LMFDB - Embedded newform 2736.2.bm.t.559.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (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:	$16$
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} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561 $ x^16 + 28x^14 + 542x^12 + 5488x^10 + 40451x^8 + 151312x^6 + 395134x^4 + 52164*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.1
Root			$-1.86197 + 3.22503i$ of defining polynomial
Character	$\chi$	$=$	2736.559
Dual form			2736.2.bm.t.1855.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.38255 + 2.39464i) q^{5} -3.26278i q^{7} +O(q^{10})$	$q+(-1.38255 + 2.39464i) q^{5} -3.26278i q^{7} +1.94197i q^{11} +(3.96863 - 2.29129i) q^{13} +(2.27533 - 3.94099i) q^{17} +(-3.43388 - 2.68486i) q^{19} +(-7.81320 + 4.51095i) q^{23} +(-1.32288 - 2.29129i) q^{25} +(-1.78556 + 1.03090i) q^{29} -1.21645 q^{31} +(7.81320 + 4.51095i) q^{35} +1.73205i q^{37} +(-3.25486 - 1.87919i) q^{41} +(-1.82467 - 1.05347i) q^{43} +(-9.49499 + 5.48194i) q^{47} -3.64575 q^{49} +(7.40250 - 4.27384i) q^{53} +(-4.65032 - 2.68486i) q^{55} +(1.68179 - 2.91295i) q^{59} +(0.322876 + 0.559237i) q^{61} +12.6713i q^{65} +(-4.04210 - 7.00112i) q^{67} +(6.13140 - 10.6199i) q^{71} +(-3.79150 + 6.56708i) q^{73} +6.33622 q^{77} +(5.04308 - 8.73488i) q^{79} -10.9639i q^{83} +(6.29150 + 10.8972i) q^{85} +(-5.93321 + 3.42554i) q^{89} +(-7.47597 - 12.9488i) q^{91} +(11.1768 - 4.51095i) q^{95} +(-16.4059 - 9.47194i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100}) $ 16 * q - 16 * q^49 - 16 * q^61 + 24 * q^73 + 16 * q^85 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.38255	+	2.39464i	−0.618294	+	1.07092i	0.371503	+	0.928432i	$0.378843\pi$
$5$	−1.38255	+	2.39464i	−0.618294	+	1.07092i	−0.989797	+	0.142485i	$0.954491\pi$
$6$		0			0
$7$		−	3.26278i		−	1.23322i	−0.787270	−	0.616608i	$-0.788506\pi$
$7$		−	3.26278i		−	1.23322i	0.787270	−	0.616608i	$-0.211494\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			1.94197i			0.585526i	0.956185	+	0.292763i	$0.0945746\pi$
$11$			1.94197i			0.585526i	−0.956185	+	0.292763i	$0.905425\pi$
$12$		0			0
$13$	3.96863	−	2.29129i	1.10070	−	0.635489i	0.164295	−	0.986411i	$-0.447465\pi$
$13$	3.96863	−	2.29129i	1.10070	−	0.635489i	0.936405	+	0.350922i	$0.114132\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.27533	−	3.94099i	0.551848	−	0.955830i	−0.446293	−	0.894887i	$-0.647256\pi$
$17$	2.27533	−	3.94099i	0.551848	−	0.955830i	0.998141	−	0.0609426i	$-0.0194107\pi$
$18$		0			0
$19$	−3.43388	−	2.68486i	−0.787785	−	0.615950i
$19$	−3.43388	−	2.68486i	−0.787785	−	0.615950i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.81320	+	4.51095i	−1.62916	+	0.940599i	−0.644822	+	0.764333i	$0.723068\pi$
$23$	−7.81320	+	4.51095i	−1.62916	+	0.940599i	−0.984343	+	0.176266i	$0.943598\pi$
$24$		0			0
$25$	−1.32288	−	2.29129i	−0.264575	−	0.458258i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.78556	+	1.03090i	−0.331571	+	0.191433i	−0.656538	−	0.754293i	$-0.727980\pi$
$29$	−1.78556	+	1.03090i	−0.331571	+	0.191433i	0.324967	+	0.945725i	$0.394647\pi$
$30$		0			0
$31$	−1.21645			−0.218480			−0.109240	−	0.994015i	$-0.534842\pi$
$31$	−1.21645			−0.218480			−0.109240	+	0.994015i	$0.534842\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.81320	+	4.51095i	1.32067	+	0.762490i
$36$		0			0
$37$			1.73205i			0.284747i	0.989813	+	0.142374i	$0.0454735\pi$
$37$			1.73205i			0.284747i	−0.989813	+	0.142374i	$0.954527\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.25486	−	1.87919i	−0.508324	−	0.293481i	0.223821	−	0.974630i	$-0.428147\pi$
$41$	−3.25486	−	1.87919i	−0.508324	−	0.293481i	−0.732144	+	0.681149i	$0.761480\pi$
$42$		0			0
$43$	−1.82467	−	1.05347i	−0.278259	−	0.160653i	0.354376	−	0.935103i	$-0.384693\pi$
$43$	−1.82467	−	1.05347i	−0.278259	−	0.160653i	−0.632635	+	0.774450i	$0.718027\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.49499	+	5.48194i	−1.38499	+	0.799623i	−0.992745	−	0.120239i	$-0.961634\pi$
$47$	−9.49499	+	5.48194i	−1.38499	+	0.799623i	−0.392242	+	0.919862i	$0.628300\pi$
$48$		0			0
$49$	−3.64575			−0.520822
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.40250	−	4.27384i	1.01681	−	0.587057i	0.103633	−	0.994616i	$-0.466953\pi$
$53$	7.40250	−	4.27384i	1.01681	−	0.587057i	0.913179	+	0.407559i	$0.133620\pi$
$54$		0			0
$55$	−4.65032	−	2.68486i	−0.627049	−	0.362027i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.68179	−	2.91295i	0.218951	−	0.379234i	−0.735537	−	0.677485i	$-0.763070\pi$
$59$	1.68179	−	2.91295i	0.218951	−	0.379234i	0.954488	+	0.298251i	$0.0964032\pi$
$60$		0			0
$61$	0.322876	+	0.559237i	0.0413400	+	0.0716030i	0.885955	−	0.463771i	$-0.153504\pi$
$61$	0.322876	+	0.559237i	0.0413400	+	0.0716030i	−0.844615	+	0.535374i	$0.820171\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			12.6713i			1.57168i
$66$		0			0
$67$	−4.04210	−	7.00112i	−0.493821	−	0.855323i	0.506154	−	0.862443i	$-0.331067\pi$
$67$	−4.04210	−	7.00112i	−0.493821	−	0.855323i	−0.999975	+	0.00712031i	$0.997734\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.13140	−	10.6199i	0.727664	−	1.26035i	−0.230204	−	0.973142i	$-0.573939\pi$
$71$	6.13140	−	10.6199i	0.727664	−	1.26035i	0.957868	−	0.287208i	$-0.0927272\pi$
$72$		0			0
$73$	−3.79150	+	6.56708i	−0.443762	+	0.768618i	−0.997965	−	0.0637634i	$-0.979690\pi$
$73$	−3.79150	+	6.56708i	−0.443762	+	0.768618i	0.554203	+	0.832381i	$0.313023\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.33622			0.722080
$78$		0			0
$79$	5.04308	−	8.73488i	0.567391	−	0.982750i	−0.429432	−	0.903099i	$-0.641286\pi$
