LMFDB - Embedded newform 2736.2.dc.a.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(449,2736)]

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, 3, 5]))

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

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (of order $6$, 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.a.1889.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 0.707107i) q^{5} +3.44949 q^{7} +O(q^{10})$	$q+(-1.22474 + 0.707107i) q^{5} +3.44949 q^{7} -6.29253i q^{11} +(-2.17423 - 1.25529i) q^{13} +(-4.22474 + 2.43916i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(-4.89898 - 2.82843i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(1.22474 - 2.12132i) q^{29} +9.43879i q^{31} +(-4.22474 + 2.43916i) q^{35} +5.97469i q^{37} +(-2.94949 - 5.10867i) q^{43} +(-4.22474 - 2.43916i) q^{47} +4.89898 q^{49} +(-2.44949 + 4.24264i) q^{53} +(4.44949 + 7.70674i) q^{55} +(4.77526 + 8.27098i) q^{59} +(-0.724745 + 1.25529i) q^{61} +3.55051 q^{65} +(-5.84847 - 3.37662i) q^{67} +(3.00000 + 5.19615i) q^{71} +(5.94949 + 10.3048i) q^{73} -21.7060i q^{77} +(-8.17423 + 4.71940i) q^{79} -8.97809i q^{83} +(3.44949 - 5.97469i) q^{85} +(-6.12372 + 10.6066i) q^{89} +(-7.50000 - 4.33013i) q^{91} +(3.67423 - 4.94975i) q^{95} +(-1.65153 + 0.953512i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 16 q^{19} - 6 q^{25} - 12 q^{35} - 2 q^{43} - 12 q^{47} + 8 q^{55} + 24 q^{59} + 2 q^{61} + 24 q^{65} + 6 q^{67} + 12 q^{71} + 14 q^{73} - 18 q^{79} + 4 q^{85} - 30 q^{91} - 36 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 16 * q^19 - 6 * q^25 - 12 * q^35 - 2 * q^43 - 12 * q^47 + 8 * q^55 + 24 * q^59 + 2 * q^61 + 24 * q^65 + 6 * q^67 + 12 * q^71 + 14 * q^73 - 18 * q^79 + 4 * q^85 - 30 * q^91 - 36 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$1$

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
$5$	−1.22474	+	0.707107i	−0.547723	+	0.316228i	−0.748203	−	0.663470i	$-0.769083\pi$
$5$	−1.22474	+	0.707107i	−0.547723	+	0.316228i	0.200480	+	0.979698i	$0.435750\pi$
$6$		0			0
$7$	3.44949			1.30378			0.651892	−	0.758312i	$-0.273975\pi$
$7$	3.44949			1.30378			0.651892	+	0.758312i	$0.273975\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	6.29253i		−	1.89727i	−0.316374	−	0.948634i	$-0.602466\pi$
$11$		−	6.29253i		−	1.89727i	0.316374	−	0.948634i	$-0.397534\pi$
$12$		0			0
$13$	−2.17423	−	1.25529i	−0.603024	−	0.348156i	0.167206	−	0.985922i	$-0.446525\pi$
$13$	−2.17423	−	1.25529i	−0.603024	−	0.348156i	−0.770230	+	0.637766i	$0.779859\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.22474	+	2.43916i	−1.02465	+	0.591583i	−0.915448	−	0.402437i	$-0.868163\pi$
$17$	−4.22474	+	2.43916i	−1.02465	+	0.591583i	−0.109203	+	0.994019i	$0.534830\pi$
$18$		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
$23$	−4.89898	−	2.82843i	−1.02151	−	0.589768i	−0.106967	−	0.994263i	$-0.534114\pi$
$23$	−4.89898	−	2.82843i	−1.02151	−	0.589768i	−0.914540	+	0.404495i	$0.867447\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.22474	−	2.12132i	0.227429	−	0.393919i	−0.729616	−	0.683857i	$-0.760301\pi$
$29$	1.22474	−	2.12132i	0.227429	−	0.393919i	0.957046	+	0.289938i	$0.0936346\pi$
$30$		0			0
$31$			9.43879i			1.69526i	0.530590	+	0.847629i	$0.321970\pi$
$31$			9.43879i			1.69526i	−0.530590	+	0.847629i	$0.678030\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.22474	+	2.43916i	−0.714112	+	0.412293i
$36$		0			0
$37$			5.97469i			0.982233i	0.871094	+	0.491117i	$0.163411\pi$
$37$			5.97469i			0.982233i	−0.871094	+	0.491117i	$0.836589\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	−2.94949	−	5.10867i	−0.449793	−	0.779064i	0.548579	−	0.836099i	$-0.315169\pi$
$43$	−2.94949	−	5.10867i	−0.449793	−	0.779064i	−0.998372	+	0.0570343i	$0.981836\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.22474	−	2.43916i	−0.616242	−	0.355788i	0.159162	−	0.987252i	$-0.449121\pi$
$47$	−4.22474	−	2.43916i	−0.616242	−	0.355788i	−0.775405	+	0.631465i	$0.782454\pi$
$48$		0			0
$49$	4.89898			0.699854
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.44949	+	4.24264i	−0.336463	+	0.582772i	−0.983765	−	0.179463i	$-0.942564\pi$
$53$	−2.44949	+	4.24264i	−0.336463	+	0.582772i	0.647302	+	0.762234i	$0.275897\pi$
$54$		0			0
$55$	4.44949	+	7.70674i	0.599969	+	1.03918i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.77526	+	8.27098i	0.621685	+	1.07679i	0.989172	+	0.146762i	$0.0468850\pi$
$59$	4.77526	+	8.27098i	0.621685	+	1.07679i	−0.367487	+	0.930029i	$0.619782\pi$
$60$		0			0
$61$	−0.724745	+	1.25529i	−0.0927941	+	0.160724i	−0.908686	−	0.417481i	$-0.862913\pi$
$61$	−0.724745	+	1.25529i	−0.0927941	+	0.160724i	0.815892	+	0.578205i	$0.196246\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.55051			0.440387
$66$		0			0
$67$	−5.84847	−	3.37662i	−0.714504	−	0.412519i	0.0982223	−	0.995164i	$-0.468684\pi$
$67$	−5.84847	−	3.37662i	−0.714504	−	0.412519i	−0.812727	+	0.582645i	$0.802018\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	0.987294	−	0.158901i	$-0.0507952\pi$
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	−0.631260	+	0.775571i	$0.717462\pi$
$72$		0			0
$73$	5.94949	+	10.3048i	0.696335	+	1.20609i	0.969729	+	0.244185i	$0.0785206\pi$
$73$	5.94949	+	10.3048i	0.696335	+	1.20609i	−0.273393	+	0.961902i	$0.588146\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	21.7060i		−	2.47363i
$78$		0			0
$79$	−8.17423	+	4.71940i	−0.919673	+	0.530974i	−0.883531	−	0.468373i	$-0.844840\pi$
$79$	−8.17423	+	4.71940i	−0.919673	+	0.530974i	−0.0361424	+	0.999347i	$0.511507\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.97809i		−	0.985474i	−0.870178	−	0.492737i	$-0.835997\pi$
