LMFDB - Embedded newform 2736.2.cg.c.1151.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1151,2736)]

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([3, 0, 3, 4]))

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

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

chi := DirichletCharacter("2736.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.cg (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1151.9
Character	$\chi$	$=$	2736.1151
Dual form			2736.2.cg.c.2591.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.34408 + 0.776006i) q^{5} -2.95486i q^{7} +O(q^{10})$	$q+(1.34408 + 0.776006i) q^{5} -2.95486i q^{7} -3.57750 q^{11} +(2.39307 + 4.14491i) q^{13} +(-3.37866 - 1.95067i) q^{17} +(-2.61248 + 3.48926i) q^{19} +(-2.64157 - 4.57534i) q^{23} +(-1.29563 - 2.24410i) q^{25} +(7.08654 - 4.09142i) q^{29} -7.52123i q^{31} +(2.29299 - 3.97157i) q^{35} +2.32246 q^{37} +(-3.70788 - 2.14074i) q^{41} +(3.48460 + 2.01183i) q^{43} +(-4.08174 - 7.06978i) q^{47} -1.73120 q^{49} +(2.03458 - 1.17467i) q^{53} +(-4.80845 - 2.77616i) q^{55} +(3.14582 - 5.44871i) q^{59} +(-6.24917 - 10.8239i) q^{61} +7.42814i q^{65} +(9.02949 - 5.21318i) q^{67} +(-2.72467 + 4.71927i) q^{71} +(-5.18870 + 8.98709i) q^{73} +10.5710i q^{77} +(8.60257 + 4.96669i) q^{79} -17.9858 q^{83} +(-3.02747 - 5.24373i) q^{85} +(-7.70648 + 4.44934i) q^{89} +(12.2476 - 7.07118i) q^{91} +(-6.21907 + 2.66256i) q^{95} +(6.08794 - 10.5446i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{13} + 24 q^{25} - 16 q^{37} - 96 q^{49} - 8 q^{61} - 8 q^{73} + 16 q^{85} + 48 q^{97}+O(q^{100}) $ 32 * q - 16 * q^13 + 24 * q^25 - 16 * q^37 - 96 * q^49 - 8 * q^61 - 8 * q^73 + 16 * q^85 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.34408	+	0.776006i	0.601092	+	0.347040i	0.769471	−	0.638682i	$-0.220520\pi$
$5$	1.34408	+	0.776006i	0.601092	+	0.347040i	−0.168379	+	0.985722i	$0.553853\pi$
$6$		0			0
$7$		−	2.95486i		−	1.11683i	−0.829561	−	0.558416i	$-0.811409\pi$
$7$		−	2.95486i		−	1.11683i	0.829561	−	0.558416i	$-0.188591\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.57750			−1.07866			−0.539328	−	0.842096i	$-0.681322\pi$
$11$	−3.57750			−1.07866			−0.539328	+	0.842096i	$0.681322\pi$
$12$		0			0
$13$	2.39307	+	4.14491i	0.663718	+	1.14959i	0.979631	+	0.200805i	$0.0643556\pi$
$13$	2.39307	+	4.14491i	0.663718	+	1.14959i	−0.315914	+	0.948788i	$0.602311\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.37866	−	1.95067i	−0.819446	−	0.473108i	0.0307791	−	0.999526i	$-0.490201\pi$
$17$	−3.37866	−	1.95067i	−0.819446	−	0.473108i	−0.850226	+	0.526419i	$0.823534\pi$
$18$		0			0
$19$	−2.61248	+	3.48926i	−0.599343	+	0.800492i
$19$	−2.61248	+	3.48926i	−0.599343	+	0.800492i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.64157	−	4.57534i	−0.550806	−	0.954025i	−0.998217	−	0.0596959i	$-0.980987\pi$
$23$	−2.64157	−	4.57534i	−0.550806	−	0.954025i	0.447410	−	0.894329i	$-0.352346\pi$
$24$		0			0
$25$	−1.29563	−	2.24410i	−0.259126	−	0.448819i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.08654	−	4.09142i	1.31594	−	0.759757i	0.332865	−	0.942975i	$-0.391985\pi$
$29$	7.08654	−	4.09142i	1.31594	−	0.759757i	0.983072	+	0.183218i	$0.0586513\pi$
$30$		0			0
$31$		−	7.52123i		−	1.35085i	−0.737427	−	0.675427i	$-0.763959\pi$
$31$		−	7.52123i		−	1.35085i	0.737427	−	0.675427i	$-0.236041\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.29299	−	3.97157i	0.387586	−	0.671319i
$36$		0			0
$37$	2.32246			0.381810			0.190905	−	0.981609i	$-0.438858\pi$
$37$	2.32246			0.381810			0.190905	+	0.981609i	$0.438858\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.70788	−	2.14074i	−0.579073	−	0.334328i	0.181692	−	0.983355i	$-0.441843\pi$
$41$	−3.70788	−	2.14074i	−0.579073	−	0.334328i	−0.760765	+	0.649028i	$0.775176\pi$
$42$		0			0
$43$	3.48460	+	2.01183i	0.531396	+	0.306802i	0.741585	−	0.670859i	$-0.234074\pi$
$43$	3.48460	+	2.01183i	0.531396	+	0.306802i	−0.210189	+	0.977661i	$0.567408\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.08174	−	7.06978i	−0.595383	−	1.03123i	−0.993493	−	0.113896i	$-0.963667\pi$
$47$	−4.08174	−	7.06978i	−0.595383	−	1.03123i	0.398110	−	0.917338i	$-0.369666\pi$
$48$		0			0
$49$	−1.73120			−0.247314
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.03458	−	1.17467i	0.279471	−	0.161353i	−0.353713	−	0.935354i	$-0.615081\pi$
$53$	2.03458	−	1.17467i	0.279471	−	0.161353i	0.633184	+	0.774001i	$0.281748\pi$
$54$		0			0
$55$	−4.80845	−	2.77616i	−0.648371	−	0.374337i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.14582	−	5.44871i	0.409550	−	0.709362i	−0.585289	−	0.810825i	$-0.699019\pi$
$59$	3.14582	−	5.44871i	0.409550	−	0.709362i	0.994839	+	0.101463i	$0.0323523\pi$
$60$		0			0
$61$	−6.24917	−	10.8239i	−0.800125	−	1.38586i	−0.919534	−	0.393011i	$-0.871433\pi$
$61$	−6.24917	−	10.8239i	−0.800125	−	1.38586i	0.119409	−	0.992845i	$-0.461900\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			7.42814i			0.921347i
$66$		0			0
$67$	9.02949	−	5.21318i	1.10313	−	0.636891i	0.166087	−	0.986111i	$-0.446887\pi$
$67$	9.02949	−	5.21318i	1.10313	−	0.636891i	0.937041	+	0.349220i	$0.113553\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.72467	+	4.71927i	−0.323359	+	0.560074i	−0.981179	−	0.193101i	$-0.938146\pi$
$71$	−2.72467	+	4.71927i	−0.323359	+	0.560074i	0.657820	+	0.753175i	$0.271479\pi$
$72$		0			0
$73$	−5.18870	+	8.98709i	−0.607291	+	1.05186i	0.384394	+	0.923169i	$0.374410\pi$
$73$	−5.18870	+	8.98709i	−0.607291	+	1.05186i	−0.991685	+	0.128690i	$0.958923\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			10.5710i			1.20468i
$78$		0			0
$79$	8.60257	+	4.96669i	0.967864	+	0.558797i	0.898585	−	0.438801i	$-0.144597\pi$
