0.652495	−	1.13015i	−0.330021	−	0.943974i	$-0.607056\pi$
$367$	12.5000	−	21.6506i	0.652495	−	1.13015i	0.982516	−	0.186180i	$-0.0596109\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.50000	−	6.06218i	−0.180259	−	0.312218i
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.5000	+	25.1147i	0.740915	+	1.28330i	0.952079	+	0.305852i	$0.0989414\pi$
$383$	14.5000	+	25.1147i	0.740915	+	1.28330i	−0.211164	+	0.977451i	$0.567725\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	−0.825491	−	0.564416i	$-0.809102\pi$
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	0.901544	+	0.432688i	$0.142435\pi$
$390$		0			0
$391$	−15.0000			−0.758583
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.50000	+	11.2583i	−0.327050	+	0.566468i
$396$		0			0
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	0.988080	+	0.153941i	$0.0491966\pi$
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	−0.360723	+	0.932673i	$0.617470\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.50000	+	16.4545i	0.474407	+	0.821698i	0.999571	−	0.0293039i	$-0.00932905\pi$
$401$	9.50000	+	16.4545i	0.474407	+	0.821698i	−0.525163	+	0.851002i	$0.675996\pi$
$402$		0			0
$403$	−2.00000	+	3.46410i	−0.0996271	+	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−40.0000			−1.98273
$408$		0			0
$409$	8.50000	−	14.7224i	0.420298	−	0.727977i	−0.575670	−	0.817682i	$-0.695259\pi$
$409$	8.50000	−	14.7224i	0.420298	−	0.727977i	0.995968	+	0.0897044i	$0.0285922\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	0.877952	−	0.478748i	$-0.158909\pi$
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	−0.853584	+	0.520955i	$0.825576\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.0000			0.582086
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	0.494211	+	0.869342i	$0.335457\pi$
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	−0.999978	+	0.00667224i	$0.997876\pi$
$432$		0			0
$433$	12.5000	−	21.6506i	0.600712	−	1.04046i	−0.392002	−	0.919964i	$-0.628217\pi$
$433$	12.5000	−	21.6506i	0.600712	−	1.04046i	0.992713	−	0.120499i	$-0.0384494\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.5000	+	12.9904i	−0.837139	+	0.621414i
$438$		0			0
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.5000	+	21.6506i	−0.593893	+	1.02865i	0.399809	+	0.916598i	$0.369076\pi$
$443$	−12.5000	+	21.6506i	−0.593893	+	1.02865i	−0.993702	+	0.112054i	$0.964257\pi$
$444$		0			0
$445$	−3.00000			−0.142214
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.0000			−1.03824			−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000			−1.03824			−0.519122	+	0.854700i	$0.673741\pi$
$450$		0			0
$451$	10.0000	+	17.3205i	0.470882	+	0.815591i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.50000	+	9.52628i	0.256161	+	0.443683i	0.965210	−	0.261476i	$-0.0842091\pi$
$461$	5.50000	+	9.52628i	0.256161	+	0.443683i	−0.709050	+	0.705159i	$0.750876\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.0000	+	17.3205i	0.459800	+	0.796398i
$474$		0			0
$475$	14.0000	−	10.3923i	0.642364	−	0.476832i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.5000	−	19.9186i	0.525448	−	0.910103i	−0.474112	−	0.880464i	$-0.657231\pi$
$479$	11.5000	−	19.9186i	0.525448	−	0.910103i	0.999561	−	0.0296389i	$-0.00943575\pi$
$480$		0			0
$481$	5.00000	−	8.66025i	0.227980	−	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.50000	−	4.33013i	0.113519	−	0.196621i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	+	0.866025i	0.0225647	+	0.0390832i	0.877087	−	0.480331i	$-0.159483\pi$
$491$	0.500000	+	0.866025i	0.0225647	+	0.0390832i	−0.854523	+	0.519414i	$0.826150\pi$
$492$		0			0
$493$	21.0000			0.945792
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		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