$79$	5.04308	−	8.73488i	0.567391	−	0.982750i	0.996823	−	0.0796509i	$-0.0253806\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	10.9639i		−	1.20344i	−0.798707	−	0.601721i	$-0.794482\pi$
$83$		−	10.9639i		−	1.20344i	0.798707	−	0.601721i	$-0.205518\pi$
$84$		0			0
$85$	6.29150	+	10.8972i	0.682409	+	1.18197i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.93321	+	3.42554i	−0.628919	+	0.363106i	−0.780333	−	0.625364i	$-0.784951\pi$
$89$	−5.93321	+	3.42554i	−0.628919	+	0.363106i	0.151415	+	0.988470i	$0.451617\pi$
$90$		0			0
$91$	−7.47597	−	12.9488i	−0.783695	−	1.35740i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.1768	−	4.51095i	1.14671	−	0.462814i
$96$		0			0
$97$	−16.4059	−	9.47194i	−1.66576	−	0.961730i	−0.969882	−	0.243574i	$-0.921680\pi$
$97$	−16.4059	−	9.47194i	−1.66576	−	0.961730i	−0.695883	−	0.718156i	$-0.744987\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.06089	−	7.03367i	−0.404074	−	0.699877i	0.590139	−	0.807301i	$-0.299073\pi$
$101$	−4.06089	−	7.03367i	−0.404074	−	0.699877i	−0.994213	+	0.107425i	$0.965740\pi$
$102$		0			0
$103$	−19.3868			−1.91024			−0.955120	−	0.296221i	$-0.904274\pi$
$103$	−19.3868			−1.91024			−0.955120	+	0.296221i	$0.904274\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.72718			0.650341			0.325170	−	0.945655i	$-0.394578\pi$
$107$	6.72718			0.650341			0.325170	+	0.945655i	$0.394578\pi$
$108$		0			0
$109$	−2.46863	−	1.42526i	−0.236452	−	0.136515i	0.377093	−	0.926175i	$-0.376924\pi$
$109$	−2.46863	−	1.42526i	−0.236452	−	0.136515i	−0.613545	+	0.789660i	$0.710257\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	14.7330i		−	1.38597i	−0.720953	−	0.692984i	$-0.756295\pi$
$113$		−	14.7330i		−	1.38597i	0.720953	−	0.692984i	$-0.243705\pi$
$114$		0			0
$115$		−	24.9464i		−	2.32627i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.8586	−	7.42391i	−1.17874	−	0.680548i
$120$		0			0
$121$	7.22876			0.657160
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.50972			−0.582247
$126$		0			0
$127$	−1.21645	−	2.10695i	−0.107942	−	0.186961i	0.806994	−	0.590559i	$-0.201093\pi$
$127$	−1.21645	−	2.10695i	−0.107942	−	0.186961i	−0.914936	+	0.403598i	$0.867759\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.76782	+	1.59800i	0.241825	+	0.139618i	0.616015	−	0.787734i	$-0.288746\pi$
$131$	2.76782	+	1.59800i	0.241825	+	0.139618i	−0.374190	+	0.927352i	$0.622079\pi$
$132$		0			0
$133$	−8.76013	+	11.2040i	−0.759600	+	0.971509i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.53019	−	9.57857i	−0.472476	−	0.818353i	0.527028	−	0.849848i	$-0.323306\pi$
$137$	−5.53019	−	9.57857i	−0.472476	−	0.818353i	−0.999504	+	0.0314955i	$0.989973\pi$
$138$		0			0
$139$	9.47794	−	5.47209i	0.803909	−	0.464137i	−0.0409275	−	0.999162i	$-0.513031\pi$
$139$	9.47794	−	5.47209i	0.803909	−	0.464137i	0.844836	+	0.535025i	$0.179698\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.44961	+	7.70695i	0.372095	+	0.644488i
$144$		0			0
$145$		−	5.70105i		−	0.473446i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.42297	+	11.1249i	−0.526190	+	0.911388i	0.473344	+	0.880878i	$0.343047\pi$
$149$	−6.42297	+	11.1249i	−0.526190	+	0.911388i	−0.999534	+	0.0305109i	$0.990287\pi$
$150$		0			0
$151$	−13.7355			−1.11778			−0.558890	−	0.829242i	$-0.688772\pi$
$151$	−13.7355			−1.11778			−0.558890	+	0.829242i	$0.688772\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.68179	−	2.91295i	0.135085	−	0.233974i
$156$		0			0
$157$	5.79150	−	10.0312i	0.462212	−	0.800575i	−0.536859	−	0.843672i	$-0.680389\pi$
$157$	5.79150	−	10.0312i	0.462212	−	0.800575i	0.999071	+	0.0430971i	$0.0137225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	14.7183	+	25.4928i	1.15996	+	2.00911i
$162$		0			0
$163$		−	7.47668i		−	0.585618i	−0.956171	−	0.292809i	$-0.905410\pi$
$163$		−	7.47668i		−	0.585618i	0.956171	−	0.292809i	$-0.0945901\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.81320	−	13.5329i	−0.604603	−	1.04720i	−0.992114	−	0.125339i	$-0.959998\pi$
$167$	−7.81320	−	13.5329i	−0.604603	−	1.04720i	0.387511	−	0.921865i	$-0.373335\pi$
$168$		0			0
$169$	4.00000	−	6.92820i	0.307692	−	0.532939i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	20.1617	+	11.6404i	1.53287	+	0.885000i	0.999228	+	0.0392871i	$0.0125087\pi$
$173$	20.1617	+	11.6404i	1.53287	+	0.885000i	0.533638	+	0.845713i	$0.320825\pi$
$174$		0			0
$175$	−7.47597	+	4.31626i	−0.565131	+	0.326278i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.36359			0.251406			0.125703	−	0.992068i	$-0.459881\pi$
$179$	3.36359			0.251406			0.125703	+	0.992068i	$0.459881\pi$
$180$		0			0
$181$	7.40588	−	4.27579i	0.550475	−	0.317817i	−0.198839	−	0.980032i	$-0.563717\pi$
$181$	7.40588	−	4.27579i	0.550475	−	0.317817i	0.749313	+	0.662215i	$0.230384\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.14764	−	2.39464i	−0.304941	−	0.176058i
$186$		0			0
$187$	7.65327	+	4.41862i	0.559663	+	0.323121i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			16.7898i			1.21487i	0.794371	+	0.607433i	$0.207801\pi$
$191$			16.7898i			1.21487i	−0.794371	+	0.607433i	$0.792199\pi$
$192$		0			0
$193$	12.9686	+	7.48744i	0.933502	+	0.538958i	0.887917	−	0.460003i	$-0.152152\pi$
$193$	12.9686	+	7.48744i	0.933502	+	0.538958i	0.0455847	+	0.998960i	$0.485485\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−19.3557			−1.37903			−0.689517	−	0.724269i	$-0.742177\pi$
$197$	−19.3557			−1.37903			−0.689517	+	0.724269i	$0.742177\pi$
$198$		0			0
$199$	−1.82467	+	1.05347i	−0.129347	+	0.0746787i	−0.563277	−	0.826268i	$-0.690460\pi$
$199$	−1.82467	+	1.05347i	−0.129347	+	0.0746787i	0.433930	+	0.900947i	$0.357126\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.36359	+	5.82591i	0.236078	+	0.408899i
$204$		0			0
