LMFDB - Embedded newform 2736.2.dc.d.1889.8

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} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 $ x^16 + 10x^14 + 46x^12 + 126x^10 + 315x^8 + 1134x^6 + 3726x^4 + 7290*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1889.8
Root			$-1.56478 - 0.742611i$ of defining polynomial
Character	$\chi$	$=$	2736.1889
Dual form			2736.2.dc.d.449.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.99029 + 1.72644i) q^{5} -0.128302 q^{7} +O(q^{10})$	$q+(2.99029 + 1.72644i) q^{5} -0.128302 q^{7} +6.15544i q^{11} +(2.67404 - 1.54386i) q^{13} +(4.91297 + 2.83650i) q^{17} +(3.96121 - 1.81902i) q^{19} +(-2.18882 + 1.26372i) q^{23} +(3.46121 + 5.99499i) q^{25} +(-3.40809 - 5.90299i) q^{29} -10.0357i q^{31} +(-0.383660 - 0.221506i) q^{35} +10.1899i q^{37} +(2.34048 - 4.05384i) q^{41} +(-3.02536 + 5.24008i) q^{43} +(-0.266142 + 0.153657i) q^{47} -6.98354 q^{49} +(4.05789 + 7.02848i) q^{53} +(-10.6270 + 18.4065i) q^{55} +(-4.05789 + 7.02848i) q^{59} +(-2.20460 - 3.81849i) q^{61} +10.6615 q^{65} +(4.51713 - 2.60797i) q^{67} +(3.67423 - 6.36396i) q^{71} +(3.15886 - 5.47131i) q^{73} -0.789755i q^{77} +(-2.95818 - 1.70791i) q^{79} +0.267885i q^{83} +(9.79411 + 16.9639i) q^{85} +(4.52931 + 7.84499i) q^{89} +(-0.343085 + 0.198080i) q^{91} +(14.9856 + 1.39941i) q^{95} +(-16.2083 - 9.35787i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) $ 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

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
$5$	2.99029	+	1.72644i	1.33730	+	0.772089i	0.986406	−	0.164328i	$-0.0525455\pi$
$5$	2.99029	+	1.72644i	1.33730	+	0.772089i	0.350891	+	0.936416i	$0.385879\pi$
$6$		0			0
$7$	−0.128302			−0.0484936			−0.0242468	−	0.999706i	$-0.507719\pi$
$7$	−0.128302			−0.0484936			−0.0242468	+	0.999706i	$0.507719\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.15544i			1.85594i	0.372660	+	0.927968i	$0.378446\pi$
$11$			6.15544i			1.85594i	−0.372660	+	0.927968i	$0.621554\pi$
$12$		0			0
$13$	2.67404	−	1.54386i	0.741646	−	0.428190i	−0.0810213	−	0.996712i	$-0.525818\pi$
$13$	2.67404	−	1.54386i	0.741646	−	0.428190i	0.822668	+	0.568523i	$0.192485\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.91297	+	2.83650i	1.19157	+	0.687953i	0.958662	−	0.284546i	$-0.0918430\pi$
$17$	4.91297	+	2.83650i	1.19157	+	0.687953i	0.232907	+	0.972499i	$0.425176\pi$
$18$		0			0
$19$	3.96121	−	1.81902i	0.908764	−	0.417311i
$19$	3.96121	−	1.81902i	0.908764	−	0.417311i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.18882	+	1.26372i	−0.456401	+	0.263503i	−0.710530	−	0.703667i	$-0.751545\pi$
$23$	−2.18882	+	1.26372i	−0.456401	+	0.263503i	0.254129	+	0.967170i	$0.418211\pi$
$24$		0			0
$25$	3.46121	+	5.99499i	0.692242	+	1.19900i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.40809	−	5.90299i	−0.632867	−	1.09616i	−0.986963	−	0.160948i	$-0.948545\pi$
$29$	−3.40809	−	5.90299i	−0.632867	−	1.09616i	0.354096	−	0.935209i	$-0.384789\pi$
$30$		0			0
$31$		−	10.0357i		−	1.80247i	−0.433336	−	0.901233i	$-0.642664\pi$
$31$		−	10.0357i		−	1.80247i	0.433336	−	0.901233i	$-0.357336\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.383660	−	0.221506i	−0.0648503	−	0.0374413i
$36$		0			0
$37$			10.1899i			1.67520i	0.546282	+	0.837601i	$0.316043\pi$
$37$			10.1899i			1.67520i	−0.546282	+	0.837601i	$0.683957\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.34048	−	4.05384i	0.365522	−	0.633103i	−0.623338	−	0.781953i	$-0.714224\pi$
$41$	2.34048	−	4.05384i	0.365522	−	0.633103i	0.988860	+	0.148850i	$0.0475571\pi$
$42$		0			0
$43$	−3.02536	+	5.24008i	−0.461363	+	0.799104i	−0.999029	−	0.0440537i	$-0.985973\pi$
$43$	−3.02536	+	5.24008i	−0.461363	+	0.799104i	0.537666	+	0.843158i	$0.319306\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.266142	+	0.153657i	−0.0388208	+	0.0224132i	−0.519285	−	0.854601i	$-0.673802\pi$
$47$	−0.266142	+	0.153657i	−0.0388208	+	0.0224132i	0.480464	+	0.877014i	$0.340468\pi$
$48$		0			0
$49$	−6.98354			−0.997648
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.05789	+	7.02848i	0.557395	+	0.965436i	0.997713	+	0.0675941i	$0.0215323\pi$
$53$	4.05789	+	7.02848i	0.557395	+	0.965436i	−0.440318	+	0.897842i	$0.645134\pi$
$54$		0			0
$55$	−10.6270	+	18.4065i	−1.43295	+	2.48194i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.05789	+	7.02848i	−0.528293	+	0.915030i	0.471163	+	0.882046i	$0.343834\pi$
$59$	−4.05789	+	7.02848i	−0.528293	+	0.915030i	−0.999456	+	0.0329839i	$0.989499\pi$
$60$		0			0
$61$	−2.20460	−	3.81849i	−0.282271	−	0.488907i	0.689673	−	0.724121i	$-0.257754\pi$
$61$	−2.20460	−	3.81849i	−0.282271	−	0.488907i	−0.971944	+	0.235214i	$0.924421\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.6615			1.32240
$66$		0			0
$67$	4.51713	−	2.60797i	0.551855	−	0.318614i	−0.198015	−	0.980199i	$-0.563449\pi$
$67$	4.51713	−	2.60797i	0.551855	−	0.318614i	0.749870	+	0.661585i	$0.230116\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.67423	−	6.36396i	0.436051	−	0.755263i	−0.561329	−	0.827592i	$-0.689710\pi$
$71$	3.67423	−	6.36396i	0.436051	−	0.755263i	0.997381	+	0.0723293i	$0.0230432\pi$
$72$		0			0
$73$	3.15886	−	5.47131i	0.369717	−	0.640368i	−0.619804	−	0.784756i	$-0.712788\pi$
