LMFDB - Embedded newform 2736.2.cg.c.2591.4

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			2591.4
Character	$\chi$	$=$	2736.2591
Dual form			2736.2.cg.c.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.03864 + 1.17701i) q^{5} -4.52875i q^{7} +O(q^{10})$	$q+(-2.03864 + 1.17701i) q^{5} -4.52875i q^{7} -2.20981 q^{11} +(-1.93525 + 3.35196i) q^{13} +(-5.29030 + 3.05436i) q^{17} +(2.75880 - 3.37477i) q^{19} +(0.557743 - 0.966039i) q^{23} +(0.270710 - 0.468883i) q^{25} +(-3.51564 - 2.02976i) q^{29} +6.92346i q^{31} +(5.33038 + 9.23249i) q^{35} +10.9681 q^{37} +(8.80594 - 5.08411i) q^{41} +(9.76727 - 5.63914i) q^{43} +(-6.43529 + 11.1462i) q^{47} -13.5095 q^{49} +(7.32894 + 4.23137i) q^{53} +(4.50500 - 2.60097i) q^{55} +(3.66774 + 6.35271i) q^{59} +(-0.719712 + 1.24658i) q^{61} -9.11126i q^{65} +(-5.28819 - 3.05314i) q^{67} +(-3.87245 - 6.70728i) q^{71} +(0.705965 + 1.22277i) q^{73} +10.0076i q^{77} +(1.92326 - 1.11039i) q^{79} +11.4689 q^{83} +(7.19002 - 12.4535i) q^{85} +(2.44169 + 1.40971i) q^{89} +(15.1802 + 8.76428i) q^{91} +(-1.65206 + 10.1271i) q^{95} +(-3.76434 - 6.52004i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.03864	+	1.17701i	−0.911709	+	0.526375i	−0.880981	−	0.473152i	$-0.843116\pi$
$5$	−2.03864	+	1.17701i	−0.911709	+	0.526375i	−0.0307282	+	0.999528i	$0.509783\pi$
$6$		0			0
$7$		−	4.52875i		−	1.71170i	−0.517220	−	0.855852i	$-0.673033\pi$
$7$		−	4.52875i		−	1.71170i	0.517220	−	0.855852i	$-0.326967\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.20981			−0.666281			−0.333141	−	0.942877i	$-0.608108\pi$
$11$	−2.20981			−0.666281			−0.333141	+	0.942877i	$0.608108\pi$
$12$		0			0
$13$	−1.93525	+	3.35196i	−0.536743	+	0.929666i	0.462334	+	0.886706i	$0.347012\pi$
$13$	−1.93525	+	3.35196i	−0.536743	+	0.929666i	−0.999077	+	0.0429603i	$0.986321\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.29030	+	3.05436i	−1.28309	+	0.740790i	−0.977411	−	0.211346i	$-0.932215\pi$
$17$	−5.29030	+	3.05436i	−1.28309	+	0.740790i	−0.305675	+	0.952136i	$0.598882\pi$
$18$		0			0
$19$	2.75880	−	3.37477i	0.632911	−	0.774224i
$19$	2.75880	−	3.37477i	0.632911	−	0.774224i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.557743	−	0.966039i	0.116297	−	0.201433i	−0.802000	−	0.597324i	$-0.796231\pi$
$23$	0.557743	−	0.966039i	0.116297	−	0.201433i	0.918298	+	0.395891i	$0.129564\pi$
$24$		0			0
$25$	0.270710	−	0.468883i	0.0541420	−	0.0937766i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.51564	−	2.02976i	−0.652838	−	0.376916i	0.136705	−	0.990612i	$-0.456349\pi$
$29$	−3.51564	−	2.02976i	−0.652838	−	0.376916i	−0.789543	+	0.613696i	$0.789682\pi$
$30$		0			0
$31$			6.92346i			1.24349i	0.783220	+	0.621745i	$0.213576\pi$
$31$			6.92346i			1.24349i	−0.783220	+	0.621745i	$0.786424\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.33038	+	9.23249i	0.900999	+	1.56058i
$36$		0			0
$37$	10.9681			1.80315			0.901573	−	0.432626i	$-0.142413\pi$
$37$	10.9681			1.80315			0.901573	+	0.432626i	$0.142413\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.80594	−	5.08411i	1.37526	−	0.794005i	0.383673	−	0.923469i	$-0.374659\pi$
$41$	8.80594	−	5.08411i	1.37526	−	0.794005i	0.991584	+	0.129464i	$0.0413257\pi$
$42$		0			0
$43$	9.76727	−	5.63914i	1.48950	−	0.859961i	0.489567	−	0.871965i	$-0.337155\pi$
$43$	9.76727	−	5.63914i	1.48950	−	0.859961i	0.999928	+	0.0120048i	$0.00382135\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.43529	+	11.1462i	−0.938683	+	1.62585i	−0.170752	+	0.985314i	$0.554620\pi$
$47$	−6.43529	+	11.1462i	−0.938683	+	1.62585i	−0.767931	+	0.640532i	$0.778714\pi$
$48$		0			0
$49$	−13.5095			−1.92993
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.32894	+	4.23137i	1.00671	+	0.581223i	0.910226	−	0.414112i	$-0.135908\pi$
$53$	7.32894	+	4.23137i	1.00671	+	0.581223i	0.0964816	+	0.995335i	$0.469241\pi$
$54$		0			0
$55$	4.50500	−	2.60097i	0.607455	−	0.350714i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.66774	+	6.35271i	0.477499	+	0.827052i	0.999667	−	0.0257902i	$-0.00821019\pi$
$59$	3.66774	+	6.35271i	0.477499	+	0.827052i	−0.522169	+	0.852842i	$0.674877\pi$
$60$		0			0
$61$	−0.719712	+	1.24658i	−0.0921497	+	0.159608i	−0.908415	−	0.418069i	$-0.862707\pi$
$61$	−0.719712	+	1.24658i	−0.0921497	+	0.159608i	0.816266	+	0.577677i	$0.196040\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.11126i		−	1.13011i
$66$		0			0
$67$	−5.28819	−	3.05314i	−0.646056	−	0.373000i	0.140888	−	0.990026i	$-0.455004\pi$
$67$	−5.28819	−	3.05314i	−0.646056	−	0.373000i	−0.786943	+	0.617025i	$0.788338\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.87245	−	6.70728i	−0.459575	−	0.796008i	0.539363	−	0.842073i	$-0.318665\pi$
$71$	−3.87245	−	6.70728i	−0.459575	−	0.796008i	−0.998938	+	0.0460654i	$0.985332\pi$
$72$		0			0
$73$	0.705965	+	1.22277i	0.0826269	+	0.143114i	0.904378	−	0.426733i	$-0.140336\pi$
$73$	0.705965	+	1.22277i	0.0826269	+	0.143114i	−0.821751	+	0.569847i	$0.807002\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			10.0076i			1.14048i
$78$		0			0
$79$	1.92326	−	1.11039i	0.216383	−	0.124929i	−0.387891	−	0.921705i	$-0.626796\pi$
$79$	1.92326	−	1.11039i	0.216383	−	0.124929i	0.604275	+	0.796776i	$0.293463\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.4689			1.25888			0.629440	−	0.777049i	$-0.283284\pi$
$83$	11.4689			1.25888			0.629440	+	0.777049i	$0.283284\pi$
$84$		0			0
$85$	7.19002	−	12.4535i	0.779867	−	1.35077i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.44169	+	1.40971i	0.258819	+	0.149429i	0.623796	−	0.781587i	$-0.285590\pi$
$89$	2.44169	+	1.40971i	0.258819	+	0.149429i	−0.364977	+	0.931017i	$0.618923\pi$
$90$		0			0