$205$	9.00000	−	5.19615i	0.628587	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.21392	−	6.66848i	0.360655	−	0.461268i
$210$		0			0
$211$	0.608223	−	1.05347i	0.0418718	−	0.0725241i	−0.844330	−	0.535824i	$-0.820001\pi$
$211$	0.608223	−	1.05347i	0.0418718	−	0.0725241i	0.886202	+	0.463299i	$0.153335\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.04538	−	2.91295i	0.344092	−	0.198662i
$216$		0			0
$217$			3.96900i			0.269433i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	20.8537i		−	1.40277i
$222$		0			0
$223$	−11.9108	+	20.6302i	−0.797608	+	1.38150i	0.123561	+	0.992337i	$0.460568\pi$
$223$	−11.9108	+	20.6302i	−0.797608	+	1.38150i	−0.921170	+	0.389161i	$0.872765\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−22.3536			−1.48366			−0.741829	−	0.670589i	$-0.766042\pi$
$227$	−22.3536			−1.48366			−0.741829	+	0.670589i	$0.766042\pi$
$228$		0			0
$229$	6.64575			0.439164			0.219582	−	0.975594i	$-0.429531\pi$
$229$	6.64575			0.439164			0.219582	+	0.975594i	$0.429531\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.01996	−	10.4269i	0.394380	−	0.683087i	−0.598641	−	0.801017i	$-0.704293\pi$
$233$	6.01996	−	10.4269i	0.394380	−	0.683087i	0.993022	+	0.117930i	$0.0376259\pi$
$234$		0			0
$235$		−	30.3162i		−	1.97761i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.70984i			0.628078i	0.949410	+	0.314039i	$0.101682\pi$
$239$			9.70984i			0.628078i	−0.949410	+	0.314039i	$0.898318\pi$
$240$		0			0
$241$	17.3745	−	10.0312i	1.11919	−	0.646165i	0.177996	−	0.984031i	$-0.443038\pi$
$241$	17.3745	−	10.0312i	1.11919	−	0.646165i	0.941194	+	0.337866i	$0.109705\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.04042	−	8.73027i	0.322021	−	0.557757i
$246$		0			0
$247$	−19.7796	−	2.78723i	−1.25854	−	0.177347i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.6264	−	9.02190i	0.986329	−	0.569458i	0.0821544	−	0.996620i	$-0.473820\pi$
$251$	15.6264	−	9.02190i	0.986329	−	0.569458i	0.904175	+	0.427162i	$0.140487\pi$
$252$		0			0
$253$	−8.76013	−	15.1730i	−0.550745	−	0.953918i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.9736	+	6.33563i	−0.684516	+	0.395206i	−0.801554	−	0.597922i	$-0.795993\pi$
$257$	−10.9736	+	6.33563i	−0.684516	+	0.395206i	0.117038	+	0.993127i	$0.462660\pi$
$258$		0			0
$259$	5.65131			0.351155
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.49499	−	5.48194i	−0.585486	−	0.338031i	0.177824	−	0.984062i	$-0.443094\pi$
$263$	−9.49499	−	5.48194i	−0.585486	−	0.338031i	−0.763311	+	0.646032i	$0.776427\pi$
$264$		0			0
$265$			23.6351i			1.45189i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.9736	−	6.33563i	−0.669074	−	0.386290i	0.126652	−	0.991947i	$-0.459577\pi$
$269$	−10.9736	−	6.33563i	−0.669074	−	0.386290i	−0.795726	+	0.605657i	$0.792910\pi$
$270$		0			0
$271$	16.9539	+	9.78835i	1.02988	+	0.594600i	0.916949	−	0.399003i	$-0.130644\pi$
$271$	16.9539	+	9.78835i	1.02988	+	0.594600i	0.112928	+	0.993603i	$0.463977\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.44961	−	2.56898i	0.268322	−	0.154916i
$276$		0			0
$277$	−24.2288			−1.45576			−0.727882	−	0.685702i	$-0.759495\pi$
$277$	−24.2288			−1.45576			−0.727882	+	0.685702i	$0.759495\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.0949	−	15.0659i	1.55669	−	0.898756i	0.559121	−	0.829086i	$-0.311139\pi$
$281$	26.0949	−	15.0659i	1.55669	−	0.898756i	0.997570	−	0.0696705i	$-0.0221948\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.13140	+	10.6199i	−0.361925	+	0.626873i
$288$		0			0
$289$	−1.85425	−	3.21165i	−0.109073	−	0.188921i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			26.6739i			1.55831i	0.626833	+	0.779154i	$0.284351\pi$
$293$			26.6739i			1.55831i	−0.626833	+	0.779154i	$0.715649\pi$
$294$		0			0
$295$	4.65032	+	8.05459i	0.270752	+	0.468957i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.6718	+	35.8046i	−1.19548	+	2.07063i
$300$		0			0
$301$	−3.43725	+	5.95350i	−0.198120	+	0.343154i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.78556			−0.102241
$306$		0			0
$307$	17.1694	−	29.7382i	0.979908	−	1.69725i	0.317226	−	0.948350i	$-0.397249\pi$
$307$	17.1694	−	29.7382i	0.979908	−	1.69725i	0.662682	−	0.748901i	$-0.269418\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.1818i			1.31452i	0.753665	+	0.657259i	$0.228284\pi$
$311$			23.1818i			1.31452i	−0.753665	+	0.657259i	$0.771716\pi$
$312$		0			0
$313$	1.82288	+	3.15731i	0.103035	+	0.178462i	0.912934	−	0.408108i	$-0.133811\pi$
$313$	1.82288	+	3.15731i	0.103035	+	0.178462i	−0.809899	+	0.586570i	$0.800478\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.61694	+	3.24294i	−0.315479	+	0.182142i	−0.649376	−	0.760468i	$-0.724970\pi$
$317$	−5.61694	+	3.24294i	−0.315479	+	0.182142i	0.333897	+	0.942610i	$0.391636\pi$
$318$		0			0
$319$	−2.00197	−	3.46751i	−0.112089	−	0.194143i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−18.3942	+	7.42391i	−1.02348	+	0.413077i
$324$		0			0
$325$	−10.5000	−	6.06218i	−0.582435	−	0.336269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	17.8864	+	30.9801i	0.986107	+	1.70799i
$330$		0			0
$331$	−21.3888			−1.17563			−0.587817	−	0.808994i	$-0.700012\pi$
$331$	−21.3888			−1.17563			−0.587817	+	0.808994i	$0.700012\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	22.3536			1.22131
$336$		0			0
$337$	−5.90588	−	3.40976i	−0.321714	−	0.185742i	0.330442	−	0.943826i	$-0.392802\pi$
$337$	−5.90588	−	3.40976i	−0.321714	−	0.185742i	−0.652156	+	0.758085i	$0.726135\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.36230i		−	0.127926i
$342$		0			0
$343$		−	10.9442i		−	0.590930i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.5810	+	6.10895i	0.568019	+	0.327946i	0.756358	−	0.654158i	$-0.226977\pi$
