LMFDB - Embedded newform 2736.2.dc.c.449.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 $ x^16 + 16x^14 + 174x^12 + 1012x^10 + 4243x^8 + 9708x^6 + 15858x^4 + 12150*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 171)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.7
Root			$1.13921 + 1.97317i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.c.1889.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.90154 - 1.67520i) q^{5} -3.54697 q^{7} +O(q^{10})$	$q+(2.90154 - 1.67520i) q^{5} -3.54697 q^{7} -0.251548i q^{11} +(-4.29049 - 2.47712i) q^{13} +(-3.11939 + 1.80098i) q^{17} +(-4.15959 - 1.30301i) q^{19} +(3.89208 + 2.24709i) q^{23} +(3.11262 - 5.39122i) q^{25} +(-2.27843 + 3.94635i) q^{29} +5.28544i q^{31} +(-10.2917 + 5.94190i) q^{35} +6.47474i q^{37} +(4.96212 + 8.59464i) q^{41} +(3.38610 + 5.86490i) q^{43} +(-8.96412 - 5.17544i) q^{47} +5.58098 q^{49} +(0.217847 - 0.377322i) q^{53} +(-0.421394 - 0.729876i) q^{55} +(2.06058 + 3.56903i) q^{59} +(-6.46836 + 11.2035i) q^{61} -16.5987 q^{65} +(-0.273084 - 0.157665i) q^{67} +(-4.10993 - 7.11860i) q^{71} +(0.356142 + 0.616857i) q^{73} +0.892232i q^{77} +(7.57733 - 4.37477i) q^{79} +1.31440i q^{83} +(-6.03401 + 10.4512i) q^{85} +(1.20839 - 2.09299i) q^{89} +(15.2182 + 8.78625i) q^{91} +(-14.2520 + 3.18743i) q^{95} +(9.36417 - 5.40640i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7}+O(q^{10}) $ 16 * q - 8 * q^7	$ 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) $ 16 * q - 8 * q^7 - 24 * q^13 + 12 * q^19 + 20 * q^25 + 4 * q^55 - 44 * q^61 + 24 * q^67 - 20 * q^73 + 48 * q^79 - 56 * q^85 + 24 * q^91 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.90154	−	1.67520i	1.29761	−	0.749174i	0.317617	−	0.948219i	$-0.397117\pi$
$5$	2.90154	−	1.67520i	1.29761	−	0.749174i	0.979990	+	0.199045i	$0.0637839\pi$
$6$		0			0
$7$	−3.54697			−1.34063			−0.670314	−	0.742078i	$-0.733841\pi$
$7$	−3.54697			−1.34063			−0.670314	+	0.742078i	$0.733841\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.251548i		−	0.0758445i	−0.999281	−	0.0379223i	$-0.987926\pi$
$11$		−	0.251548i		−	0.0758445i	0.999281	−	0.0379223i	$-0.0120739\pi$
$12$		0			0
$13$	−4.29049	−	2.47712i	−1.18997	−	0.687028i	−0.231669	−	0.972795i	$-0.574419\pi$
$13$	−4.29049	−	2.47712i	−1.18997	−	0.687028i	−0.958299	+	0.285766i	$0.907752\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.11939	+	1.80098i	−0.756562	+	0.436801i	−0.828060	−	0.560639i	$-0.810555\pi$
$17$	−3.11939	+	1.80098i	−0.756562	+	0.436801i	0.0714979	+	0.997441i	$0.477222\pi$
$18$		0			0
$19$	−4.15959	−	1.30301i	−0.954275	−	0.298931i
$19$	−4.15959	−	1.30301i	−0.954275	−	0.298931i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.89208	+	2.24709i	0.811555	+	0.468552i	0.847496	−	0.530802i	$-0.178109\pi$
$23$	3.89208	+	2.24709i	0.811555	+	0.468552i	−0.0359405	+	0.999354i	$0.511443\pi$
$24$		0			0
$25$	3.11262	−	5.39122i	0.622524	−	1.07824i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.27843	+	3.94635i	−0.423093	+	0.732819i	−0.996240	−	0.0866337i	$-0.972389\pi$
$29$	−2.27843	+	3.94635i	−0.423093	+	0.732819i	0.573147	+	0.819452i	$0.305722\pi$
$30$		0			0
$31$			5.28544i			0.949294i	0.880176	+	0.474647i	$0.157424\pi$
$31$			5.28544i			0.949294i	−0.880176	+	0.474647i	$0.842576\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.2917	+	5.94190i	−1.73961	+	1.00436i
$36$		0			0
$37$			6.47474i			1.06444i	0.846606	+	0.532220i	$0.178642\pi$
$37$			6.47474i			1.06444i	−0.846606	+	0.532220i	$0.821358\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.96212	+	8.59464i	0.774953	+	1.34226i	0.934821	+	0.355119i	$0.115560\pi$
$41$	4.96212	+	8.59464i	0.774953	+	1.34226i	−0.159868	+	0.987138i	$0.551107\pi$
$42$		0			0
$43$	3.38610	+	5.86490i	0.516376	+	0.894389i	0.999819	+	0.0190137i	$0.00605261\pi$
$43$	3.38610	+	5.86490i	0.516376	+	0.894389i	−0.483443	+	0.875376i	$0.660614\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.96412	−	5.17544i	−1.30755	−	0.754915i	−0.325865	−	0.945416i	$-0.605655\pi$
$47$	−8.96412	−	5.17544i	−1.30755	−	0.754915i	−0.981687	+	0.190501i	$0.938989\pi$
$48$		0			0
$49$	5.58098			0.797283
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.217847	−	0.377322i	0.0299236	−	0.0518291i	−0.850676	−	0.525691i	$-0.823807\pi$
$53$	0.217847	−	0.377322i	0.0299236	−	0.0518291i	0.880599	+	0.473862i	$0.157140\pi$
$54$		0			0
$55$	−0.421394	−	0.729876i	−0.0568207	−	0.0984164i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.06058	+	3.56903i	0.268265	+	0.464648i	0.968414	−	0.249349i	$-0.0802165\pi$
$59$	2.06058	+	3.56903i	0.268265	+	0.464648i	−0.700149	+	0.713997i	$0.746883\pi$
$60$		0			0
$61$	−6.46836	+	11.2035i	−0.828189	+	1.43447i	0.0712687	+	0.997457i	$0.477295\pi$
$61$	−6.46836	+	11.2035i	−0.828189	+	1.43447i	−0.899458	+	0.437008i	$0.856038\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−16.5987			−2.05882
$66$		0			0
$67$	−0.273084	−	0.157665i	−0.0333625	−	0.0192618i	0.483226	−	0.875496i	$-0.339465\pi$
$67$	−0.273084	−	0.157665i	−0.0333625	−	0.0192618i	−0.516588	+	0.856234i	$0.672798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.10993	−	7.11860i	−0.487759	−	0.844823i	0.512142	−	0.858901i	$-0.328852\pi$
$71$	−4.10993	−	7.11860i	−0.487759	−	0.844823i	−0.999901	+	0.0140778i	$0.995519\pi$
$72$		0			0
$73$	0.356142	+	0.616857i	0.0416833	+	0.0721976i	0.886114	−	0.463467i	$-0.153395\pi$
$73$	0.356142	+	0.616857i	0.0416833	+	0.0721976i	−0.844431	+	0.535664i	$0.820061\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.892232i			0.101679i
$78$		0			0
$79$	7.57733	−	4.37477i	0.852516	−	0.492200i	−0.00898288	−	0.999960i	$-0.502859\pi$
$79$	7.57733	−	4.37477i	0.852516	−	0.492200i	0.861499	+	0.507759i	$0.169526\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.31440i			0.144274i	0.997395	+	0.0721369i	$0.0229819\pi$