$73$	3.15886	−	5.47131i	0.369717	−	0.640368i	0.989521	+	0.144388i	$0.0461213\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.789755i		−	0.0900010i
$78$		0			0
$79$	−2.95818	−	1.70791i	−0.332821	−	0.192154i	0.324272	−	0.945964i	$-0.394881\pi$
$79$	−2.95818	−	1.70791i	−0.332821	−	0.192154i	−0.657093	+	0.753810i	$0.728214\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.267885i			0.0294042i	0.999892	+	0.0147021i	$0.00468000\pi$
$83$			0.267885i			0.0294042i	−0.999892	+	0.0147021i	$0.995320\pi$
$84$		0			0
$85$	9.79411	+	16.9639i	1.06232	+	1.83999i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.52931	+	7.84499i	0.480105	+	0.831567i	0.999740	−	0.0228219i	$-0.00726507\pi$
$89$	4.52931	+	7.84499i	0.480105	+	0.831567i	−0.519634	+	0.854389i	$0.673932\pi$
$90$		0			0
$91$	−0.343085	+	0.198080i	−0.0359651	+	0.0207644i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	14.9856	+	1.39941i	1.53749	+	0.143577i
$96$		0			0
$97$	−16.2083	−	9.35787i	−1.64570	−	0.950147i	−0.978752	−	0.205046i	$-0.934266\pi$
$97$	−16.2083	−	9.35787i	−1.64570	−	0.950147i	−0.666951	−	0.745101i	$-0.732401\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.78204	+	3.91561i	−0.674838	+	0.389618i	−0.797907	−	0.602780i	$-0.794060\pi$
$101$	−6.78204	+	3.91561i	−0.674838	+	0.389618i	0.123069	+	0.992398i	$0.460726\pi$
$102$		0			0
$103$			8.23558i			0.811476i	0.913989	+	0.405738i	$0.132985\pi$
$103$			8.23558i			0.811476i	−0.913989	+	0.405738i	$0.867015\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.1744			1.66031			0.830156	−	0.557531i	$-0.188251\pi$
$107$	17.1744			1.66031			0.830156	+	0.557531i	$0.188251\pi$
$108$		0			0
$109$	2.49872	+	1.44264i	0.239334	+	0.138180i	0.614871	−	0.788628i	$-0.289208\pi$
$109$	2.49872	+	1.44264i	0.239334	+	0.138180i	−0.375537	+	0.926808i	$0.622542\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.45803			−0.795665			−0.397832	−	0.917458i	$-0.630237\pi$
$113$	−8.45803			−0.795665			−0.397832	+	0.917458i	$0.630237\pi$
$114$		0			0
$115$	−8.72693			−0.813791
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.630343	−	0.363929i	−0.0577834	−	0.0333613i
$120$		0			0
$121$	−26.8895			−2.44450
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.63789i			0.593710i
$126$		0			0
$127$	1.02341	−	0.590867i	0.0908131	−	0.0524310i	−0.453906	−	0.891050i	$-0.649970\pi$
$127$	1.02341	−	0.590867i	0.0908131	−	0.0524310i	0.544719	+	0.838619i	$0.316636\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.266142	+	0.153657i	0.0232529	+	0.0134251i	0.511581	−	0.859235i	$-0.329060\pi$
$131$	0.266142	+	0.153657i	0.0232529	+	0.0134251i	−0.488328	+	0.872660i	$0.662393\pi$
$132$		0			0
$133$	−0.508231	+	0.233384i	−0.0440692	+	0.0202369i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.4904	−	6.05665i	0.896257	−	0.517454i	0.0202731	−	0.999794i	$-0.493546\pi$
$137$	10.4904	−	6.05665i	0.896257	−	0.517454i	0.875984	+	0.482340i	$0.160213\pi$
$138$		0			0
$139$	3.55592	+	6.15903i	0.301609	+	0.522402i	0.976501	−	0.215515i	$-0.0691429\pi$
$139$	3.55592	+	6.15903i	0.301609	+	0.522402i	−0.674891	+	0.737917i	$0.735810\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.50314	+	16.4599i	0.794693	+	1.37645i
$144$		0			0
$145$		−	23.5355i		−	1.95452i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.69193	−	3.28624i	−0.466301	−	0.269219i	0.248389	−	0.968660i	$-0.420099\pi$
$149$	−5.69193	−	3.28624i	−0.466301	−	0.269219i	−0.714690	+	0.699441i	$0.753432\pi$
$150$		0			0
$151$		−	11.4775i		−	0.934022i	−0.884251	−	0.467011i	$-0.845331\pi$
$151$		−	11.4775i		−	0.934022i	0.884251	−	0.467011i	$-0.154669\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	17.3261	−	30.0096i	1.39166	−	2.41043i
$156$		0			0
$157$	−9.42242	+	16.3201i	−0.751991	+	1.30249i	0.194866	+	0.980830i	$0.437573\pi$
$157$	−9.42242	+	16.3201i	−0.751991	+	1.30249i	−0.946857	+	0.321656i	$0.895760\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.280830	−	0.162137i	0.0221325	−	0.0127782i
$162$		0			0
$163$	1.23625			0.0968303			0.0484152	−	0.998827i	$-0.484583\pi$
$163$	1.23625			0.0968303			0.0484152	+	0.998827i	$0.484583\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.35818	−	7.54859i	−0.337246	−	0.584128i	0.646667	−	0.762772i	$-0.276162\pi$
$167$	−4.35818	−	7.54859i	−0.337246	−	0.584128i	−0.983914	+	0.178644i	$0.942829\pi$
$168$		0			0
$169$	−1.73299	+	3.00163i	−0.133307	+	0.230895i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.98057	−	10.3587i	0.454695	−	0.787554i	−0.543976	−	0.839101i	$-0.683082\pi$
$173$	5.98057	−	10.3587i	0.454695	−	0.787554i	0.998671	+	0.0515467i	$0.0164151\pi$
$174$		0			0
$175$	−0.444080	−	0.769169i	−0.0335693	−	0.0581437i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.2645			0.916690			0.458345	−	0.888774i	$-0.348442\pi$
$179$	12.2645			0.916690			0.458345	+	0.888774i	$0.348442\pi$
$180$		0			0
$181$	4.82596	−	2.78627i	0.358711	−	0.207102i	−0.309804	−	0.950800i	$-0.600264\pi$
$181$	4.82596	−	2.78627i	0.358711	−	0.207102i	0.668515	+	0.743699i	$0.266930\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−17.5922	+	30.4706i	−1.29340	+	2.24024i
$186$		0			0
$187$	−17.4599	+	30.2415i	−1.27680	+	2.21148i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			0.552731i			0.0399942i	0.999800	+	0.0199971i	$0.00636570\pi$
$191$			0.552731i			0.0399942i	−0.999800	+	0.0199971i	$0.993634\pi$