$83$		−	8.97809i		−	0.985474i	0.870178	−	0.492737i	$-0.164003\pi$
$84$		0			0
$85$	3.44949	−	5.97469i	0.374150	−	0.648046i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.12372	+	10.6066i	−0.649113	+	1.12430i	0.334221	+	0.942495i	$0.391527\pi$
$89$	−6.12372	+	10.6066i	−0.649113	+	1.12430i	−0.983335	+	0.181803i	$0.941807\pi$
$90$		0			0
$91$	−7.50000	−	4.33013i	−0.786214	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.67423	−	4.94975i	0.376969	−	0.507833i
$96$		0			0
$97$	−1.65153	+	0.953512i	−0.167688	+	0.0968144i	−0.581495	−	0.813550i	$-0.697532\pi$
$97$	−1.65153	+	0.953512i	−0.167688	+	0.0968144i	0.413807	+	0.910364i	$0.364199\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.55051	+	3.78194i	0.651800	+	0.376317i	0.789146	−	0.614206i	$-0.210524\pi$
$101$	6.55051	+	3.78194i	0.651800	+	0.376317i	−0.137345	+	0.990523i	$0.543857\pi$
$102$		0			0
$103$		−	10.9959i		−	1.08346i	−0.840554	−	0.541728i	$-0.817770\pi$
$103$		−	10.9959i		−	1.08346i	0.840554	−	0.541728i	$-0.182230\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.79796			−0.367163			−0.183581	−	0.983005i	$-0.558769\pi$
$107$	−3.79796			−0.367163			−0.183581	+	0.983005i	$0.558769\pi$
$108$		0			0
$109$	14.6969	−	8.48528i	1.40771	−	0.812743i	0.412544	−	0.910938i	$-0.364640\pi$
$109$	14.6969	−	8.48528i	1.40771	−	0.812743i	0.995167	+	0.0981950i	$0.0313069\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.1464			−1.61300			−0.806500	−	0.591234i	$-0.798641\pi$
$113$	−17.1464			−1.61300			−0.806500	+	0.591234i	$0.798641\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.5732	+	8.41385i	−1.33592	+	0.771296i
$120$		0			0
$121$	−28.5959			−2.59963
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$	7.34847	+	4.24264i	0.652071	+	0.376473i	0.789249	−	0.614073i	$-0.210470\pi$
$127$	7.34847	+	4.24264i	0.652071	+	0.376473i	−0.137178	+	0.990546i	$0.543803\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.79796	+	3.92480i	−0.593940	+	0.342912i	−0.766654	−	0.642060i	$-0.778080\pi$
$131$	−6.79796	+	3.92480i	−0.593940	+	0.342912i	0.172714	+	0.984972i	$0.444746\pi$
$132$		0			0
$133$	−13.7980	+	5.97469i	−1.19643	+	0.518071i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.550510	+	0.317837i	0.0470333	+	0.0271547i	0.523332	−	0.852129i	$-0.324689\pi$
$137$	0.550510	+	0.317837i	0.0470333	+	0.0271547i	−0.476299	+	0.879283i	$0.658022\pi$
$138$		0			0
$139$	−3.50000	+	6.06218i	−0.296866	+	0.514187i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−3.50000	+	6.06218i	−0.296866	+	0.514187i	0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.89898	+	13.6814i	−0.660546	+	1.14410i
$144$		0			0
$145$			3.46410i			0.287678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.5505	+	7.24604i	−1.02818	+	0.593619i	−0.916462	−	0.400121i	$-0.868968\pi$
$149$	−12.5505	+	7.24604i	−1.02818	+	0.593619i	−0.111716	+	0.993740i	$0.535635\pi$
$150$		0			0
$151$		−	15.4135i		−	1.25433i	−0.778886	−	0.627166i	$-0.784215\pi$
$151$		−	15.4135i		−	1.25433i	0.778886	−	0.627166i	$-0.215785\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.67423	−	11.5601i	−0.536087	−	0.928531i
$156$		0			0
$157$	1.17423	+	2.03383i	0.0937141	+	0.162318i	0.909071	−	0.416641i	$-0.136793\pi$
$157$	1.17423	+	2.03383i	0.0937141	+	0.162318i	−0.815357	+	0.578958i	$0.803459\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.8990	−	9.75663i	−1.33183	−	0.768930i
$162$		0			0
$163$	21.6969			1.69944			0.849718	−	0.527238i	$-0.176772\pi$
$163$	21.6969			1.69944			0.849718	+	0.527238i	$0.176772\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.77526	−	3.07483i	0.137373	−	0.237938i	−0.789128	−	0.614228i	$-0.789467\pi$
$167$	1.77526	−	3.07483i	0.137373	−	0.237938i	0.926502	+	0.376291i	$0.122801\pi$
$168$		0			0
$169$	−3.34847	−	5.79972i	−0.257575	−	0.446132i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.12372	−	10.6066i	−0.465578	−	0.806405i	0.533649	−	0.845706i	$-0.320820\pi$
$173$	−6.12372	−	10.6066i	−0.465578	−	0.806405i	−0.999227	+	0.0393009i	$0.987487\pi$
$174$		0			0
$175$	−5.17423	+	8.96204i	−0.391135	+	0.677466i
$176$		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
$181$	−19.3485	−	11.1708i	−1.43816	−	0.830322i	−0.440438	−	0.897783i	$-0.645177\pi$
$181$	−19.3485	−	11.1708i	−1.43816	−	0.830322i	−0.997722	+	0.0674605i	$0.978510\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.22474	−	7.31747i	−0.310609	−	0.537991i
$186$		0			0
$187$	15.3485	+	26.5843i	1.12239	+	1.94404i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.4422i		−	0.900285i	−0.892957	−	0.450143i	$-0.851373\pi$
$191$		−	12.4422i		−	0.900285i	0.892957	−	0.450143i	$-0.148627\pi$
$192$		0			0
$193$	0.151531	−	0.0874863i	0.0109074	−	0.00629740i	−0.494536	−	0.869157i	$-0.664662\pi$
$193$	0.151531	−	0.0874863i	0.0109074	−	0.00629740i	0.505444	+	0.862860i	$0.331329\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			0.492810i			0.0351113i	0.999846	+	0.0175556i	$0.00558842\pi$
$197$			0.492810i			0.0351113i	−0.999846	+	0.0175556i	$0.994412\pi$
$198$		0			0
$199$	−3.62372	+	6.27647i	−0.256879	+	0.444927i	−0.965404	−	0.260758i	$-0.916028\pi$
$199$	−3.62372	+	6.27647i	−0.256879	+	0.444927i	0.708525	+	0.705686i	$0.249361\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.22474	−	7.31747i	0.296519	−	0.513586i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.8990	+	25.1701i	0.753898	+	1.74105i
$210$		0			0
$211$	−17.8485	+	10.3048i	−1.22874	+	0.709413i	−0.966766	−	0.255662i	$-0.917707\pi$