$79$	8.60257	+	4.96669i	0.967864	+	0.558797i	0.0692799	+	0.997597i	$0.477930\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−17.9858			−1.97419			−0.987097	−	0.160126i	$-0.948810\pi$
$83$	−17.9858			−1.97419			−0.987097	+	0.160126i	$0.948810\pi$
$84$		0			0
$85$	−3.02747	−	5.24373i	−0.328375	−	0.568762i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.70648	+	4.44934i	−0.816885	+	0.471629i	−0.849341	−	0.527844i	$-0.823000\pi$
$89$	−7.70648	+	4.44934i	−0.816885	+	0.471629i	0.0324559	+	0.999473i	$0.489667\pi$
$90$		0			0
$91$	12.2476	−	7.07118i	1.28390	−	0.741261i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.21907	+	2.66256i	−0.638063	+	0.273173i
$96$		0			0
$97$	6.08794	−	10.5446i	0.618137	−	1.07064i	−0.371688	−	0.928358i	$-0.621221\pi$
$97$	6.08794	−	10.5446i	0.618137	−	1.07064i	0.989825	−	0.142287i	$-0.0454457\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.47913	+	5.47278i	−0.943209	+	0.544562i	−0.890965	−	0.454072i	$-0.849971\pi$
$101$	−9.47913	+	5.47278i	−0.943209	+	0.544562i	−0.0522441	+	0.998634i	$0.516637\pi$
$102$		0			0
$103$		−	10.4264i		−	1.02734i	−0.857988	−	0.513670i	$-0.828286\pi$
$103$		−	10.4264i		−	1.02734i	0.857988	−	0.513670i	$-0.171714\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.6914			−1.90364			−0.951820	−	0.306657i	$-0.900790\pi$
$107$	−19.6914			−1.90364			−0.951820	+	0.306657i	$0.900790\pi$
$108$		0			0
$109$	−7.94405	+	13.7595i	−0.760902	+	1.31792i	0.181484	+	0.983394i	$0.441910\pi$
$109$	−7.94405	+	13.7595i	−0.760902	+	1.31792i	−0.942386	+	0.334527i	$0.891423\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.66966i		−	0.627429i	−0.949517	−	0.313715i	$-0.898426\pi$
$113$		−	6.66966i		−	0.627429i	0.949517	−	0.313715i	$-0.101574\pi$
$114$		0			0
$115$		−	8.19951i		−	0.764608i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.76397	+	9.98348i	−0.528382	+	0.915184i
$120$		0			0
$121$	1.79849			0.163500
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.7817i		−	1.05379i
$126$		0			0
$127$	−2.29254	+	1.32360i	−0.203430	+	0.117450i	−0.598254	−	0.801306i	$-0.704139\pi$
$127$	−2.29254	+	1.32360i	−0.203430	+	0.117450i	0.394825	+	0.918757i	$0.370805\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.36489	−	16.2205i	0.818214	−	1.41719i	−0.0887828	−	0.996051i	$-0.528298\pi$
$131$	9.36489	−	16.2205i	0.818214	−	1.41719i	0.906997	−	0.421137i	$-0.138369\pi$
$132$		0			0
$133$	10.3103	+	7.71951i	0.894015	+	0.669366i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.1197	−	7.57468i	1.12089	−	0.647148i	0.179265	−	0.983801i	$-0.442628\pi$
$137$	13.1197	−	7.57468i	1.12089	−	0.647148i	0.941629	+	0.336652i	$0.109295\pi$
$138$		0			0
$139$	7.86612	−	4.54151i	0.667196	−	0.385206i	−0.127818	−	0.991798i	$-0.540797\pi$
$139$	7.86612	−	4.54151i	0.667196	−	0.385206i	0.795013	+	0.606592i	$0.207464\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.56120	−	14.8284i	−0.715923	−	1.24002i
$144$		0			0
$145$	12.6999			1.05467
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.72619	+	5.03807i	0.714877	+	0.412735i	0.812864	−	0.582453i	$-0.197907\pi$
$149$	8.72619	+	5.03807i	0.714877	+	0.412735i	−0.0979869	+	0.995188i	$0.531240\pi$
$150$		0			0
$151$		−	15.1780i		−	1.23517i	−0.786504	−	0.617586i	$-0.788111\pi$
$151$		−	15.1780i		−	1.23517i	0.786504	−	0.617586i	$-0.211889\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.83652	−	10.1092i	0.468801	−	0.811987i
$156$		0			0
$157$	3.25867	−	5.64418i	0.260070	−	0.450454i	−0.706190	−	0.708022i	$-0.749588\pi$
$157$	3.25867	−	5.64418i	0.260070	−	0.450454i	0.966260	+	0.257568i	$0.0829210\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.5195	+	7.80549i	−1.06549	+	0.615158i
$162$		0			0
$163$			5.36702i			0.420377i	0.977661	+	0.210189i	$0.0674078\pi$
$163$			5.36702i			0.420377i	−0.977661	+	0.210189i	$0.932592\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.55271	+	13.0817i	0.584447	+	1.01229i	0.994944	+	0.100429i	$0.0320217\pi$
$167$	7.55271	+	13.0817i	0.584447	+	1.01229i	−0.410498	+	0.911862i	$0.634645\pi$
$168$		0			0
$169$	−4.95354	+	8.57979i	−0.381042	+	0.659984i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.97214	−	1.13862i	−0.149939	−	0.0865674i	0.423153	−	0.906058i	$-0.360923\pi$
$173$	−1.97214	−	1.13862i	−0.149939	−	0.0865674i	−0.573093	+	0.819491i	$0.694256\pi$
$174$		0			0
$175$	−6.63099	+	3.82840i	−0.501256	+	0.289400i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.3998			−1.00155			−0.500773	−	0.865579i	$-0.666951\pi$
$179$	−13.3998			−1.00155			−0.500773	+	0.865579i	$0.666951\pi$
$180$		0			0
$181$	4.72566	+	8.18508i	0.351255	+	0.608392i	0.986470	−	0.163944i	$-0.0524215\pi$
$181$	4.72566	+	8.18508i	0.351255	+	0.608392i	−0.635214	+	0.772336i	$0.719088\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.12157	+	1.80224i	0.229503	+	0.132503i
$186$		0			0
$187$	12.0872	+	6.97853i	0.883901	+	0.510321i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.87185			0.135442			0.0677210	−	0.997704i	$-0.478427\pi$
$191$	1.87185			0.135442			0.0677210	+	0.997704i	$0.478427\pi$
$192$		0			0
$193$	8.91040	−	15.4333i	0.641385	−	1.11091i	−0.343739	−	0.939065i	$-0.611694\pi$
$193$	8.91040	−	15.4333i	0.641385	−	1.11091i	0.985124	−	0.171846i	$-0.0549731\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3.51742i			0.250606i	0.992119	+	0.125303i	$0.0399903\pi$
$197$			3.51742i			0.250606i	−0.992119	+	0.125303i	$0.960010\pi$
$198$		0			0
$199$	−0.896925	+	0.517840i	−0.0635813	+	0.0367087i	−0.531454	−	0.847087i	$-0.678354\pi$
$199$	−0.896925	+	0.517840i	−0.0635813	+	0.0367087i	0.467872	+	0.883796i	$0.345021\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0896	−	20.9397i	−0.848521	−	1.46968i
$204$		0			0
$205$	−3.32246	−	5.75467i	−0.232051	−	0.401923i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.34613	−	12.4828i	0.646486	−	0.863456i