$347$	10.5810	+	6.10895i	0.568019	+	0.327946i	−0.188339	+	0.982104i	$0.560310\pi$
$348$		0			0
$349$	−29.9373			−1.60250			−0.801252	−	0.598327i	$-0.795832\pi$
$349$	−29.9373			−1.60250			−0.801252	+	0.598327i	$0.795832\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.8654			−1.37668			−0.688338	−	0.725390i	$-0.741659\pi$
$353$	−25.8654			−1.37668			−0.688338	+	0.725390i	$0.741659\pi$
$354$		0			0
$355$	16.9539	+	29.3650i	0.899820	+	1.55853i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.6264	−	9.02190i	−0.824730	−	0.476158i	0.0273150	−	0.999627i	$-0.491304\pi$
$359$	−15.6264	−	9.02190i	−0.824730	−	0.476158i	−0.852045	+	0.523469i	$0.824638\pi$
$360$		0			0
$361$	4.58301	+	18.4390i	0.241211	+	0.970473i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.4839	−	18.1586i	−0.548751	−	0.950464i
$366$		0			0
$367$	1.82467	−	1.05347i	0.0952469	−	0.0549908i	−0.451620	−	0.892210i	$-0.649154\pi$
$367$	1.82467	−	1.05347i	0.0952469	−	0.0549908i	0.546867	+	0.837220i	$0.315820\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.9446	−	24.1528i	−0.723968	−	1.25395i
$372$		0			0
$373$			2.85052i			0.147595i	0.997273	+	0.0737973i	$0.0235118\pi$
$373$			2.85052i			0.147595i	−0.997273	+	0.0737973i	$0.976488\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.72416	+	8.18248i	−0.243307	+	0.421419i
$378$		0			0
$379$	12.5191			0.643061			0.321530	−	0.946899i	$-0.395803\pi$
$379$	12.5191			0.643061			0.321530	+	0.946899i	$0.395803\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.81320	−	13.5329i	0.399236	−	0.691497i	−0.594396	−	0.804172i	$-0.702609\pi$
$383$	7.81320	−	13.5329i	0.399236	−	0.691497i	0.993632	+	0.112676i	$0.0359421\pi$
$384$		0			0
$385$	−8.76013	+	15.1730i	−0.446458	+	0.773287i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.91274	+	11.9732i	0.350490	+	0.607066i	0.986335	−	0.164750i	$-0.0526818\pi$
$389$	6.91274	+	11.9732i	0.350490	+	0.607066i	−0.635846	+	0.771816i	$0.719348\pi$
$390$		0			0
$391$			41.0556i			2.07627i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.9446	+	24.1528i	0.701629	+	1.21526i
$396$		0			0
$397$	5.38562	−	9.32817i	0.270297	−	0.468167i	−0.698641	−	0.715472i	$-0.746212\pi$
$397$	5.38562	−	9.32817i	0.270297	−	0.468167i	0.968938	+	0.247305i	$0.0795449\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.50433	−	5.48733i	−0.474624	−	0.274024i	0.243549	−	0.969888i	$-0.421688\pi$
$401$	−9.50433	−	5.48733i	−0.474624	−	0.274024i	−0.718173	+	0.695864i	$0.755022\pi$
$402$		0			0
$403$	−4.82762	+	2.78723i	−0.240481	+	0.138842i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.36359			−0.166727
$408$		0			0
$409$	−11.4686	+	6.62141i	−0.567087	+	0.327408i	−0.755985	−	0.654589i	$-0.772842\pi$
$409$	−11.4686	+	6.62141i	−0.567087	+	0.327408i	0.188898	+	0.981997i	$0.439508\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.50433	−	5.48733i	−0.467678	−	0.270014i
$414$		0			0
$415$	26.2546	+	15.1581i	1.28879	+	0.744081i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	16.7898i		−	0.820234i	−0.912033	−	0.410117i	$-0.865488\pi$
$419$		−	16.7898i		−	0.820234i	0.912033	−	0.410117i	$-0.134512\pi$
$420$		0			0
$421$	25.4059	+	14.6681i	1.23821	+	0.714879i	0.968727	−	0.248127i	$-0.0798151\pi$
$421$	25.4059	+	14.6681i	1.23821	+	0.714879i	0.269479	+	0.963006i	$0.413148\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0399			−0.584022
$426$		0			0
$427$	1.82467	−	1.05347i	0.0883019	−	0.0509811i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.49499	−	16.4458i	−0.457358	−	0.792167i	0.541463	−	0.840725i	$-0.317871\pi$
$431$	−9.49499	−	16.4458i	−0.457358	−	0.792167i	−0.998820	+	0.0485580i	$0.984537\pi$
$432$		0			0
$433$	12.9686	−	7.48744i	0.623232	−	0.359823i	−0.154894	−	0.987931i	$-0.549504\pi$
$433$	12.9686	−	7.48744i	0.623232	−	0.359823i	0.778126	+	0.628108i	$0.216170\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	38.9408	+	5.48733i	1.86279	+	0.262495i
$438$		0			0
$439$	−16.5612	+	28.6848i	−0.790421	+	1.36905i	0.135286	+	0.990807i	$0.456805\pi$
$439$	−16.5612	+	28.6848i	−0.790421	+	1.36905i	−0.925707	+	0.378242i	$0.876529\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.72718	+	3.88394i	−0.319618	+	0.184531i	−0.651222	−	0.758887i	$-0.725743\pi$
$443$	−6.72718	+	3.88394i	−0.319618	+	0.184531i	0.331604	+	0.943419i	$0.392410\pi$
$444$		0			0
$445$		−	18.9439i		−	0.898026i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	23.2807i		−	1.09869i	−0.835597	−	0.549343i	$-0.814878\pi$
$449$		−	23.2807i		−	1.09869i	0.835597	−	0.549343i	$-0.185122\pi$
$450$		0			0
$451$	3.64934	−	6.32084i	0.171841	−	0.297637i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	41.3436			1.93822
$456$		0			0
$457$	2.06275			0.0964912			0.0482456	−	0.998836i	$-0.484637\pi$
$457$	2.06275			0.0964912			0.0482456	+	0.998836i	$0.484637\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.7183	+	25.4928i	−0.685498	+	1.18732i	0.287783	+	0.957696i	$0.407082\pi$
$461$	−14.7183	+	25.4928i	−0.685498	+	1.18732i	−0.973280	+	0.229621i	$0.926251\pi$
$462$		0			0
$463$			37.7928i			1.75638i	0.478310	+	0.878191i	$0.341249\pi$
$463$			37.7928i			1.75638i	−0.478310	+	0.878191i	$0.658751\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10.9639i		−	0.507348i	−0.967290	−	0.253674i	$-0.918361\pi$
$467$		−	10.9639i		−	0.507348i	0.967290	−	0.253674i	$-0.0816390\pi$
$468$		0			0
$469$	−22.8431	+	13.1885i	−1.05480	+	0.608988i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.04581	−	3.54345i	0.0940665	−	0.162928i
$474$		0			0
$475$	−1.60921	+	11.4197i	−0.0738355	+	0.523974i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.8586	−	7.42391i	0.587524	−	0.339207i	−0.176594	−	0.984284i	$-0.556508\pi$