$211$	−17.8485	+	10.3048i	−1.22874	+	0.709413i	−0.261973	+	0.965075i	$0.584373\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.22474	+	4.17121i	0.492724	+	0.284474i
$216$		0			0
$217$			32.5590i			2.21025i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.2474			0.823853
$222$		0			0
$223$	−5.17423	+	2.98735i	−0.346492	+	0.200047i	−0.663139	−	0.748496i	$-0.730776\pi$
$223$	−5.17423	+	2.98735i	−0.346492	+	0.200047i	0.316647	+	0.948543i	$0.397443\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.1464			0.739814			0.369907	−	0.929069i	$-0.379389\pi$
$227$	11.1464			0.739814			0.369907	+	0.929069i	$0.379389\pi$
$228$		0			0
$229$	5.24745			0.346761			0.173381	−	0.984855i	$-0.444531\pi$
$229$	5.24745			0.346761			0.173381	+	0.984855i	$0.444531\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.24745	+	5.33902i	−0.605821	+	0.349771i	−0.771328	−	0.636438i	$-0.780407\pi$
$233$	−9.24745	+	5.33902i	−0.605821	+	0.349771i	0.165507	+	0.986209i	$0.447074\pi$
$234$		0			0
$235$	6.89898			0.450040
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	8.62815i		−	0.558108i	−0.960275	−	0.279054i	$-0.909979\pi$
$239$		−	8.62815i		−	0.558108i	0.960275	−	0.279054i	$-0.0900209\pi$
$240$		0			0
$241$	26.8485	+	15.5010i	1.72946	+	0.998505i	0.892058	+	0.451920i	$0.149261\pi$
$241$	26.8485	+	15.5010i	1.72946	+	0.998505i	0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	+	3.46410i	−0.383326	+	0.221313i
$246$		0			0
$247$	10.8712	+	1.25529i	0.691716	+	0.0798725i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.22474	−	0.707107i	−0.0773052	−	0.0446322i	0.460849	−	0.887478i	$-0.347545\pi$
$251$	−1.22474	−	0.707107i	−0.0773052	−	0.0446322i	−0.538154	+	0.842846i	$0.680878\pi$
$252$		0			0
$253$	−17.7980	+	30.8270i	−1.11895	+	1.93807i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	−0.615948	−	0.787787i	$-0.711227\pi$
$257$	6.00000	−	10.3923i	0.374270	−	0.648254i	0.990217	+	0.139533i	$0.0445601\pi$
$258$		0			0
$259$			20.6096i			1.28062i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.7980	+	10.8530i	−1.15913	+	0.669225i	−0.951096	−	0.308896i	$-0.900040\pi$
$263$	−18.7980	+	10.8530i	−1.15913	+	0.669225i	−0.208036	+	0.978121i	$0.566707\pi$
$264$		0			0
$265$		−	6.92820i		−	0.425596i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.77526	−	13.4671i	−0.474066	−	0.821106i	0.525493	−	0.850798i	$-0.323881\pi$
$269$	−7.77526	−	13.4671i	−0.474066	−	0.821106i	−0.999559	+	0.0296918i	$0.990547\pi$
$270$		0			0
$271$	−13.6969	−	23.7238i	−0.832030	−	1.44112i	−0.896426	−	0.443193i	$-0.853845\pi$
$271$	−13.6969	−	23.7238i	−0.832030	−	1.44112i	0.0643963	−	0.997924i	$-0.479488\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.3485	+	9.43879i	0.985850	+	0.569181i
$276$		0			0
$277$	18.8990			1.13553			0.567765	−	0.823191i	$-0.307808\pi$
$277$	18.8990			1.13553			0.567765	+	0.823191i	$0.307808\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.57321	+	9.65309i	−0.332470	+	0.575855i	−0.982996	−	0.183629i	$-0.941215\pi$
$281$	−5.57321	+	9.65309i	−0.332470	+	0.575855i	0.650525	+	0.759484i	$0.274549\pi$
$282$		0			0
$283$	−2.55051	−	4.41761i	−0.151612	−	0.262600i	0.780208	−	0.625520i	$-0.215113\pi$
$283$	−2.55051	−	4.41761i	−0.151612	−	0.262600i	−0.931820	+	0.362920i	$0.881780\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	3.39898	−	5.88721i	0.199940	−	0.346306i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	25.3485			1.48087			0.740437	−	0.672126i	$-0.234619\pi$
$293$	25.3485			1.48087			0.740437	+	0.672126i	$0.234619\pi$
$294$		0			0
$295$	−11.6969	−	6.75323i	−0.681022	−	0.393188i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.10102	+	12.2993i	0.410663	+	0.711289i
$300$		0			0
$301$	−10.1742	−	17.6223i	−0.586433	−	1.01573i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.04989i		−	0.117376i
$306$		0			0
$307$	−1.34847	+	0.778539i	−0.0769612	+	0.0444336i	−0.537987	−	0.842953i	$-0.680815\pi$
$307$	−1.34847	+	0.778539i	−0.0769612	+	0.0444336i	0.461026	+	0.887387i	$0.347482\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	25.9487i		−	1.47141i	−0.677300	−	0.735707i	$-0.736850\pi$
$311$		−	25.9487i		−	1.47141i	0.677300	−	0.735707i	$-0.263150\pi$
$312$		0			0
$313$	3.34847	−	5.79972i	0.189267	−	0.327819i	−0.755739	−	0.654873i	$-0.772722\pi$
$313$	3.34847	−	5.79972i	0.189267	−	0.327819i	0.945006	+	0.327053i	$0.106056\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.67423	−	6.36396i	0.206366	−	0.357436i	−0.744201	−	0.667955i	$-0.767170\pi$
$317$	3.67423	−	6.36396i	0.206366	−	0.357436i	0.950567	+	0.310520i	$0.100503\pi$
$318$		0			0
$319$	−13.3485	−	7.70674i	−0.747371	−	0.431495i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.6742	−	17.0741i	0.705213	−	0.950029i
$324$		0			0
$325$	6.52270	−	3.76588i	0.361815	−	0.208894i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.5732	−	8.41385i	−0.803447	−	0.463871i
$330$		0			0
$331$		−	15.2385i		−	0.837584i	−0.908082	−	0.418792i	$-0.862454\pi$
$331$		−	15.2385i		−	0.837584i	0.908082	−	0.418792i	$-0.137546\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.55051			0.521800
$336$		0			0
$337$	31.1969	−	18.0116i	1.69941	−	0.981152i	0.753085	−	0.657924i	$-0.228565\pi$
$337$	31.1969	−	18.0116i	1.69941	−	0.981152i	0.946321	−	0.323229i	$-0.104768\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	59.3939			3.21636
$342$		0			0
$343$	−7.24745			−0.391325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.7753	+	14.8814i	−1.38369	+	0.798873i	−0.992594	−	0.121478i	$-0.961237\pi$
$347$	−25.7753	+	14.8814i	−1.38369	+	0.798873i	−0.391094	+	0.920351i	$0.627903\pi$
$348$		0			0