$192$		0			0
$193$	3.05895	+	1.76609i	0.220188	+	0.127126i	0.606037	−	0.795436i	$-0.292758\pi$
$193$	3.05895	+	1.76609i	0.220188	+	0.127126i	−0.385849	+	0.922562i	$0.626092\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	17.8123i		−	1.26907i	−0.772893	−	0.634536i	$-0.781191\pi$
$197$		−	17.8123i		−	1.26907i	0.772893	−	0.634536i	$-0.218809\pi$
$198$		0			0
$199$	7.03231	+	12.1803i	0.498507	+	0.863440i	0.999999	−	0.00172286i	$-0.000548403\pi$
$199$	7.03231	+	12.1803i	0.498507	+	0.863440i	−0.501491	+	0.865163i	$0.667215\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.437265	+	0.757365i	0.0306900	+	0.0531566i
$204$		0			0
$205$	13.9974	−	8.08143i	0.977624	−	0.564431i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.1969	+	24.3830i	0.774503	+	1.68661i
$210$		0			0
$211$	−5.38991	−	3.11187i	−0.371057	−	0.214230i	0.302863	−	0.953034i	$-0.402057\pi$
$211$	−5.38991	−	3.11187i	−0.371057	−	0.214230i	−0.673920	+	0.738804i	$0.735391\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−18.0934	+	10.4462i	−1.23396	+	0.712426i
$216$		0			0
$217$			1.28760i			0.0874080i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.5166			1.17830
$222$		0			0
$223$	21.7146	+	12.5369i	1.45412	+	0.839534i	0.998711	−	0.0507502i	$-0.0161612\pi$
$223$	21.7146	+	12.5369i	1.45412	+	0.839534i	0.455405	+	0.890285i	$0.349495\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.767319			0.0509288			0.0254644	−	0.999676i	$-0.491894\pi$
$227$	0.767319			0.0509288			0.0254644	+	0.999676i	$0.491894\pi$
$228$		0			0
$229$	17.8613			1.18031			0.590154	−	0.807291i	$-0.299067\pi$
$229$	17.8613			1.18031			0.590154	+	0.807291i	$0.299067\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.84166	−	1.64063i	−0.186163	−	0.107482i	0.404022	−	0.914749i	$-0.367612\pi$
$233$	−2.84166	−	1.64063i	−0.186163	−	0.107482i	−0.590185	+	0.807268i	$0.700945\pi$
$234$		0			0
$235$	−1.06112			−0.0692199
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3.14557i		−	0.203470i	−0.994812	−	0.101735i	$-0.967561\pi$
$239$		−	3.14557i		−	0.203470i	0.994812	−	0.101735i	$-0.0324394\pi$
$240$		0			0
$241$	−12.2096	+	7.04921i	−0.786488	+	0.454079i	−0.838725	−	0.544556i	$-0.816698\pi$
$241$	−12.2096	+	7.04921i	−0.786488	+	0.454079i	0.0522365	+	0.998635i	$0.483365\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−20.8828	−	12.0567i	−1.33415	−	0.770273i
$246$		0			0
$247$	7.78414	−	10.9797i	0.495293	−	0.698621i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.4984	+	8.94801i	−0.978251	+	0.564793i	−0.901742	−	0.432276i	$-0.857711\pi$
$251$	−15.4984	+	8.94801i	−0.978251	+	0.564793i	−0.0765091	+	0.997069i	$0.524377\pi$
$252$		0			0
$253$	−7.77874	−	13.4732i	−0.489045	−	0.847051i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.96287	+	6.86390i	0.247197	+	0.428158i	0.962747	−	0.270403i	$-0.0871571\pi$
$257$	3.96287	+	6.86390i	0.247197	+	0.428158i	−0.715550	+	0.698562i	$0.753824\pi$
$258$		0			0
$259$		−	1.30738i		−	0.0812365i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.5199	+	13.0018i	1.38863	+	0.801728i	0.993161	−	0.116750i	$-0.0372475\pi$
$263$	22.5199	+	13.0018i	1.38863	+	0.801728i	0.395473	+	0.918478i	$0.370581\pi$
$264$		0			0
$265$			28.0229i			1.72143i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.18882	−	3.79115i	0.133455	−	0.231150i	−0.791551	−	0.611103i	$-0.790726\pi$
$269$	2.18882	−	3.79115i	0.133455	−	0.231150i	0.925006	+	0.379952i	$0.124060\pi$
$270$		0			0
$271$	1.00000	−	1.73205i	0.0607457	−	0.105215i	−0.834053	−	0.551684i	$-0.813985\pi$
$271$	1.00000	−	1.73205i	0.0607457	−	0.105215i	0.894799	+	0.446469i	$0.147319\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−36.9018	+	21.3053i	−2.22526	+	1.28476i
$276$		0			0
$277$	2.86130			0.171919			0.0859593	−	0.996299i	$-0.472605\pi$
$277$	2.86130			0.171919			0.0859593	+	0.996299i	$0.472605\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.1841	+	24.5676i	0.846153	+	1.46558i	0.884616	+	0.466320i	$0.154421\pi$
$281$	14.1841	+	24.5676i	0.846153	+	1.46558i	−0.0384627	+	0.999260i	$0.512246\pi$
$282$		0			0
$283$	13.3316	−	23.0911i	0.792483	−	1.37262i	−0.131942	−	0.991257i	$-0.542121\pi$
$283$	13.3316	−	23.0911i	0.792483	−	1.37262i	0.924425	−	0.381363i	$-0.124545\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.300289	+	0.520115i	−0.0177255	+	0.0307014i
$288$		0			0
$289$	7.59148	+	13.1488i	0.446558	+	0.773461i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.4011			−1.42553			−0.712764	−	0.701404i	$-0.752557\pi$
$293$	−24.4011			−1.42553			−0.712764	+	0.701404i	$0.752557\pi$
$294$		0			0
$295$	−24.2685	+	14.0114i	−1.41297	+	0.815778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.90200	+	6.75847i	−0.225659	+	0.390852i
$300$		0			0
$301$	0.388159	−	0.672312i	0.0223731	−	0.0387514i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	15.2245i		−	0.871752i
$306$		0			0
$307$	−22.8811	−	13.2104i	−1.30589	−	0.753957i	−0.324484	−	0.945891i	$-0.605191\pi$
$307$	−22.8811	−	13.2104i	−1.30589	−	0.753957i	−0.981408	+	0.191935i	$0.938524\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.45845i		−	0.309520i	−0.987952	−	0.154760i	$-0.950540\pi$
$311$		−	5.45845i		−	0.309520i	0.987952	−	0.154760i	$-0.0494605\pi$
$312$		0			0
$313$	9.57630	+	16.5866i	0.541285	+	0.937532i	0.998831	+	0.0483465i	$0.0153952\pi$