$83$			1.31440i			0.144274i	−0.997395	+	0.0721369i	$0.977018\pi$
$84$		0			0
$85$	−6.03401	+	10.4512i	−0.654481	+	1.13359i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.20839	−	2.09299i	0.128089	−	0.221857i	−0.794847	−	0.606810i	$-0.792449\pi$
$89$	1.20839	−	2.09299i	0.128089	−	0.221857i	0.922936	+	0.384953i	$0.125782\pi$
$90$		0			0
$91$	15.2182	+	8.78625i	1.59530	+	0.921049i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.2520	+	3.18743i	−1.46223	+	0.327023i
$96$		0			0
$97$	9.36417	−	5.40640i	0.950787	−	0.548937i	0.0574618	−	0.998348i	$-0.481699\pi$
$97$	9.36417	−	5.40640i	0.950787	−	0.548937i	0.893325	+	0.449410i	$0.148366\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.40904	−	3.12291i	−0.538220	−	0.310741i	0.206137	−	0.978523i	$-0.433911\pi$
$101$	−5.40904	−	3.12291i	−0.538220	−	0.310741i	−0.744357	+	0.667782i	$0.767244\pi$
$102$		0			0
$103$			4.38110i			0.431683i	0.976428	+	0.215841i	$0.0692494\pi$
$103$			4.38110i			0.431683i	−0.976428	+	0.215841i	$0.930751\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.98108			0.191519			0.0957593	−	0.995405i	$-0.469472\pi$
$107$	1.98108			0.191519			0.0957593	+	0.995405i	$0.469472\pi$
$108$		0			0
$109$	−0.897781	+	0.518334i	−0.0859918	+	0.0496474i	−0.542379	−	0.840134i	$-0.682476\pi$
$109$	−0.897781	+	0.518334i	−0.0859918	+	0.0496474i	0.456387	+	0.889781i	$0.349143\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.32954			−0.125073			−0.0625364	−	0.998043i	$-0.519919\pi$
$113$	−1.32954			−0.125073			−0.0625364	+	0.998043i	$0.519919\pi$
$114$		0			0
$115$	15.0574			1.40411
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.0644	−	6.38801i	1.01427	−	0.585588i
$120$		0			0
$121$	10.9367			0.994248
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	4.10505i		−	0.367167i
$126$		0			0
$127$	−7.10729	−	4.10340i	−0.630670	−	0.364118i	0.150341	−	0.988634i	$-0.451963\pi$
$127$	−7.10729	−	4.10340i	−0.630670	−	0.364118i	−0.781012	+	0.624517i	$0.785296\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.70643	+	2.71726i	−0.411203	+	0.237408i	−0.691307	−	0.722562i	$-0.742965\pi$
$131$	−4.70643	+	2.71726i	−0.411203	+	0.237408i	0.280103	+	0.959970i	$0.409631\pi$
$132$		0			0
$133$	14.7539	+	4.62173i	1.27933	+	0.400755i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.8941	−	6.86706i	−1.01618	−	0.586692i	−0.103185	−	0.994662i	$-0.532903\pi$
$137$	−11.8941	−	6.86706i	−1.01618	−	0.586692i	−0.912995	+	0.407970i	$0.866237\pi$
$138$		0			0
$139$	1.17422	−	2.03381i	0.0995959	−	0.172505i	−0.811922	−	0.583767i	$-0.801578\pi$
$139$	1.17422	−	2.03381i	0.0995959	−	0.172505i	0.911517	+	0.411261i	$0.134912\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.623113	+	1.07926i	−0.0521073	+	0.0902525i
$144$		0			0
$145$			15.2673i			1.26788i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.31058	+	4.79812i	−0.680829	+	0.393077i	−0.800167	−	0.599777i	$-0.795256\pi$
$149$	−8.31058	+	4.79812i	−0.680829	+	0.393077i	0.119338	+	0.992854i	$0.461923\pi$
$150$		0			0
$151$			7.06053i			0.574577i	0.957844	+	0.287289i	$0.0927539\pi$
$151$			7.06053i			0.574577i	−0.957844	+	0.287289i	$0.907246\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.85420	+	15.3359i	0.711186	+	1.23181i
$156$		0			0
$157$	−8.17492	−	14.1594i	−0.652429	−	1.13004i	−0.982532	−	0.186096i	$-0.940417\pi$
$157$	−8.17492	−	14.1594i	−0.652429	−	1.13004i	0.330102	−	0.943945i	$-0.392917\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−13.8051	−	7.97037i	−1.08799	−	0.628153i
$162$		0			0
$163$	−12.3119			−0.964340			−0.482170	−	0.876078i	$-0.660151\pi$
$163$	−12.3119			−0.964340			−0.482170	+	0.876078i	$0.660151\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.623113	−	1.07926i	0.0482179	−	0.0835159i	−0.840909	−	0.541176i	$-0.817979\pi$
$167$	0.623113	−	1.07926i	0.0482179	−	0.0835159i	0.889127	+	0.457660i	$0.151312\pi$
$168$		0			0
$169$	5.77221	+	9.99776i	0.444016	+	0.769058i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.24623	−	2.15853i	−0.0947488	−	0.164110i	0.814755	−	0.579805i	$-0.196871\pi$
$173$	−1.24623	−	2.15853i	−0.0947488	−	0.164110i	−0.909504	+	0.415696i	$0.863538\pi$
$174$		0			0
$175$	−11.0404	+	19.1225i	−0.834573	+	1.44552i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−21.0722			−1.57501			−0.787507	−	0.616306i	$-0.788628\pi$
$179$	−21.0722			−1.57501			−0.787507	+	0.616306i	$0.788628\pi$
$180$		0			0
$181$	−18.4387	−	10.6456i	−1.37054	−	0.791280i	−0.379542	−	0.925174i	$-0.623918\pi$
$181$	−18.4387	−	10.6456i	−1.37054	−	0.791280i	−0.990996	+	0.133894i	$0.957252\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.8465	+	18.7867i	0.797452	+	1.38123i
$186$		0			0
$187$	0.453032	+	0.784675i	0.0331290	+	0.0573811i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.1888i			0.737237i	0.929581	+	0.368618i	$0.120169\pi$
$191$			10.1888i			0.737237i	−0.929581	+	0.368618i	$0.879831\pi$
$192$		0			0
$193$	−8.93139	+	5.15654i	−0.642896	+	0.371176i	−0.785729	−	0.618571i	$-0.787712\pi$
$193$	−8.93139	+	5.15654i	−0.642896	+	0.371176i	0.142833	+	0.989747i	$0.454379\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.3901i		−	1.02525i	−0.858613	−	0.512625i	$-0.828673\pi$
$197$		−	14.3901i		−	1.02525i	0.858613	−	0.512625i	$-0.171327\pi$
$198$		0			0
$199$	11.3022	−	19.5759i	0.801189	−	1.38770i	−0.117644	−	0.993056i	$-0.537534\pi$
$199$	11.3022	−	19.5759i	0.801189	−	1.38770i	0.918834	−	0.394645i	$-0.129132\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.08150	−	13.9976i	0.567210	−	0.982437i
$204$		0			0
$205$	28.7956	+	16.6251i	2.01117	+	1.16115i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.327769	+	1.04633i	−0.0226723	+	0.0723765i