$349$	−1.24745			−0.0667744			−0.0333872	−	0.999442i	$-0.510629\pi$
$349$	−1.24745			−0.0667744			−0.0333872	+	0.999442i	$0.510629\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			17.4634i			0.929482i	0.885447	+	0.464741i	$0.153852\pi$
$353$			17.4634i			0.929482i	−0.885447	+	0.464741i	$0.846148\pi$
$354$		0			0
$355$	−7.34847	−	4.24264i	−0.390016	−	0.225176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.79796	+	5.65685i	−0.517116	+	0.298557i	−0.735754	−	0.677249i	$-0.763172\pi$
$359$	−9.79796	+	5.65685i	−0.517116	+	0.298557i	0.218638	+	0.975806i	$0.429839\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.5732	−	8.41385i	−0.762797	−	0.440401i
$366$		0			0
$367$	0.174235	−	0.301783i	0.00909497	−	0.0157530i	−0.861442	−	0.507856i	$-0.830438\pi$
$367$	0.174235	−	0.301783i	0.00909497	−	0.0157530i	0.870537	+	0.492103i	$0.163772\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.44949	+	14.6349i	−0.438676	+	0.759809i
$372$		0			0
$373$		−	29.2699i		−	1.51554i	−0.652523	−	0.757769i	$-0.726290\pi$
$373$		−	29.2699i		−	1.51554i	0.652523	−	0.757769i	$-0.273710\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.32577	+	3.07483i	−0.274291	+	0.158362i
$378$		0			0
$379$		−	19.0526i		−	0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$		−	19.0526i		−	0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.550510	−	0.953512i	−0.0281298	−	0.0487222i	0.851618	−	0.524163i	$-0.175622\pi$
$383$	−0.550510	−	0.953512i	−0.0281298	−	0.0487222i	−0.879748	+	0.475441i	$0.842288\pi$
$384$		0			0
$385$	15.3485	+	26.5843i	0.782230	+	1.35486i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.1010	−	7.56388i	−0.664248	−	0.383504i	0.129646	−	0.991560i	$-0.458616\pi$
$389$	−13.1010	−	7.56388i	−0.664248	−	0.383504i	−0.793894	+	0.608057i	$0.791949\pi$
$390$		0			0
$391$	27.5959			1.39559
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.67423	−	11.5601i	0.335817	−	0.581652i
$396$		0			0
$397$	−18.1742	−	31.4787i	−0.912139	−	1.57987i	−0.811037	−	0.584994i	$-0.801097\pi$
$397$	−18.1742	−	31.4787i	−0.912139	−	1.57987i	−0.101101	−	0.994876i	$-0.532237\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.1237	+	20.9989i	0.605430	+	1.04864i	0.991983	+	0.126368i	$0.0403321\pi$
$401$	12.1237	+	20.9989i	0.605430	+	1.04864i	−0.386553	+	0.922267i	$0.626335\pi$
$402$		0			0
$403$	11.8485	−	20.5222i	0.590214	−	1.02228i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	37.5959			1.86356
$408$		0			0
$409$	19.0454	+	10.9959i	0.941735	+	0.543711i	0.890504	−	0.454976i	$-0.150352\pi$
$409$	19.0454	+	10.9959i	0.941735	+	0.543711i	0.0512311	+	0.998687i	$0.483685\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16.4722	+	28.5307i	0.810544	+	1.40390i
$414$		0			0
$415$	6.34847	+	10.9959i	0.311634	+	0.539766i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			24.5344i			1.19859i	0.800530	+	0.599293i	$0.204552\pi$
$419$			24.5344i			1.19859i	−0.800530	+	0.599293i	$0.795448\pi$
$420$		0			0
$421$	−13.3485	+	7.70674i	−0.650565	+	0.375604i	−0.788672	−	0.614814i	$-0.789231\pi$
$421$	−13.3485	+	7.70674i	−0.650565	+	0.375604i	0.138108	+	0.990417i	$0.455898\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	14.6349i		−	0.709899i
$426$		0			0
$427$	−2.50000	+	4.33013i	−0.120983	+	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.0227	−	24.2880i	0.675450	−	1.16991i	−0.300887	−	0.953660i	$-0.597283\pi$
$431$	14.0227	−	24.2880i	0.675450	−	1.16991i	0.976337	−	0.216254i	$-0.0693839\pi$
$432$		0			0
$433$	−4.19694	−	2.42310i	−0.201692	−	0.116447i	0.395752	−	0.918357i	$-0.370484\pi$
$433$	−4.19694	−	2.42310i	−0.201692	−	0.116447i	−0.597444	+	0.801910i	$0.703817\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.4949	+	2.82843i	1.17175	+	0.135302i
$438$		0			0
$439$	−9.52270	+	5.49794i	−0.454494	+	0.262402i	−0.709726	−	0.704478i	$-0.751181\pi$
$439$	−9.52270	+	5.49794i	−0.454494	+	0.262402i	0.255232	+	0.966880i	$0.417848\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−28.2247	−	16.2956i	−1.34100	−	0.774226i	−0.354044	−	0.935229i	$-0.615194\pi$
$443$	−28.2247	−	16.2956i	−1.34100	−	0.774226i	−0.986954	+	0.161003i	$0.948527\pi$
$444$		0			0
$445$		−	17.3205i		−	0.821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.24745			−0.294835			−0.147418	−	0.989074i	$-0.547096\pi$
$449$	−6.24745			−0.294835			−0.147418	+	0.989074i	$0.547096\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	12.2474			0.574169
$456$		0			0
$457$	−9.69694			−0.453604			−0.226802	−	0.973941i	$-0.572827\pi$
$457$	−9.69694			−0.453604			−0.226802	+	0.973941i	$0.572827\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.12372	−	1.80348i	0.145486	−	0.0839966i	−0.425490	−	0.904963i	$-0.639898\pi$
$461$	3.12372	−	1.80348i	0.145486	−	0.0839966i	0.570976	+	0.820967i	$0.306565\pi$
$462$		0			0
$463$	−5.85357			−0.272039			−0.136019	−	0.990706i	$-0.543431\pi$
$463$	−5.85357			−0.272039			−0.136019	+	0.990706i	$0.543431\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			7.21393i			0.333821i	0.985972	+	0.166910i	$0.0533791\pi$
$467$			7.21393i			0.333821i	−0.985972	+	0.166910i	$0.946621\pi$
$468$		0			0
$469$	−20.1742	−	11.6476i	−0.931560	−	0.537836i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−32.1464	+	18.5597i	−1.47809	+	0.853378i
$474$		0			0
$475$	1.50000	−	12.9904i	0.0688247	−	0.596040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	26.1464	+	15.0956i	1.19466	+	0.689738i	0.959360	−	0.282185i	$-0.0910592\pi$
$479$	26.1464	+	15.0956i	1.19466	+	0.689738i	0.235301	+	0.971923i	$0.424392\pi$
$480$		0			0
$481$	7.50000	−	12.9904i	0.341971	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.34847	−	2.33562i	0.0612308	−	0.106055i