$91$	15.1802	+	8.76428i	1.59131	+	0.918746i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.65206	+	10.1271i	−0.169498	+	1.03902i
$96$		0			0
$97$	−3.76434	−	6.52004i	−0.382211	−	0.662009i	0.609167	−	0.793042i	$-0.291504\pi$
$97$	−3.76434	−	6.52004i	−0.382211	−	0.662009i	−0.991378	+	0.131033i	$0.958171\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.97686	+	3.45074i	0.594720	+	0.343362i	0.766962	−	0.641693i	$-0.221768\pi$
$101$	5.97686	+	3.45074i	0.594720	+	0.343362i	−0.172242	+	0.985055i	$0.555101\pi$
$102$		0			0
$103$		−	6.10628i		−	0.601670i	−0.953676	−	0.300835i	$-0.902735\pi$
$103$		−	6.10628i		−	0.601670i	0.953676	−	0.300835i	$-0.0972653\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.7942			1.43021			0.715106	−	0.699016i	$-0.246378\pi$
$107$	14.7942			1.43021			0.715106	+	0.699016i	$0.246378\pi$
$108$		0			0
$109$	3.10938	+	5.38560i	0.297824	+	0.515847i	0.975638	−	0.219387i	$-0.0704057\pi$
$109$	3.10938	+	5.38560i	0.297824	+	0.515847i	−0.677814	+	0.735234i	$0.737072\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.1164i		−	1.51610i	−0.652197	−	0.758050i	$-0.726152\pi$
$113$		−	16.1164i		−	1.51610i	0.652197	−	0.758050i	$-0.273848\pi$
$114$		0			0
$115$			2.62588i			0.244864i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.8324	+	23.9584i	1.26801	+	2.19626i
$120$		0			0
$121$	−6.11676			−0.556069
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.4956i		−	0.938755i
$126$		0			0
$127$	−6.77908	−	3.91390i	−0.601546	−	0.347303i	0.168103	−	0.985769i	$-0.446236\pi$
$127$	−6.77908	−	3.91390i	−0.601546	−	0.347303i	−0.769650	+	0.638467i	$0.779569\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.31980	+	9.21416i	0.464793	+	0.805045i	0.999192	−	0.0401871i	$-0.0127954\pi$
$131$	5.31980	+	9.21416i	0.464793	+	0.805045i	−0.534399	+	0.845232i	$0.679462\pi$
$132$		0			0
$133$	−15.2835	−	12.4939i	−1.32524	−	1.08336i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.1775	+	5.87601i	0.869527	+	0.502021i	0.867191	−	0.497976i	$-0.165923\pi$
$137$	10.1775	+	5.87601i	0.869527	+	0.502021i	0.00233571	+	0.999997i	$0.499257\pi$
$138$		0			0
$139$	12.4737	+	7.20171i	1.05801	+	0.610841i	0.924880	−	0.380258i	$-0.124165\pi$
$139$	12.4737	+	7.20171i	1.05801	+	0.610841i	0.133127	+	0.991099i	$0.457498\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.27654	−	7.40718i	0.357622	−	0.619419i
$144$		0			0
$145$	9.55618			0.793598
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.17035	+	4.13980i	−0.587418	+	0.339146i	−0.764076	−	0.645126i	$-0.776805\pi$
$149$	−7.17035	+	4.13980i	−0.587418	+	0.339146i	0.176658	+	0.984272i	$0.443471\pi$
$150$		0			0
$151$			9.37541i			0.762960i	0.924377	+	0.381480i	$0.124585\pi$
$151$			9.37541i			0.762960i	−0.924377	+	0.381480i	$0.875415\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.14899	−	14.1145i	−0.654542	−	1.13370i
$156$		0			0
$157$	4.81951	+	8.34764i	0.384639	+	0.666214i	0.991719	−	0.128427i	$-0.0409926\pi$
$157$	4.81951	+	8.34764i	0.384639	+	0.666214i	−0.607080	+	0.794641i	$0.707659\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.37494	−	2.52588i	−0.344794	−	0.199067i
$162$		0			0
$163$			9.23142i			0.723060i	0.932360	+	0.361530i	$0.117746\pi$
$163$			9.23142i			0.723060i	−0.932360	+	0.361530i	$0.882254\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.7275	+	22.0447i	−0.984883	+	1.70587i	−0.342429	+	0.939544i	$0.611250\pi$
$167$	−12.7275	+	22.0447i	−0.984883	+	1.70587i	−0.642454	+	0.766324i	$0.722084\pi$
$168$		0			0
$169$	−0.990422	−	1.71546i	−0.0761863	−	0.131959i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−15.5030	+	8.95068i	−1.17867	+	0.680508i	−0.955708	−	0.294318i	$-0.904907\pi$
$173$	−15.5030	+	8.95068i	−1.17867	+	0.680508i	−0.222967	+	0.974826i	$0.571574\pi$
$174$		0			0
$175$	−2.12345	−	1.22598i	−0.160518	−	0.0926751i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.1297			1.65405			0.827026	−	0.562164i	$-0.190031\pi$
$179$	22.1297			1.65405			0.827026	+	0.562164i	$0.190031\pi$
$180$		0			0
$181$	−4.29619	+	7.44121i	−0.319333	+	0.553101i	−0.980349	−	0.197271i	$-0.936792\pi$
$181$	−4.29619	+	7.44121i	−0.319333	+	0.553101i	0.661016	+	0.750372i	$0.270126\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−22.3601	+	12.9096i	−1.64394	+	0.949132i
$186$		0			0
$187$	11.6905	−	6.74953i	0.854896	−	0.493575i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.53510			0.400506			0.200253	−	0.979744i	$-0.435824\pi$
$191$	5.53510			0.400506			0.200253	+	0.979744i	$0.435824\pi$
$192$		0			0
$193$	7.70377	+	13.3433i	0.554529	+	0.960473i	0.997940	+	0.0641545i	$0.0204350\pi$
$193$	7.70377	+	13.3433i	0.554529	+	0.960473i	−0.443411	+	0.896319i	$0.646232\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			16.0128i			1.14086i	0.821346	+	0.570431i	$0.193224\pi$
$197$			16.0128i			1.14086i	−0.821346	+	0.570431i	$0.806776\pi$
$198$		0			0
$199$	7.06083	+	4.07657i	0.500529	+	0.288980i	0.728932	−	0.684586i	$-0.240017\pi$
$199$	7.06083	+	4.07657i	0.500529	+	0.288980i	−0.228403	+	0.973567i	$0.573350\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.19225	+	15.9214i	−0.645170	+	1.11747i
$204$		0			0
$205$	−11.9681	+	20.7294i	−0.835889	+	1.44780i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.09640	+	7.45758i	−0.421697	+	0.515851i
$210$		0			0
$211$	5.99589	−	3.46173i	0.412774	−	0.238315i	−0.279207	−	0.960231i	$-0.590072\pi$
$211$	5.99589	−	3.46173i	0.412774	−	0.238315i	0.691981	+	0.721916i	$0.256738\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.2747	+	22.9924i	−0.905324	+	1.56807i
$216$		0			0
$217$	31.3546			2.12849
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	23.6438i		−	1.59046i