$210$		0			0
$211$	8.09839	−	4.67561i	0.557516	−	0.321882i	−0.194632	−	0.980876i	$-0.562351\pi$
$211$	8.09839	−	4.67561i	0.557516	−	0.321882i	0.752148	+	0.658994i	$0.229018\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.6498	+	11.3448i	1.34011	+	0.773711i
$216$		0			0
$217$		−	18.7473i		−	1.27265i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.8449			1.20038
$222$		0			0
$223$	2.63200	−	1.51959i	0.176252	−	0.101759i	−0.409279	−	0.912409i	$-0.634220\pi$
$223$	2.63200	−	1.51959i	0.176252	−	0.101759i	0.585530	+	0.810650i	$0.300886\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.35592			0.156368			0.0781841	−	0.996939i	$-0.475088\pi$
$227$	2.35592			0.156368			0.0781841	+	0.996939i	$0.475088\pi$
$228$		0			0
$229$	−3.15801			−0.208687			−0.104344	−	0.994541i	$-0.533274\pi$
$229$	−3.15801			−0.208687			−0.104344	+	0.994541i	$0.533274\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.4642	+	11.8150i	−1.34065	+	0.774026i	−0.986903	−	0.161314i	$-0.948427\pi$
$233$	−20.4642	+	11.8150i	−1.34065	+	0.774026i	−0.353750	+	0.935340i	$0.615094\pi$
$234$		0			0
$235$	−34.6797			−2.26225
$236$		0			0
$237$		0			0
$238$		0			0
$239$			20.5878i			1.33171i	0.746080	+	0.665857i	$0.231934\pi$
$239$			20.5878i			1.33171i	−0.746080	+	0.665857i	$0.768066\pi$
$240$		0			0
$241$	3.12835	+	1.80615i	0.201515	+	0.116345i	0.597362	−	0.801972i	$-0.296216\pi$
$241$	3.12835	+	1.80615i	0.201515	+	0.116345i	−0.395847	+	0.918316i	$0.629549\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	16.1934	−	9.34928i	1.03456	−	0.597304i
$246$		0			0
$247$	14.6190	+	15.8943i	0.930183	+	1.01133i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.28043	−	3.62601i	−0.396417	−	0.228872i	0.288520	−	0.957474i	$-0.406837\pi$
$251$	−6.28043	−	3.62601i	−0.396417	−	0.228872i	−0.684937	+	0.728602i	$0.740170\pi$
$252$		0			0
$253$	0.565252	−	0.979044i	0.0355371	−	0.0615520i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.40905	−	16.2970i	0.586920	−	1.01658i	−0.407713	−	0.913110i	$-0.633674\pi$
$257$	9.40905	−	16.2970i	0.586920	−	1.01658i	0.994633	−	0.103466i	$-0.0329932\pi$
$258$		0			0
$259$		−	22.9657i		−	1.42702i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.21963	+	3.01355i	−0.321856	+	0.185824i	−0.652220	−	0.758030i	$-0.726162\pi$
$263$	−5.21963	+	3.01355i	−0.321856	+	0.185824i	0.330364	+	0.943854i	$0.392829\pi$
$264$		0			0
$265$		−	1.45975i		−	0.0896718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.29557	+	5.70810i	0.200935	+	0.348029i	0.948830	−	0.315788i	$-0.102269\pi$
$269$	3.29557	+	5.70810i	0.200935	+	0.348029i	−0.747895	+	0.663817i	$0.768935\pi$
$270$		0			0
$271$	−10.4497	−	18.0994i	−0.634773	−	1.09946i	−0.986563	−	0.163380i	$-0.947760\pi$
$271$	−10.4497	−	18.0994i	−0.634773	−	1.09946i	0.351790	−	0.936079i	$-0.385573\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.35615	−	0.782973i	−0.0817788	−	0.0472150i
$276$		0			0
$277$	−25.8223			−1.55151			−0.775755	−	0.631034i	$-0.782631\pi$
$277$	−25.8223			−1.55151			−0.775755	+	0.631034i	$0.782631\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.43948	+	16.3497i	−0.563112	+	0.975338i	0.434111	+	0.900860i	$0.357063\pi$
$281$	−9.43948	+	16.3497i	−0.563112	+	0.975338i	−0.997223	+	0.0744789i	$0.976271\pi$
$282$		0			0
$283$	8.38245	+	14.5188i	0.498285	+	0.863055i	0.999998	−	0.00197917i	$-0.000629990\pi$
$283$	8.38245	+	14.5188i	0.498285	+	0.863055i	−0.501713	+	0.865034i	$0.667297\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.6005	−	30.4849i	−1.03892	−	1.79947i
$288$		0			0
$289$	−2.01295	+	3.48654i	−0.118409	+	0.205091i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.0683			−1.63977			−0.819885	−	0.572529i	$-0.805963\pi$
$293$	−28.0683			−1.63977			−0.819885	+	0.572529i	$0.805963\pi$
$294$		0			0
$295$	11.9577	+	6.90378i	0.696204	+	0.401954i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.1326	−	19.2823i	−0.643816	−	1.11512i
$300$		0			0
$301$	−12.0104	−	20.8026i	−0.692268	−	1.19904i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			43.3433i			2.48183i
$306$		0			0
$307$	−19.6261	+	11.3311i	−1.12012	+	0.646703i	−0.941432	−	0.337203i	$-0.890519\pi$
$307$	−19.6261	+	11.3311i	−1.12012	+	0.646703i	−0.178690	+	0.983905i	$0.557186\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			17.8544i			1.01243i	0.862407	+	0.506216i	$0.168956\pi$
$311$			17.8544i			1.01243i	−0.862407	+	0.506216i	$0.831044\pi$
$312$		0			0
$313$	7.18630	−	12.4470i	0.406193	−	0.703548i	−0.588266	−	0.808667i	$-0.700189\pi$
$313$	7.18630	−	12.4470i	0.406193	−	0.703548i	0.994460	+	0.105120i	$0.0335226\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.89586	−	15.4081i	0.499641	−	0.865404i	−0.500359	−	0.865818i	$-0.666799\pi$
$317$	8.89586	−	15.4081i	0.499641	−	0.865404i	1.00000		0.000414314i	$0.000131880\pi$
$318$		0			0
$319$	0.992696	+	0.573133i	0.0555803	+	0.0320893i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.3220	−	3.42674i	0.852542	−	0.190669i
$324$		0			0
$325$	−26.7093	+	15.4206i	−1.48157	+	0.855383i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	31.7955	+	18.3571i	1.75294	+	1.01206i
$330$		0			0
$331$			27.4202i			1.50715i	0.657361	+	0.753575i	$0.271673\pi$
$331$			27.4202i			1.50715i	−0.657361	+	0.753575i	$0.728327\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.05648			−0.0577219
$336$		0			0
$337$	−14.9188	+	8.61340i	−0.812681	+	0.469202i	−0.847886	−	0.530179i	$-0.822125\pi$
$337$	−14.9188	+	8.61340i	−0.812681	+	0.469202i	0.0352050	+	0.999380i	$0.488792\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.32954			0.0719987
$342$		0			0
$343$	5.03321			0.271768
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.28635	−	4.20677i	0.391152	−	0.225832i	−0.291507	−	0.956569i	$-0.594157\pi$