$479$	12.8586	−	7.42391i	0.587524	−	0.339207i	0.764118	+	0.645077i	$0.223175\pi$
$480$		0			0
$481$	3.96863	+	6.87386i	0.180954	+	0.313421i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	45.3638	−	26.1908i	2.05987	−	1.18926i
$486$		0			0
$487$	25.0381			1.13459			0.567293	−	0.823516i	$-0.307991\pi$
$487$	25.0381			1.13459			0.567293	+	0.823516i	$0.307991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−21.7578	−	12.5619i	−0.981916	−	0.566909i	−0.0790678	−	0.996869i	$-0.525194\pi$
$491$	−21.7578	−	12.5619i	−0.981916	−	0.566909i	−0.902848	+	0.429960i	$0.858528\pi$
$492$		0			0
$493$			9.38251i			0.422567i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−34.6504	−	20.0054i	−1.55428	−	0.897367i
$498$		0			0
$499$	21.4269	+	12.3708i	0.959202	+	0.553795i	0.895927	−	0.444201i	$-0.146512\pi$
$499$	21.4269	+	12.3708i	0.959202	+	0.553795i	0.0632743	+	0.997996i	$0.479846\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.8892	−	16.1018i	1.24352	−	0.717946i	0.273710	−	0.961812i	$-0.411749\pi$
$503$	27.8892	−	16.1018i	1.24352	−	0.717946i	0.969809	+	0.243867i	$0.0784160\pi$
$504$		0			0
$505$	22.4575			0.999346
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.6100	−	17.0953i	1.31244	−	0.757738i	0.329941	−	0.944002i	$-0.392971\pi$
$509$	29.6100	−	17.0953i	1.31244	−	0.757738i	0.982500	+	0.186264i	$0.0596378\pi$
$510$		0			0
$511$	21.4269	+	12.3708i	0.947872	+	0.547254i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26.8032	−	46.4245i	1.18109	−	2.04571i
$516$		0			0
$517$	−10.6458	−	18.4390i	−0.468200	−	0.810946i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			16.0645i			0.703797i	0.936038	+	0.351898i	$0.114464\pi$
$521$			16.0645i			0.703797i	−0.936038	+	0.351898i	$0.885536\pi$
$522$		0			0
$523$	1.60921	+	2.78723i	0.0703657	+	0.121877i	0.899062	−	0.437822i	$-0.144250\pi$
$523$	1.60921	+	2.78723i	0.0703657	+	0.121877i	−0.828696	+	0.559699i	$0.810917\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.76782	+	4.79400i	−0.120568	+	0.208830i
$528$		0			0
$529$	29.1974	−	50.5714i	1.26945	−	2.19875i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−17.2231			−0.746015
$534$		0			0
$535$	−9.30064	+	16.1092i	−0.402102	+	0.696461i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	7.07994i		−	0.304954i
$540$		0			0
$541$	5.20850	+	9.02138i	0.223931	+	0.387860i	0.955998	−	0.293373i	$-0.0947777\pi$
$541$	5.20850	+	9.02138i	0.223931	+	0.387860i	−0.732067	+	0.681232i	$0.761444\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.82599	−	3.94099i	0.292393	−	0.168813i
$546$		0			0
$547$	10.6944	+	18.5232i	0.457259	+	0.791996i	0.998815	−	0.0486689i	$-0.0154979\pi$
$547$	10.6944	+	18.5232i	0.457259	+	0.791996i	−0.541556	+	0.840665i	$0.682165\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.89922	+	1.25403i	0.379120	+	0.0534234i
$552$		0			0
$553$	−28.5000	−	16.4545i	−1.21194	−	0.699716i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	6.33622	+	10.9747i	0.268474	+	0.465011i	0.968468	−	0.249138i	$-0.0801472\pi$
$557$	6.33622	+	10.9747i	0.268474	+	0.465011i	−0.699994	+	0.714149i	$0.746814\pi$
$558$		0			0
$559$	−9.65524			−0.408373
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−37.9800			−1.60067			−0.800333	−	0.599556i	$-0.795344\pi$
$563$	−37.9800			−1.60067			−0.800333	+	0.599556i	$0.795344\pi$
$564$		0			0
$565$	35.2804	+	20.3691i	1.48426	+	0.856936i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7.88197i			0.330429i	0.986258	+	0.165215i	$0.0528317\pi$
$569$			7.88197i			0.330429i	−0.986258	+	0.165215i	$0.947168\pi$
$570$		0			0
$571$		−	5.57446i		−	0.233284i	−0.993174	−	0.116642i	$-0.962787\pi$
$571$		−	5.57446i		−	0.233284i	0.993174	−	0.116642i	$-0.0372130\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	20.6718	+	11.9349i	0.862073	+	0.497718i
$576$		0			0
$577$	−27.6458			−1.15091			−0.575454	−	0.817834i	$-0.695175\pi$
$577$	−27.6458			−1.15091			−0.575454	+	0.817834i	$0.695175\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−35.7727			−1.48410
$582$		0			0
$583$	8.29966	+	14.3754i	0.343737	+	0.595370i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.81320	−	4.51095i	−0.322485	−	0.186187i	0.330015	−	0.943976i	$-0.392946\pi$
$587$	−7.81320	−	4.51095i	−0.322485	−	0.186187i	−0.652500	+	0.757789i	$0.726280\pi$
$588$		0			0
$589$	4.17712	+	3.26599i	0.172115	+	0.134573i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.0345	+	26.0406i	0.617394	+	1.06936i	0.989959	+	0.141352i	$0.0451449\pi$
$593$	15.0345	+	26.0406i	0.617394	+	1.06936i	−0.372566	+	0.928006i	$0.621522\pi$
$594$		0			0
$595$	35.5552	−	20.5278i	1.45762	−	0.841558i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.1768	−	19.3588i	−0.456671	−	0.790978i	0.542111	−	0.840307i	$-0.317625\pi$
$599$	−11.1768	−	19.3588i	−0.456671	−	0.790978i	−0.998783	+	0.0493290i	$0.984292\pi$
$600$		0			0
$601$			3.57278i			0.145737i	0.997342	+	0.0728684i	$0.0232153\pi$
$601$			3.57278i			0.145737i	−0.997342	+	0.0728684i	$0.976785\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.99410	+	17.3103i	−0.406318	+	0.703763i
$606$		0			0
$607$	−16.5230			−0.670648			−0.335324	−	0.942103i	$-0.608846\pi$
$607$	−16.5230			−0.670648			−0.335324	+	0.942103i	$0.608846\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−25.1214	+	43.5115i	−1.01630	+	1.76029i
$612$		0			0
$613$	9.76013	−	16.9050i	0.394208	−	0.682788i	−0.598792	−	0.800905i	$-0.704352\pi$
$613$	9.76013	−	16.9050i	0.394208	−	0.682788i	0.993000	+	0.118117i	$0.0376856\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.38203	+	14.5181i	0.337448	+	0.584477i	0.983952	−	0.178434i	$-0.0571029\pi$