$503$	−10.5000	+	18.1865i	−0.468172	+	0.810897i	−0.999338	−	0.0363700i	$-0.988421\pi$
$503$	−10.5000	+	18.1865i	−0.468172	+	0.810897i	0.531167	+	0.847267i	$0.321754\pi$
$504$		0			0
$505$	1.00000			0.0444994
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.50000	−	12.9904i	0.332432	−	0.575789i	−0.650556	−	0.759458i	$-0.725464\pi$
$509$	7.50000	−	12.9904i	0.332432	−	0.575789i	0.982988	+	0.183669i	$0.0587976\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−14.0000	+	24.2487i	−0.615719	+	1.06646i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.0000			−1.13908			−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000			−1.13908			−0.569540	+	0.821963i	$0.692879\pi$
$522$		0			0
$523$	−9.50000	+	16.4545i	−0.415406	+	0.719504i	−0.995471	−	0.0950659i	$-0.969694\pi$
$523$	−9.50000	+	16.4545i	−0.415406	+	0.719504i	0.580065	+	0.814570i	$0.303027\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	−	10.3923i	−0.261364	−	0.452696i
$528$		0			0
$529$	−1.00000	−	1.73205i	−0.0434783	−	0.0753066i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.00000			−0.216574
$534$		0			0
$535$	−10.0000	−	17.3205i	−0.432338	−	0.748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.0000			1.20605
$540$		0			0
$541$	0.500000	−	0.866025i	0.0214967	−	0.0372333i	−0.855077	−	0.518501i	$-0.826490\pi$
$541$	0.500000	−	0.866025i	0.0214967	−	0.0372333i	0.876574	+	0.481268i	$0.159824\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.50000	+	2.59808i	−0.0642529	+	0.111289i
$546$		0			0
$547$	12.5000	−	21.6506i	0.534461	−	0.925714i	−0.464728	−	0.885454i	$-0.653848\pi$
$547$	12.5000	−	21.6506i	0.534461	−	0.925714i	0.999189	−	0.0402607i	$-0.0128188\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	24.5000	−	18.1865i	1.04374	−	0.774772i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.50000	+	4.33013i	−0.105928	+	0.183473i	−0.914117	−	0.405450i	$-0.867115\pi$
$557$	−2.50000	+	4.33013i	−0.105928	+	0.183473i	0.808189	+	0.588924i	$0.200448\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$		0			0
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$	7.00000	+	12.1244i	0.294492	+	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.0000			−0.922288			−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000			−0.922288			−0.461144	+	0.887325i	$0.652561\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.0000	+	17.3205i	0.417029	+	0.722315i
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−22.0000	+	38.1051i	−0.911147	+	1.57815i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.50000	−	12.9904i	−0.309558	−	0.536170i	0.668708	−	0.743525i	$-0.266848\pi$
$587$	−7.50000	−	12.9904i	−0.309558	−	0.536170i	−0.978266	+	0.207355i	$0.933514\pi$
$588$		0			0
$589$	−16.0000	−	6.92820i	−0.659269	−	0.285472i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.50000	+	4.33013i	−0.102663	+	0.177817i	−0.912781	−	0.408450i	$-0.866070\pi$
$593$	−2.50000	+	4.33013i	−0.102663	+	0.177817i	0.810118	+	0.586267i	$0.199403\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.5000	+	38.9711i	−0.919325	+	1.59232i	−0.118882	+	0.992908i	$0.537931\pi$
$599$	−22.5000	+	38.9711i	−0.919325	+	1.59232i	−0.800443	+	0.599409i	$0.795402\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.50000	−	4.33013i	−0.101639	−	0.176045i
$606$		0			0
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.50000	−	6.06218i	−0.141595	−	0.245249i
$612$		0			0
$613$	14.5000	+	25.1147i	0.585649	+	1.01437i	0.994794	+	0.101905i	$0.0324938\pi$
$613$	14.5000	+	25.1147i	0.585649	+	1.01437i	−0.409145	+	0.912470i	$0.634173\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.0859976	+	0.996295i	$0.527408\pi$
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.819818	+	0.572624i	$0.805926\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.0000	+	25.9808i	0.598089	+	1.03592i