$486$		0			0
$487$		−	22.6916i		−	1.02826i	−0.857713	−	0.514128i	$-0.828116\pi$
$487$		−	22.6916i		−	1.02826i	0.857713	−	0.514128i	$-0.171884\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.14643	+	4.70334i	−0.367643	+	0.212259i	−0.672428	−	0.740162i	$-0.734749\pi$
$491$	−8.14643	+	4.70334i	−0.367643	+	0.212259i	0.304785	+	0.952421i	$0.401415\pi$
$492$		0			0
$493$			11.9494i			0.538173i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3485	+	17.9241i	0.464192	+	0.804005i
$498$		0			0
$499$	2.74745	+	4.75872i	0.122993	+	0.213030i	0.920947	−	0.389689i	$-0.127417\pi$
$499$	2.74745	+	4.75872i	0.122993	+	0.213030i	−0.797954	+	0.602719i	$0.794084\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	35.5176	+	20.5061i	1.58365	+	0.914322i	0.994320	+	0.106430i	$0.0339421\pi$
$503$	35.5176	+	20.5061i	1.58365	+	0.914322i	0.589331	+	0.807891i	$0.299391\pi$
$504$		0			0
$505$	−10.6969			−0.476008
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.674235	+	1.16781i	−0.0298849	+	0.0517622i	−0.880581	−	0.473896i	$-0.842847\pi$
$509$	−0.674235	+	1.16781i	−0.0298849	+	0.0517622i	0.850696	+	0.525658i	$0.176181\pi$
$510$		0			0
$511$	20.5227	+	35.5464i	0.907871	+	1.57248i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.77526	+	13.4671i	0.342619	+	0.593433i
$516$		0			0
$517$	−15.3485	+	26.5843i	−0.675025	+	1.16918i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	27.7980			1.21785			0.608925	−	0.793228i	$-0.291601\pi$
$521$	27.7980			1.21785			0.608925	+	0.793228i	$0.291601\pi$
$522$		0			0
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−23.0227	−	39.8765i	−1.00288	−	1.73705i
$528$		0			0
$529$	4.50000	+	7.79423i	0.195652	+	0.338880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.65153	−	2.68556i	0.201103	−	0.116107i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	30.8270i		−	1.32781i
$540$		0			0
$541$	−7.82577	+	13.5546i	−0.336456	+	0.582759i	−0.983763	−	0.179470i	$-0.942562\pi$
$541$	−7.82577	+	13.5546i	−0.336456	+	0.582759i	0.647307	+	0.762229i	$0.275895\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	+	20.7846i	−0.514024	+	0.890315i
$546$		0			0
$547$	31.1969	+	18.0116i	1.33388	+	0.770119i	0.985893	−	0.167379i	$-0.0535303\pi$
$547$	31.1969	+	18.0116i	1.33388	+	0.770119i	0.347992	+	0.937497i	$0.386864\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.22474	+	10.6066i	−0.0521759	+	0.451856i
$552$		0			0
$553$	−28.1969	+	16.2795i	−1.19906	+	0.692275i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.5959	+	9.58166i	0.703192	+	0.405988i	0.808535	−	0.588448i	$-0.200261\pi$
$557$	16.5959	+	9.58166i	0.703192	+	0.405988i	−0.105343	+	0.994436i	$0.533594\pi$
$558$		0			0
$559$			14.8099i			0.626393i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.5959			0.573000			0.286500	−	0.958080i	$-0.407508\pi$
$563$	13.5959			0.573000			0.286500	+	0.958080i	$0.407508\pi$
$564$		0			0
$565$	21.0000	−	12.1244i	0.883477	−	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−40.8990			−1.71457			−0.857287	−	0.514838i	$-0.827852\pi$
$569$	−40.8990			−1.71457			−0.857287	+	0.514838i	$0.827852\pi$
$570$		0			0
$571$	−25.6969			−1.07538			−0.537692	−	0.843142i	$-0.680704\pi$
$571$	−25.6969			−1.07538			−0.537692	+	0.843142i	$0.680704\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	14.6969	−	8.48528i	0.612905	−	0.353861i
$576$		0			0
$577$	−13.7980			−0.574417			−0.287208	−	0.957868i	$-0.592727\pi$
$577$	−13.7980			−0.574417			−0.287208	+	0.957868i	$0.592727\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	30.9698i		−	1.28485i
$582$		0			0
$583$	26.6969	+	15.4135i	1.10567	+	0.638361i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.8763	−	6.85677i	0.490186	−	0.283009i	−0.234465	−	0.972124i	$-0.575334\pi$
$587$	11.8763	−	6.85677i	0.490186	−	0.283009i	0.724652	+	0.689115i	$0.242001\pi$
$588$		0			0
$589$	−16.3485	−	37.7552i	−0.673627	−	1.55567i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13.4722	−	7.77817i	−0.553237	−	0.319411i	0.197190	−	0.980365i	$-0.436818\pi$
$593$	−13.4722	−	7.77817i	−0.553237	−	0.319411i	−0.750426	+	0.660954i	$0.770152\pi$
$594$		0			0
$595$	11.8990	−	20.6096i	0.487811	−	0.844913i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.797959	+	1.38211i	−0.0326037	+	0.0564713i	−0.881867	−	0.471499i	$-0.843713\pi$
$599$	−0.797959	+	1.38211i	−0.0326037	+	0.0564713i	0.849263	+	0.527970i	$0.177047\pi$
$600$		0			0
$601$			6.75323i			0.275470i	0.990469	+	0.137735i	$0.0439822\pi$
$601$			6.75323i			0.275470i	−0.990469	+	0.137735i	$0.956018\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	35.0227	−	20.2204i	1.42388	−	0.822075i
$606$		0			0
$607$		−	40.2658i		−	1.63434i	−0.576399	−	0.817168i	$-0.695543\pi$
$607$		−	40.2658i		−	1.63434i	0.576399	−	0.817168i	$-0.304457\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.12372	+	10.6066i	0.247739	+	0.429097i
$612$		0			0
$613$	−4.55051	−	7.88171i	−0.183793	−	0.318339i	0.759376	−	0.650652i	$-0.225504\pi$
$613$	−4.55051	−	7.88171i	−0.183793	−	0.318339i	−0.943169	+	0.332313i	$0.892171\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	37.8990	+	21.8810i	1.52576	+	0.880895i	0.999533	+	0.0305482i	$0.00972531\pi$
$617$	37.8990	+	21.8810i	1.52576	+	0.880895i	0.526222	+	0.850347i	$0.323608\pi$
$618$		0			0
$619$	−22.3939			−0.900086			−0.450043	−	0.893007i	$-0.648591\pi$
$619$	−22.3939			−0.900086			−0.450043	+	0.893007i	$0.648591\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−21.1237	+	36.5874i	−0.846304	+	1.46584i