$617$	8.38203	+	14.5181i	0.337448	+	0.584477i	−0.646504	+	0.762911i	$0.723770\pi$
$618$		0			0
$619$			7.47668i			0.300513i	0.988647	+	0.150256i	$0.0480099\pi$
$619$			7.47668i			0.300513i	−0.988647	+	0.150256i	$0.951990\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.1768	+	19.3588i	0.447789	+	0.775593i
$624$		0			0
$625$	15.6144	−	27.0449i	0.624575	−	1.08180i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.82599	+	3.94099i	0.272170	+	0.157137i
$630$		0			0
$631$	14.1283	−	8.15696i	0.562437	−	0.324723i	−0.191686	−	0.981456i	$-0.561395\pi$
$631$	14.1283	−	8.15696i	0.562437	−	0.324723i	0.754123	+	0.656733i	$0.228062\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.72718			0.266960
$636$		0			0
$637$	−14.4686	+	8.35347i	−0.573268	+	0.330976i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.2021	+	14.5505i	0.995424	+	0.574708i	0.906891	−	0.421365i	$-0.138449\pi$
$641$	25.2021	+	14.5505i	0.995424	+	0.574708i	0.0885329	+	0.996073i	$0.471782\pi$
$642$		0			0
$643$	32.7296	+	18.8964i	1.29073	+	0.745202i	0.978783	−	0.204899i	$-0.0656864\pi$
$643$	32.7296	+	18.8964i	1.29073	+	0.745202i	0.311944	+	0.950100i	$0.399020\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32.2037i		−	1.26606i	−0.774128	−	0.633029i	$-0.781812\pi$
$647$		−	32.2037i		−	1.26606i	0.774128	−	0.633029i	$-0.218188\pi$
$648$		0			0
$649$	5.65687	+	3.26599i	0.222051	+	0.128201i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.1412			−0.827320			−0.413660	−	0.910431i	$-0.635750\pi$
$653$	−21.1412			−0.827320			−0.413660	+	0.910431i	$0.635750\pi$
$654$		0			0
$655$	−7.65327	+	4.41862i	−0.299038	+	0.172650i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.5810	+	18.3269i	0.412178	+	0.713913i	0.995128	−	0.0985953i	$-0.0314349\pi$
$659$	10.5810	+	18.3269i	0.412178	+	0.713913i	−0.582950	+	0.812508i	$0.698102\pi$
$660$		0			0
$661$	27.0000	−	15.5885i	1.05018	−	0.606321i	0.127479	−	0.991841i	$-0.459311\pi$
$661$	27.0000	−	15.5885i	1.05018	−	0.606321i	0.922699	+	0.385521i	$0.125978\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−14.7183	−	36.4674i	−0.570750	−	1.41415i
$666$		0			0
$667$	9.30064	−	16.1092i	0.360122	−	0.623750i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.08602	+	0.627015i	−0.0419254	+	0.0242056i
$672$		0			0
$673$			5.59237i			0.215570i	0.994174	+	0.107785i	$0.0343758\pi$
$673$			5.59237i			0.215570i	−0.994174	+	0.107785i	$0.965624\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	25.3425i		−	0.973992i	−0.873404	−	0.486996i	$-0.838093\pi$
$677$		−	25.3425i		−	0.973992i	0.873404	−	0.486996i	$-0.161907\pi$
$678$		0			0
$679$	−30.9049	+	53.5288i	−1.18602	+	2.05425i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.89922			0.340519			0.170260	−	0.985399i	$-0.445539\pi$
$683$	8.89922			0.340519			0.170260	+	0.985399i	$0.445539\pi$
$684$		0			0
$685$	30.5830			1.16852
$686$		0			0
$687$		0			0
$688$		0			0
$689$	19.5852	−	33.9225i	0.746136	−	1.29235i
$690$		0			0
$691$		−	19.5767i		−	0.744733i	−0.928086	−	0.372366i	$-0.878547\pi$
$691$		−	19.5767i		−	0.744733i	0.928086	−	0.372366i	$-0.121453\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			30.2617i			1.14789i
$696$		0			0
$697$	−14.8118	+	8.55157i	−0.561035	+	0.323914i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.4225	+	23.2484i	−0.506959	+	0.878079i	0.493008	+	0.870025i	$0.335897\pi$
$701$	−13.4225	+	23.2484i	−0.506959	+	0.878079i	−0.999968	+	0.00805464i	$0.997436\pi$
$702$		0			0
$703$	4.65032	−	5.94765i	0.175390	−	0.224320i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.9493	+	13.2498i	−0.863099	+	0.498310i
$708$		0			0
$709$	17.5516	+	30.4003i	0.659165	+	1.14171i	0.980832	+	0.194855i	$0.0624236\pi$
$709$	17.5516	+	30.4003i	0.659165	+	1.14171i	−0.321667	+	0.946853i	$0.604243\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.50433	−	5.48733i	0.355940	−	0.205502i
$714$		0			0
$715$	−24.6072			−0.920257
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.68179	+	0.970984i	0.0627203	+	0.0362116i	0.531032	−	0.847352i	$-0.321804\pi$
$719$	1.68179	+	0.970984i	0.0627203	+	0.0362116i	−0.468312	+	0.883563i	$0.655138\pi$
$720$		0			0
$721$			63.2549i			2.35574i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.72416	+	2.72749i	0.175451	+	0.101297i
$726$		0			0
$727$	18.7786	+	10.8418i	0.696459	+	0.402101i	0.806027	−	0.591878i	$-0.201613\pi$
$727$	18.7786	+	10.8418i	0.696459	+	0.402101i	−0.109568	+	0.993979i	$0.534947\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.30345	+	4.79400i	−0.307114	+	0.177312i
$732$		0			0
$733$	−30.3542			−1.12116			−0.560580	−	0.828100i	$-0.689422\pi$
$733$	−30.3542			−1.12116			−0.560580	+	0.828100i	$0.689422\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.5960	−	7.84963i	0.500814	−	0.289145i
$738$		0			0
$739$	14.1283	+	8.15696i	0.519717	+	0.300059i	0.736819	−	0.676090i	$-0.236327\pi$
$739$	14.1283	+	8.15696i	0.519717	+	0.300059i	−0.217102	+	0.976149i	$0.569660\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.9040	+	31.0106i	−0.656833	+	1.13767i	0.324598	+	0.945852i	$0.394771\pi$
$743$	−17.9040	+	31.0106i	−0.656833	+	1.13767i	−0.981431	+	0.191816i	$0.938562\pi$
$744$		0			0
$745$	−17.7601	−	30.7614i	−0.650681	−	1.12701i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	21.9493i		−	0.802011i
$750$		0			0
$751$	18.9940	+	32.8987i	0.693103	+	1.20049i	0.970816	+	0.239825i	$0.0770901\pi$
$751$	18.9940	+	32.8987i	0.693103	+	1.20049i	−0.277713	+	0.960664i	$0.589577\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.9900	−	32.8916i	0.691116	−	1.19705i
$756$		0			0
$757$	16.6144	−	28.7769i	0.603860	−	1.04592i	−0.388371	−	0.921503i	$-0.626962\pi$