$210$		0			0
$211$	6.51358	+	3.76062i	0.448413	+	0.258892i	0.707160	−	0.707054i	$-0.249976\pi$
$211$	6.51358	+	3.76062i	0.448413	+	0.258892i	−0.258747	+	0.965945i	$0.583309\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.12239	+	5.40814i	0.212945	+	0.368832i
$216$		0			0
$217$	−22.2242			−1.50868
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	18.6724i		−	1.25604i
$222$		0			0
$223$	21.7471	+	12.5557i	1.45630	+	0.840793i	0.998826	−	0.0484325i	$-0.0154226\pi$
$223$	21.7471	+	12.5557i	1.45630	+	0.840793i	0.457469	+	0.889225i	$0.348756\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.58598			0.304382			0.152191	−	0.988351i	$-0.451367\pi$
$227$	4.58598			0.304382			0.152191	+	0.988351i	$0.451367\pi$
$228$		0			0
$229$	−10.3584			−0.684503			−0.342251	−	0.939608i	$-0.611189\pi$
$229$	−10.3584			−0.684503			−0.342251	+	0.939608i	$0.611189\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.5657	+	9.56419i	1.08525	+	0.626571i	0.932309	−	0.361664i	$-0.117791\pi$
$233$	16.5657	+	9.56419i	1.08525	+	0.626571i	0.152945	+	0.988235i	$0.451124\pi$
$234$		0			0
$235$		−	12.6698i		−	0.826488i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.95891			0.579504			0.289752	−	0.957102i	$-0.406427\pi$
$239$	8.95891			0.579504			0.289752	+	0.957102i	$0.406427\pi$
$240$		0			0
$241$	−1.04028	−	1.80181i	−0.0670101	−	0.116065i	0.830574	−	0.556909i	$-0.188013\pi$
$241$	−1.04028	−	1.80181i	−0.0670101	−	0.116065i	−0.897584	+	0.440844i	$0.854679\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.32687	−	1.34342i	−0.148659	−	0.0858281i
$246$		0			0
$247$	−20.7145	−	2.47845i	−1.31803	−	0.157700i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.49712	−	16.4495i	−0.599453	−	1.03828i	−0.992902	−	0.118936i	$-0.962052\pi$
$251$	−9.49712	−	16.4495i	−0.599453	−	1.03828i	0.393449	−	0.919346i	$-0.371282\pi$
$252$		0			0
$253$	9.45023	+	16.3683i	0.594131	+	1.02906i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.1578	−	11.0607i	1.19503	−	0.689950i	0.235585	−	0.971854i	$-0.424299\pi$
$257$	19.1578	−	11.0607i	1.19503	−	0.689950i	0.959443	+	0.281904i	$0.0909660\pi$
$258$		0			0
$259$		−	6.86254i		−	0.426418i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.31065	+	12.6624i	−0.450794	+	0.780798i	−0.998436	−	0.0559148i	$-0.982192\pi$
$263$	−7.31065	+	12.6624i	−0.450794	+	0.780798i	0.547641	+	0.836713i	$0.315526\pi$
$264$		0			0
$265$	3.64619			0.223984
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.7954	−	7.38740i	−0.780147	−	0.450418i	0.0563356	−	0.998412i	$-0.482058\pi$
$269$	−12.7954	−	7.38740i	−0.780147	−	0.450418i	−0.836482	+	0.547994i	$0.815392\pi$
$270$		0			0
$271$	−25.3521	−	14.6370i	−1.54003	−	0.889136i	−0.998836	−	0.0482382i	$-0.984639\pi$
$271$	−25.3521	−	14.6370i	−1.54003	−	0.889136i	−0.541193	−	0.840898i	$-0.682027\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.63511	+	8.02825i	0.279508	+	0.484122i
$276$		0			0
$277$	−7.08628			−0.425773			−0.212887	−	0.977077i	$-0.568287\pi$
$277$	−7.08628			−0.425773			−0.212887	+	0.977077i	$0.568287\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.7896	+	6.22936i	−0.643652	+	0.371613i	−0.786020	−	0.618201i	$-0.787862\pi$
$281$	−10.7896	+	6.22936i	−0.643652	+	0.371613i	0.142368	+	0.989814i	$0.454528\pi$
$282$		0			0
$283$	−16.6722	−	9.62572i	−0.991062	−	0.572190i	−0.0854701	−	0.996341i	$-0.527239\pi$
$283$	−16.6722	−	9.62572i	−0.991062	−	0.572190i	−0.905591	+	0.424151i	$0.860573\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.32560	+	10.9563i	−0.373388	+	0.646727i
$288$		0			0
$289$	−0.889753	−	1.54110i	−0.0523384	−	0.0906528i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.82121i			0.106396i	0.998584	+	0.0531980i	$0.0169415\pi$
$293$			1.82121i			0.106396i	−0.998584	+	0.0531980i	$0.983059\pi$
$294$		0			0
$295$	8.45647	−	4.88234i	0.492355	−	0.284261i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.6429	−	21.8982i	0.731160	−	1.26641i
$300$		0			0
$301$	5.94469	−	10.2965i	0.342646	−	0.593480i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	19.3976i		−	1.11070i
$306$		0			0
$307$	18.0734	+	10.4347i	1.03150	+	0.595538i	0.917414	−	0.397933i	$-0.130272\pi$
$307$	18.0734	+	10.4347i	1.03150	+	0.595538i	0.114087	+	0.993471i	$0.463606\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−21.2309			−1.20389			−0.601946	−	0.798537i	$-0.705608\pi$
$311$	−21.2309			−1.20389			−0.601946	+	0.798537i	$0.705608\pi$
$312$		0			0
$313$	1.62822	+	2.82016i	0.0920324	+	0.159405i	0.908366	−	0.418176i	$-0.137330\pi$
$313$	1.62822	+	2.82016i	0.0920324	+	0.159405i	−0.816334	+	0.577580i	$0.803997\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0712	+	6.96932i	−0.677987	+	0.391436i	−0.799096	−	0.601203i	$-0.794688\pi$
$317$	−12.0712	+	6.96932i	−0.677987	+	0.391436i	0.121109	+	0.992639i	$0.461355\pi$
$318$		0			0
$319$	−25.3521	+	14.6370i	−1.41944	+	0.819516i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.6331	−	6.69296i	0.869849	−	0.372406i
$324$		0			0
$325$	6.20106	−	10.7405i	0.343973	−	0.595778i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.8902	+	12.0610i	−1.15171	+	0.664943i
$330$		0			0
$331$			7.14723i			0.392847i	0.980519	+	0.196424i	$0.0629328\pi$
$331$			7.14723i			0.392847i	−0.980519	+	0.196424i	$0.937067\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.1818			0.884108
$336$		0			0
$337$	−6.47152	+	11.2090i	−0.352526	+	0.610593i	−0.986691	−	0.162605i	$-0.948011\pi$
$337$	−6.47152	+	11.2090i	−0.352526	+	0.610593i	0.634165	+	0.773197i	$0.281344\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			26.9072i			1.45711i
$342$		0			0
$343$		−	15.5686i		−	0.840624i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.70837	+	13.3513i	−0.413807	+	0.716735i	−0.995302	−	0.0968150i	$-0.969134\pi$