$313$	9.57630	+	16.5866i	0.541285	+	0.937532i	−0.457546	+	0.889186i	$0.651271\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.60365	−	11.4379i	−0.370898	−	0.642414i	0.618806	−	0.785544i	$-0.287617\pi$
$317$	−6.60365	−	11.4379i	−0.370898	−	0.642414i	−0.989704	+	0.143130i	$0.954283\pi$
$318$		0			0
$319$	36.3355	−	20.9783i	2.03440	−	1.17456i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.6209	+	2.29920i	1.36995	+	0.127931i
$324$		0			0
$325$	18.5108	+	10.6872i	1.02680	+	0.592822i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.0341466	−	0.0197145i	0.00188256	−	0.00108690i
$330$		0			0
$331$		−	7.76881i		−	0.427013i	−0.976942	−	0.213506i	$-0.931512\pi$
$331$		−	7.76881i		−	0.427013i	0.976942	−	0.213506i	$-0.0684884\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	18.0100			0.983992
$336$		0			0
$337$	−7.26573	−	4.19487i	−0.395789	−	0.228509i	0.288876	−	0.957366i	$-0.406718\pi$
$337$	−7.26573	−	4.19487i	−0.395789	−	0.228509i	−0.684666	+	0.728857i	$0.740052\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	61.7742			3.34526
$342$		0			0
$343$	1.79411			0.0968731
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.5491	−	14.7508i	−1.37155	−	0.791862i	−0.380423	−	0.924813i	$-0.624222\pi$
$347$	−25.5491	−	14.7508i	−1.37155	−	0.791862i	−0.991123	+	0.132950i	$0.957555\pi$
$348$		0			0
$349$	0.753350			0.0403259			0.0201630	−	0.999797i	$-0.493581\pi$
$349$	0.753350			0.0403259			0.0201630	+	0.999797i	$0.493581\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	13.9472i		−	0.742337i	−0.928566	−	0.371168i	$-0.878957\pi$
$353$		−	13.9472i		−	0.742337i	0.928566	−	0.371168i	$-0.121043\pi$
$354$		0			0
$355$	21.9740	−	12.6867i	1.16626	−	0.673341i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.73687	−	1.00278i	−0.0916684	−	0.0529248i	0.453465	−	0.891274i	$-0.350188\pi$
$359$	−1.73687	−	1.00278i	−0.0916684	−	0.0529248i	−0.545133	+	0.838349i	$0.683521\pi$
$360$		0			0
$361$	12.3823	−	14.4110i	0.651702	−	0.758475i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.8918	−	10.9072i	0.988842	−	0.570908i
$366$		0			0
$367$	−2.98159	−	5.16427i	−0.155638	−	0.269573i	0.777653	−	0.628693i	$-0.216410\pi$
$367$	−2.98159	−	5.16427i	−0.155638	−	0.269573i	−0.933291	+	0.359121i	$0.883077\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.520636	−	0.901767i	−0.0270301	−	0.0468174i
$372$		0			0
$373$		−	18.2317i		−	0.944003i	−0.881598	−	0.472001i	$-0.843532\pi$
$373$		−	18.2317i		−	0.944003i	0.881598	−	0.472001i	$-0.156468\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.2268	−	10.5232i	−0.938727	−	0.541974i
$378$		0			0
$379$			24.2245i			1.24433i	0.782887	+	0.622164i	$0.213746\pi$
$379$			24.2245i			1.24433i	−0.782887	+	0.622164i	$0.786254\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.43857	+	14.6160i	−0.431191	+	0.746845i	−0.996976	−	0.0777079i	$-0.975240\pi$
$383$	−8.43857	+	14.6160i	−0.431191	+	0.746845i	0.565785	+	0.824553i	$0.308573\pi$
$384$		0			0
$385$	1.36347	−	2.36159i	0.0694887	−	0.120358i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.0649	−	12.7392i	1.11873	−	0.645901i	0.177656	−	0.984093i	$-0.443149\pi$
$389$	22.0649	−	12.7392i	1.11873	−	0.645901i	0.941077	+	0.338192i	$0.109815\pi$
$390$		0			0
$391$	−14.3381			−0.725111
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.89720	−	10.2143i	−0.296720	−	0.513935i
$396$		0			0
$397$	16.4070	−	28.4178i	0.823446	−	1.42625i	−0.0796559	−	0.996822i	$-0.525382\pi$
$397$	16.4070	−	28.4178i	0.823446	−	1.42625i	0.903101	−	0.429427i	$-0.141285\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−7.43184	+	12.8723i	−0.371128	+	0.642813i	−0.989739	−	0.142884i	$-0.954362\pi$
$401$	−7.43184	+	12.8723i	−0.371128	+	0.642813i	0.618611	+	0.785697i	$0.287696\pi$
$402$		0			0
$403$	−15.4937	−	26.8359i	−0.771797	−	1.33679i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−62.7231			−3.10907
$408$		0			0
$409$	19.8468	−	11.4586i	0.981361	−	0.566589i	0.0786806	−	0.996900i	$-0.474929\pi$
$409$	19.8468	−	11.4586i	0.981361	−	0.566589i	0.902681	+	0.430311i	$0.141596\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.520636	−	0.901767i	0.0256188	−	0.0443731i
$414$		0			0
$415$	−0.462489	+	0.801054i	−0.0227027	+	0.0393222i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			39.0255i			1.90652i	0.302152	+	0.953260i	$0.402295\pi$
$419$			39.0255i			1.90652i	−0.302152	+	0.953260i	$0.597705\pi$
$420$		0			0
$421$	−23.5330	−	13.5868i	−1.14693	−	0.662179i	−0.198791	−	0.980042i	$-0.563701\pi$
$421$	−23.5330	−	13.5868i	−1.14693	−	0.662179i	−0.948137	+	0.317863i	$0.897035\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			39.2709i			1.90492i
$426$		0			0
$427$	0.282855	+	0.489919i	0.0136883	+	0.0237089i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.2273	−	21.1783i	−0.588968	−	1.02012i	−0.994368	−	0.105983i	$-0.966201\pi$
$431$	−12.2273	−	21.1783i	−0.588968	−	1.02012i	0.405400	−	0.914139i	$-0.367132\pi$
$432$		0			0
$433$	−12.0455	+	6.95450i	−0.578872	+	0.334212i	−0.760685	−	0.649121i	$-0.775137\pi$
$433$	−12.0455	+	6.95450i	−0.578872	+	0.334212i	0.181813	+	0.983333i	$0.441803\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.37165	+	8.98735i	−0.304797	+	0.429923i
$438$		0			0
$439$	−32.3212	−	18.6607i	−1.54261	−	0.890624i	−0.998673	−	0.0514959i	$-0.983601\pi$