$347$	7.28635	−	4.20677i	0.391152	−	0.225832i	0.682659	+	0.730737i	$0.260823\pi$
$348$		0			0
$349$	−11.5389			−0.617661			−0.308831	−	0.951117i	$-0.599938\pi$
$349$	−11.5389			−0.617661			−0.308831	+	0.951117i	$0.599938\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12.0544i			0.641592i	0.947148	+	0.320796i	$0.103950\pi$
$353$			12.0544i			0.641592i	−0.947148	+	0.320796i	$0.896050\pi$
$354$		0			0
$355$	−23.8502	−	13.7699i	−1.26584	−	0.730833i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−27.7710	+	16.0336i	−1.46570	+	0.846220i	−0.999265	−	0.0383367i	$-0.987794\pi$
$359$	−27.7710	+	16.0336i	−1.46570	+	0.846220i	−0.466432	+	0.884557i	$0.654461\pi$
$360$		0			0
$361$	15.6043	+	10.8400i	0.821281	+	0.570524i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.06672	+	1.19322i	0.108177	+	0.0624561i
$366$		0			0
$367$	1.47004	−	2.54618i	0.0767354	−	0.132910i	−0.825104	−	0.564981i	$-0.808884\pi$
$367$	1.47004	−	2.54618i	0.0767354	−	0.132910i	0.901840	+	0.432071i	$0.142217\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.772695	+	1.33835i	−0.0401164	+	0.0694836i
$372$		0			0
$373$		−	22.4374i		−	1.16177i	−0.813987	−	0.580883i	$-0.802707\pi$
$373$		−	22.4374i		−	1.16177i	0.813987	−	0.580883i	$-0.197293\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.5511	−	11.2879i	1.00693	−	0.581354i
$378$		0			0
$379$			9.54429i			0.490257i	0.969491	+	0.245129i	$0.0788302\pi$
$379$			9.54429i			0.490257i	−0.969491	+	0.245129i	$0.921170\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.61920	−	13.1968i	−0.389323	−	0.674327i	0.603035	−	0.797714i	$-0.293958\pi$
$383$	−7.61920	−	13.1968i	−0.389323	−	0.674327i	−0.992359	+	0.123387i	$0.960624\pi$
$384$		0			0
$385$	1.49467	+	2.58885i	0.0761755	+	0.131940i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.6174	−	9.01671i	−0.791833	−	0.457165i	0.0487743	−	0.998810i	$-0.484469\pi$
$389$	−15.6174	−	9.01671i	−0.791833	−	0.457165i	−0.840608	+	0.541645i	$0.817802\pi$
$390$		0			0
$391$	−16.1879			−0.818656
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.6573	−	25.3872i	0.737488	−	1.27737i
$396$		0			0
$397$	7.09927	+	12.2963i	0.356302	+	0.617133i	0.987340	−	0.158619i	$-0.0507041\pi$
$397$	7.09927	+	12.2963i	0.356302	+	0.617133i	−0.631038	+	0.775752i	$0.717371\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.24623	−	9.08674i	−0.261984	−	0.453770i	0.704785	−	0.709421i	$-0.251044\pi$
$401$	−5.24623	−	9.08674i	−0.261984	−	0.453770i	−0.966769	+	0.255651i	$0.917710\pi$
$402$		0			0
$403$	13.0927	−	22.6772i	0.652192	−	1.12963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.62871			0.0807320
$408$		0			0
$409$	10.9650	+	6.33065i	0.542185	+	0.313031i	0.745964	−	0.665986i	$-0.231989\pi$
$409$	10.9650	+	6.33065i	0.542185	+	0.313031i	−0.203779	+	0.979017i	$0.565322\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.30881	−	12.6592i	−0.359643	−	0.622920i
$414$		0			0
$415$	2.20188	+	3.81378i	0.108086	+	0.187211i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			27.3392i			1.33561i	0.744338	+	0.667804i	$0.232765\pi$
$419$			27.3392i			1.33561i	−0.744338	+	0.667804i	$0.767235\pi$
$420$		0			0
$421$	22.3913	−	12.9276i	1.09129	−	0.630055i	0.157368	−	0.987540i	$-0.449699\pi$
$421$	22.3913	−	12.9276i	1.09129	−	0.630055i	0.933919	+	0.357485i	$0.116366\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			22.4230i			1.08768i
$426$		0			0
$427$	22.9431	−	39.7386i	1.11029	−	1.92308i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.09274	+	14.0170i	−0.389813	+	0.675176i	−0.992424	−	0.122859i	$-0.960794\pi$
$431$	−8.09274	+	14.0170i	−0.389813	+	0.675176i	0.602611	+	0.798035i	$0.294127\pi$
$432$		0			0
$433$	14.4504	+	8.34294i	0.694442	+	0.400936i	0.805274	−	0.592903i	$-0.202018\pi$
$433$	14.4504	+	8.34294i	0.694442	+	0.400936i	−0.110832	+	0.993839i	$0.535352\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13.2615	−	14.4184i	−0.634382	−	0.689726i
$438$		0			0
$439$	29.2233	−	16.8721i	1.39475	−	0.805261i	0.400916	−	0.916115i	$-0.368692\pi$
$439$	29.2233	−	16.8721i	1.39475	−	0.805261i	0.993837	+	0.110854i	$0.0353587\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.3770	−	9.45529i	−0.778096	−	0.449234i	0.0576588	−	0.998336i	$-0.481636\pi$
$443$	−16.3770	−	9.45529i	−0.778096	−	0.449234i	−0.835755	+	0.549102i	$0.814970\pi$
$444$		0			0
$445$		−	8.09719i		−	0.383844i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.0530			1.51267			0.756336	−	0.654183i	$-0.226987\pi$
$449$	32.0530			1.51267			0.756336	+	0.654183i	$0.226987\pi$
$450$		0			0
$451$	2.16196	−	1.24821i	0.101803	−	0.0587759i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	58.8751			2.76011
$456$		0			0
$457$	27.5605			1.28922			0.644612	−	0.764510i	$-0.277019\pi$
$457$	27.5605			1.28922			0.644612	+	0.764510i	$0.277019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.4207	−	8.90317i	0.718215	−	0.414662i	−0.0958800	−	0.995393i	$-0.530567\pi$
$461$	15.4207	−	8.90317i	0.718215	−	0.414662i	0.814096	+	0.580731i	$0.197233\pi$
$462$		0			0
$463$	−31.5005			−1.46395			−0.731977	−	0.681329i	$-0.761402\pi$
$463$	−31.5005			−1.46395			−0.731977	+	0.681329i	$0.761402\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	34.7591i		−	1.60846i	−0.594317	−	0.804231i	$-0.702578\pi$
$467$		−	34.7591i		−	1.60846i	0.594317	−	0.804231i	$-0.297422\pi$
$468$		0			0
$469$	0.968620	+	0.559233i	0.0447267	+	0.0258230i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.47530	−	0.851767i	0.0678345	−	0.0391643i
$474$		0			0
$475$	−19.9720	+	18.3695i	−0.916379	+	0.842849i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.62312	−	2.66916i	−0.211236	−	0.121957i	0.390650	−	0.920539i	$-0.372250\pi$
$479$	−4.62312	−	2.66916i	−0.211236	−	0.121957i	−0.601886	+	0.798582i	$0.705584\pi$