$347$	−7.70837	+	13.3513i	−0.413807	+	0.716735i	0.581495	+	0.813550i	$0.302468\pi$
$348$		0			0
$349$	29.4727			1.57764			0.788820	−	0.614625i	$-0.210693\pi$
$349$	29.4727			1.57764			0.788820	+	0.614625i	$0.210693\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			19.7000i			1.04853i	0.851556	+	0.524264i	$0.175659\pi$
$353$			19.7000i			1.04853i	−0.851556	+	0.524264i	$0.824341\pi$
$354$		0			0
$355$	−7.32437	+	4.22873i	−0.388737	+	0.224437i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.64022	+	11.5012i	−0.350457	+	0.607010i	−0.986330	−	0.164785i	$-0.947307\pi$
$359$	−6.64022	+	11.5012i	−0.350457	+	0.607010i	0.635872	+	0.771794i	$0.280640\pi$
$360$		0			0
$361$	−5.34993	−	18.2312i	−0.281575	−	0.959539i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.9481	+	8.05292i	−0.730075	+	0.421509i
$366$		0			0
$367$	−14.4570	+	8.34674i	−0.754648	+	0.435696i	−0.827371	−	0.561656i	$-0.810165\pi$
$367$	−14.4570	+	8.34674i	−0.754648	+	0.435696i	0.0727227	+	0.997352i	$0.476831\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.47098	−	6.01191i	−0.180204	−	0.312123i
$372$		0			0
$373$	−30.7828			−1.59387			−0.796937	−	0.604062i	$-0.793548\pi$
$373$	−30.7828			−1.59387			−0.796937	+	0.604062i	$0.793548\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	33.9171	+	19.5821i	1.74682	+	1.00853i
$378$		0			0
$379$			14.9927i			0.770125i	0.922890	+	0.385063i	$0.125820\pi$
$379$			14.9927i			0.770125i	−0.922890	+	0.385063i	$0.874180\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.26956	−	3.93100i	0.115969	−	0.200865i	−0.802197	−	0.597059i	$-0.796336\pi$
$383$	2.26956	−	3.93100i	0.115969	−	0.200865i	0.918167	+	0.396194i	$0.129669\pi$
$384$		0			0
$385$	−8.20317	+	14.2083i	−0.418072	+	0.724122i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.79236	−	1.61217i	0.141578	−	0.0817404i	−0.427538	−	0.903998i	$-0.640619\pi$
$389$	2.79236	−	1.61217i	0.141578	−	0.0817404i	0.569116	+	0.822257i	$0.307286\pi$
$390$		0			0
$391$			20.6114i			1.04236i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.70837	+	13.3513i	0.387850	+	0.671776i
$396$		0			0
$397$	6.15174	−	10.6551i	0.308747	−	0.534765i	−0.669342	−	0.742955i	$-0.733424\pi$
$397$	6.15174	−	10.6551i	0.308747	−	0.534765i	0.978088	+	0.208190i	$0.0667571\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−21.1635	−	12.2188i	−1.05686	−	0.610176i	−0.132296	−	0.991210i	$-0.542235\pi$
$401$	−21.1635	−	12.2188i	−1.05686	−	0.610176i	−0.924561	+	0.381034i	$0.875568\pi$
$402$		0			0
$403$	31.1749	−	17.9988i	1.55293	−	0.896585i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.30859			−0.411842
$408$		0			0
$409$	7.96304	+	13.7924i	0.393747	+	0.681990i	0.992940	−	0.118615i	$-0.0378453\pi$
$409$	7.96304	+	13.7924i	0.393747	+	0.681990i	−0.599193	+	0.800604i	$0.704512\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.1002	−	9.29544i	−0.792238	−	0.457399i
$414$		0			0
$415$	−24.1743	−	13.9571i	−1.18667	−	0.685125i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	38.2550			1.86888			0.934439	−	0.356123i	$-0.115901\pi$
$419$	38.2550			1.86888			0.934439	+	0.356123i	$0.115901\pi$
$420$		0			0
$421$	6.06443	−	10.5039i	0.295562	−	0.511929i	−0.679553	−	0.733626i	$-0.737826\pi$
$421$	6.06443	−	10.5039i	0.295562	−	0.511929i	0.975116	+	0.221697i	$0.0711597\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			10.1094i			0.490378i
$426$		0			0
$427$	−31.9831	+	18.4654i	−1.54777	+	0.893605i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.9424	+	22.4169i	0.623413	+	1.07978i	0.988845	+	0.148945i	$0.0475877\pi$
$431$	12.9424	+	22.4169i	0.623413	+	1.07978i	−0.365433	+	0.930838i	$0.619079\pi$
$432$		0			0
$433$	11.3701	+	19.6936i	0.546413	+	0.946416i	0.998516	+	0.0544501i	$0.0173406\pi$
$433$	11.3701	+	19.6936i	0.546413	+	0.946416i	−0.452103	+	0.891966i	$0.649326\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	22.8656	+	2.73582i	1.09381	+	0.130872i
$438$		0			0
$439$	30.1175	+	17.3883i	1.43743	+	0.829900i	0.997671	−	0.0682169i	$-0.0217310\pi$
$439$	30.1175	+	17.3883i	1.43743	+	0.829900i	0.439758	+	0.898116i	$0.355064\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.93456	+	8.54692i	0.234448	+	0.406076i	0.959112	−	0.283026i	$-0.0913383\pi$
$443$	4.93456	+	8.54692i	0.234448	+	0.406076i	−0.724664	+	0.689102i	$0.758005\pi$
$444$		0			0
$445$	−13.8109			−0.654697
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.88038i			0.371898i	0.982559	+	0.185949i	$0.0595359\pi$
$449$			7.88038i			0.371898i	−0.982559	+	0.185949i	$0.940464\pi$
$450$		0			0
$451$	13.2649	+	7.65850i	0.624621	+	0.360625i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21.9491			1.02899
$456$		0			0
$457$	7.32909			0.342840			0.171420	−	0.985198i	$-0.445164\pi$
$457$	7.32909			0.342840			0.171420	+	0.985198i	$0.445164\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.1717	−	10.4914i	−0.846340	−	0.488634i	0.0130745	−	0.999915i	$-0.495838\pi$
$461$	−18.1717	−	10.4914i	−0.846340	−	0.488634i	−0.859414	+	0.511280i	$0.829171\pi$
$462$		0			0
$463$			15.7207i			0.730605i	0.930889	+	0.365302i	$0.119034\pi$
$463$			15.7207i			0.730605i	−0.930889	+	0.365302i	$0.880966\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	27.8549			1.28897			0.644485	−	0.764617i	$-0.277072\pi$
$467$	27.8549			1.28897			0.644485	+	0.764617i	$0.277072\pi$
$468$		0			0
$469$	−15.4042	−	26.6809i	−0.711301	−	1.23201i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−12.4661	−	7.19733i	−0.573194	−	0.330934i
$474$		0			0
$475$	11.2150	+	1.34186i	0.514582	+	0.0615686i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.78739	+	10.0241i	0.264433	+	0.458011i	0.967415	−	0.253197i	$-0.0814820\pi$
$479$	5.78739	+	10.0241i	0.264433	+	0.458011i	−0.702982	+	0.711207i	$0.748149\pi$