$630$		0			0
$631$	20.5000	−	35.5070i	0.816092	−	1.41351i	−0.0924489	−	0.995717i	$-0.529469\pi$
$631$	20.5000	−	35.5070i	0.816092	−	1.41351i	0.908541	−	0.417796i	$-0.137197\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.00000			0.119051
$636$		0			0
$637$	−3.50000	+	6.06218i	−0.138675	+	0.240192i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	19.5000	+	33.7750i	0.770204	+	1.33403i	0.937451	+	0.348117i	$0.113179\pi$
$641$	19.5000	+	33.7750i	0.770204	+	1.33403i	−0.167247	+	0.985915i	$0.553488\pi$
$642$		0			0
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	0.990274	−	0.139134i	$-0.0444318\pi$
$643$	9.50000	+	16.4545i	0.374643	+	0.648901i	−0.615630	+	0.788035i	$0.711098\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$	6.00000	+	10.3923i	0.235521	+	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	30.0000			1.17399			0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000			1.17399			0.586995	+	0.809590i	$0.300311\pi$
$654$		0			0
$655$	−7.50000	+	12.9904i	−0.293049	+	0.507576i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.50000	+	4.33013i	−0.0973862	+	0.168678i	−0.910602	−	0.413284i	$-0.864382\pi$
$659$	−2.50000	+	4.33013i	−0.0973862	+	0.168678i	0.813216	+	0.581962i	$0.197715\pi$
$660$		0			0
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	0.389503	+	0.921025i	$0.372647\pi$
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	−0.992383	+	0.123194i	$0.960686\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.5000	+	30.3109i	0.677603	+	1.17364i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.0000	−	38.1051i	0.849301	−	1.47103i
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10.0000			−0.384331			−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000			−0.384331			−0.192166	+	0.981363i	$0.561551\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.0000			0.612223			0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000			0.612223			0.306111	+	0.951996i	$0.400972\pi$
$684$		0			0
$685$	5.00000			0.191040
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.50000	−	9.52628i	−0.209533	−	0.362922i
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.00000			−0.341389
$696$		0			0
$697$	7.50000	−	12.9904i	0.284083	−	0.492046i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.5000	+	30.3109i	0.660966	+	1.14483i	0.980362	+	0.197205i	$0.0631865\pi$
$701$	17.5000	+	30.3109i	0.660966	+	1.14483i	−0.319396	+	0.947621i	$0.603480\pi$
$702$		0			0
$703$	40.0000	+	17.3205i	1.50863	+	0.653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.50000	+	2.59808i	−0.0563337	+	0.0975728i	−0.892817	−	0.450420i	$-0.851274\pi$
$709$	−1.50000	+	2.59808i	−0.0563337	+	0.0975728i	0.836483	+	0.547992i	$0.184608\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.0000	−	17.3205i	0.374503	−	0.648658i
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.50000	−	6.06218i	−0.130528	−	0.226081i	0.793352	−	0.608763i	$-0.208334\pi$
$719$	−3.50000	−	6.06218i	−0.130528	−	0.226081i	−0.923880	+	0.382682i	$0.875001\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.0000	−	24.2487i	−0.519947	−	0.900575i
$726$		0			0
$727$	3.50000	+	6.06218i	0.129808	+	0.224834i	0.923602	−	0.383353i	$-0.125231\pi$
$727$	3.50000	+	6.06218i	0.129808	+	0.224834i	−0.793794	+	0.608186i	$0.791897\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.50000	−	12.9904i	0.277398	−	0.480467i
$732$		0			0
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.00000	−	10.3923i	0.221013	−	0.382805i
$738$		0			0
$739$	−6.50000	−	11.2583i	−0.239106	−	0.414144i	0.721352	−	0.692569i	$-0.243521\pi$
$739$	−6.50000	−	11.2583i	−0.239106	−	0.414144i	−0.960458	+	0.278425i	$0.910188\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.5000	+	21.6506i	0.458581	+	0.794285i	0.998886	−	0.0471840i	$-0.0150247\pi$