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.5732	−	25.2415i	−0.581072	−	1.00645i
$630$		0			0
$631$	9.72474	−	16.8438i	0.387136	−	0.670539i	−0.604927	−	0.796281i	$-0.706798\pi$
$631$	9.72474	−	16.8438i	0.387136	−	0.670539i	0.992063	+	0.125742i	$0.0401311\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000			−0.476205
$636$		0			0
$637$	−10.6515	−	6.14966i	−0.422029	−	0.243659i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.977296	−	1.69273i	−0.0386009	−	0.0668587i	0.846080	−	0.533057i	$-0.178957\pi$
$641$	−0.977296	−	1.69273i	−0.0386009	−	0.0668587i	−0.884680	+	0.466198i	$0.845623\pi$
$642$		0			0
$643$	1.70204	+	2.94802i	0.0671219	+	0.116259i	0.897633	−	0.440743i	$-0.145285\pi$
$643$	1.70204	+	2.94802i	0.0671219	+	0.116259i	−0.830511	+	0.557002i	$0.811952\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			16.8277i			0.661565i	0.943707	+	0.330783i	$0.107313\pi$
$647$			16.8277i			0.661565i	−0.943707	+	0.330783i	$0.892687\pi$
$648$		0			0
$649$	52.0454	−	30.0484i	2.04296	−	1.17950i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			10.1066i			0.395501i	0.980252	+	0.197750i	$0.0633636\pi$
$653$			10.1066i			0.395501i	−0.980252	+	0.197750i	$0.936636\pi$
$654$		0			0
$655$	5.55051	−	9.61377i	0.216876	−	0.375641i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.6969	+	41.0443i	−0.923102	+	1.59886i	−0.128516	+	0.991707i	$0.541021\pi$
$659$	−23.6969	+	41.0443i	−0.923102	+	1.59886i	−0.794586	+	0.607151i	$0.792312\pi$
$660$		0			0
$661$	5.69694	+	3.28913i	0.221585	+	0.127932i	0.606684	−	0.794943i	$-0.292499\pi$
$661$	5.69694	+	3.28913i	0.221585	+	0.127932i	−0.385099	+	0.922875i	$0.625833\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.6742	−	17.0741i	0.491486	−	0.662105i
$666$		0			0
$667$	−12.0000	+	6.92820i	−0.464642	+	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.89898	+	4.56048i	0.304937	+	0.176055i
$672$		0			0
$673$			19.0526i			0.734422i	0.930138	+	0.367211i	$0.119687\pi$
$673$			19.0526i			0.734422i	−0.930138	+	0.367211i	$0.880313\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.6515			−0.870569			−0.435285	−	0.900293i	$-0.643352\pi$
$677$	−22.6515			−0.870569			−0.435285	+	0.900293i	$0.643352\pi$
$678$		0			0
$679$	−5.69694	+	3.28913i	−0.218628	+	0.126225i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.0454			0.613960			0.306980	−	0.951716i	$-0.400681\pi$
$683$	16.0454			0.613960			0.306980	+	0.951716i	$0.400681\pi$
$684$		0			0
$685$	−0.898979			−0.0343482
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.6515	−	6.14966i	0.405791	−	0.234284i
$690$		0			0
$691$	−28.6969			−1.09168			−0.545841	−	0.837888i	$-0.683790\pi$
$691$	−28.6969			−1.09168			−0.545841	+	0.837888i	$0.683790\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	9.89949i		−	0.375509i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	13.4722	−	7.77817i	0.508838	−	0.293778i	−0.223518	−	0.974700i	$-0.571754\pi$
$701$	13.4722	−	7.77817i	0.508838	−	0.293778i	0.732356	+	0.680922i	$0.238421\pi$
$702$		0			0
$703$	−10.3485	−	23.8988i	−0.390300	−	0.901359i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	22.5959	+	13.0458i	0.849807	+	0.490636i
$708$		0			0
$709$	4.17423	−	7.22999i	0.156767	−	0.271528i	−0.776934	−	0.629582i	$-0.783226\pi$
$709$	4.17423	−	7.22999i	0.156767	−	0.271528i	0.933701	+	0.358054i	$0.116560\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	26.6969	−	46.2405i	0.999808	−	1.73172i
$714$		0			0
$715$		−	22.3417i		−	0.835532i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.1464	+	9.89949i	−0.639454	+	0.369189i	−0.784404	−	0.620250i	$-0.787031\pi$
$719$	−17.1464	+	9.89949i	−0.639454	+	0.369189i	0.144950	+	0.989439i	$0.453698\pi$
$720$		0			0
$721$		−	37.9301i		−	1.41259i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.67423	+	6.36396i	0.136458	+	0.236352i
$726$		0			0
$727$	12.1742	+	21.0864i	0.451517	+	0.782051i	0.998481	−	0.0551057i	$-0.0175496\pi$
$727$	12.1742	+	21.0864i	0.451517	+	0.782051i	−0.546963	+	0.837157i	$0.684216\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24.9217	+	14.3885i	0.921762	+	0.532179i
$732$		0			0
$733$	−1.30306			−0.0481297			−0.0240648	−	0.999710i	$-0.507661\pi$
$733$	−1.30306			−0.0481297			−0.0240648	+	0.999710i	$0.507661\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.2474	+	36.8017i	−0.782660	+	1.35561i
$738$		0			0
$739$	−6.19694	−	10.7334i	−0.227958	−	0.394835i	0.729245	−	0.684253i	$-0.239872\pi$
$739$	−6.19694	−	10.7334i	−0.227958	−	0.394835i	−0.957203	+	0.289418i	$0.906538\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.69694	+	9.86739i	0.209000	+	0.361999i	0.951400	−	0.307958i	$-0.0996457\pi$
$743$	5.69694	+	9.86739i	0.209000	+	0.361999i	−0.742400	+	0.669957i	$0.766312\pi$
$744$		0			0
$745$	10.2474	−	17.7491i	0.375437	−	0.650277i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−13.1010			−0.478701
$750$		0			0
$751$	27.2196	+	15.7153i	0.993259	+	0.573458i	0.906247	−	0.422749i	$-0.138935\pi$
$751$	27.2196	+	15.7153i	0.993259	+	0.573458i	0.0870120	+	0.996207i	$0.472268\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.8990	+	18.8776i	0.396654	+	0.687026i
$756$		0			0
$757$	−4.82577	−	8.35847i	−0.175395	−	0.303794i	0.764903	−	0.644146i	$-0.222787\pi$
$757$	−4.82577	−	8.35847i	−0.175395	−	0.303794i	−0.940298	+	0.340352i	$0.889454\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			3.25702i			0.118067i	0.998256	+	0.0590335i	$0.0188019\pi$
$761$			3.25702i			0.118067i	−0.998256	+	0.0590335i	$0.981198\pi$
$762$		0			0
$763$	50.6969	−	29.2699i	1.83535	−	1.05964i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	23.9774i		−	0.865774i