$480$		0			0
$481$	16.0387	−	27.7798i	0.731301	−	1.26665i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.1137	−	31.3738i	0.822499	−	1.42461i
$486$		0			0
$487$			37.8144i			1.71353i	0.515705	+	0.856766i	$0.327530\pi$
$487$			37.8144i			1.71353i	−0.515705	+	0.856766i	$0.672470\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.95830	+	5.17208i	−0.404283	+	0.233413i	−0.688330	−	0.725398i	$-0.741656\pi$
$491$	−8.95830	+	5.17208i	−0.404283	+	0.233413i	0.284048	+	0.958810i	$0.408323\pi$
$492$		0			0
$493$		−	16.4136i		−	0.739231i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.5778	+	25.2495i	0.653903	+	1.13259i
$498$		0			0
$499$	15.6603	+	27.1244i	0.701051	+	1.21426i	0.968098	+	0.250572i	$0.0806187\pi$
$499$	15.6603	+	27.1244i	0.701051	+	1.21426i	−0.267047	+	0.963683i	$0.586048\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.1154	+	11.0363i	0.852315	+	0.492084i	0.861431	−	0.507874i	$-0.169568\pi$
$503$	19.1154	+	11.0363i	0.852315	+	0.492084i	−0.00911608	+	0.999958i	$0.502902\pi$
$504$		0			0
$505$	−20.9261			−0.931198
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.73681	−	9.93645i	0.254280	−	0.440425i	−0.710420	−	0.703778i	$-0.751495\pi$
$509$	5.73681	−	9.93645i	0.254280	−	0.440425i	0.964700	+	0.263353i	$0.0848283\pi$
$510$		0			0
$511$	−1.26323	−	2.18797i	−0.0558818	−	0.0967901i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.33924	+	12.7119i	0.323405	+	0.560155i
$516$		0			0
$517$	−1.30187	+	2.25490i	−0.0572562	+	0.0991706i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.0265			−0.702133			−0.351066	−	0.936351i	$-0.614181\pi$
$521$	−16.0265			−0.702133			−0.351066	+	0.936351i	$0.614181\pi$
$522$		0			0
$523$	10.1731	+	5.87344i	0.444839	+	0.256828i	0.705648	−	0.708563i	$-0.250656\pi$
$523$	10.1731	+	5.87344i	0.444839	+	0.256828i	−0.260809	+	0.965390i	$0.583989\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.51897	−	16.4873i	−0.414653	−	0.718200i
$528$		0			0
$529$	−1.40114	−	2.42684i	−0.0609189	−	0.105515i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	49.1670i		−	2.12966i
$534$		0			0
$535$	5.74819	−	3.31872i	0.248516	−	0.143481i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	1.40388i		−	0.0604695i
$540$		0			0
$541$	19.5906	−	33.9319i	0.842265	−	1.45885i	−0.0457105	−	0.998955i	$-0.514555\pi$
$541$	19.5906	−	33.9319i	0.842265	−	1.45885i	0.887975	−	0.459891i	$-0.152111\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.73663	+	3.00793i	−0.0743891	+	0.128846i
$546$		0			0
$547$	−8.36275	−	4.82824i	−0.357565	−	0.206440i	0.310447	−	0.950591i	$-0.399521\pi$
$547$	−8.36275	−	4.82824i	−0.357565	−	0.206440i	−0.668012	+	0.744150i	$0.732855\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.6194	−	13.4464i	0.622809	−	0.572835i
$552$		0			0
$553$	−26.8765	+	15.5172i	−1.14291	+	0.659858i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.1760	−	15.1127i	−1.10911	−	0.640346i	−0.170512	−	0.985356i	$-0.554542\pi$
$557$	−26.1760	−	15.1127i	−1.10911	−	0.640346i	−0.938599	+	0.345010i	$0.887876\pi$
$558$		0			0
$559$		−	33.5511i		−	1.41906i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	39.7803			1.67654			0.838269	−	0.545257i	$-0.183568\pi$
$563$	39.7803			1.67654			0.838269	+	0.545257i	$0.183568\pi$
$564$		0			0
$565$	−3.85772	+	2.22725i	−0.162295	+	0.0937013i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7383			0.534018			0.267009	−	0.963694i	$-0.413965\pi$
$569$	12.7383			0.534018			0.267009	+	0.963694i	$0.413965\pi$
$570$		0			0
$571$	4.21174			0.176256			0.0881280	−	0.996109i	$-0.471912\pi$
$571$	4.21174			0.176256			0.0881280	+	0.996109i	$0.471912\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.2291	−	13.9887i	1.01042	−	0.583369i
$576$		0			0
$577$	−4.82818			−0.201000			−0.100500	−	0.994937i	$-0.532044\pi$
$577$	−4.82818			−0.201000			−0.100500	+	0.994937i	$0.532044\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	4.66212i		−	0.193418i
$582$		0			0
$583$	−0.0949144	−	0.0547989i	−0.00393095	−	0.00226954i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.4624	−	8.92722i	0.638201	−	0.368466i	−0.145720	−	0.989326i	$-0.546550\pi$
$587$	15.4624	−	8.92722i	0.638201	−	0.368466i	0.783921	+	0.620860i	$0.213216\pi$
$588$		0			0
$589$	6.88698	−	21.9853i	0.283773	−	0.905887i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−20.4379	−	11.7998i	−0.839284	−	0.484561i	0.0177369	−	0.999843i	$-0.494354\pi$
$593$	−20.4379	−	11.7998i	−0.839284	−	0.484561i	−0.857021	+	0.515282i	$0.827687\pi$
$594$		0			0
$595$	21.4025	−	37.0701i	0.877415	−	1.51973i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	23.1857	−	40.1588i	0.947342	−	1.64084i	0.196350	−	0.980534i	$-0.437091\pi$
$599$	23.1857	−	40.1588i	0.947342	−	1.64084i	0.750992	−	0.660311i	$-0.229576\pi$
$600$		0			0
$601$			20.8985i			0.852466i	0.904613	+	0.426233i	$0.140160\pi$
$601$			20.8985i			0.852466i	−0.904613	+	0.426233i	$0.859840\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	31.7333	−	18.3212i	1.29014	−	0.744865i
$606$		0			0
$607$		−	47.0730i		−	1.91064i	−0.295580	−	0.955318i	$-0.595513\pi$
$607$		−	47.0730i		−	1.91064i	0.295580	−	0.955318i	$-0.404487\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.6403	+	44.4103i	1.03730	+	1.79665i
$612$		0			0
$613$	0.887781	+	1.53768i	0.0358571	+	0.0621063i	0.883397	−	0.468625i	$-0.155251\pi$
$613$	0.887781	+	1.53768i	0.0358571	+	0.0621063i	−0.847540	+	0.530732i	$0.821917\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.39218	+	3.69052i	0.257339	+	0.148575i	0.623120	−	0.782126i	$-0.285865\pi$
$617$	6.39218	+	3.69052i	0.257339	+	0.148575i	−0.365781	+	0.930701i	$0.619198\pi$
$618$		0			0
$619$	−25.0914			−1.00851			−0.504254	−	0.863555i	$-0.668232\pi$