$743$	12.5000	+	21.6506i	0.458581	+	0.794285i	−0.540306	+	0.841469i	$0.681691\pi$
$744$		0			0
$745$	1.50000	−	2.59808i	0.0549557	−	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	14.5000	−	25.1147i	0.529113	−	0.916450i	−0.470311	−	0.882501i	$-0.655858\pi$
$751$	14.5000	−	25.1147i	0.529113	−	0.916450i	0.999424	−	0.0339490i	$-0.0108084\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	−	13.8564i	−0.291150	−	0.504286i
$756$		0			0
$757$	8.50000	+	14.7224i	0.308938	+	0.535096i	0.978130	−	0.207993i	$-0.0666932\pi$
$757$	8.50000	+	14.7224i	0.308938	+	0.535096i	−0.669193	+	0.743089i	$0.733360\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.00000			−0.108324
$768$		0			0
$769$	−3.50000	+	6.06218i	−0.126213	+	0.218608i	−0.922207	−	0.386698i	$-0.873616\pi$
$769$	−3.50000	+	6.06218i	−0.126213	+	0.218608i	0.795993	+	0.605305i	$0.206949\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−10.5000	+	18.1865i	−0.377659	+	0.654124i	−0.990721	−	0.135910i	$-0.956604\pi$
$773$	−10.5000	+	18.1865i	−0.377659	+	0.654124i	0.613062	+	0.790034i	$0.289937\pi$
$774$		0			0
$775$	−8.00000	+	13.8564i	−0.287368	+	0.497737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.50000	−	21.6506i	−0.0895718	−	0.775715i
$780$		0			0
$781$	22.0000	+	38.1051i	0.787222	+	1.36351i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.50000	+	6.06218i	−0.124920	+	0.216368i
$786$		0			0
$787$	−52.0000			−1.85360			−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000			−1.85360			−0.926800	+	0.375555i	$0.877452\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.50000	+	9.52628i	0.195311	+	0.338288i
$794$		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
$799$	21.0000			0.742927
$800$		0			0
$801$		0			0
$802$		0			0
$803$	30.0000	+	51.9615i	1.05868	+	1.83368i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	10.5000	−	18.1865i	0.368705	−	0.638616i	−0.620658	−	0.784081i	$-0.713135\pi$
$811$	10.5000	−	18.1865i	0.368705	−	0.638616i	0.989363	+	0.145465i	$0.0464680\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.00000	−	3.46410i	−0.0700569	−	0.121342i
$816$		0			0
$817$	−2.50000	−	21.6506i	−0.0874639	−	0.757460i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.50000	+	7.79423i	−0.157051	+	0.272020i	−0.933804	−	0.357785i	$-0.883532\pi$
$821$	−4.50000	+	7.79423i	−0.157051	+	0.272020i	0.776753	+	0.629805i	$0.216865\pi$
$822$		0			0
$823$	−15.5000	+	26.8468i	−0.540296	+	0.935820i	0.458591	+	0.888648i	$0.348354\pi$
$823$	−15.5000	+	26.8468i	−0.540296	+	0.935820i	−0.998887	+	0.0471726i	$0.984979\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.50000	−	6.06218i	0.121707	−	0.210803i	−0.798734	−	0.601684i	$-0.794497\pi$
$827$	3.50000	−	6.06218i	0.121707	−	0.210803i	0.920441	+	0.390882i	$0.127830\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.5000	−	18.1865i	−0.363803	−	0.630126i
$834$		0			0
$835$	−15.0000			−0.519096
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.5000	+	21.6506i	0.431548	+	0.747463i	0.997007	−	0.0773135i	$-0.0246342\pi$
$839$	12.5000	+	21.6506i	0.431548	+	0.747463i	−0.565459	+	0.824776i	$0.691301\pi$
$840$		0			0
$841$	−10.0000	−	17.3205i	−0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.00000	−	10.3923i	0.206406	−	0.357506i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.0000	+	43.3013i	−0.856989	+	1.48435i
$852$		0			0
$853$	−19.5000	−	33.7750i	−0.667667	−	1.15643i	−0.978555	−	0.205987i	$-0.933960\pi$
$853$	−19.5000	−	33.7750i	−0.667667	−	1.15643i	0.310887	−	0.950447i	$-0.399374\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.50000	+	12.9904i	0.256195	+	0.443743i	0.965219	−	0.261441i	$-0.0841977\pi$