$439$	−32.3212	−	18.6607i	−1.54261	−	0.890624i	−0.543933	−	0.839128i	$-0.683066\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.7798	−	9.11046i	0.749720	−	0.432851i	−0.0758729	−	0.997118i	$-0.524174\pi$
$443$	15.7798	−	9.11046i	0.749720	−	0.432851i	0.825593	+	0.564267i	$0.190841\pi$
$444$		0			0
$445$			31.2783i			1.48274i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.75528			−0.413187			−0.206594	−	0.978427i	$-0.566238\pi$
$449$	−8.75528			−0.413187			−0.206594	+	0.978427i	$0.566238\pi$
$450$		0			0
$451$	24.9532	+	14.4067i	1.17500	+	0.678386i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.36790			−0.0641280
$456$		0			0
$457$	−11.9974			−0.561217			−0.280608	−	0.959822i	$-0.590536\pi$
$457$	−11.9974			−0.561217			−0.280608	+	0.959822i	$0.590536\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.25947	−	1.88185i	−0.151809	−	0.0876467i	0.422172	−	0.906516i	$-0.361268\pi$
$461$	−3.25947	−	1.88185i	−0.151809	−	0.0876467i	−0.573980	+	0.818869i	$0.694601\pi$
$462$		0			0
$463$	−40.9277			−1.90207			−0.951036	−	0.309081i	$-0.899979\pi$
$463$	−40.9277			−1.90207			−0.951036	+	0.309081i	$0.899979\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	23.9556i		−	1.10853i	−0.832339	−	0.554267i	$-0.812998\pi$
$467$		−	23.9556i		−	1.10853i	0.832339	−	0.554267i	$-0.187002\pi$
$468$		0			0
$469$	−0.579556	+	0.334607i	−0.0267614	+	0.0154507i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−32.2550	−	18.6224i	−1.48309	−	0.856260i
$474$		0			0
$475$	24.6156	+	17.4514i	1.12944	+	0.800725i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6.82400	−	3.93984i	0.311796	−	0.180016i	−0.335934	−	0.941886i	$-0.609052\pi$
$479$	6.82400	−	3.93984i	0.311796	−	0.180016i	0.647730	+	0.761870i	$0.275718\pi$
$480$		0			0
$481$	15.7317	+	27.2481i	0.717304	+	1.24241i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−32.3116	−	55.9654i	−1.46720	−	2.54126i
$486$		0			0
$487$		−	31.9933i		−	1.44976i	−0.688878	−	0.724878i	$-0.741896\pi$
$487$		−	31.9933i		−	1.44976i	0.688878	−	0.724878i	$-0.258104\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.45502	−	3.72681i	−0.291311	−	0.168189i	0.347222	−	0.937783i	$-0.387125\pi$
$491$	−6.45502	−	3.72681i	−0.291311	−	0.168189i	−0.638533	+	0.769594i	$0.720458\pi$
$492$		0			0
$493$		−	38.6682i		−	1.74153i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.471411	+	0.816508i	−0.0211457	+	0.0366254i
$498$		0			0
$499$	11.7230	−	20.3049i	0.524794	−	0.908970i	−0.474789	−	0.880100i	$-0.657476\pi$
$499$	11.7230	−	20.3049i	0.524794	−	0.908970i	0.999583	−	0.0288705i	$-0.00919104\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.0950195	+	0.0548595i	−0.00423671	+	0.00244606i	−0.502117	−	0.864800i	$-0.667445\pi$
$503$	−0.0950195	+	0.0548595i	−0.00423671	+	0.00244606i	0.497880	+	0.867246i	$0.334112\pi$
$504$		0			0
$505$	−27.0403			−1.20328
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.55979	−	16.5580i	−0.423730	−	0.733922i	0.572571	−	0.819855i	$-0.305946\pi$
$509$	−9.55979	−	16.5580i	−0.423730	−	0.733922i	−0.996301	+	0.0859332i	$0.972613\pi$
$510$		0			0
$511$	−0.405288	+	0.701980i	−0.0179289	+	0.0310538i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.2183	+	24.6267i	−0.626531	+	1.08518i
$516$		0			0
$517$	−0.945829	−	1.63822i	−0.0415975	−	0.0720490i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.4449			−1.46525			−0.732624	−	0.680634i	$-0.761704\pi$
$521$	−33.4449			−1.46525			−0.732624	+	0.680634i	$0.761704\pi$
$522$		0			0
$523$	−32.4569	+	18.7390i	−1.41924	+	0.819399i	−0.996232	−	0.0867263i	$-0.972359\pi$
$523$	−32.4569	+	18.7390i	−1.41924	+	0.819399i	−0.423009	+	0.906126i	$0.639026\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.4663	−	49.3051i	1.24001	−	2.14776i
$528$		0			0
$529$	−8.30604	+	14.3865i	−0.361132	+	0.625499i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	14.4535i		−	0.626052i
$534$		0			0
$535$	51.3564	+	29.6506i	2.22033	+	1.28191i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	42.9868i		−	1.85157i
$540$		0			0
$541$	18.5343	+	32.1023i	0.796850	+	1.38018i	0.921658	+	0.388004i	$0.126835\pi$
$541$	18.5343	+	32.1023i	0.796850	+	1.38018i	−0.124808	+	0.992181i	$0.539831\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.98126	+	8.62779i	0.213374	+	0.369574i
$546$		0			0
$547$	33.3773	−	19.2704i	1.42711	−	0.823944i	0.430220	−	0.902724i	$-0.358436\pi$
$547$	33.3773	−	19.2704i	1.42711	−	0.823944i	0.996892	+	0.0787805i	$0.0251026\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.2378	−	17.1836i	−1.03257	−	0.732045i
$552$		0			0
$553$	0.379540	+	0.219128i	0.0161397	+	0.00931825i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.6367	−	17.1108i	1.25575	−	0.725007i	0.283503	−	0.958971i	$-0.408503\pi$
$557$	29.6367	−	17.1108i	1.25575	−	0.725007i	0.972245	+	0.233965i	$0.0751700\pi$
$558$		0			0
$559$			18.6829i			0.790203i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.0940			0.551847			0.275923	−	0.961180i	$-0.411016\pi$
$563$	13.0940			0.551847			0.275923	+	0.961180i	$0.411016\pi$
$564$		0			0
$565$	−25.2919	−	14.6023i	−1.06404	−	0.614324i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.2587			1.14274			0.571371	−	0.820692i	$-0.306412\pi$
$569$	27.2587			1.14274			0.571371	+	0.820692i	$0.306412\pi$