$619$	−25.0914			−1.00851			−0.504254	+	0.863555i	$0.668232\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.28612	+	7.42377i	−0.171720	+	0.297427i
$624$		0			0
$625$	8.68630	+	15.0451i	0.347452	+	0.601804i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−11.6609	−	20.1972i	−0.464949	−	0.805316i
$630$		0			0
$631$	2.73692	−	4.74048i	0.108955	−	0.188715i	−0.806392	−	0.591381i	$-0.798583\pi$
$631$	2.73692	−	4.74048i	0.108955	−	0.188715i	0.915347	+	0.402666i	$0.131916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.4961			−1.09115
$636$		0			0
$637$	−23.9451	−	13.8247i	−0.948741	−	0.547756i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20.6782	+	35.8157i	0.816740	+	1.41464i	0.908072	+	0.418815i	$0.137554\pi$
$641$	20.6782	+	35.8157i	0.816740	+	1.41464i	−0.0913319	+	0.995821i	$0.529112\pi$
$642$		0			0
$643$	−21.1035	−	36.5523i	−0.832239	−	1.44148i	−0.896259	−	0.443532i	$-0.853725\pi$
$643$	−21.1035	−	36.5523i	−0.832239	−	1.44148i	0.0640194	−	0.997949i	$-0.479608\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.7852i			0.502639i	0.967904	+	0.251319i	$0.0808645\pi$
$647$			12.7852i			0.502639i	−0.967904	+	0.251319i	$0.919136\pi$
$648$		0			0
$649$	0.897781	−	0.518334i	0.0352410	−	0.0203464i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12.4197i		−	0.486021i	−0.970024	−	0.243011i	$-0.921865\pi$
$653$		−	12.4197i		−	0.486021i	0.970024	−	0.243011i	$-0.0781350\pi$
$654$		0			0
$655$	−9.10394	+	15.7685i	−0.355720	+	0.616126i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.38658	+	2.40162i	−0.0540133	+	0.0935538i	−0.891768	−	0.452493i	$-0.850535\pi$
$659$	−1.38658	+	2.40162i	−0.0540133	+	0.0935538i	0.837755	+	0.546047i	$0.183868\pi$
$660$		0			0
$661$	7.90526	+	4.56411i	0.307479	+	0.177523i	0.645798	−	0.763508i	$-0.276525\pi$
$661$	7.90526	+	4.56411i	0.307479	+	0.177523i	−0.338319	+	0.941032i	$0.609858\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	50.5514	−	11.3057i	1.96030	−	0.438416i
$666$		0			0
$667$	−17.7356	+	10.2397i	−0.686727	+	0.396482i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.81822	+	1.62710i	0.108796	+	0.0628136i
$672$		0			0
$673$			3.33862i			0.128694i	0.997928	+	0.0643472i	$0.0204965\pi$
$673$			3.33862i			0.128694i	−0.997928	+	0.0643472i	$0.979503\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	45.3794			1.74407			0.872037	−	0.489441i	$-0.162799\pi$
$677$	45.3794			1.74407			0.872037	+	0.489441i	$0.162799\pi$
$678$		0			0
$679$	−33.2144	+	19.1763i	−1.27465	+	0.735921i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	33.7656			1.29201			0.646003	−	0.763335i	$-0.276439\pi$
$683$	33.7656			1.29201			0.646003	+	0.763335i	$0.276439\pi$
$684$		0			0
$685$	−46.0149			−1.75814
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.86934	+	1.07926i	−0.0712161	+	0.0411167i
$690$		0			0
$691$	47.6057			1.81101			0.905504	−	0.424338i	$-0.139493\pi$
$691$	47.6057			1.81101			0.905504	+	0.424338i	$0.139493\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.86823i		−	0.298459i
$696$		0			0
$697$	−30.9575	−	17.8733i	−1.17260	−	0.677001i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.81431	+	5.66630i	−0.370682	+	0.214013i	−0.673756	−	0.738954i	$-0.735320\pi$
$701$	−9.81431	+	5.66630i	−0.370682	+	0.214013i	0.303075	+	0.952967i	$0.401987\pi$
$702$		0			0
$703$	8.43664	−	26.9323i	0.318194	−	1.01577i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.1857	+	11.0769i	0.721553	+	0.416589i
$708$		0			0
$709$	8.10276	−	14.0344i	0.304306	−	0.527073i	−0.672801	−	0.739824i	$-0.734909\pi$
$709$	8.10276	−	14.0344i	0.304306	−	0.527073i	0.977106	+	0.212751i	$0.0682423\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−11.8769	+	20.5714i	−0.444793	+	0.770404i
$714$		0			0
$715$			4.17537i			0.156150i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.73659	+	3.88937i	−0.251232	+	0.145049i	−0.620328	−	0.784342i	$-0.713001\pi$
$719$	−6.73659	+	3.88937i	−0.251232	+	0.145049i	0.369096	+	0.929391i	$0.379667\pi$
$720$		0			0
$721$		−	15.5396i		−	0.578726i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.1837	+	24.5670i	0.526771	+	0.912394i
$726$		0			0
$727$	−15.7320	−	27.2486i	−0.583467	−	1.01059i	−0.995065	−	0.0992290i	$-0.968362\pi$
$727$	−15.7320	−	27.2486i	−0.583467	−	1.01059i	0.411598	−	0.911366i	$-0.364971\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−21.1251	−	12.1966i	−0.781341	−	0.451107i
$732$		0			0
$733$	−8.80147			−0.325090			−0.162545	−	0.986701i	$-0.551970\pi$
$733$	−8.80147			−0.325090			−0.162545	+	0.986701i	$0.551970\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.0396603	+	0.0686936i	−0.00146091	+	0.00253036i
$738$		0			0
$739$	−3.08996	−	5.35197i	−0.113666	−	0.196876i	0.803580	−	0.595197i	$-0.202926\pi$
$739$	−3.08996	−	5.35197i	−0.113666	−	0.196876i	−0.917246	+	0.398322i	$0.869593\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.3126	−	35.1825i	−0.745197	−	1.29072i	−0.950103	−	0.311937i	$-0.899022\pi$
$743$	−20.3126	−	35.1825i	−0.745197	−	1.29072i	0.204906	−	0.978782i	$-0.434311\pi$
$744$		0			0
$745$	−16.0757	+	27.8438i	−0.588966	+	1.02012i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.02684			−0.256755
$750$		0			0
$751$	10.4204	+	6.01624i	0.380247	+	0.219536i	0.677926	−	0.735130i	$-0.262879\pi$
$751$	10.4204	+	6.01624i	0.380247	+	0.219536i	−0.297679	+	0.954666i	$0.596212\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.8278	+	20.4864i	0.430459	+	0.745576i
$756$		0			0
$757$	4.43515	+	7.68190i	0.161198	+	0.279204i	0.935299	−	0.353859i	$-0.115131\pi$
$757$	4.43515	+	7.68190i	0.161198	+	0.279204i	−0.774100	+	0.633063i	$0.781797\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	10.3745i		−	0.376076i	−0.982162	−	0.188038i	$-0.939787\pi$
$761$		−	10.3745i		−	0.376076i	0.982162	−	0.188038i	$-0.0602127\pi$