$222$		0			0
$223$	10.0426	−	5.79810i	0.672502	−	0.388269i	−0.124522	−	0.992217i	$-0.539740\pi$
$223$	10.0426	−	5.79810i	0.672502	−	0.388269i	0.797024	+	0.603947i	$0.206406\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.6608			0.707580			0.353790	−	0.935325i	$-0.384893\pi$
$227$	10.6608			0.707580			0.353790	+	0.935325i	$0.384893\pi$
$228$		0			0
$229$	15.6115			1.03164			0.515819	−	0.856697i	$-0.327488\pi$
$229$	15.6115			1.03164			0.515819	+	0.856697i	$0.327488\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.49250	+	5.48050i	−0.621875	+	0.359039i	−0.777598	−	0.628761i	$-0.783562\pi$
$233$	−9.49250	+	5.48050i	−0.621875	+	0.359039i	0.155724	+	0.987801i	$0.450229\pi$
$234$		0			0
$235$		−	30.2976i		−	1.97640i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.4236			−1.38578			−0.692890	−	0.721043i	$-0.743663\pi$
$239$	−21.4236			−1.38578			−0.692890	+	0.721043i	$0.743663\pi$
$240$		0			0
$241$	−4.63270	+	8.02408i	−0.298419	+	0.516876i	−0.975774	−	0.218779i	$-0.929792\pi$
$241$	−4.63270	+	8.02408i	−0.298419	+	0.516876i	0.677356	+	0.735656i	$0.263126\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	27.5411	−	15.9009i	1.75954	−	1.01587i
$246$		0			0
$247$	5.97311	+	15.7784i	0.380060	+	1.00396i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.50899	−	2.61364i	0.0952465	−	0.164972i	−0.814465	−	0.580213i	$-0.802969\pi$
$251$	1.50899	−	2.61364i	0.0952465	−	0.164972i	0.909711	+	0.415241i	$0.136303\pi$
$252$		0			0
$253$	−1.23250	+	2.13476i	−0.0774868	+	0.134211i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.08448	+	4.09023i	0.441918	+	0.255141i	0.704411	−	0.709792i	$-0.251211\pi$
$257$	7.08448	+	4.09023i	0.441918	+	0.255141i	−0.262493	+	0.964934i	$0.584545\pi$
$258$		0			0
$259$		−	49.6718i		−	3.08645i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.5332	−	25.1723i	−0.896157	−	1.55219i	−0.832367	−	0.554225i	$-0.813015\pi$
$263$	−14.5332	−	25.1723i	−0.896157	−	1.55219i	−0.0637898	−	0.997963i	$-0.520319\pi$
$264$		0			0
$265$	−19.9215			−1.22377
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.48754	+	4.32293i	−0.456523	+	0.263574i	−0.710581	−	0.703615i	$-0.751568\pi$
$269$	−7.48754	+	4.32293i	−0.456523	+	0.263574i	0.254058	+	0.967189i	$0.418235\pi$
$270$		0			0
$271$	7.76888	−	4.48537i	0.471926	−	0.272467i	−0.245120	−	0.969493i	$-0.578827\pi$
$271$	7.76888	−	4.48537i	0.471926	−	0.272467i	0.717046	+	0.697026i	$0.245494\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.598216	+	1.03614i	−0.0360738	+	0.0624816i
$276$		0			0
$277$	−1.57331			−0.0945308			−0.0472654	−	0.998882i	$-0.515051\pi$
$277$	−1.57331			−0.0945308			−0.0472654	+	0.998882i	$0.515051\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.46467	−	2.57768i	−0.266340	−	0.153771i	0.360883	−	0.932611i	$-0.382475\pi$
$281$	−4.46467	−	2.57768i	−0.266340	−	0.153771i	−0.627223	+	0.778840i	$0.715809\pi$
$282$		0			0
$283$	−25.2487	+	14.5773i	−1.50088	+	0.866533i	−0.500880	+	0.865517i	$0.666990\pi$
$283$	−25.2487	+	14.5773i	−1.50088	+	0.866533i	−0.999999	+	0.00101602i	$0.999677\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−23.0247	−	39.8799i	−1.35910	−	2.35403i
$288$		0			0
$289$	10.1582	−	17.5945i	0.597540	−	1.03497i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5.21854i		−	0.304871i	−0.988313	−	0.152435i	$-0.951288\pi$
$293$		−	5.21854i		−	0.304871i	0.988313	−	0.152435i	$-0.0487116\pi$
$294$		0			0
$295$	−14.9544	−	8.63393i	−0.870680	−	0.502687i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.15875	+	3.73906i	0.124844	+	0.216236i
$300$		0			0
$301$	−25.5382	−	44.2335i	−1.47200	−	2.54958i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.38844i		−	0.194021i
$306$		0			0
$307$	−23.9644	+	13.8359i	−1.36772	+	0.789655i	−0.990637	−	0.136524i	$-0.956407\pi$
$307$	−23.9644	+	13.8359i	−1.36772	+	0.789655i	−0.377085	+	0.926179i	$0.623074\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	26.9799			1.52989			0.764945	−	0.644096i	$-0.222766\pi$
$311$	26.9799			1.52989			0.764945	+	0.644096i	$0.222766\pi$
$312$		0			0
$313$	−4.63164	+	8.02224i	−0.261796	+	0.453444i	−0.966719	−	0.255840i	$-0.917648\pi$
$313$	−4.63164	+	8.02224i	−0.261796	+	0.453444i	0.704923	+	0.709283i	$0.250981\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.6001	−	6.11999i	−0.595362	−	0.343733i	0.171853	−	0.985123i	$-0.445025\pi$
$317$	−10.6001	−	6.11999i	−0.595362	−	0.343733i	−0.767215	+	0.641390i	$0.778358\pi$
$318$		0			0
$319$	7.76888	+	4.48537i	0.434974	+	0.251132i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.28712	+	26.2799i	−0.238542	+	1.46225i
$324$		0			0
$325$	1.04778	+	1.81482i	0.0581206	+	0.100668i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	50.4785	+	29.1438i	2.78297	+	1.60675i
$330$		0			0
$331$		−	26.2382i		−	1.44218i	−0.692841	−	0.721091i	$-0.743641\pi$
$331$		−	26.2382i		−	1.44218i	0.692841	−	0.721091i	$-0.256359\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14.3743			0.785353
$336$		0			0
$337$	14.7994	+	25.6333i	0.806175	+	1.39634i	0.915495	+	0.402329i	$0.131799\pi$
$337$	14.7994	+	25.6333i	0.806175	+	1.39634i	−0.109321	+	0.994007i	$0.534868\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	15.2995i		−	0.828514i
$342$		0			0
$343$			29.4800i			1.59177i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.61389	+	4.52739i	0.140321	+	0.243043i	0.927618	−	0.373531i	$-0.121853\pi$
$347$	2.61389	+	4.52739i	0.140321	+	0.243043i	−0.787296	+	0.616575i	$0.788520\pi$
$348$		0			0
$349$	−9.20603			−0.492787			−0.246394	−	0.969170i	$-0.579246\pi$
$349$	−9.20603			−0.492787			−0.246394	+	0.969170i	$0.579246\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.15636i		−	0.487344i	−0.969858	−	0.243672i	$-0.921648\pi$