$480$		0			0
$481$	5.55780	+	9.62639i	0.253414	+	0.438926i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.3654	−	9.44856i	0.743114	−	0.429037i
$486$		0			0
$487$			4.00714i			0.181581i	0.995870	+	0.0907905i	$0.0289394\pi$
$487$			4.00714i			0.181581i	−0.995870	+	0.0907905i	$0.971061\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.72467	−	4.71927i	0.122963	−	0.212978i	−0.797972	−	0.602695i	$-0.794094\pi$
$491$	2.72467	−	4.71927i	0.122963	−	0.212978i	0.920935	+	0.389717i	$0.127427\pi$
$492$		0			0
$493$	−31.9240			−1.43779
$494$		0			0
$495$		0			0
$496$		0			0
$497$	13.9448	+	8.05103i	0.625509	+	0.361138i
$498$		0			0
$499$	2.76253	+	1.59495i	0.123668	+	0.0713997i	0.560558	−	0.828115i	$-0.310587\pi$
$499$	2.76253	+	1.59495i	0.123668	+	0.0713997i	−0.436890	+	0.899515i	$0.643920\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.2446	+	33.3326i	0.858072	+	1.48623i	0.873765	+	0.486348i	$0.161671\pi$
$503$	19.2446	+	33.3326i	0.858072	+	1.48623i	−0.0156927	+	0.999877i	$0.504995\pi$
$504$		0			0
$505$	−16.9876			−0.755940
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8.72135	−	5.03527i	0.386567	−	0.223185i	−0.294105	−	0.955773i	$-0.595021\pi$
$509$	8.72135	−	5.03527i	0.386567	−	0.223185i	0.680672	+	0.732589i	$0.261688\pi$
$510$		0			0
$511$	26.5556	+	15.3319i	1.17475	+	0.678242i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.09092	−	14.0139i	0.356529	−	0.617526i
$516$		0			0
$517$	14.6024	+	25.2921i	0.642213	+	1.11235i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	34.7688i		−	1.52325i	−0.648018	−	0.761625i	$-0.724402\pi$
$521$		−	34.7688i		−	1.52325i	0.648018	−	0.761625i	$-0.275598\pi$
$522$		0			0
$523$	−4.64797	+	2.68351i	−0.203242	+	0.117342i	−0.598167	−	0.801372i	$-0.704104\pi$
$523$	−4.64797	+	2.68351i	−0.203242	+	0.117342i	0.394925	+	0.918713i	$0.370771\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.6715	+	25.4117i	−0.639099	+	1.10695i
$528$		0			0
$529$	−2.45584	+	4.25363i	−0.106775	+	0.184941i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	20.4918i		−	0.887597i
$534$		0			0
$535$	−26.4669	−	15.2807i	−1.14426	−	0.660640i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.19337			0.266767
$540$		0			0
$541$	−6.23120	−	10.7928i	−0.267900	−	0.464017i	0.700419	−	0.713732i	$-0.252996\pi$
$541$	−6.23120	−	10.7928i	−0.267900	−	0.464017i	−0.968319	+	0.249715i	$0.919663\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−21.3549	+	12.3293i	−0.914744	+	0.528128i
$546$		0			0
$547$	1.10047	−	0.635358i	0.0470528	−	0.0271659i	−0.476289	−	0.879289i	$-0.658018\pi$
$547$	1.10047	−	0.635358i	0.0470528	−	0.0271659i	0.523342	+	0.852123i	$0.324685\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.23739	+	35.4155i	−0.180519	+	1.50875i
$552$		0			0
$553$	14.6759	−	25.4194i	0.624082	−	1.08094i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	37.0099	−	21.3677i	1.56816	−	0.905378i	0.571776	−	0.820409i	$-0.306254\pi$
$557$	37.0099	−	21.3677i	1.56816	−	0.905378i	0.996384	−	0.0849682i	$-0.0270789\pi$
$558$		0			0
$559$			19.2578i			0.814519i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.0540			0.634451			0.317226	−	0.948350i	$-0.397249\pi$
$563$	15.0540			0.634451			0.317226	+	0.948350i	$0.397249\pi$
$564$		0			0
$565$	5.17570	−	8.96457i	0.217743	−	0.377142i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	27.3871i		−	1.14813i	−0.818811	−	0.574064i	$-0.805366\pi$
$569$		−	27.3871i		−	1.14813i	0.818811	−	0.574064i	$-0.194634\pi$
$570$		0			0
$571$			20.0687i			0.839848i	0.907559	+	0.419924i	$0.137943\pi$
$571$			20.0687i			0.839848i	−0.907559	+	0.419924i	$0.862057\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.84500	+	11.8559i	−0.285456	+	0.494425i
$576$		0			0
$577$	27.6632			1.15163			0.575816	−	0.817579i	$-0.304684\pi$
$577$	27.6632			1.15163			0.575816	+	0.817579i	$0.304684\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			53.1454i			2.20484i
$582$		0			0
$583$	−7.27871	+	4.20237i	−0.301454	+	0.174044i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.24250	+	10.8123i	−0.257655	+	0.446272i	−0.965613	−	0.259982i	$-0.916283\pi$
$587$	−6.24250	+	10.8123i	−0.257655	+	0.446272i	0.707958	+	0.706255i	$0.249617\pi$
$588$		0			0
$589$	26.2436	+	19.6490i	1.08135	+	0.809625i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	22.7999	−	13.1635i	0.936280	−	0.540562i	0.0474878	−	0.998872i	$-0.484878\pi$
$593$	22.7999	−	13.1635i	0.936280	−	0.540562i	0.888792	+	0.458310i	$0.151545\pi$
$594$		0			0
$595$	−15.4945	+	8.94574i	−0.635212	+	0.366740i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.90978	−	15.4322i	−0.364044	−	0.630542i	0.624578	−	0.780962i	$-0.285271\pi$
$599$	−8.90978	−	15.4322i	−0.364044	−	0.630542i	−0.988622	+	0.150420i	$0.951937\pi$
$600$		0			0
$601$	4.14657			0.169142			0.0845711	−	0.996417i	$-0.473048\pi$
$601$	4.14657			0.169142			0.0845711	+	0.996417i	$0.473048\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.41732	+	1.39564i	0.0982782	+	0.0567409i
$606$		0			0
$607$			9.01676i			0.365979i	0.983115	+	0.182989i	$0.0585774\pi$
$607$			9.01676i			0.365979i	−0.983115	+	0.182989i	$0.941423\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	19.5358	−	33.8369i	0.790332	−	1.36890i
$612$		0			0
$613$	9.07558	−	15.7194i	0.366559	−	0.634900i	−0.622466	−	0.782647i	$-0.713869\pi$
$613$	9.07558	−	15.7194i	0.366559	−	0.634900i	0.989025	+	0.147748i	$0.0472023\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.3186	−	9.99891i	0.697221	−	0.402541i	−0.109090	−	0.994032i	$-0.534794\pi$
$617$	17.3186	−	9.99891i	0.697221	−	0.402541i	0.806312	+	0.591491i	$0.201460\pi$
$618$		0			0
$619$		−	6.82304i		−	0.274241i	−0.990554	−	0.137120i	$-0.956215\pi$