$757$	16.6144	−	28.7769i	0.603860	−	1.04592i	0.992231	−	0.124413i	$-0.0397047\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−29.4365			−1.06707			−0.533536	−	0.845777i	$-0.679137\pi$
$761$	−29.4365			−1.06707			−0.533536	+	0.845777i	$0.679137\pi$
$762$		0			0
$763$	−4.65032	+	8.05459i	−0.168353	+	0.291596i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	15.4139i		−	0.556564i
$768$		0			0
$769$	−21.0203	−	36.4082i	−0.758010	−	1.31291i	−0.943864	−	0.330334i	$-0.892838\pi$
$769$	−21.0203	−	36.4082i	−0.758010	−	1.31291i	0.185854	−	0.982577i	$-0.440495\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29.9263	+	17.2779i	−1.07637	+	0.621445i	−0.929916	−	0.367772i	$-0.880121\pi$
$773$	−29.9263	+	17.2779i	−1.07637	+	0.621445i	−0.146458	+	0.989217i	$0.546787\pi$
$774$		0			0
$775$	1.60921	+	2.78723i	0.0578044	+	0.100120i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.13140	+	15.1918i	0.219680	+	0.544302i
$780$		0			0
$781$	20.6235	+	11.9070i	0.737968	+	0.426066i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	16.0141	+	27.7372i	0.571566	+	0.989982i
$786$		0			0
$787$	41.9920			1.49685			0.748427	−	0.663217i	$-0.230809\pi$
$787$	41.9920			1.49685			0.748427	+	0.663217i	$0.230809\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−48.0707			−1.70920
$792$		0			0
$793$	2.56275	+	1.47960i	0.0910058	+	0.0525422i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.75839i			0.133129i	0.997782	+	0.0665645i	$0.0212038\pi$
$797$			3.75839i			0.133129i	−0.997782	+	0.0665645i	$0.978796\pi$
$798$		0			0
$799$			49.8929i			1.76508i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.7531	−	7.36298i	−0.450046	−	0.259834i
$804$		0			0
$805$	−81.3948			−2.86879
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−26.4979			−0.931617			−0.465809	−	0.884885i	$-0.654236\pi$
$809$	−26.4979			−0.931617			−0.465809	+	0.884885i	$0.654236\pi$
$810$		0			0
$811$	3.21841	+	5.57446i	0.113014	+	0.195746i	0.916984	−	0.398924i	$-0.130616\pi$
$811$	3.21841	+	5.57446i	0.113014	+	0.195746i	−0.803970	+	0.594670i	$0.797283\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17.9040	+	10.3369i	0.627149	+	0.362084i
$816$		0			0
$817$	3.43725	+	8.51648i	0.120254	+	0.297954i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.7845	−	27.3396i	−0.550884	−	0.954159i	−0.998211	−	0.0597889i	$-0.980957\pi$
$821$	−15.7845	−	27.3396i	−0.550884	−	0.954159i	0.447327	−	0.894371i	$-0.352376\pi$
$822$		0			0
$823$	−35.5552	+	20.5278i	−1.23938	+	0.715554i	−0.968967	−	0.247191i	$-0.920492\pi$
$823$	−35.5552	+	20.5278i	−1.23938	+	0.715554i	−0.270410	+	0.962745i	$0.587159\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.7172	+	44.5434i	0.894273	+	1.54893i	0.834701	+	0.550703i	$0.185640\pi$
$827$	25.7172	+	44.5434i	0.894273	+	1.54893i	0.0595722	+	0.998224i	$0.481026\pi$
$828$		0			0
$829$			48.1714i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$829$			48.1714i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8.29529	+	14.3679i	−0.287415	+	0.497817i
$834$		0			0
$835$	43.2085			1.49529
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.81320	−	13.5329i	0.269742	−	0.467206i	−0.699053	−	0.715069i	$-0.746395\pi$
$839$	7.81320	−	13.5329i	0.269742	−	0.467206i	0.968795	+	0.247863i	$0.0797284\pi$
$840$		0			0
$841$	−12.3745	+	21.4333i	−0.426707	+	0.739078i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.0604	+	19.1571i	0.380489	+	0.659026i
$846$		0			0
$847$		−	23.5859i		−	0.810420i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.81320	−	13.5329i	−0.267833	−	0.463900i
$852$		0			0
$853$	13.8542	−	23.9963i	0.474360	−	0.821616i	−0.525209	−	0.850974i	$-0.676013\pi$
$853$	13.8542	−	23.9963i	0.474360	−	0.821616i	0.999569	+	0.0293572i	$0.00934602\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.66275	+	4.42409i	0.261755	+	0.151124i	0.625135	−	0.780517i	$-0.285044\pi$
$857$	7.66275	+	4.42409i	0.261755	+	0.151124i	−0.363380	+	0.931641i	$0.618377\pi$
$858$		0			0
$859$	−40.3828	+	23.3150i	−1.37784	+	0.795499i	−0.991900	−	0.127023i	$-0.959458\pi$
$859$	−40.3828	+	23.3150i	−1.37784	+	0.795499i	−0.385944	+	0.922522i	$0.626124\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.4443			−1.10442			−0.552209	−	0.833706i	$-0.686215\pi$
$863$	−32.4443			−1.10442			−0.552209	+	0.833706i	$0.686215\pi$
$864$		0			0
$865$	−55.7490	+	32.1867i	−1.89552	+	1.09438i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.9629	+	9.79351i	0.575425	+	0.332222i
$870$		0			0
$871$	−32.0832	−	18.5232i	−1.08710	−	0.627635i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			21.2398i			0.718037i
$876$		0			0
$877$	0.625492	+	0.361128i	0.0211214	+	0.0121944i	0.510524	−	0.859864i	$-0.329452\pi$
$877$	0.625492	+	0.361128i	0.0211214	+	0.0121944i	−0.489402	+	0.872058i	$0.662785\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.37716			−0.147470			−0.0737351	−	0.997278i	$-0.523492\pi$
$881$	−4.37716			−0.147470			−0.0737351	+	0.997278i	$0.523492\pi$
$882$		0			0
$883$	16.7766	−	9.68598i	0.564578	−	0.325959i	−0.190403	−	0.981706i	$-0.560979\pi$
$883$	16.7766	−	9.68598i	0.564578	−	0.325959i	0.754981	+	0.655747i	$0.227646\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.6718	+	35.8046i	0.694090	+	1.20220i	0.970486	+	0.241156i	$0.0775266\pi$
$887$	20.6718	+	35.8046i	0.694090	+	1.20220i	−0.276396	+	0.961044i	$0.589140\pi$
$888$		0			0
$889$	−6.87451	+	3.96900i	−0.230564	+	0.133116i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	47.3229	+	6.66848i	1.58360	+	0.223152i
$894$		0			0
$895$	−4.65032	+	8.05459i	−0.155443	+	0.269235i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.17204	−	1.25403i	0.0724417	−	0.0418242i