$353$		−	9.15636i		−	0.487344i	0.969858	−	0.243672i	$-0.0783520\pi$
$354$		0			0
$355$	15.7891	+	9.11583i	0.837998	+	0.483818i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.44735	−	2.50688i	−0.0763881	−	0.132308i	0.825301	−	0.564693i	$-0.191005\pi$
$359$	−1.44735	−	2.50688i	−0.0763881	−	0.132308i	−0.901689	+	0.432385i	$0.857672\pi$
$360$		0			0
$361$	−3.77809	−	18.6206i	−0.198847	−	0.980031i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.87842	−	1.66186i	−0.150663	−	0.0869855i
$366$		0			0
$367$	16.4712	+	9.50967i	0.859791	+	0.496400i	0.863942	−	0.503591i	$-0.167988\pi$
$367$	16.4712	+	9.50967i	0.859791	+	0.496400i	−0.00415139	+	0.999991i	$0.501321\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.1628	−	33.1909i	0.994882	−	1.72319i
$372$		0			0
$373$	−0.00833803			−0.000431727			−0.000215863	−	1.00000i	$-0.500069\pi$
$373$	−0.00833803			−0.000431727			−0.000215863		1.00000i	$0.500069\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.6073	−	7.85619i	0.700813	−	0.404614i
$378$		0			0
$379$		−	5.34592i		−	0.274602i	−0.990529	−	0.137301i	$-0.956157\pi$
$379$		−	5.34592i		−	0.274602i	0.990529	−	0.137301i	$-0.0438427\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.6120	−	20.1126i	−0.593346	−	1.02771i	−0.993778	−	0.111379i	$-0.964473\pi$
$383$	−11.6120	−	20.1126i	−0.593346	−	1.02771i	0.400432	−	0.916326i	$-0.368860\pi$
$384$		0			0
$385$	−11.7791	−	20.4020i	−0.600319	−	1.03978i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.26382	−	4.77112i	−0.418993	−	0.241905i	0.275654	−	0.961257i	$-0.411106\pi$
$389$	−8.26382	−	4.77112i	−0.418993	−	0.241905i	−0.694646	+	0.719352i	$0.744439\pi$
$390$		0			0
$391$			6.81418i			0.344608i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.61389	+	4.52739i	−0.131519	+	0.227798i
$396$		0			0
$397$	3.38426	+	5.86170i	0.169851	+	0.294191i	0.938367	−	0.345640i	$-0.112338\pi$
$397$	3.38426	+	5.86170i	0.169851	+	0.294191i	−0.768516	+	0.639830i	$0.779005\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.1073	+	5.83548i	−0.504737	+	0.291410i	−0.730668	−	0.682733i	$-0.760791\pi$
$401$	−10.1073	+	5.83548i	−0.504737	+	0.291410i	0.225931	+	0.974143i	$0.427458\pi$
$402$		0			0
$403$	−23.2072	−	13.3987i	−1.15603	−	0.667435i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.2374			−1.20140
$408$		0			0
$409$	11.0902	−	19.2088i	0.548376	−	0.949815i	−0.450010	−	0.893024i	$-0.648580\pi$
$409$	11.0902	−	19.2088i	0.548376	−	0.949815i	0.998386	−	0.0567918i	$-0.0180871\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.7698	−	16.6102i	1.41567	−	0.817337i
$414$		0			0
$415$	−23.3811	+	13.4991i	−1.14773	+	0.662643i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.0150			−0.635825			−0.317913	−	0.948120i	$-0.602982\pi$
$419$	−13.0150			−0.635825			−0.317913	+	0.948120i	$0.602982\pi$
$420$		0			0
$421$	−7.28024	−	12.6097i	−0.354817	−	0.614562i	0.632269	−	0.774749i	$-0.282124\pi$
$421$	−7.28024	−	12.6097i	−0.354817	−	0.614562i	−0.987087	+	0.160187i	$0.948790\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.30738i			0.160431i
$426$		0			0
$427$	5.64543	+	3.25939i	0.273202	+	0.157733i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.52961	−	13.0417i	0.362688	−	0.628195i	−0.625714	−	0.780053i	$-0.715192\pi$
$431$	7.52961	−	13.0417i	0.362688	−	0.628195i	0.988402	+	0.151858i	$0.0485256\pi$
$432$		0			0
$433$	6.57106	−	11.3814i	0.315785	−	0.546956i	−0.663819	−	0.747893i	$-0.731065\pi$
$433$	6.57106	−	11.3814i	0.315785	−	0.546956i	0.979604	+	0.200937i	$0.0643988\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.72146	−	4.54736i	−0.0823485	−	0.217530i
$438$		0			0
$439$	−19.6360	+	11.3368i	−0.937173	+	0.541077i	−0.889073	−	0.457765i	$-0.848650\pi$
$439$	−19.6360	+	11.3368i	−0.937173	+	0.541077i	−0.0481003	+	0.998843i	$0.515317\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.77264	−	8.26645i	0.226755	−	0.392751i	−0.730090	−	0.683351i	$-0.760522\pi$
$443$	4.77264	−	8.26645i	0.226755	−	0.392751i	0.956845	+	0.290600i	$0.0938550\pi$
$444$		0			0
$445$	−6.63699			−0.314623
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16.6043i			0.783606i	0.920049	+	0.391803i	$0.128148\pi$
$449$			16.6043i			0.783606i	−0.920049	+	0.391803i	$0.871852\pi$
$450$		0			0
$451$	−19.4594	+	11.2349i	−0.916308	+	0.529031i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−41.2626			−1.93442
$456$		0			0
$457$	29.4140			1.37593			0.687964	−	0.725745i	$-0.258505\pi$
$457$	29.4140			1.37593			0.687964	+	0.725745i	$0.258505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.667036	−	0.385113i	0.0310670	−	0.0179365i	−0.484386	−	0.874854i	$-0.660957\pi$
$461$	0.667036	−	0.385113i	0.0310670	−	0.0179365i	0.515453	+	0.856918i	$0.327624\pi$
$462$		0			0
$463$		−	9.54934i		−	0.443795i	−0.975070	−	0.221898i	$-0.928775\pi$
$463$		−	9.54934i		−	0.443795i	0.975070	−	0.221898i	$-0.0712251\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.92366			−0.0890162			−0.0445081	−	0.999009i	$-0.514172\pi$
$467$	−1.92366			−0.0890162			−0.0445081	+	0.999009i	$0.514172\pi$
$468$		0			0
$469$	−13.8269	+	23.9489i	−0.638467	+	1.10586i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.5838	+	12.4614i	−0.992423	+	0.572976i
$474$		0			0
$475$	−0.835538	−	2.20714i	−0.0383371	−	0.101270i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.10999	−	5.38667i	0.142099	−	0.246123i	−0.786188	−	0.617988i	$-0.787948\pi$
$479$	3.10999	−	5.38667i	0.142099	−	0.246123i	0.928287	+	0.371865i	$0.121281\pi$
$480$		0			0
$481$	−21.2261	+	36.7647i	−0.967826	+	1.67632i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	15.3483	+	8.86135i	0.696931	+	0.402373i