$619$		−	6.82304i		−	0.274241i	0.990554	−	0.137120i	$-0.0437848\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.1472	+	22.7716i	0.526730	+	0.912324i
$624$		0			0
$625$	2.66454	−	4.61513i	0.106582	−	0.184605i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7.84681	−	4.53036i	−0.312873	−	0.180637i
$630$		0			0
$631$	−33.3500	+	19.2546i	−1.32764	+	0.766515i	−0.984935	−	0.172927i	$-0.944677\pi$
$631$	−33.3500	+	19.2546i	−1.32764	+	0.766515i	−0.342708	+	0.939442i	$0.611344\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.10847			−0.163040
$636$		0			0
$637$	−4.14288	−	7.17568i	−0.164147	−	0.284311i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.7653	−	11.9889i	−0.820181	−	0.473532i	0.0302978	−	0.999541i	$-0.490354\pi$
$641$	−20.7653	−	11.9889i	−0.820181	−	0.473532i	−0.850479	+	0.526009i	$0.823688\pi$
$642$		0			0
$643$	−3.05767	−	1.76535i	−0.120583	−	0.0696185i	0.438495	−	0.898733i	$-0.355512\pi$
$643$	−3.05767	−	1.76535i	−0.120583	−	0.0696185i	−0.559078	+	0.829115i	$0.688845\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.83957			−0.347519			−0.173760	−	0.984788i	$-0.555592\pi$
$647$	−8.83957			−0.347519			−0.173760	+	0.984788i	$0.555592\pi$
$648$		0			0
$649$	−11.2541	+	19.4928i	−0.441764	+	0.765158i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.10915i		−	0.0434042i	−0.999764	−	0.0217021i	$-0.993091\pi$
$653$		−	1.10915i		−	0.0434042i	0.999764	−	0.0217021i	$-0.00690854\pi$
$654$		0			0
$655$	25.1744	−	14.5344i	0.983643	−	0.567907i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.73179	+	13.3919i	0.301188	+	0.521673i	0.976405	−	0.215946i	$-0.0692835\pi$
$659$	7.73179	+	13.3919i	0.301188	+	0.521673i	−0.675217	+	0.737619i	$0.735950\pi$
$660$		0			0
$661$	5.32246	+	9.21877i	0.207020	+	0.358569i	0.950774	−	0.309884i	$-0.100290\pi$
$661$	5.32246	+	9.21877i	0.207020	+	0.358569i	−0.743755	+	0.668453i	$0.766957\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.86749	+	18.3765i	0.305088	+	0.712610i
$666$		0			0
$667$	−37.4392	−	21.6156i	−1.44965	−	0.836958i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.3564	+	38.7224i	0.863060	+	1.49486i
$672$		0			0
$673$	−39.8299			−1.53533			−0.767664	−	0.640852i	$-0.778581\pi$
$673$	−39.8299			−1.53533			−0.767664	+	0.640852i	$0.778581\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	30.7547i		−	1.18200i	−0.806671	−	0.591000i	$-0.798733\pi$
$677$		−	30.7547i		−	1.18200i	0.806671	−	0.591000i	$-0.201267\pi$
$678$		0			0
$679$	−31.1579	−	17.9890i	−1.19573	−	0.690355i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−20.2903			−0.776387			−0.388194	−	0.921578i	$-0.626901\pi$
$683$	−20.2903			−0.776387			−0.388194	+	0.921578i	$0.626901\pi$
$684$		0			0
$685$	23.5120			0.898347
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.73778	+	5.62211i	0.370980	+	0.214185i
$690$		0			0
$691$			8.66267i			0.329544i	0.986332	+	0.164772i	$0.0526888\pi$
$691$			8.66267i			0.329544i	−0.986332	+	0.164772i	$0.947311\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0969			0.534728
$696$		0			0
$697$	8.35178	+	14.4657i	0.316346	+	0.547927i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.2447	+	11.6883i	0.764634	+	0.441461i	0.830957	−	0.556337i	$-0.187794\pi$
$701$	20.2447	+	11.6883i	0.764634	+	0.441461i	−0.0663234	+	0.997798i	$0.521127\pi$
$702$		0			0
$703$	−6.06737	+	8.10367i	−0.228835	+	0.305636i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.1713	+	28.0095i	0.608184	+	1.05341i
$708$		0			0
$709$	26.0163	+	45.0616i	0.977063	+	1.69232i	0.672951	+	0.739687i	$0.265027\pi$
$709$	26.0163	+	45.0616i	0.977063	+	1.69232i	0.304113	+	0.952636i	$0.401640\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−34.4122	+	19.8679i	−1.28875	+	0.744059i
$714$		0			0
$715$		−	26.5742i		−	0.993817i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.2859	−	19.5477i	0.420892	−	0.729006i	−0.575135	−	0.818058i	$-0.695051\pi$
$719$	11.2859	−	19.5477i	0.420892	−	0.729006i	0.996027	+	0.0890524i	$0.0283839\pi$
$720$		0			0
$721$	−30.8084			−1.14737
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.3631	−	10.6019i	−0.681987	−	0.393745i
$726$		0			0
$727$	−2.24947	−	1.29873i	−0.0834281	−	0.0481672i	0.457706	−	0.889104i	$-0.348671\pi$
$727$	−2.24947	−	1.29873i	−0.0834281	−	0.0481672i	−0.541134	+	0.840937i	$0.682005\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.84886	−	13.5946i	−0.290300	−	0.502815i
$732$		0			0
$733$	−24.7135			−0.912813			−0.456407	−	0.889771i	$-0.650864\pi$
$733$	−24.7135			−0.912813			−0.456407	+	0.889771i	$0.650864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32.3030	+	18.6501i	−1.18990	+	0.686987i
$738$		0			0
$739$	33.8487	+	19.5425i	1.24514	+	0.718884i	0.970137	−	0.242558i	$-0.0779865\pi$
$739$	33.8487	+	19.5425i	1.24514	+	0.718884i	0.275007	+	0.961442i	$0.411320\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.14989	−	8.91988i	0.188931	−	0.327238i	−0.755963	−	0.654614i	$-0.772831\pi$
$743$	5.14989	−	8.91988i	0.188931	−	0.327238i	0.944894	+	0.327376i	$0.106164\pi$
$744$		0			0
$745$	7.81914	+	13.5432i	0.286471	+	0.496183i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			58.1854i			2.12605i
$750$		0			0
$751$	−37.7889	+	21.8174i	−1.37894	+	0.796129i	−0.992031	−	0.125990i	$-0.959789\pi$
$751$	−37.7889	+	21.8174i	−1.37894	+	0.796129i	−0.386905	+	0.922120i	$0.626456\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.7783	−	20.4005i	0.428654	−	0.742451i
$756$		0			0
$757$	−9.69201	+	16.7871i	−0.352262	+	0.610136i	−0.986645	−	0.162883i	$-0.947921\pi$
$757$	−9.69201	+	16.7871i	−0.352262	+	0.610136i	0.634383	+	0.773019i	$0.281254\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.1351i		−	1.49115i	−0.666423	−	0.745574i	$-0.732176\pi$
$761$		−	41.1351i		−	1.49115i	0.666423	−	0.745574i	$-0.267824\pi$