$570$		0			0
$571$	15.6836			0.656339			0.328169	−	0.944619i	$-0.393568\pi$
$571$	15.6836			0.656339			0.328169	+	0.944619i	$0.393568\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15.1519	−	8.74797i	−0.631879	−	0.364816i
$576$		0			0
$577$	−5.95534			−0.247924			−0.123962	−	0.992287i	$-0.539560\pi$
$577$	−5.95534			−0.247924			−0.123962	+	0.992287i	$0.539560\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	0.0343702i		−	0.00142592i
$582$		0			0
$583$	−43.2634	+	24.9781i	−1.79179	+	1.03449i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.1425	+	8.16520i	0.583725	+	0.337014i	0.762612	−	0.646856i	$-0.223916\pi$
$587$	14.1425	+	8.16520i	0.583725	+	0.337014i	−0.178887	+	0.983870i	$0.557250\pi$
$588$		0			0
$589$	−18.2551	−	39.7535i	−0.752189	−	1.63801i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−26.5930	+	15.3535i	−1.09204	+	0.630491i	−0.934120	−	0.356960i	$-0.883813\pi$
$593$	−26.5930	+	15.3535i	−1.09204	+	0.630491i	−0.157923	+	0.987451i	$0.550480\pi$
$594$		0			0
$595$	−1.25660	−	2.17650i	−0.0515157	−	0.0892279i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14.5825	+	25.2576i	0.595823	+	1.03200i	0.993430	+	0.114440i	$0.0365074\pi$
$599$	14.5825	+	25.2576i	0.595823	+	1.03200i	−0.397607	+	0.917556i	$0.630159\pi$
$600$		0			0
$601$		−	9.18206i		−	0.374544i	−0.982308	−	0.187272i	$-0.940035\pi$
$601$		−	9.18206i		−	0.374544i	0.982308	−	0.187272i	$-0.0599646\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−80.4073	−	46.4232i	−3.26902	−	1.88737i
$606$		0			0
$607$		−	22.1903i		−	0.900678i	−0.892858	−	0.450339i	$-0.851303\pi$
$607$		−	22.1903i		−	0.900678i	0.892858	−	0.450339i	$-0.148697\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.474451	+	0.821772i	−0.0191942	+	0.0332454i
$612$		0			0
$613$	11.9682	−	20.7295i	0.483389	−	0.837255i	−0.516429	−	0.856330i	$-0.672739\pi$
$613$	11.9682	−	20.7295i	0.483389	−	0.837255i	0.999818	+	0.0190752i	$0.00607221\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.193621	+	0.111787i	−0.00779487	+	0.00450037i	−0.503892	−	0.863766i	$-0.668099\pi$
$617$	−0.193621	+	0.111787i	−0.00779487	+	0.00450037i	0.496098	+	0.868267i	$0.334766\pi$
$618$		0			0
$619$	−33.5284			−1.34762			−0.673811	−	0.738904i	$-0.735344\pi$
$619$	−33.5284			−1.34762			−0.673811	+	0.738904i	$0.735344\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.581119	−	1.00653i	−0.0232820	−	0.0403257i
$624$		0			0
$625$	5.84611	−	10.1258i	0.233845	−	0.405031i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−28.9035	+	50.0624i	−1.15246	+	1.99612i
$630$		0			0
$631$	−16.6017	−	28.7549i	−0.660902	−	1.14472i	−0.980379	−	0.197121i	$-0.936841\pi$
$631$	−16.6017	−	28.7549i	−0.660902	−	1.14472i	0.319477	−	0.947594i	$-0.396493\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.08039			0.161925
$636$		0			0
$637$	−18.6743	+	10.7816i	−0.739902	+	0.427183i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.5276	+	21.6984i	−0.494810	+	0.857035i	−0.999982	−	0.00598312i	$-0.998096\pi$
$641$	−12.5276	+	21.6984i	−0.494810	+	0.857035i	0.505173	+	0.863018i	$0.331429\pi$
$642$		0			0
$643$	3.49677	−	6.05659i	0.137899	−	0.238848i	−0.788802	−	0.614647i	$-0.789298\pi$
$643$	3.49677	−	6.05659i	0.137899	−	0.238848i	0.926701	+	0.375799i	$0.122632\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	33.0282i		−	1.29847i	−0.760587	−	0.649236i	$-0.775089\pi$
$647$		−	33.0282i		−	1.29847i	0.760587	−	0.649236i	$-0.224911\pi$
$648$		0			0
$649$	−43.2634	−	24.9781i	−1.69824	−	0.980478i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.3504i			1.03117i	0.856838	+	0.515586i	$0.172426\pi$
$653$			26.3504i			1.03117i	−0.856838	+	0.515586i	$0.827574\pi$
$654$		0			0
$655$	0.530561	+	0.918958i	0.0207307	+	0.0359067i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.27115	−	16.0581i	−0.361153	−	0.625535i	0.626998	−	0.779021i	$-0.284283\pi$
$659$	−9.27115	−	16.0581i	−0.361153	−	0.625535i	−0.988151	+	0.153486i	$0.950950\pi$
$660$		0			0
$661$	9.41916	−	5.43816i	0.366363	−	0.211520i	−0.305505	−	0.952190i	$-0.598825\pi$
$661$	9.41916	−	5.43816i	0.366363	−	0.211520i	0.671868	+	0.740671i	$0.265492\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.92268	−	0.179547i	−0.0745583	−	0.00696255i
$666$		0			0
$667$	14.9194	+	8.61372i	0.577682	+	0.333525i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	23.5045	−	13.5703i	0.907381	−	0.523876i
$672$		0			0
$673$		−	18.1654i		−	0.700225i	−0.936708	−	0.350113i	$-0.886143\pi$
$673$		−	18.1654i		−	0.700225i	0.936708	−	0.350113i	$-0.113857\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.7104			−0.642233			−0.321117	−	0.947040i	$-0.604058\pi$
$677$	−16.7104			−0.642233			−0.321117	+	0.947040i	$0.604058\pi$
$678$		0			0
$679$	2.07956	+	1.20063i	0.0798060	+	0.0460760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−24.6896			−0.944722			−0.472361	−	0.881405i	$-0.656598\pi$
$683$	−24.6896			−0.944722			−0.472361	+	0.881405i	$0.656598\pi$
$684$		0			0
$685$	41.8258			1.59808
$686$		0			0
$687$		0			0
$688$		0			0
$689$	21.7020	+	12.5296i	0.826779	+	0.477341i
$690$		0			0
$691$	−8.76981			−0.333619			−0.166810	−	0.985989i	$-0.553347\pi$
$691$	−8.76981			−0.333619			−0.166810	+	0.985989i	$0.553347\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24.5564i			0.931476i