$857$	7.50000	+	12.9904i	0.256195	+	0.443743i	−0.709024	+	0.705184i	$0.750864\pi$
$858$		0			0
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$	7.50000	−	12.9904i	0.255008	−	0.441686i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.0000	+	45.0333i	0.881990	+	1.52765i
$870$		0			0
$871$	1.50000	+	2.59808i	0.0508256	+	0.0880325i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.50000	+	4.33013i	0.0844190	+	0.146218i	0.905143	−	0.425106i	$-0.139763\pi$
$877$	2.50000	+	4.33013i	0.0844190	+	0.146218i	−0.820724	+	0.571324i	$0.806430\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$	12.5000	−	21.6506i	0.420658	−	0.728602i	−0.575346	−	0.817910i	$-0.695132\pi$
$883$	12.5000	−	21.6506i	0.420658	−	0.728602i	0.996004	+	0.0893086i	$0.0284657\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.5000	+	21.6506i	−0.419709	+	0.726957i	−0.995910	−	0.0903508i	$-0.971201\pi$
$887$	−12.5000	+	21.6506i	−0.419709	+	0.726957i	0.576201	+	0.817308i	$0.304535\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	24.5000	−	18.1865i	0.819861	−	0.608589i
$894$		0			0
$895$	6.00000	+	10.3923i	0.200558	+	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−14.0000	+	24.2487i	−0.466926	+	0.808740i
$900$		0			0
$901$	33.0000			1.09939
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.00000			−0.166206
$906$		0			0
$907$	17.5000	+	30.3109i	0.581078	+	1.00646i	0.995352	+	0.0963043i	$0.0307022\pi$
$907$	17.5000	+	30.3109i	0.581078	+	1.00646i	−0.414274	+	0.910152i	$0.635964\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.0000			−0.362069
$924$		0			0
$925$	20.0000	−	34.6410i	0.657596	−	1.13899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.5000	+	19.9186i	0.377303	+	0.653508i	0.990669	−	0.136291i	$-0.0435183\pi$
$929$	11.5000	+	19.9186i	0.377303	+	0.653508i	−0.613366	+	0.789799i	$0.710185\pi$
$930$		0			0
$931$	−28.0000	−	12.1244i	−0.917663	−	0.397360i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.00000	+	10.3923i	−0.196221	+	0.339865i
$936$		0			0
$937$	−11.5000	+	19.9186i	−0.375689	+	0.650712i	−0.990430	−	0.138017i	$-0.955927\pi$
$937$	−11.5000	+	19.9186i	−0.375689	+	0.650712i	0.614741	+	0.788729i	$0.289260\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.50000	−	16.4545i	0.309691	−	0.536401i	−0.668604	−	0.743619i	$-0.733108\pi$
$941$	9.50000	−	16.4545i	0.309691	−	0.536401i	0.978295	+	0.207218i	$0.0664410\pi$
$942$		0			0
$943$	25.0000			0.814112
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5000	−	23.3827i	−0.438691	−	0.759835i	0.558898	−	0.829237i	$-0.311224\pi$
$947$	−13.5000	−	23.3827i	−0.438691	−	0.759835i	−0.997589	+	0.0694014i	$0.977891\pi$
$948$		0			0
$949$	−15.0000			−0.486921
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.5000	−	21.6506i	−0.404915	−	0.701333i	0.589397	−	0.807844i	$-0.299365\pi$
$953$	−12.5000	−	21.6506i	−0.404915	−	0.701333i	−0.994312	+	0.106511i	$0.966032\pi$
$954$		0			0
$955$	−8.00000	−	13.8564i	−0.258874	−	0.448383i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.50000	+	12.9904i	−0.241434	+	0.418175i
$966$		0			0
$967$	−22.5000	−	38.9711i	−0.723551	−	1.25323i	−0.959568	−	0.281478i	$-0.909175\pi$
$967$	−22.5000	−	38.9711i	−0.723551	−	1.25323i	0.236016	−	0.971749i	$-0.424158\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17.5000	−	30.3109i	−0.561602	−	0.972723i	−0.997357	−	0.0726575i	$-0.976852\pi$
$971$	−17.5000	−	30.3109i	−0.561602	−	0.972723i	0.435755	−	0.900065i	$-0.356481\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.0000			0.319928			0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000			0.319928			0.159964	+	0.987123i	$0.448862\pi$
$978$		0			0
$979$	−6.00000	+	10.3923i	−0.191761	+	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.5000	+	38.9711i	0.717639	+	1.24299i	0.961933	+	0.273285i	$0.0881103\pi$