$762$		0			0
$763$	40.6574	+	23.4736i	1.47190	+	0.849800i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	30.1126			1.08730
$768$		0			0
$769$	−20.9257	−	36.2444i	−0.754600	−	1.30701i	−0.945573	−	0.325410i	$-0.894498\pi$
$769$	−20.9257	−	36.2444i	−0.754600	−	1.30701i	0.190973	−	0.981595i	$-0.438836\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.7846	+	16.0414i	−0.999342	+	0.576970i	−0.908054	−	0.418854i	$-0.862432\pi$
$773$	−27.7846	+	16.0414i	−0.999342	+	0.576970i	−0.0912886	+	0.995824i	$0.529099\pi$
$774$		0			0
$775$	−16.8784	+	9.74473i	−0.606289	+	0.350041i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.1564	−	7.34511i	0.614690	−	0.263166i
$780$		0			0
$781$	9.74751	−	16.8832i	0.348793	−	0.604128i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.75983	−	5.05749i	0.312652	−	0.180510i
$786$		0			0
$787$		−	9.20538i		−	0.328136i	−0.986449	−	0.164068i	$-0.947538\pi$
$787$		−	9.20538i		−	0.328136i	0.986449	−	0.164068i	$-0.0524617\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−19.7079			−0.700733
$792$		0			0
$793$	29.9094	−	51.8046i	1.06211	−	1.83963i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			23.2175i			0.822404i	0.911544	+	0.411202i	$0.134891\pi$
$797$			23.2175i			0.822404i	−0.911544	+	0.411202i	$0.865109\pi$
$798$		0			0
$799$			31.8485i			1.12672i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.5626	−	32.1513i	0.655058	−	1.13459i
$804$		0			0
$805$	−24.2284			−0.853939
$806$		0			0
$807$		0			0
$808$		0			0
$809$			53.1510i			1.86869i	0.356371	+	0.934345i	$0.384014\pi$
$809$			53.1510i			1.86869i	−0.356371	+	0.934345i	$0.615986\pi$
$810$		0			0
$811$	−4.76280	+	2.74980i	−0.167244	+	0.0965586i	−0.581286	−	0.813699i	$-0.697450\pi$
$811$	−4.76280	+	2.74980i	−0.167244	+	0.0965586i	0.414041	+	0.910258i	$0.364117\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.16484	+	7.21371i	−0.145888	+	0.252685i
$816$		0			0
$817$	−16.1233	+	6.90281i	−0.564081	+	0.241499i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.0362	−	20.2282i	1.22277	−	0.705968i	0.257264	−	0.966341i	$-0.417179\pi$
$821$	35.0362	−	20.2282i	1.22277	−	0.705968i	0.965508	+	0.260373i	$0.0838457\pi$
$822$		0			0
$823$	−19.3542	+	11.1741i	−0.674643	+	0.389506i	−0.797834	−	0.602878i	$-0.794021\pi$
$823$	−19.3542	+	11.1741i	−0.674643	+	0.389506i	0.123190	+	0.992383i	$0.460687\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	23.6175	+	40.9067i	0.821261	+	1.42247i	0.904744	+	0.425956i	$0.140062\pi$
$827$	23.6175	+	40.9067i	0.821261	+	1.42247i	−0.0834832	+	0.996509i	$0.526605\pi$
$828$		0			0
$829$	23.4010			0.812750			0.406375	−	0.913706i	$-0.366793\pi$
$829$	23.4010			0.812750			0.406375	+	0.913706i	$0.366793\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.84914	+	3.37700i	0.202661	+	0.117006i
$834$		0			0
$835$			23.4438i			0.811306i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.1780	+	26.2890i	−0.524002	+	0.907599i	0.475607	+	0.879658i	$0.342228\pi$
$839$	−15.1780	+	26.2890i	−0.524002	+	0.907599i	−0.999610	+	0.0279408i	$0.991105\pi$
$840$		0			0
$841$	18.9794	−	32.8732i	0.654461	−	1.13356i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.3159	+	7.68796i	−0.458082	+	0.264474i
$846$		0			0
$847$		−	5.31430i		−	0.182602i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.13495	−	10.6260i	−0.210303	−	0.364256i
$852$		0			0
$853$	−3.86624	+	6.69652i	−0.132377	+	0.229285i	−0.924593	−	0.380957i	$-0.875594\pi$
$853$	−3.86624	+	6.69652i	−0.132377	+	0.229285i	0.792215	+	0.610242i	$0.208928\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.16835	−	1.25190i	−0.0740695	−	0.0427641i	0.462508	−	0.886615i	$-0.346950\pi$
$857$	−2.16835	−	1.25190i	−0.0740695	−	0.0427641i	−0.536577	+	0.843851i	$0.680283\pi$
$858$		0			0
$859$	14.8497	−	8.57348i	0.506665	−	0.292523i	−0.224797	−	0.974406i	$-0.572172\pi$
$859$	14.8497	−	8.57348i	0.506665	−	0.292523i	0.731462	+	0.681882i	$0.238838\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	56.7183			1.93071			0.965356	−	0.260937i	$-0.0840314\pi$
$863$	56.7183			1.93071			0.965356	+	0.260937i	$0.0840314\pi$
$864$		0			0
$865$	−1.76715	−	3.06079i	−0.0600848	−	0.104070i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.7757	−	17.7683i	−1.04399	−	0.602750i
$870$		0			0
$871$	43.2164	+	24.9510i	1.46433	+	0.845432i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−34.8134			−1.17691
$876$		0			0
$877$	−19.0768	+	33.0420i	−0.644178	+	1.11575i	0.340313	+	0.940312i	$0.389467\pi$
$877$	−19.0768	+	33.0420i	−0.644178	+	1.11575i	−0.984491	+	0.175436i	$0.943866\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	23.3654i		−	0.787202i	−0.919281	−	0.393601i	$-0.871229\pi$
$881$		−	23.3654i		−	0.787202i	0.919281	−	0.393601i	$-0.128771\pi$
$882$		0			0
$883$	−10.0755	+	5.81707i	−0.339066	+	0.195760i	−0.659859	−	0.751389i	$-0.729384\pi$
$883$	−10.0755	+	5.81707i	−0.339066	+	0.195760i	0.320793	+	0.947149i	$0.396051\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.63103	−	4.55708i	−0.0883415	−	0.153012i	0.818469	−	0.574551i	$-0.194823\pi$
$887$	−2.63103	−	4.55708i	−0.0883415	−	0.153012i	−0.906810	+	0.421539i	$0.861490\pi$
$888$		0			0
$889$	3.91104	+	6.77412i	0.131172	+	0.227197i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	35.3318	+	4.22737i	1.18233	+	0.141464i
$894$		0			0
$895$	−18.0104	−	10.3983i	−0.602021	−	0.347577i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30.7725	−	53.2995i	−1.02632	−	1.77764i
$900$		0			0
$901$	−9.16556			−0.305349
$902$		0			0
$903$		0			0
$904$		0			0
$905$			14.6686i			0.487599i
$906$		0			0
$907$	−27.4669	−	15.8580i	−0.912023	−	0.526557i	−0.0309416	−	0.999521i	$-0.509851\pi$