$696$		0			0
$697$	22.9974	−	13.2776i	0.871090	−	0.502924i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.40318	+	2.54218i	0.166306	+	0.0960167i	0.580843	−	0.814016i	$-0.302723\pi$
$701$	4.40318	+	2.54218i	0.166306	+	0.0960167i	−0.414537	+	0.910032i	$0.636056\pi$
$702$		0			0
$703$	18.5355	+	40.3642i	0.699081	+	1.52236i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.870149	−	0.502381i	0.0327253	−	0.0188940i
$708$		0			0
$709$	−3.03554	−	5.25771i	−0.114002	−	0.197457i	0.803378	−	0.595469i	$-0.203034\pi$
$709$	−3.03554	−	5.25771i	−0.114002	−	0.197457i	−0.917380	+	0.398012i	$0.869700\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.6823	+	21.9664i	0.474955	+	0.822646i
$714$		0			0
$715$			65.6265i			2.45429i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.0422	+	6.37520i	0.411803	+	0.237755i	0.691564	−	0.722315i	$-0.256922\pi$
$719$	11.0422	+	6.37520i	0.411803	+	0.237755i	−0.279761	+	0.960070i	$0.590255\pi$
$720$		0			0
$721$		−	1.05664i		−	0.0393514i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	23.5922	−	40.8630i	0.876194	−	1.51761i
$726$		0			0
$727$	6.69117	−	11.5895i	0.248162	−	0.429829i	−0.714854	−	0.699274i	$-0.753507\pi$
$727$	6.69117	−	11.5895i	0.248162	−	0.429829i	0.963016	+	0.269445i	$0.0868402\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−29.7270	+	17.1629i	−1.09949	+	0.634792i
$732$		0			0
$733$	29.7204			1.09775			0.548875	−	0.835905i	$-0.315056\pi$
$733$	29.7204			1.09775			0.548875	+	0.835905i	$0.315056\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16.0532	+	27.8049i	0.591327	+	1.02421i
$738$		0			0
$739$	24.0889	−	41.7232i	0.886124	−	1.53481i	0.0417050	−	0.999130i	$-0.486721\pi$
$739$	24.0889	−	41.7232i	0.886124	−	1.53481i	0.844419	−	0.535683i	$-0.179946\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.6384	+	20.1582i	−0.426970	+	0.739534i	−0.996602	−	0.0823661i	$-0.973752\pi$
$743$	−11.6384	+	20.1582i	−0.426970	+	0.739534i	0.569632	+	0.821900i	$0.307086\pi$
$744$		0			0
$745$	−11.3470	−	19.6536i	−0.415722	−	0.720052i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.20351			−0.0805145
$750$		0			0
$751$	33.2476	−	19.1955i	1.21322	−	0.700453i	0.249761	−	0.968307i	$-0.419648\pi$
$751$	33.2476	−	19.1955i	1.21322	−	0.700453i	0.963459	+	0.267854i	$0.0863145\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	19.8152	−	34.3209i	0.721148	−	1.24906i
$756$		0			0
$757$	20.1942	−	34.9774i	0.733971	−	1.27127i	−0.221203	−	0.975228i	$-0.570998\pi$
$757$	20.1942	−	34.9774i	0.733971	−	1.27127i	0.955173	−	0.296047i	$-0.0956684\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41.5304i		−	1.50548i	−0.658319	−	0.752739i	$-0.728732\pi$
$761$		−	41.5304i		−	1.50548i	0.658319	−	0.752739i	$-0.271268\pi$
$762$		0			0
$763$	−0.320591	−	0.185093i	−0.0116062	−	0.00670082i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			25.0593i			0.904838i
$768$		0			0
$769$	4.89531	+	8.47892i	0.176529	+	0.305758i	0.940689	−	0.339269i	$-0.110180\pi$
$769$	4.89531	+	8.47892i	0.176529	+	0.305758i	−0.764160	+	0.645027i	$0.776846\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.935403	−	1.62016i	−0.0336441	−	0.0582733i	0.848713	−	0.528854i	$-0.177378\pi$
$773$	−0.935403	−	1.62016i	−0.0336441	−	0.0582733i	−0.882357	+	0.470580i	$0.844045\pi$
$774$		0			0
$775$	60.1639	−	34.7357i	2.16115	−	1.24774i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.89714	−	20.3155i	0.0679722	−	0.727878i
$780$		0			0
$781$	39.1730	+	22.6165i	1.40172	+	0.809284i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−56.3515	+	32.5345i	−2.01127	+	1.16121i
$786$		0			0
$787$			15.6323i			0.557232i	0.960403	+	0.278616i	$0.0898756\pi$
$787$			15.6323i			0.557232i	−0.960403	+	0.278616i	$0.910124\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.08518			0.0385846
$792$		0			0
$793$	−11.7904	−	6.80720i	−0.418690	−	0.241731i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	9.67372			0.342661			0.171330	−	0.985214i	$-0.445193\pi$
$797$	9.67372			0.342661			0.171330	+	0.985214i	$0.445193\pi$
$798$		0			0
$799$	−1.74340			−0.0616769
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33.6783	+	19.4442i	1.18848	+	0.686171i
$804$		0			0
$805$	1.11968			0.0394636
$806$		0			0
$807$		0			0
$808$		0			0
$809$			39.9424i			1.40430i	0.712030	+	0.702149i	$0.247776\pi$
$809$			39.9424i			1.40430i	−0.712030	+	0.702149i	$0.752224\pi$
$810$		0			0
$811$	34.7438	−	20.0594i	1.22002	−	0.704380i	0.255099	−	0.966915i	$-0.417892\pi$
$811$	34.7438	−	20.0594i	1.22002	−	0.704380i	0.964922	+	0.262535i	$0.0845587\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.69673	+	2.13431i	0.129491	+	0.0747616i
$816$		0			0
$817$	−2.45228	+	26.2602i	−0.0857946	+	0.918729i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.1307	−	11.6225i	0.702566	−	0.405627i	−0.105736	−	0.994394i	$-0.533720\pi$
$821$	20.1307	−	11.6225i	0.702566	−	0.405627i	0.808302	+	0.588768i	$0.200387\pi$
$822$		0			0
$823$	8.56092	+	14.8280i	0.298415	+	0.516870i	0.975774	−	0.218783i	$-0.0702087\pi$
$823$	8.56092	+	14.8280i	0.298415	+	0.516870i	−0.677359	+	0.735653i	$0.736875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.8529	+	43.0465i	0.864221	+	1.49687i	0.867819	+	0.496881i	$0.165521\pi$
$827$	24.8529	+	43.0465i	0.864221	+	1.49687i	−0.00359800	+	0.999994i	$0.501145\pi$