$486$		0			0
$487$			28.1411i			1.27519i	0.770371	+	0.637596i	$0.220071\pi$
$487$			28.1411i			1.27519i	−0.770371	+	0.637596i	$0.779929\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.87245	+	6.70728i	0.174761	+	0.302695i	0.940079	−	0.340958i	$-0.110751\pi$
$491$	3.87245	+	6.70728i	0.174761	+	0.302695i	−0.765317	+	0.643653i	$0.777418\pi$
$492$		0			0
$493$	24.7984			1.11686
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−30.3756	+	17.5373i	−1.36253	+	0.786657i
$498$		0			0
$499$	6.92971	−	4.00087i	0.310216	−	0.179103i	−0.336807	−	0.941574i	$-0.609347\pi$
$499$	6.92971	−	4.00087i	0.310216	−	0.179103i	0.647023	+	0.762470i	$0.276014\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	13.6119	−	23.5764i	0.606923	−	1.05122i	−0.384822	−	0.922991i	$-0.625737\pi$
$503$	13.6119	−	23.5764i	0.606923	−	1.05122i	0.991744	−	0.128230i	$-0.0409297\pi$
$504$		0			0
$505$	−16.2463			−0.722949
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.61590	+	5.55174i	0.426217	+	0.246077i	0.697734	−	0.716357i	$-0.254192\pi$
$509$	9.61590	+	5.55174i	0.426217	+	0.246077i	−0.271516	+	0.962434i	$0.587525\pi$
$510$		0			0
$511$	5.53760	−	3.19713i	0.244969	−	0.141433i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.18716	+	12.4485i	0.316704	+	0.548548i
$516$		0			0
$517$	14.2207	−	24.6310i	0.625427	−	1.08327i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	39.3354i		−	1.72332i	−0.507489	−	0.861658i	$-0.669426\pi$
$521$		−	39.3354i		−	1.72332i	0.507489	−	0.861658i	$-0.330574\pi$
$522$		0			0
$523$	7.99464	+	4.61571i	0.349581	+	0.201831i	0.664501	−	0.747288i	$-0.268644\pi$
$523$	7.99464	+	4.61571i	0.349581	+	0.201831i	−0.314920	+	0.949118i	$0.601978\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.1467	−	36.6272i	−0.921165	−	1.59550i
$528$		0			0
$529$	10.8778	+	18.8410i	0.472950	+	0.819173i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			39.3562i			1.70471i
$534$		0			0
$535$	−30.1601	+	17.4130i	−1.30394	+	0.752828i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	29.8534			1.28588
$540$		0			0
$541$	−18.0095	+	31.1934i	−0.774290	+	1.34111i	0.160902	+	0.986970i	$0.448560\pi$
$541$	−18.0095	+	31.1934i	−0.774290	+	1.34111i	−0.935193	+	0.354140i	$0.884774\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.6778	−	7.31954i	−0.543058	−	0.313535i
$546$		0			0
$547$	3.79089	+	2.18867i	0.162087	+	0.0935808i	0.578849	−	0.815435i	$-0.303502\pi$
$547$	3.79089	+	2.18867i	0.162087	+	0.0935808i	−0.416763	+	0.909015i	$0.636835\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.5489	+	6.26478i	−0.705006	+	0.266889i
$552$		0			0
$553$	−5.02869	−	8.70995i	−0.213842	−	0.370385i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.0588	−	7.53951i	−0.553320	−	0.319459i	0.197140	−	0.980375i	$-0.436835\pi$
$557$	−13.0588	−	7.53951i	−0.553320	−	0.319459i	−0.750460	+	0.660916i	$0.770168\pi$
$558$		0			0
$559$			43.6527i			1.84631i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	30.9478			1.30429			0.652146	−	0.758093i	$-0.273869\pi$
$563$	30.9478			1.30429			0.652146	+	0.758093i	$0.273869\pi$
$564$		0			0
$565$	18.9691	+	32.8555i	0.798037	+	1.38224i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			13.9393i			0.584364i	0.956363	+	0.292182i	$0.0943813\pi$
$569$			13.9393i			0.584364i	−0.956363	+	0.292182i	$0.905619\pi$
$570$		0			0
$571$			6.98811i			0.292443i	0.989252	+	0.146222i	$0.0467113\pi$
$571$			6.98811i			0.292443i	−0.989252	+	0.146222i	$0.953289\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.301973	−	0.523032i	−0.0125931	−	0.0218120i
$576$		0			0
$577$	−44.7007			−1.86091			−0.930457	−	0.366400i	$-0.880590\pi$
$577$	−44.7007			−1.86091			−0.930457	+	0.366400i	$0.880590\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	51.9399i		−	2.15483i
$582$		0			0
$583$	−16.1955	−	9.35050i	−0.670751	−	0.387258i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.5945	−	32.2065i	−0.767475	−	1.32931i	−0.938928	−	0.344114i	$-0.888179\pi$
$587$	−18.5945	−	32.2065i	−0.767475	−	1.32931i	0.171453	−	0.985192i	$-0.445154\pi$
$588$		0			0
$589$	23.3651	+	19.1004i	0.962740	+	0.787019i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−26.8013	−	15.4738i	−1.10060	−	0.635431i	−0.164221	−	0.986424i	$-0.552511\pi$
$593$	−26.8013	−	15.4738i	−1.10060	−	0.635431i	−0.936378	+	0.350992i	$0.885844\pi$
$594$		0			0
$595$	−56.3986	−	32.5618i	−2.31212	−	1.33490i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.1647	−	17.6057i	0.415317	−	0.719350i	−0.580145	−	0.814513i	$-0.697004\pi$
$599$	10.1647	−	17.6057i	0.415317	−	0.719350i	0.995462	+	0.0951635i	$0.0303374\pi$
$600$		0			0
$601$	32.4968			1.32557			0.662786	−	0.748809i	$-0.269374\pi$
$601$	32.4968			1.32557			0.662786	+	0.748809i	$0.269374\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.4699	−	7.19949i	0.506973	−	0.292701i
$606$		0			0
$607$			33.5169i			1.36041i	0.733022	+	0.680205i	$0.238109\pi$
$607$			33.5169i			1.36041i	−0.733022	+	0.680205i	$0.761891\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−24.9078	−	43.1416i	−1.00766	−	1.74532i
$612$		0			0
$613$	−1.51809	−	2.62942i	−0.0613152	−	0.106201i	0.833738	−	0.552160i	$-0.186196\pi$
$613$	−1.51809	−	2.62942i	−0.0613152	−	0.106201i	−0.895054	+	0.445959i	$0.852863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.29902	−	4.79144i	−0.334106	−	0.192896i	0.323557	−	0.946209i	$-0.395121\pi$
$617$	−8.29902	−	4.79144i	−0.334106	−	0.192896i	−0.657663	+	0.753313i	$0.728455\pi$
$618$		0			0
$619$		−	0.824590i		−	0.0331431i	−0.999863	−	0.0165715i	$-0.994725\pi$