$768$		0			0
$769$	−18.2980	+	31.6930i	−0.659841	+	1.14288i	0.320815	+	0.947142i	$0.396043\pi$
$769$	−18.2980	+	31.6930i	−0.659841	+	1.14288i	−0.980656	+	0.195737i	$0.937290\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.4722	−	38.9230i	0.808269	−	1.39996i	−0.105793	−	0.994388i	$-0.533738\pi$
$773$	22.4722	−	38.9230i	0.808269	−	1.39996i	0.914062	−	0.405574i	$-0.132928\pi$
$774$		0			0
$775$	−24.5227	−	14.1582i	−0.880882	−	0.508577i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	32.6969	−	18.8776i	1.16999	−	0.675493i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.87628	−	1.66062i	−0.102659	−	0.0592700i
$786$		0			0
$787$		−	7.10318i		−	0.253201i	−0.991954	−	0.126600i	$-0.959593\pi$
$787$		−	7.10318i		−	0.253201i	0.991954	−	0.126600i	$-0.0404066\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−59.1464			−2.10300
$792$		0			0
$793$	3.15153	−	1.81954i	0.111914	−	0.0646137i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	50.0908			1.77431			0.887154	−	0.461474i	$-0.152679\pi$
$797$	50.0908			1.77431			0.887154	+	0.461474i	$0.152679\pi$
$798$		0			0
$799$	23.7980			0.841911
$800$		0			0
$801$		0			0
$802$		0			0
$803$	64.8434	−	37.4373i	2.28827	−	1.32113i
$804$		0			0
$805$	27.5959			0.972628
$806$		0			0
$807$		0			0
$808$		0			0
$809$			25.9487i			0.912306i	0.889901	+	0.456153i	$0.150773\pi$
$809$			25.9487i			0.912306i	−0.889901	+	0.456153i	$0.849227\pi$
$810$		0			0
$811$	11.6969	+	6.75323i	0.410735	+	0.237138i	0.691106	−	0.722754i	$-0.257124\pi$
$811$	11.6969	+	6.75323i	0.410735	+	0.237138i	−0.280370	+	0.959892i	$0.590457\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−26.5732	+	15.3421i	−0.930819	+	0.537409i
$816$		0			0
$817$	20.6464	+	15.3260i	0.722327	+	0.536189i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.1691	−	12.7994i	−0.773708	−	0.446701i	0.0604877	−	0.998169i	$-0.480734\pi$
$821$	−22.1691	−	12.7994i	−0.773708	−	0.446701i	−0.834196	+	0.551468i	$0.814068\pi$
$822$		0			0
$823$	2.65153	−	4.59259i	0.0924266	−	0.160087i	−0.816105	−	0.577903i	$-0.803871\pi$
$823$	2.65153	−	4.59259i	0.0924266	−	0.160087i	0.908532	+	0.417816i	$0.137204\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17.8207	−	30.8663i	0.619685	−	1.07333i	−0.369858	−	0.929088i	$-0.620594\pi$
$827$	17.8207	−	30.8663i	0.619685	−	1.07333i	0.989543	−	0.144238i	$-0.0460730\pi$
$828$		0			0
$829$			14.4600i			0.502216i	0.967959	+	0.251108i	$0.0807949\pi$
$829$			14.4600i			0.502216i	−0.967959	+	0.251108i	$0.919205\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−20.6969	+	11.9494i	−0.717106	+	0.414022i
$834$		0			0
$835$			5.02118i			0.173765i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.0227	+	29.4842i	0.587689	+	1.01791i	0.994534	+	0.104410i	$0.0332955\pi$
$839$	17.0227	+	29.4842i	0.587689	+	1.01791i	−0.406845	+	0.913497i	$0.633371\pi$
$840$		0			0
$841$	11.5000	+	19.9186i	0.396552	+	0.686848i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.20204	+	4.73545i	0.282159	+	0.162904i
$846$		0			0
$847$	−98.6413			−3.38936
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16.8990	−	29.2699i	0.579290	−	1.00336i
$852$		0			0
$853$	22.1742	+	38.4069i	0.759231	+	1.31503i	0.943243	+	0.332103i	$0.107758\pi$
$853$	22.1742	+	38.4069i	0.759231	+	1.31503i	−0.184012	+	0.982924i	$0.558908\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.5732	−	35.6339i	−0.702768	−	1.21723i	−0.967491	−	0.252905i	$-0.918614\pi$
$857$	−20.5732	−	35.6339i	−0.702768	−	1.21723i	0.264724	−	0.964324i	$-0.414719\pi$
$858$		0			0
$859$	−2.94949	+	5.10867i	−0.100635	+	0.174305i	−0.911947	−	0.410309i	$-0.865421\pi$
$859$	−2.94949	+	5.10867i	−0.100635	+	0.174305i	0.811311	+	0.584614i	$0.198754\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.6515			0.771067			0.385534	−	0.922694i	$-0.374017\pi$
$863$	22.6515			0.771067			0.385534	+	0.922694i	$0.374017\pi$
$864$		0			0
$865$	15.0000	+	8.66025i	0.510015	+	0.294457i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	29.6969	+	51.4366i	1.00740	+	1.74487i
$870$		0			0
$871$	8.47730	+	14.6831i	0.287242	+	0.497518i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	39.0265i		−	1.31934i
$876$		0			0
$877$	2.47730	−	1.43027i	0.0836523	−	0.0482967i	−0.457590	−	0.889163i	$-0.651287\pi$
$877$	2.47730	−	1.43027i	0.0836523	−	0.0482967i	0.541243	+	0.840866i	$0.317954\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	4.87832i		−	0.164355i	−0.996618	−	0.0821773i	$-0.973813\pi$
$881$		−	4.87832i		−	0.164355i	0.996618	−	0.0821773i	$-0.0261874\pi$
$882$		0			0
$883$	1.94949	−	3.37662i	0.0656056	−	0.113632i	−0.831357	−	0.555739i	$-0.812435\pi$
$883$	1.94949	−	3.37662i	0.0656056	−	0.113632i	0.896962	+	0.442107i	$0.145769\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.6742	+	32.3447i	−0.627019	+	1.08603i	0.361127	+	0.932517i	$0.382392\pi$
$887$	−18.6742	+	32.3447i	−0.627019	+	1.08603i	−0.988147	+	0.153513i	$0.950941\pi$
$888$		0			0
$889$	25.3485	+	14.6349i	0.850160	+	0.490840i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.1237	+	2.43916i	0.706878	+	0.0816233i
$894$		0			0
$895$	14.6969	−	8.48528i	0.491264	−	0.283632i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0227	+	11.5601i	0.667795	+	0.385551i
$900$		0			0
$901$		−	23.8988i		−	0.796183i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	31.5959			1.05028
$906$		0			0
$907$	−6.30306	+	3.63907i	−0.209290	+	0.120833i	−0.600981	−	0.799263i	$-0.705223\pi$
$907$	−6.30306	+	3.63907i	−0.209290	+	0.120833i	0.391692	+	0.920097i	$0.371890\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	−56.4949			−1.86971