$907$	−27.4669	−	15.8580i	−0.912023	−	0.526557i	−0.881082	+	0.472964i	$0.843184\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.4605			0.876677			0.438338	−	0.898810i	$-0.355567\pi$
$911$	26.4605			0.876677			0.438338	+	0.898810i	$0.355567\pi$
$912$		0			0
$913$	64.3440			2.12948
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−47.9292	−	27.6719i	−1.58276	−	0.913808i
$918$		0			0
$919$		−	53.0421i		−	1.74970i	−0.484395	−	0.874849i	$-0.660960\pi$
$919$		−	53.0421i		−	1.74970i	0.484395	−	0.874849i	$-0.339040\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−26.0813			−0.858477
$924$		0			0
$925$	−3.00905	−	5.21182i	−0.0989368	−	0.171364i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	8.56206	+	4.94331i	0.280912	+	0.162185i	0.633836	−	0.773467i	$-0.281479\pi$
$929$	8.56206	+	4.94331i	0.280912	+	0.162185i	−0.352924	+	0.935652i	$0.614813\pi$
$930$		0			0
$931$	4.52272	−	6.04061i	0.148226	−	0.197973i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10.8308	+	18.7594i	0.354204	+	0.613499i
$936$		0			0
$937$	−11.5017	−	19.9215i	−0.375743	−	0.650806i	0.614695	−	0.788765i	$-0.289279\pi$
$937$	−11.5017	−	19.9215i	−0.375743	−	0.650806i	−0.990438	+	0.137959i	$0.955946\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.2656	−	23.8247i	1.34522	−	0.776663i	0.357652	−	0.933855i	$-0.383577\pi$
$941$	41.2656	−	23.8247i	1.34522	−	0.776663i	0.987568	+	0.157192i	$0.0502440\pi$
$942$		0			0
$943$			22.6197i			0.736600i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.94511	−	8.56517i	0.160694	−	0.278331i	−0.774424	−	0.632667i	$-0.781960\pi$
$947$	4.94511	−	8.56517i	0.160694	−	0.278331i	0.935118	+	0.354337i	$0.115293\pi$
$948$		0			0
$949$	−49.6676			−1.61228
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.5341	−	15.8968i	−0.891915	−	0.514948i	−0.0173469	−	0.999850i	$-0.505522\pi$
$953$	−27.5341	−	15.8968i	−0.891915	−	0.514948i	−0.874569	+	0.484902i	$0.838855\pi$
$954$		0			0
$955$	2.51592	+	1.45256i	0.0814131	+	0.0470039i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.3821	−	38.7670i	−0.722756	−	1.25185i
$960$		0			0
$961$	−25.5689			−0.824805
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23.9526	−	13.8291i	0.771062	−	0.445173i
$966$		0			0
$967$	−39.3678	−	22.7290i	−1.26598	−	0.730915i	−0.291757	−	0.956493i	$-0.594240\pi$
$967$	−39.3678	−	22.7290i	−1.26598	−	0.730915i	−0.974225	+	0.225578i	$0.927573\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.8498	+	20.5244i	−0.380278	+	0.658660i	−0.991102	−	0.133106i	$-0.957505\pi$
$971$	−11.8498	+	20.5244i	−0.380278	+	0.658660i	0.610824	+	0.791766i	$0.290838\pi$
$972$		0			0
$973$	−13.4195	−	23.2433i	−0.430210	−	0.745146i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			21.4168i			0.685184i	0.939484	+	0.342592i	$0.111305\pi$
$977$			21.4168i			0.685184i	−0.939484	+	0.342592i	$0.888695\pi$
$978$		0			0
$979$	27.5699	−	15.9175i	0.881138	−	0.508726i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1.17570	−	2.03638i	0.0374991	−	0.0649504i	−0.846667	−	0.532124i	$-0.821394\pi$
$983$	1.17570	−	2.03638i	0.0374991	−	0.0649504i	0.884166	+	0.467173i	$0.154728\pi$
$984$		0			0
$985$	−2.72954	+	4.72770i	−0.0869703	+	0.150637i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	21.2576i		−	0.675954i
$990$		0			0
$991$	−46.7179	−	26.9726i	−1.48404	−	0.856813i	−0.484209	−	0.874952i	$-0.660893\pi$
$991$	−46.7179	−	26.9726i	−1.48404	−	0.856813i	−0.999835	+	0.0181390i	$0.994226\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.60739			−0.0509576
$996$		0			0
$997$	15.0931	+	26.1420i	0.478004	+	0.827926i	0.999682	−	0.0252157i	$-0.00802727\pi$
$997$	15.0931	+	26.1420i	0.478004	+	0.827926i	−0.521678	+	0.853142i	$0.674694\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.cg.c.1151.9	yes	32
3.2	odd	2	inner	2736.2.cg.c.1151.7	✓	32
4.3	odd	2	inner	2736.2.cg.c.1151.10	yes	32
12.11	even	2	inner	2736.2.cg.c.1151.8	yes	32
19.7	even	3	inner	2736.2.cg.c.2591.8	yes	32
57.26	odd	6	inner	2736.2.cg.c.2591.10	yes	32
76.7	odd	6	inner	2736.2.cg.c.2591.7	yes	32
228.83	even	6	inner	2736.2.cg.c.2591.9	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.c.1151.7	✓	32	3.2	odd	2	inner
2736.2.cg.c.1151.8	yes	32	12.11	even	2	inner
2736.2.cg.c.1151.9	yes	32	1.1	even	1	trivial
2736.2.cg.c.1151.10	yes	32	4.3	odd	2	inner
2736.2.cg.c.2591.7	yes	32	76.7	odd	6	inner
2736.2.cg.c.2591.8	yes	32	19.7	even	3	inner
2736.2.cg.c.2591.9	yes	32	228.83	even	6	inner
2736.2.cg.c.2591.10	yes	32	57.26	odd	6	inner

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.cg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.34408 + 0.776006i) q^{5} -2.95486i q^{7} +O(q^{10})\)	\(q+(1.34408 + 0.776006i) q^{5} -2.95486i q^{7} -3.57750 q^{11} +(2.39307 + 4.14491i) q^{13} +(-3.37866 - 1.95067i) q^{17} +(-2.61248 + 3.48926i) q^{19} +(-2.64157 - 4.57534i) q^{23} +(-1.29563 - 2.24410i) q^{25} +(7.08654 - 4.09142i) q^{29} -7.52123i q^{31} +(2.29299 - 3.97157i) q^{35} +2.32246 q^{37} +(-3.70788 - 2.14074i) q^{41} +(3.48460 + 2.01183i) q^{43} +(-4.08174 - 7.06978i) q^{47} -1.73120 q^{49} +(2.03458 - 1.17467i) q^{53} +(-4.80845 - 2.77616i) q^{55} +(3.14582 - 5.44871i) q^{59} +(-6.24917 - 10.8239i) q^{61} +7.42814i q^{65} +(9.02949 - 5.21318i) q^{67} +(-2.72467 + 4.71927i) q^{71} +(-5.18870 + 8.98709i) q^{73} +10.5710i q^{77} +(8.60257 + 4.96669i) q^{79} -17.9858 q^{83} +(-3.02747 - 5.24373i) q^{85} +(-7.70648 + 4.44934i) q^{89} +(12.2476 - 7.07118i) q^{91} +(-6.21907 + 2.66256i) q^{95} +(6.08794 - 10.5446i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{13} + 24 q^{25} - 16 q^{37} - 96 q^{49} - 8 q^{61} - 8 q^{73} + 16 q^{85} + 48 q^{97}+O(q^{100}) \) 32 * q - 16 * q^13 + 24 * q^25 - 16 * q^37 - 96 * q^49 - 8 * q^61 - 8 * q^73 + 16 * q^85 + 48 * q^97

Introduction

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