$619$		−	0.824590i		−	0.0331431i	0.999863	−	0.0165715i	$-0.00527513\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.38423	−	11.0578i	0.255779	−	0.443022i
$624$		0			0
$625$	13.7070	+	23.7412i	0.548279	+	0.949648i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−58.0246	+	33.5005i	−2.31359	+	1.33575i
$630$		0			0
$631$	12.5556	+	7.24900i	0.499832	+	0.288578i	0.728644	−	0.684893i	$-0.240151\pi$
$631$	12.5556	+	7.24900i	0.499832	+	0.288578i	−0.228812	+	0.973471i	$0.573484\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.4268			0.731247
$636$		0			0
$637$	26.1444	−	45.2834i	1.03588	−	1.79419i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	34.1303	−	19.7051i	1.34807	−	0.778306i	0.360090	−	0.932918i	$-0.382746\pi$
$641$	34.1303	−	19.7051i	1.34807	−	0.778306i	0.987975	+	0.154612i	$0.0494126\pi$
$642$		0			0
$643$	−16.9787	+	9.80267i	−0.669576	+	0.386580i	−0.795916	−	0.605407i	$-0.793010\pi$
$643$	−16.9787	+	9.80267i	−0.669576	+	0.386580i	0.126340	+	0.991987i	$0.459677\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.60076			−0.141560			−0.0707802	−	0.997492i	$-0.522549\pi$
$647$	−3.60076			−0.141560			−0.0707802	+	0.997492i	$0.522549\pi$
$648$		0			0
$649$	−8.10499	−	14.0382i	−0.318149	−	0.551049i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			42.3769i			1.65833i	0.559001	+	0.829167i	$0.311185\pi$
$653$			42.3769i			1.65833i	−0.559001	+	0.829167i	$0.688815\pi$
$654$		0			0
$655$	−21.6903	−	12.5229i	−0.847512	−	0.489311i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.3285	−	24.8177i	0.558159	−	0.966760i	−0.439491	−	0.898247i	$-0.644841\pi$
$659$	14.3285	−	24.8177i	0.558159	−	0.966760i	0.997650	−	0.0685131i	$-0.0218255\pi$
$660$		0			0
$661$	13.9681	−	24.1935i	0.543297	−	0.941017i	−0.455415	−	0.890279i	$-0.650509\pi$
$661$	13.9681	−	24.1935i	0.543297	−	0.941017i	0.998712	−	0.0507383i	$-0.0161574\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	45.8629	+	7.48177i	1.77849	+	0.290131i
$666$		0			0
$667$	−3.92165	+	2.26416i	−0.151847	+	0.0876688i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.59042	−	2.75469i	0.0613976	−	0.106344i
$672$		0			0
$673$	−16.0401			−0.618302			−0.309151	−	0.951013i	$-0.600045\pi$
$673$	−16.0401			−0.618302			−0.309151	+	0.951013i	$0.600045\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			36.1869i			1.39077i	0.718635	+	0.695387i	$0.244767\pi$
$677$			36.1869i			1.39077i	−0.718635	+	0.695387i	$0.755233\pi$
$678$		0			0
$679$	−29.5276	+	17.0478i	−1.13316	+	0.654233i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.05252			0.155065			0.0775326	−	0.996990i	$-0.475296\pi$
$683$	4.05252			0.155065			0.0775326	+	0.996990i	$0.475296\pi$
$684$		0			0
$685$	−27.6645			−1.05701
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28.3667	+	16.3775i	−1.08069	+	0.623935i
$690$		0			0
$691$			2.15655i			0.0820392i	0.999158	+	0.0410196i	$0.0130606\pi$
$691$			2.15655i			0.0820392i	−0.999158	+	0.0410196i	$0.986939\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−33.9060			−1.28613
$696$		0			0
$697$	−31.0574	+	53.7930i	−1.17638	+	2.03755i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.6045	+	13.0507i	−0.853761	+	0.492919i	−0.861918	−	0.507048i	$-0.830737\pi$
$701$	−22.6045	+	13.0507i	−0.853761	+	0.492919i	0.00815702	+	0.999967i	$0.497404\pi$
$702$		0			0
$703$	30.2588	−	37.0148i	1.14123	−	1.39604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.6275	−	27.0677i	0.587734	−	1.01799i
$708$		0			0
$709$	−2.35040	+	4.07101i	−0.0882710	+	0.152890i	−0.906780	−	0.421603i	$-0.861467\pi$
$709$	−2.35040	+	4.07101i	−0.0882710	+	0.152890i	0.818509	+	0.574493i	$0.194801\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.68833	+	3.86151i	0.250480	+	0.144615i
$714$		0			0
$715$			20.1341i			0.752973i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.404085	−	0.699896i	−0.0150698	−	0.0261017i	0.858392	−	0.512994i	$-0.171464\pi$
$719$	−0.404085	−	0.699896i	−0.0150698	−	0.0261017i	−0.873462	+	0.486892i	$0.838130\pi$
$720$		0			0
$721$	−27.6538			−1.02988
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.90344	+	1.09895i	−0.0706919	+	0.0408140i
$726$		0			0
$727$	0.582996	−	0.336593i	0.0216221	−	0.0124835i	−0.489150	−	0.872200i	$-0.662693\pi$
$727$	0.582996	−	0.336593i	0.0216221	−	0.0124835i	0.510772	+	0.859716i	$0.329360\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−34.4479	+	59.6655i	−1.27410	+	2.20681i
$732$		0			0
$733$	18.5478			0.685077			0.342538	−	0.939504i	$-0.388713\pi$
$733$	18.5478			0.685077			0.342538	+	0.939504i	$0.388713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.6859	+	6.74685i	0.430455	+	0.248523i
$738$		0			0
$739$	8.34510	−	4.81805i	0.306980	−	0.177235i	−0.338595	−	0.940932i	$-0.609951\pi$
$739$	8.34510	−	4.81805i	0.306980	−	0.177235i	0.645574	+	0.763698i	$0.276618\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.37405	+	4.11197i	0.0870953	+	0.150853i	0.906282	−	0.422673i	$-0.138908\pi$
$743$	2.37405	+	4.11197i	0.0870953	+	0.150853i	−0.819187	+	0.573527i	$0.805575\pi$
$744$		0			0
$745$	9.74519	−	16.8792i	0.357036	−	0.618405i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	66.9993i		−	2.44810i
$750$		0			0
$751$	−31.6510	−	18.2737i	−1.15496	−	0.666818i	−0.204871	−	0.978789i	$-0.565677\pi$
$751$	−31.6510	−	18.2737i	−1.15496	−	0.666818i	−0.950092	+	0.311971i	$0.899011\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−11.0350	−	19.1131i	−0.401603	−	0.695597i
$756$		0			0
$757$	−10.5170	−	18.2159i	−0.382245	−	0.662069i	0.609137	−	0.793065i	$-0.291516\pi$
$757$	−10.5170	−	18.2159i	−0.382245	−	0.662069i	−0.991383	+	0.130996i	$0.958182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	17.3294i		−	0.628192i	−0.949391	−	0.314096i	$-0.898299\pi$