$762$		0			0
$763$	3.18440	−	1.83851i	0.115283	−	0.0665587i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	20.4172i		−	0.737222i
$768$		0			0
$769$	−17.1749	+	29.7478i	−0.619343	+	1.07273i	0.370262	+	0.928927i	$0.379268\pi$
$769$	−17.1749	+	29.7478i	−0.619343	+	1.07273i	−0.989606	+	0.143807i	$0.954066\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−26.4831	+	45.8701i	−0.952531	+	1.64983i	−0.212612	+	0.977137i	$0.568197\pi$
$773$	−26.4831	+	45.8701i	−0.952531	+	1.64983i	−0.739920	+	0.672695i	$0.765136\pi$
$774$		0			0
$775$	28.4950	+	16.4516i	1.02357	+	0.590958i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.44147	−	42.2158i	−0.338276	−	1.51254i
$780$		0			0
$781$	−1.79067	+	1.03384i	−0.0640752	+	0.0369938i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−47.4397	−	27.3893i	−1.69319	−	0.977567i
$786$		0			0
$787$			1.75236i			0.0624648i	0.999512	+	0.0312324i	$0.00994319\pi$
$787$			1.75236i			0.0624648i	−0.999512	+	0.0312324i	$0.990057\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	4.71584			0.167676
$792$		0			0
$793$	55.5049	−	32.0458i	1.97104	−	1.13798i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−43.7424			−1.54944			−0.774718	−	0.632307i	$-0.782108\pi$
$797$	−43.7424			−1.54944			−0.774718	+	0.632307i	$0.782108\pi$
$798$		0			0
$799$	37.2834			1.31899
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.155169	−	0.0895868i	0.00547579	−	0.00316145i
$804$		0			0
$805$	−53.4080			−1.88238
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.97955i			0.0695974i	0.999394	+	0.0347987i	$0.0110790\pi$
$809$			1.97955i			0.0695974i	−0.999394	+	0.0347987i	$0.988921\pi$
$810$		0			0
$811$	17.9304	+	10.3521i	0.629620	+	0.363511i	0.780605	−	0.625025i	$-0.214911\pi$
$811$	17.9304	+	10.3521i	0.629620	+	0.363511i	−0.150985	+	0.988536i	$0.548244\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−35.7234	+	20.6249i	−1.25134	+	0.722459i
$816$		0			0
$817$	−6.44277	−	28.8077i	−0.225404	−	1.00785i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.7618	+	13.1415i	0.794392	+	0.458643i	0.841507	−	0.540247i	$-0.181669\pi$
$821$	22.7618	+	13.1415i	0.794392	+	0.458643i	−0.0471143	+	0.998890i	$0.515002\pi$
$822$		0			0
$823$	−9.16801	+	15.8795i	−0.319577	+	0.553523i	−0.980400	−	0.197019i	$-0.936874\pi$
$823$	−9.16801	+	15.8795i	−0.319577	+	0.553523i	0.660823	+	0.750542i	$0.270207\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25.7899	+	44.6694i	−0.896803	+	1.55331i	−0.0652454	+	0.997869i	$0.520783\pi$
$827$	−25.7899	+	44.6694i	−0.896803	+	1.55331i	−0.831557	+	0.555439i	$0.812550\pi$
$828$		0			0
$829$			30.7526i			1.06808i	0.845458	+	0.534042i	$0.179327\pi$
$829$			30.7526i			1.06808i	−0.845458	+	0.534042i	$0.820673\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.4092	+	10.0512i	−0.603194	+	0.348254i
$834$		0			0
$835$		−	4.17537i		−	0.144495i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.2481	+	36.8028i	0.733566	+	1.27057i	0.955350	+	0.295478i	$0.0954788\pi$
$839$	21.2481	+	36.8028i	0.733566	+	1.27057i	−0.221784	+	0.975096i	$0.571188\pi$
$840$		0			0
$841$	4.11755	+	7.13180i	0.141984	+	0.245924i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	33.4966	+	19.3393i	1.15232	+	0.665291i
$846$		0			0
$847$	−38.7922			−1.33292
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−14.5494	+	25.2002i	−0.498745	+	0.863852i
$852$		0			0
$853$	−0.296220	−	0.513068i	−0.0101424	−	0.0175671i	0.860910	−	0.508758i	$-0.169895\pi$
$853$	−0.296220	−	0.513068i	−0.0101424	−	0.0175671i	−0.871052	+	0.491191i	$0.836562\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.8976	−	24.0714i	−0.474734	−	0.822264i	0.524847	−	0.851196i	$-0.324122\pi$
$857$	−13.8976	−	24.0714i	−0.474734	−	0.822264i	−0.999581	+	0.0289327i	$0.990789\pi$
$858$		0			0
$859$	−11.8549	+	20.5332i	−0.404483	+	0.700585i	−0.994261	−	0.106980i	$-0.965882\pi$
$859$	−11.8549	+	20.5332i	−0.404483	+	0.700585i	0.589778	+	0.807565i	$0.299215\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14.2464			0.484954			0.242477	−	0.970157i	$-0.422040\pi$
$863$	14.2464			0.484954			0.242477	+	0.970157i	$0.422040\pi$
$864$		0			0
$865$	−7.23195	−	4.17537i	−0.245894	−	0.141967i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.10046	−	1.90606i	−0.0373307	−	0.0646587i
$870$		0			0
$871$	0.781109	+	1.35292i	0.0264669	+	0.0458420i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			14.5605i			0.492234i
$876$		0			0
$877$	−16.5049	+	9.52910i	−0.557331	+	0.321775i	−0.752073	−	0.659079i	$-0.770946\pi$
$877$	−16.5049	+	9.52910i	−0.557331	+	0.321775i	0.194743	+	0.980854i	$0.437613\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			16.0700i			0.541411i	0.962662	+	0.270706i	$0.0872570\pi$
$881$			16.0700i			0.541411i	−0.962662	+	0.270706i	$0.912743\pi$
$882$		0			0
$883$	13.6274	−	23.6033i	0.458598	−	0.794316i	−0.540289	−	0.841480i	$-0.681685\pi$
$883$	13.6274	−	23.6033i	0.458598	−	0.794316i	0.998887	+	0.0471641i	$0.0150184\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.46016	−	14.6534i	0.284065	−	0.492014i	−0.688317	−	0.725410i	$-0.741650\pi$
$887$	8.46016	−	14.6534i	0.284065	−	0.492014i	0.972382	+	0.233395i	$0.0749837\pi$
$888$		0			0
$889$	25.2093	+	14.5546i	0.845494	+	0.488146i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	30.5434	+	33.2080i	1.02210	+	1.11126i
$894$		0			0
$895$	−61.1419	+	35.3003i	−2.04375	+	1.17996i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.8582	−	12.0425i	−0.695660	−	0.401640i
$900$		0			0
$901$			1.56935i			0.0522826i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−71.3341			−2.37123
$906$		0			0
$907$	−2.16196	+	1.24821i	−0.0717868	+	0.0414461i	−0.535464	−	0.844558i	$-0.679863\pi$