$983$	22.5000	+	38.9711i	0.717639	+	1.24299i	−0.244294	+	0.969701i	$0.578556\pi$
$984$		0			0
$985$	1.00000	+	1.73205i	0.0318626	+	0.0551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25.0000			0.794954
$990$		0			0
$991$	−2.50000	−	4.33013i	−0.0794151	−	0.137551i	0.823583	−	0.567196i	$-0.191972\pi$
$991$	−2.50000	−	4.33013i	−0.0794151	−	0.137551i	−0.902998	+	0.429645i	$0.858639\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.00000			0.221915
$996$		0			0
$997$	26.5000	−	45.8993i	0.839263	−	1.45365i	−0.0512480	−	0.998686i	$-0.516320\pi$
$997$	26.5000	−	45.8993i	0.839263	−	1.45365i	0.890511	−	0.454961i	$-0.150347\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.g.577.1		2
3.2	odd	2		304.2.i.a.273.1		2
4.3	odd	2		684.2.k.b.577.1		2
12.11	even	2		76.2.e.a.45.1	✓	2
19.11	even	3	inner	2736.2.s.g.1873.1		2
24.5	odd	2		1216.2.i.g.577.1		2
24.11	even	2		1216.2.i.c.577.1		2
57.11	odd	6		304.2.i.a.49.1		2
57.26	odd	6		5776.2.a.k.1.1		1
57.50	even	6		5776.2.a.f.1.1		1
60.23	odd	4		1900.2.s.a.349.1		4
60.47	odd	4		1900.2.s.a.349.2		4
60.59	even	2		1900.2.i.a.501.1		2
76.11	odd	6		684.2.k.b.505.1		2
228.11	even	6		76.2.e.a.49.1	yes	2
228.83	even	6		1444.2.a.b.1.1		1
228.107	odd	6		1444.2.a.c.1.1		1
228.179	odd	6		1444.2.e.b.429.1		2
228.227	odd	2		1444.2.e.b.653.1		2
456.11	even	6		1216.2.i.c.961.1		2
456.125	odd	6		1216.2.i.g.961.1		2
1140.239	even	6		1900.2.i.a.201.1		2
1140.467	odd	12		1900.2.s.a.49.1		4
1140.923	odd	12		1900.2.s.a.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.e.a.45.1	✓	2	12.11	even	2
76.2.e.a.49.1	yes	2	228.11	even	6
304.2.i.a.49.1		2	57.11	odd	6
304.2.i.a.273.1		2	3.2	odd	2
684.2.k.b.505.1		2	76.11	odd	6
684.2.k.b.577.1		2	4.3	odd	2
1216.2.i.c.577.1		2	24.11	even	2
1216.2.i.c.961.1		2	456.11	even	6
1216.2.i.g.577.1		2	24.5	odd	2
1216.2.i.g.961.1		2	456.125	odd	6
1444.2.a.b.1.1		1	228.83	even	6
1444.2.a.c.1.1		1	228.107	odd	6
1444.2.e.b.429.1		2	228.179	odd	6
1444.2.e.b.653.1		2	228.227	odd	2
1900.2.i.a.201.1		2	1140.239	even	6
1900.2.i.a.501.1		2	60.59	even	2
1900.2.s.a.49.1		4	1140.467	odd	12
1900.2.s.a.49.2		4	1140.923	odd	12
1900.2.s.a.349.1		4	60.23	odd	4
1900.2.s.a.349.2		4	60.47	odd	4
2736.2.s.g.577.1		2	1.1	even	1	trivial
2736.2.s.g.1873.1		2	19.11	even	3	inner
5776.2.a.f.1.1		1	57.50	even	6
5776.2.a.k.1.1		1	57.26	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} -4.00000 q^{11} +(0.500000 - 0.866025i) q^{13} +(1.50000 + 2.59808i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-2.50000 + 4.33013i) q^{23} +(2.00000 - 3.46410i) q^{25} +(3.50000 - 6.06218i) q^{29} -4.00000 q^{31} +10.0000 q^{37} +(-2.50000 - 4.33013i) q^{41} +(-2.50000 - 4.33013i) q^{43} +(3.50000 - 6.06218i) q^{47} -7.00000 q^{49} +(5.50000 - 9.52628i) q^{53} +(2.00000 + 3.46410i) q^{55} +(-1.50000 - 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} -1.00000 q^{65} +(-1.50000 + 2.59808i) q^{67} +(-5.50000 - 9.52628i) q^{71} +(-7.50000 - 12.9904i) q^{73} +(-6.50000 - 11.2583i) q^{79} +(1.50000 - 2.59808i) q^{85} +(1.50000 - 2.59808i) q^{89} +(-0.500000 - 4.33013i) q^{95} +(2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 8 q^{11} + q^{13} + 3 q^{17} + 8 q^{19} - 5 q^{23} + 4 q^{25} + 7 q^{29} - 8 q^{31} + 20 q^{37} - 5 q^{41} - 5 q^{43} + 7 q^{47} - 14 q^{49} + 11 q^{53} + 4 q^{55} - 3 q^{59} - 11 q^{61}+ \cdots + 5 q^{97}+O(q^{100}) \) 2 * q - q^5 - 8 * q^11 + q^13 + 3 * q^17 + 8 * q^19 - 5 * q^23 + 4 * q^25 + 7 * q^29 - 8 * q^31 + 20 * q^37 - 5 * q^41 - 5 * q^43 + 7 * q^47 - 14 * q^49 + 11 * q^53 + 4 * q^55 - 3 * q^59 - 11 * q^61 - 2 * q^65 - 3 * q^67 - 11 * q^71 - 15 * q^73 - 13 * q^79 + 3 * q^85 + 3 * q^89 - q^95 + 5 * q^97

Introduction

L-functions

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