$761$		−	17.3294i		−	0.628192i	0.949391	−	0.314096i	$-0.101701\pi$
$762$		0			0
$763$	24.3900	−	14.0816i	0.882978	−	0.509787i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28.3920			−1.02518
$768$		0			0
$769$	24.7378	−	42.8472i	0.892069	−	1.54511i	0.0546775	−	0.998504i	$-0.482587\pi$
$769$	24.7378	−	42.8472i	0.892069	−	1.54511i	0.837391	−	0.546604i	$-0.184080\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	12.6856	+	7.32402i	0.456269	+	0.263427i	0.710474	−	0.703723i	$-0.248481\pi$
$773$	12.6856	+	7.32402i	0.456269	+	0.263427i	−0.254205	+	0.967150i	$0.581814\pi$
$774$		0			0
$775$	3.24629	+	1.87425i	0.116610	+	0.0673250i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.13610	−	43.7440i	0.255677	−	1.56729i
$780$		0			0
$781$	8.55736	+	14.8218i	0.306207	+	0.530365i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−19.6505	−	11.3452i	−0.701357	−	0.404929i
$786$		0			0
$787$		−	1.98263i		−	0.0706730i	−0.999375	−	0.0353365i	$-0.988750\pi$
$787$		−	1.98263i		−	0.0706730i	0.999375	−	0.0353365i	$-0.0112503\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−72.9869			−2.59511
$792$		0			0
$793$	−2.78565	−	4.82489i	−0.0989214	−	0.171337i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.85639i			0.242866i	0.992600	+	0.121433i	$0.0387489\pi$
$797$			6.85639i			0.242866i	−0.992600	+	0.121433i	$0.961251\pi$
$798$		0			0
$799$		−	78.6226i		−	2.78147i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.56004	−	2.70208i	−0.0550528	−	0.0953542i
$804$		0			0
$805$	11.8919			0.419136
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	32.5568i		−	1.14464i	−0.820032	−	0.572318i	$-0.806044\pi$
$809$		−	32.5568i		−	1.14464i	0.820032	−	0.572318i	$-0.193956\pi$
$810$		0			0
$811$	−27.4796	−	15.8654i	−0.964940	−	0.557108i	−0.0672501	−	0.997736i	$-0.521423\pi$
$811$	−27.4796	−	15.8654i	−0.964940	−	0.557108i	−0.897690	+	0.440628i	$0.854756\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.8655	−	18.8196i	−0.380601	−	0.659221i
$816$		0			0
$817$	7.91514	−	48.5195i	0.276916	−	1.69748i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.4784	+	12.4006i	0.749601	+	0.432782i	0.825550	−	0.564330i	$-0.190865\pi$
$821$	21.4784	+	12.4006i	0.749601	+	0.432782i	−0.0759489	+	0.997112i	$0.524199\pi$
$822$		0			0
$823$	45.5988	+	26.3265i	1.58947	+	0.917683i	0.993394	+	0.114758i	$0.0366092\pi$
$823$	45.5988	+	26.3265i	1.58947	+	0.917683i	0.596080	+	0.802925i	$0.296724\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.4725	+	31.9954i	−0.642353	+	1.11259i	0.342553	+	0.939499i	$0.388709\pi$
$827$	−18.4725	+	31.9954i	−0.642353	+	1.11259i	−0.984906	+	0.173090i	$0.944625\pi$
$828$		0			0
$829$	−23.7453			−0.824709			−0.412355	−	0.911023i	$-0.635293\pi$
$829$	−23.7453			−0.824709			−0.412355	+	0.911023i	$0.635293\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	71.4695	−	41.2629i	2.47627	−	1.42968i
$834$		0			0
$835$		−	59.9216i		−	2.07367i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.7716	+	34.2454i	0.682591	+	1.18228i	0.974187	+	0.225741i	$0.0724802\pi$
$839$	19.7716	+	34.2454i	0.682591	+	1.18228i	−0.291597	+	0.956541i	$0.594187\pi$
$840$		0			0
$841$	−6.26018	−	10.8429i	−0.215868	−	0.373895i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.03823	+	2.33147i	0.138919	+	0.0802052i
$846$		0			0
$847$			27.7012i			0.951826i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.11739	−	10.5956i	0.209701	−	0.363213i
$852$		0			0
$853$	10.6741	+	18.4880i	0.365473	+	0.633019i	0.988852	−	0.148902i	$-0.0475738\pi$
$853$	10.6741	+	18.4880i	0.365473	+	0.633019i	−0.623379	+	0.781920i	$0.714240\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.28726	+	0.743200i	−0.0439720	+	0.0253872i	−0.521825	−	0.853053i	$-0.674749\pi$
$857$	−1.28726	+	0.743200i	−0.0439720	+	0.0253872i	0.477853	+	0.878440i	$0.341415\pi$
$858$		0			0
$859$	18.6202	+	10.7504i	0.635314	+	0.366799i	0.782807	−	0.622264i	$-0.213787\pi$
$859$	18.6202	+	10.7504i	0.635314	+	0.366799i	−0.147493	+	0.989063i	$0.547120\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.60365			0.156710			0.0783550	−	0.996926i	$-0.475033\pi$
$863$	4.60365			0.156710			0.0783550	+	0.996926i	$0.475033\pi$
$864$		0			0
$865$	21.0701	−	36.4945i	0.716405	−	1.24085i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.25003	+	2.45375i	−0.144172	+	0.0832379i
$870$		0			0
$871$	20.4680	−	11.8172i	0.693532	−	0.400411i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−47.5319			−1.60687
$876$		0			0
$877$	8.92472	+	15.4581i	0.301366	+	0.521982i	0.976446	−	0.215763i	$-0.0692239\pi$
$877$	8.92472	+	15.4581i	0.301366	+	0.521982i	−0.675079	+	0.737745i	$0.735891\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.6647i			0.426686i	0.976977	+	0.213343i	$0.0684352\pi$
$881$			12.6647i			0.426686i	−0.976977	+	0.213343i	$0.931565\pi$
$882$		0			0
$883$	19.1777	+	11.0722i	0.645380	+	0.372610i	0.786684	−	0.617356i	$-0.211796\pi$
$883$	19.1777	+	11.0722i	0.645380	+	0.372610i	−0.141304	+	0.989966i	$0.545129\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.695475	+	1.20460i	−0.0233518	+	0.0404465i	−0.877465	−	0.479640i	$-0.840767\pi$
$887$	−0.695475	+	1.20460i	−0.0233518	+	0.0404465i	0.854113	+	0.520087i	$0.174100\pi$
$888$		0			0
$889$	−17.7251	+	30.7007i	−0.594480	+	1.02967i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	19.8623	+	52.4678i	0.664667	+	1.75577i
$894$		0			0
$895$	−45.1146	+	26.0469i	−1.50801	+	0.870652i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	14.0529	−	24.3404i	0.468692	−	0.811798i
$900$		0			0
$901$	−51.6964			−1.72226
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	20.2266i		−	0.672356i