$907$	−2.16196	+	1.24821i	−0.0717868	+	0.0414461i	0.463677	+	0.886004i	$0.346530\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31.0838			−1.02985			−0.514926	−	0.857235i	$-0.672181\pi$
$911$	−31.0838			−1.02985			−0.514926	+	0.857235i	$0.672181\pi$
$912$		0			0
$913$	0.330634			0.0109424
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.6936	−	9.63804i	0.551270	−	0.318276i
$918$		0			0
$919$	−13.1750			−0.434604			−0.217302	−	0.976104i	$-0.569726\pi$
$919$	−13.1750			−0.434604			−0.217302	+	0.976104i	$0.569726\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			40.7231i			1.34042i
$924$		0			0
$925$	34.9067	+	20.1534i	1.14773	+	0.662640i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.08127	−	0.624270i	0.0354752	−	0.0204816i	−0.482157	−	0.876085i	$-0.660147\pi$
$929$	1.08127	−	0.624270i	0.0354752	−	0.0204816i	0.517633	+	0.855603i	$0.326813\pi$
$930$		0			0
$931$	−23.2146	−	7.27207i	−0.760827	−	0.238332i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.62898	+	1.51784i	0.0859769	+	0.0496388i
$936$		0			0
$937$	2.58551	−	4.47824i	0.0844649	−	0.146298i	−0.820698	−	0.571362i	$-0.806415\pi$
$937$	2.58551	−	4.47824i	0.0844649	−	0.146298i	0.905163	+	0.425064i	$0.139749\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.7406	−	32.4597i	0.610926	−	1.05816i	−0.380159	−	0.924921i	$-0.624131\pi$
$941$	18.7406	−	32.4597i	0.610926	−	1.05816i	0.991085	−	0.133234i	$-0.0425361\pi$
$942$		0			0
$943$			44.6014i			1.45242i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.42574	+	2.55520i	−0.143817	+	0.0830328i	−0.570182	−	0.821518i	$-0.693127\pi$
$947$	−4.42574	+	2.55520i	−0.143817	+	0.0830328i	0.426365	+	0.904551i	$0.359794\pi$
$948$		0			0
$949$		−	3.52882i		−	0.114550i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.30089	+	10.9135i	0.204106	+	0.353522i	0.949848	−	0.312713i	$-0.101238\pi$
$953$	6.30089	+	10.9135i	0.204106	+	0.353522i	−0.745742	+	0.666235i	$0.767905\pi$
$954$		0			0
$955$	17.0683	+	29.5632i	0.552319	+	0.956644i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	42.1880	+	24.3572i	1.36232	+	0.786536i
$960$		0			0
$961$	3.06408			0.0988412
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−17.2765	+	29.9238i	−0.556151	+	0.963282i
$966$		0			0
$967$	−3.11699	−	5.39879i	−0.100236	−	0.173613i	0.811546	−	0.584289i	$-0.198626\pi$
$967$	−3.11699	−	5.39879i	−0.100236	−	0.173613i	−0.911782	+	0.410675i	$0.865293\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.10993	−	7.11860i	−0.131894	−	0.228447i	0.792513	−	0.609855i	$-0.208772\pi$
$971$	−4.10993	−	7.11860i	−0.131894	−	0.228447i	−0.924407	+	0.381408i	$0.875439\pi$
$972$		0			0
$973$	−4.16492	+	7.21385i	−0.133521	+	0.231265i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	46.0799			1.47423			0.737113	−	0.675769i	$-0.236188\pi$
$977$	46.0799			1.47423			0.737113	+	0.675769i	$0.236188\pi$
$978$		0			0
$979$	−0.526487	−	0.303968i	−0.0168266	−	0.00971485i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.60597	−	6.24572i	−0.115013	−	0.199208i	0.802772	−	0.596286i	$-0.203358\pi$
$983$	−3.60597	−	6.24572i	−0.115013	−	0.199208i	−0.917785	+	0.397078i	$0.870024\pi$
$984$		0			0
$985$	−24.1063	−	41.7534i	−0.768091	−	1.33037i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			30.4356i			0.967795i
$990$		0			0
$991$	17.3753	−	10.0316i	0.551945	−	0.318665i	−0.197961	−	0.980210i	$-0.563432\pi$
$991$	17.3753	−	10.0316i	0.551945	−	0.318665i	0.749906	+	0.661544i	$0.230099\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	75.7338i		−	2.40092i
$996$		0			0
$997$	−0.0129544	+	0.0224377i	−0.000410270	+	0.000710608i	−0.866230	−	0.499645i	$-0.833464\pi$
$997$	−0.0129544	+	0.0224377i	−0.000410270	+	0.000710608i	0.865820	+	0.500355i	$0.166797\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.c.449.7		16
3.2	odd	2	inner	2736.2.dc.c.449.2		16
4.3	odd	2		171.2.m.a.107.2	yes	16
12.11	even	2		171.2.m.a.107.7	yes	16
19.8	odd	6	inner	2736.2.dc.c.1889.2		16
57.8	even	6	inner	2736.2.dc.c.1889.7		16
76.27	even	6		171.2.m.a.8.7	yes	16
228.179	odd	6		171.2.m.a.8.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.m.a.8.2	✓	16	228.179	odd	6
171.2.m.a.8.7	yes	16	76.27	even	6
171.2.m.a.107.2	yes	16	4.3	odd	2
171.2.m.a.107.7	yes	16	12.11	even	2
2736.2.dc.c.449.2		16	3.2	odd	2	inner
2736.2.dc.c.449.7		16	1.1	even	1	trivial
2736.2.dc.c.1889.2		16	19.8	odd	6	inner
2736.2.dc.c.1889.7		16	57.8	even	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.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.90154 - 1.67520i) q^{5} -3.54697 q^{7} +O(q^{10})\)	\(q+(2.90154 - 1.67520i) q^{5} -3.54697 q^{7} -0.251548i q^{11} +(-4.29049 - 2.47712i) q^{13} +(-3.11939 + 1.80098i) q^{17} +(-4.15959 - 1.30301i) q^{19} +(3.89208 + 2.24709i) q^{23} +(3.11262 - 5.39122i) q^{25} +(-2.27843 + 3.94635i) q^{29} +5.28544i q^{31} +(-10.2917 + 5.94190i) q^{35} +6.47474i q^{37} +(4.96212 + 8.59464i) q^{41} +(3.38610 + 5.86490i) q^{43} +(-8.96412 - 5.17544i) q^{47} +5.58098 q^{49} +(0.217847 - 0.377322i) q^{53} +(-0.421394 - 0.729876i) q^{55} +(2.06058 + 3.56903i) q^{59} +(-6.46836 + 11.2035i) q^{61} -16.5987 q^{65} +(-0.273084 - 0.157665i) q^{67} +(-4.10993 - 7.11860i) q^{71} +(0.356142 + 0.616857i) q^{73} +0.892232i q^{77} +(7.57733 - 4.37477i) q^{79} +1.31440i q^{83} +(-6.03401 + 10.4512i) q^{85} +(1.20839 - 2.09299i) q^{89} +(15.2182 + 8.78625i) q^{91} +(-14.2520 + 3.18743i) q^{95} +(9.36417 - 5.40640i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7}+O(q^{10}) \) 16 * q - 8 * q^7	\( 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) 16 * q - 8 * q^7 - 24 * q^13 + 12 * q^19 + 20 * q^25 + 4 * q^55 - 44 * q^61 + 24 * q^67 - 20 * q^73 + 48 * q^79 - 56 * q^85 + 24 * q^91 + 48 * q^97

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