$900$		0			0
$901$		−	38.8976i		−	1.29587i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			23.6459i			0.786017i
$906$		0			0
$907$	21.3888	−	37.0464i	0.710203	−	1.23011i	−0.254578	−	0.967052i	$-0.581937\pi$
$907$	21.3888	−	37.0464i	0.710203	−	1.23011i	0.964781	−	0.263055i	$-0.0847300\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.2628			−0.406285			−0.203142	−	0.979149i	$-0.565115\pi$
$911$	−12.2628			−0.406285			−0.203142	+	0.979149i	$0.565115\pi$
$912$		0			0
$913$	21.2915			0.704646
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.21392	−	9.03078i	0.172179	−	0.298223i
$918$		0			0
$919$			22.8395i			0.753405i	0.926334	+	0.376702i	$0.122942\pi$
$919$			22.8395i			0.753405i	−0.926334	+	0.376702i	$0.877058\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	56.1952i		−	1.84969i
$924$		0			0
$925$	3.96863	−	2.29129i	0.130488	−	0.0753371i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.7792	−	32.5264i	0.616124	−	1.06716i	−0.374062	−	0.927404i	$-0.622035\pi$
$929$	18.7792	−	32.5264i	0.616124	−	1.06716i	0.990186	−	0.139754i	$-0.0446313\pi$
$930$		0			0
$931$	12.5191	+	9.78835i	0.410296	+	0.320800i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.1620	+	12.2179i	−0.692072	+	0.399568i
$936$		0			0
$937$	4.90588	+	8.49723i	0.160268	+	0.277593i	0.934965	−	0.354740i	$-0.115431\pi$
$937$	4.90588	+	8.49723i	0.160268	+	0.277593i	−0.774697	+	0.632333i	$0.782097\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$	33.9078			1.10419
$944$		0			0
$945$		0			0
$946$		0			0
$947$	37.3842	+	21.5838i	1.21482	+	0.701379i	0.963806	−	0.266604i	$-0.0859016\pi$
$947$	37.3842	+	21.5838i	1.21482	+	0.701379i	0.251017	+	0.967983i	$0.419235\pi$
$948$		0			0
$949$			34.7497i			1.12802i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.576515	−	0.332851i	−0.0186752	−	0.0107821i	0.490633	−	0.871366i	$-0.336765\pi$
$953$	−0.576515	−	0.332851i	−0.0186752	−	0.0107821i	−0.509309	+	0.860584i	$0.670099\pi$
$954$		0			0
$955$	−40.2055	−	23.2127i	−1.30102	−	0.751144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−31.2528	+	18.0438i	−1.00921	+	0.582665i
$960$		0			0
$961$	−29.5203			−0.952266
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−35.8595	+	20.7035i	−1.15436	+	0.666469i
$966$		0			0
$967$	−24.4299	−	14.1046i	−0.785612	−	0.453573i	0.0528033	−	0.998605i	$-0.483184\pi$
$967$	−24.4299	−	14.1046i	−0.785612	−	0.453573i	−0.838416	+	0.545031i	$0.816518\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.4850	+	49.3374i	−0.914127	+	1.58331i	−0.105951	+	0.994371i	$0.533789\pi$
$971$	−28.4850	+	49.3374i	−0.914127	+	1.58331i	−0.808175	+	0.588942i	$0.799545\pi$
$972$		0			0
$973$	−17.8542	−	30.9245i	−0.572381	−	0.991393i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.55057i			0.209571i	0.994495	+	0.104786i	$0.0334156\pi$
$977$			6.55057i			0.209571i	−0.994495	+	0.104786i	$0.966584\pi$
$978$		0			0
$979$	−6.65229	−	11.5221i	−0.212608	−	0.368248i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.04538	−	8.73886i	0.160923	−	0.278726i	−0.774277	−	0.632847i	$-0.781886\pi$
$983$	5.04538	−	8.73886i	0.160923	−	0.278726i	0.935200	+	0.354120i	$0.115220\pi$
$984$		0			0
$985$	26.7601	−	46.3499i	0.852649	−	1.47683i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19.0087			0.604440
$990$		0			0
$991$	9.90887	−	17.1627i	0.314766	−	0.545190i	−0.664622	−	0.747180i	$-0.731407\pi$
$991$	9.90887	−	17.1627i	0.314766	−	0.545190i	0.979388	+	0.201990i	$0.0647408\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.82591i		−	0.184694i
$996$		0			0
$997$	17.7915	+	30.8158i	0.563463	+	0.975946i	0.997191	+	0.0749021i	$0.0238644\pi$
$997$	17.7915	+	30.8158i	0.563463	+	0.975946i	−0.433728	+	0.901044i	$0.642802\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.t.559.1	✓	16
3.2	odd	2	inner	2736.2.bm.t.559.7	yes	16
4.3	odd	2	inner	2736.2.bm.t.559.2	yes	16
12.11	even	2	inner	2736.2.bm.t.559.8	yes	16
19.12	odd	6	inner	2736.2.bm.t.1855.1	yes	16
57.50	even	6	inner	2736.2.bm.t.1855.7	yes	16
76.31	even	6	inner	2736.2.bm.t.1855.2	yes	16
228.107	odd	6	inner	2736.2.bm.t.1855.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.bm.t.559.1	✓	16	1.1	even	1	trivial
2736.2.bm.t.559.2	yes	16	4.3	odd	2	inner
2736.2.bm.t.559.7	yes	16	3.2	odd	2	inner
2736.2.bm.t.559.8	yes	16	12.11	even	2	inner
2736.2.bm.t.1855.1	yes	16	19.12	odd	6	inner
2736.2.bm.t.1855.2	yes	16	76.31	even	6	inner
2736.2.bm.t.1855.7	yes	16	57.50	even	6	inner
2736.2.bm.t.1855.8	yes	16	228.107	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-1.38255 + 2.39464i) q^{5} -3.26278i q^{7} +O(q^{10})\)	\(q+(-1.38255 + 2.39464i) q^{5} -3.26278i q^{7} +1.94197i q^{11} +(3.96863 - 2.29129i) q^{13} +(2.27533 - 3.94099i) q^{17} +(-3.43388 - 2.68486i) q^{19} +(-7.81320 + 4.51095i) q^{23} +(-1.32288 - 2.29129i) q^{25} +(-1.78556 + 1.03090i) q^{29} -1.21645 q^{31} +(7.81320 + 4.51095i) q^{35} +1.73205i q^{37} +(-3.25486 - 1.87919i) q^{41} +(-1.82467 - 1.05347i) q^{43} +(-9.49499 + 5.48194i) q^{47} -3.64575 q^{49} +(7.40250 - 4.27384i) q^{53} +(-4.65032 - 2.68486i) q^{55} +(1.68179 - 2.91295i) q^{59} +(0.322876 + 0.559237i) q^{61} +12.6713i q^{65} +(-4.04210 - 7.00112i) q^{67} +(6.13140 - 10.6199i) q^{71} +(-3.79150 + 6.56708i) q^{73} +6.33622 q^{77} +(5.04308 - 8.73488i) q^{79} -10.9639i q^{83} +(6.29150 + 10.8972i) q^{85} +(-5.93321 + 3.42554i) q^{89} +(-7.47597 - 12.9488i) q^{91} +(11.1768 - 4.51095i) q^{95} +(-16.4059 - 9.47194i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100}) \) 16 * q - 16 * q^49 - 16 * q^61 + 24 * q^73 + 16 * q^85 - 72 * q^97

Introduction

L-functions

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