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−23.4495	+	13.5386i	−0.774370	+	0.447083i
$918$		0			0
$919$	35.0454			1.15604			0.578021	−	0.816022i	$-0.303825\pi$
$919$	35.0454			1.15604			0.578021	+	0.816022i	$0.303825\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	15.0635i		−	0.495822i
$924$		0			0
$925$	−15.5227	−	8.96204i	−0.510383	−	0.294670i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.4268	−	10.6387i	0.604563	−	0.349045i	−0.166271	−	0.986080i	$-0.553173\pi$
$929$	18.4268	−	10.6387i	0.604563	−	0.349045i	0.770835	+	0.637035i	$0.219839\pi$
$930$		0			0
$931$	−19.5959	+	8.48528i	−0.642230	+	0.278094i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−37.5959	−	21.7060i	−1.22952	−	0.709863i
$936$		0			0
$937$	21.1969	−	36.7142i	0.692474	−	1.19940i	−0.278551	−	0.960421i	$-0.589854\pi$
$937$	21.1969	−	36.7142i	0.692474	−	1.19940i	0.971025	−	0.238978i	$-0.0768125\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.5505	+	26.9343i	−0.506932	+	0.878032i	0.493036	+	0.870009i	$0.335887\pi$
$941$	−15.5505	+	26.9343i	−0.506932	+	0.878032i	−0.999968	+	0.00802314i	$0.997446\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.4949	+	8.94598i	−0.503517	+	0.290705i	−0.730165	−	0.683271i	$-0.760557\pi$
$947$	−15.4949	+	8.94598i	−0.503517	+	0.290705i	0.226648	+	0.973977i	$0.427223\pi$
$948$		0			0
$949$		−	29.8735i		−	0.969733i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	5.44949	+	9.43879i	0.176526	+	0.305752i	0.940688	−	0.339272i	$-0.110181\pi$
$953$	5.44949	+	9.43879i	0.176526	+	0.305752i	−0.764162	+	0.645024i	$0.776847\pi$
$954$		0			0
$955$	8.79796	+	15.2385i	0.284695	+	0.493107i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.89898	+	1.09638i	0.0613212	+	0.0354038i
$960$		0			0
$961$	−58.0908			−1.87390
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.123724	+	0.214297i	−0.00398283	+	0.00689846i
$966$		0			0
$967$	−1.17423	−	2.03383i	−0.0377608	−	0.0654037i	0.846527	−	0.532345i	$-0.178689\pi$
$967$	−1.17423	−	2.03383i	−0.0377608	−	0.0654037i	−0.884288	+	0.466942i	$0.845356\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−26.8207	−	46.4548i	−0.860716	−	1.49080i	−0.871239	−	0.490859i	$-0.836683\pi$
$971$	−26.8207	−	46.4548i	−0.860716	−	1.49080i	0.0105228	−	0.999945i	$-0.496650\pi$
$972$		0			0
$973$	−12.0732	+	20.9114i	−0.387049	+	0.670389i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.2474			−0.775745			−0.387872	−	0.921713i	$-0.626790\pi$
$977$	−24.2474			−0.775745			−0.387872	+	0.921713i	$0.626790\pi$
$978$		0			0
$979$	66.7423	+	38.5337i	2.13309	+	1.23154i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.2474	+	26.4094i	0.486318	+	0.842328i	0.999876	−	0.0157271i	$-0.00500630\pi$
$983$	15.2474	+	26.4094i	0.486318	+	0.842328i	−0.513558	+	0.858055i	$0.671673\pi$
$984$		0			0
$985$	−0.348469	−	0.603566i	−0.0111032	−	0.0192312i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			33.3697i			1.06109i
$990$		0			0
$991$	−16.1288	+	9.31198i	−0.512349	+	0.295805i	−0.733799	−	0.679367i	$-0.762254\pi$
$991$	−16.1288	+	9.31198i	−0.512349	+	0.295805i	0.221450	+	0.975172i	$0.428921\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	10.2494i		−	0.324929i
$996$		0			0
$997$	−1.52270	+	2.63740i	−0.0482245	+	0.0835273i	−0.889130	−	0.457655i	$-0.848690\pi$
$997$	−1.52270	+	2.63740i	−0.0482245	+	0.0835273i	0.840906	+	0.541182i	$0.182023\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.a.449.1		4
3.2	odd	2		2736.2.dc.b.449.2		4
4.3	odd	2		342.2.s.b.107.1	yes	4
12.11	even	2		342.2.s.a.107.2	✓	4
19.8	odd	6		2736.2.dc.b.1889.2		4
57.8	even	6	inner	2736.2.dc.a.1889.1		4
76.27	even	6		342.2.s.a.179.2	yes	4
228.179	odd	6		342.2.s.b.179.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.2.s.a.107.2	✓	4	12.11	even	2
342.2.s.a.179.2	yes	4	76.27	even	6
342.2.s.b.107.1	yes	4	4.3	odd	2
342.2.s.b.179.1	yes	4	228.179	odd	6
2736.2.dc.a.449.1		4	1.1	even	1	trivial
2736.2.dc.a.1889.1		4	57.8	even	6	inner
2736.2.dc.b.449.2		4	3.2	odd	2
2736.2.dc.b.1889.2		4	19.8	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.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{5} +3.44949 q^{7} +O(q^{10})\)	\(q+(-1.22474 + 0.707107i) q^{5} +3.44949 q^{7} -6.29253i q^{11} +(-2.17423 - 1.25529i) q^{13} +(-4.22474 + 2.43916i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(-4.89898 - 2.82843i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(1.22474 - 2.12132i) q^{29} +9.43879i q^{31} +(-4.22474 + 2.43916i) q^{35} +5.97469i q^{37} +(-2.94949 - 5.10867i) q^{43} +(-4.22474 - 2.43916i) q^{47} +4.89898 q^{49} +(-2.44949 + 4.24264i) q^{53} +(4.44949 + 7.70674i) q^{55} +(4.77526 + 8.27098i) q^{59} +(-0.724745 + 1.25529i) q^{61} +3.55051 q^{65} +(-5.84847 - 3.37662i) q^{67} +(3.00000 + 5.19615i) q^{71} +(5.94949 + 10.3048i) q^{73} -21.7060i q^{77} +(-8.17423 + 4.71940i) q^{79} -8.97809i q^{83} +(3.44949 - 5.97469i) q^{85} +(-6.12372 + 10.6066i) q^{89} +(-7.50000 - 4.33013i) q^{91} +(3.67423 - 4.94975i) q^{95} +(-1.65153 + 0.953512i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 6 q^{13} - 12 q^{17} - 16 q^{19} - 6 q^{25} - 12 q^{35} - 2 q^{43} - 12 q^{47} + 8 q^{55} + 24 q^{59} + 2 q^{61} + 24 q^{65} + 6 q^{67} + 12 q^{71} + 14 q^{73} - 18 q^{79} + 4 q^{85} - 30 q^{91} - 36 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 6 * q^13 - 12 * q^17 - 16 * q^19 - 6 * q^25 - 12 * q^35 - 2 * q^43 - 12 * q^47 + 8 * q^55 + 24 * q^59 + 2 * q^61 + 24 * q^65 + 6 * q^67 + 12 * q^71 + 14 * q^73 - 18 * q^79 + 4 * q^85 - 30 * q^91 - 36 * q^97

Introduction

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