$906$		0			0
$907$	14.9113	−	8.60902i	0.495120	−	0.285858i	−0.231576	−	0.972817i	$-0.574388\pi$
$907$	14.9113	−	8.60902i	0.495120	−	0.285858i	0.726696	+	0.686959i	$0.241055\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.6288			0.716596			0.358298	−	0.933607i	$-0.383357\pi$
$911$	21.6288			0.716596			0.358298	+	0.933607i	$0.383357\pi$
$912$		0			0
$913$	−25.3441			−0.838768
$914$		0			0
$915$		0			0
$916$		0			0
$917$	41.7286	−	24.0920i	1.37800	−	0.795588i
$918$		0			0
$919$			23.1352i			0.763158i	0.924336	+	0.381579i	$0.124620\pi$
$919$			23.1352i			0.763158i	−0.924336	+	0.381579i	$0.875380\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	29.9767			0.986696
$924$		0			0
$925$	2.96918	−	5.14276i	0.0976259	−	0.169093i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	44.6055	−	25.7530i	1.46346	−	0.844928i	0.464290	−	0.885683i	$-0.346310\pi$
$929$	44.6055	−	25.7530i	1.46346	−	0.844928i	0.999169	+	0.0407550i	$0.0129763\pi$
$930$		0			0
$931$	−37.2700	+	45.5915i	−1.22148	+	1.49420i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.8885	+	27.5198i	−0.519611	+	0.899993i
$936$		0			0
$937$	−7.16235	+	12.4056i	−0.233984	+	0.405272i	−0.958977	−	0.283484i	$-0.908510\pi$
$937$	−7.16235	+	12.4056i	−0.233984	+	0.405272i	0.724993	+	0.688756i	$0.241843\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.9558	+	12.0988i	0.683139	+	0.394411i	0.801037	−	0.598615i	$-0.204282\pi$
$941$	20.9558	+	12.0988i	0.683139	+	0.394411i	−0.117897	+	0.993026i	$0.537615\pi$
$942$		0			0
$943$		−	11.3425i		−	0.369363i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.51942	+	6.09582i	0.114366	+	0.198087i	0.917526	−	0.397676i	$-0.130183\pi$
$947$	3.51942	+	6.09582i	0.114366	+	0.198087i	−0.803160	+	0.595763i	$0.796850\pi$
$948$		0			0
$949$	−5.46488			−0.177398
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.1720	+	15.1104i	−0.847795	+	0.489475i	−0.859906	−	0.510452i	$-0.829478\pi$
$953$	−26.1720	+	15.1104i	−0.847795	+	0.489475i	0.0121110	+	0.999927i	$0.496145\pi$
$954$		0			0
$955$	−11.2841	+	6.51487i	−0.365144	+	0.210816i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	26.6110	−	46.0915i	0.859312	−	1.48837i
$960$		0			0
$961$	−16.9343			−0.546267
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−31.4105	−	18.1348i	−1.01114	−	0.583781i
$966$		0			0
$967$	3.64062	−	2.10191i	0.117075	−	0.0675930i	−0.440319	−	0.897841i	$-0.645135\pi$
$967$	3.64062	−	2.10191i	0.117075	−	0.0675930i	0.557394	+	0.830248i	$0.311801\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.69080	+	15.0529i	0.278901	+	0.483071i	0.971112	−	0.238624i	$-0.0766965\pi$
$971$	8.69080	+	15.0529i	0.278901	+	0.483071i	−0.692211	+	0.721695i	$0.743363\pi$
$972$		0			0
$973$	32.6147	−	56.4903i	1.04558	−	1.81100i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			31.8258i			1.01820i	0.860708	+	0.509099i	$0.170021\pi$
$977$			31.8258i			1.01820i	−0.860708	+	0.509099i	$0.829979\pi$
$978$		0			0
$979$	−5.39567	−	3.11519i	−0.172446	−	0.0995619i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.6506	+	35.7679i	0.658652	+	1.14082i	0.980965	+	0.194185i	$0.0622064\pi$
$983$	20.6506	+	35.7679i	0.658652	+	1.14082i	−0.322313	+	0.946633i	$0.604460\pi$
$984$		0			0
$985$	−18.8472	−	32.6443i	−0.600521	−	1.04013i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	12.5808i		−	0.400045i
$990$		0			0
$991$	22.4995	−	12.9901i	0.714719	−	0.412643i	−0.0980870	−	0.995178i	$-0.531272\pi$
$991$	22.4995	−	12.9901i	0.714719	−	0.412643i	0.812806	+	0.582535i	$0.197939\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−19.1927			−0.608449
$996$		0			0
$997$	−25.8769	+	44.8201i	−0.819529	+	1.41947i	0.0865002	+	0.996252i	$0.472432\pi$
$997$	−25.8769	+	44.8201i	−0.819529	+	1.41947i	−0.906029	+	0.423215i	$0.860902\pi$
$998$		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.2591.4	yes	32
3.2	odd	2	inner	2736.2.cg.c.2591.14	yes	32
4.3	odd	2	inner	2736.2.cg.c.2591.3	yes	32
12.11	even	2	inner	2736.2.cg.c.2591.13	yes	32
19.11	even	3	inner	2736.2.cg.c.1151.13	yes	32
57.11	odd	6	inner	2736.2.cg.c.1151.3	✓	32
76.11	odd	6	inner	2736.2.cg.c.1151.14	yes	32
228.11	even	6	inner	2736.2.cg.c.1151.4	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.cg.c.1151.3	✓	32	57.11	odd	6	inner
2736.2.cg.c.1151.4	yes	32	228.11	even	6	inner
2736.2.cg.c.1151.13	yes	32	19.11	even	3	inner
2736.2.cg.c.1151.14	yes	32	76.11	odd	6	inner
2736.2.cg.c.2591.3	yes	32	4.3	odd	2	inner
2736.2.cg.c.2591.4	yes	32	1.1	even	1	trivial
2736.2.cg.c.2591.13	yes	32	12.11	even	2	inner
2736.2.cg.c.2591.14	yes	32	3.2	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-2.03864 + 1.17701i) q^{5} -4.52875i q^{7} +O(q^{10})\)	\(q+(-2.03864 + 1.17701i) q^{5} -4.52875i q^{7} -2.20981 q^{11} +(-1.93525 + 3.35196i) q^{13} +(-5.29030 + 3.05436i) q^{17} +(2.75880 - 3.37477i) q^{19} +(0.557743 - 0.966039i) q^{23} +(0.270710 - 0.468883i) q^{25} +(-3.51564 - 2.02976i) q^{29} +6.92346i q^{31} +(5.33038 + 9.23249i) q^{35} +10.9681 q^{37} +(8.80594 - 5.08411i) q^{41} +(9.76727 - 5.63914i) q^{43} +(-6.43529 + 11.1462i) q^{47} -13.5095 q^{49} +(7.32894 + 4.23137i) q^{53} +(4.50500 - 2.60097i) q^{55} +(3.66774 + 6.35271i) q^{59} +(-0.719712 + 1.24658i) q^{61} -9.11126i q^{65} +(-5.28819 - 3.05314i) q^{67} +(-3.87245 - 6.70728i) q^{71} +(0.705965 + 1.22277i) q^{73} +10.0076i q^{77} +(1.92326 - 1.11039i) q^{79} +11.4689 q^{83} +(7.19002 - 12.4535i) q^{85} +(2.44169 + 1.40971i) q^{89} +(15.1802 + 8.76428i) q^{91} +(-1.65206 + 10.1271i) q^{95} +(-3.76434 - 6.52004i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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