$828$		0			0
$829$			28.5577i			0.991851i	0.868365	+	0.495925i	$0.165171\pi$
$829$			28.5577i			0.991851i	−0.868365	+	0.495925i	$0.834829\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.3099	−	19.8088i	−1.18877	−	0.686335i
$834$		0			0
$835$		−	30.0966i		−	1.04154i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.62736	−	8.01483i	0.159754	−	0.276703i	−0.775026	−	0.631930i	$-0.782263\pi$
$839$	4.62736	−	8.01483i	0.159754	−	0.276703i	0.934780	+	0.355227i	$0.115596\pi$
$840$		0			0
$841$	−8.73019	+	15.1211i	−0.301041	+	0.521418i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.3643	+	5.98383i	−0.356543	+	0.205850i
$846$		0			0
$847$	3.44997			0.118543
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.8771	−	22.3038i	−0.441421	−	0.764563i
$852$		0			0
$853$	16.4398	−	28.4745i	0.562887	−	0.974949i	−0.434356	−	0.900741i	$-0.643024\pi$
$853$	16.4398	−	28.4745i	0.562887	−	0.974949i	0.997243	−	0.0742075i	$-0.0236427\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.7060	−	28.9357i	0.570667	−	0.988424i	−0.425831	−	0.904803i	$-0.640018\pi$
$857$	16.7060	−	28.9357i	0.570667	−	0.988424i	0.996498	−	0.0836213i	$-0.0266486\pi$
$858$		0			0
$859$	−26.0074	−	45.0462i	−0.887362	−	1.53696i	−0.842983	−	0.537941i	$-0.819202\pi$
$859$	−26.0074	−	45.0462i	−0.887362	−	1.53696i	−0.0443790	−	0.999015i	$-0.514131\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.1258			0.889333			0.444666	−	0.895696i	$-0.353322\pi$
$863$	26.1258			0.889333			0.444666	+	0.895696i	$0.353322\pi$
$864$		0			0
$865$	35.7673	−	20.6502i	1.21612	−	0.702129i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.5129	−	18.2089i	0.356626	−	0.617695i
$870$		0			0
$871$	8.05267	−	13.9476i	0.272854	−	0.472597i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	0.851653i		−	0.0287911i
$876$		0			0
$877$	41.7441	+	24.1010i	1.40960	+	0.813832i	0.995349	−	0.0963324i	$-0.0307112\pi$
$877$	41.7441	+	24.1010i	1.40960	+	0.813832i	0.414248	+	0.910164i	$0.364045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1.73279i		−	0.0583793i	−0.999574	−	0.0291897i	$-0.990707\pi$
$881$		−	1.73279i		−	0.0583793i	0.999574	−	0.0291897i	$-0.00929268\pi$
$882$		0			0
$883$	−2.13330	−	3.69499i	−0.0717914	−	0.124346i	0.827895	−	0.560883i	$-0.189538\pi$
$883$	−2.13330	−	3.69499i	−0.0717914	−	0.124346i	−0.899686	+	0.436537i	$0.856205\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−14.3211	−	24.8049i	−0.480855	−	0.832865i	0.518904	−	0.854833i	$-0.326340\pi$
$887$	−14.3211	−	24.8049i	−0.480855	−	0.832865i	−0.999759	+	0.0219673i	$0.993007\pi$
$888$		0			0
$889$	−0.131306	+	0.0758094i	−0.00440385	+	0.00254256i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.774739	+	1.09279i	−0.0259257	+	0.0365687i
$894$		0			0
$895$	36.6743	+	21.1739i	1.22589	+	0.707766i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−59.2406	+	34.2026i	−1.97579	+	1.14072i
$900$		0			0
$901$			46.0409i			1.53384i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.2413			0.639603
$906$		0			0
$907$	−17.1153	−	9.88155i	−0.568305	−	0.328111i	0.188167	−	0.982137i	$-0.439745\pi$
$907$	−17.1153	−	9.88155i	−0.568305	−	0.328111i	−0.756472	+	0.654026i	$0.773079\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.0355			0.531278			0.265639	−	0.964073i	$-0.414417\pi$
$911$	16.0355			0.531278			0.265639	+	0.964073i	$0.414417\pi$
$912$		0			0
$913$	−1.64895			−0.0545724
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.0341466	−	0.0197145i	−0.00112762	−	0.000651031i
$918$		0			0
$919$	−35.5068			−1.17126			−0.585630	−	0.810579i	$-0.699153\pi$
$919$	−35.5068			−1.17126			−0.585630	+	0.810579i	$0.699153\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	22.6900i		−	0.746851i
$924$		0			0
$925$	−61.0881	+	35.2692i	−2.00856	+	1.15964i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29.5144	−	17.0402i	−0.968337	−	0.559070i	−0.0696082	−	0.997574i	$-0.522175\pi$
$929$	−29.5144	−	17.0402i	−0.968337	−	0.559070i	−0.898729	+	0.438505i	$0.855508\pi$
$930$		0			0
$931$	−27.6633	+	12.7032i	−0.906626	+	0.416330i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−104.420	+	60.2871i	−3.41491	+	1.97160i
$936$		0			0
$937$	8.05895	+	13.9585i	0.263274	+	0.456005i	0.967110	−	0.254358i	$-0.0818642\pi$
$937$	8.05895	+	13.9585i	0.263274	+	0.456005i	−0.703836	+	0.710363i	$0.748531\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	14.8370	+	25.6984i	0.483671	+	0.837743i	0.999824	−	0.0187537i	$-0.00596984\pi$
$941$	14.8370	+	25.6984i	0.483671	+	0.837743i	−0.516153	+	0.856496i	$0.672637\pi$
$942$		0			0
$943$			11.8308i			0.385265i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−39.4627	−	22.7838i	−1.28236	−	0.740373i	−0.305084	−	0.952325i	$-0.598685\pi$
$947$	−39.4627	−	22.7838i	−1.28236	−	0.740373i	−0.977280	+	0.211952i	$0.932018\pi$
$948$		0			0
$949$		−	19.5074i		−	0.633236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−6.56950	+	11.3787i	−0.212807	+	0.368593i	−0.952592	−	0.304251i	$-0.901594\pi$
$953$	−6.56950	+	11.3787i	−0.212807	+	0.368593i	0.739785	+	0.672843i	$0.234927\pi$
$954$		0			0
$955$	−0.954258	+	1.65282i	−0.0308791	+	0.0534841i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.34594	+	0.777079i	−0.0434627	+	0.0250932i
$960$		0			0
$961$	−69.7153			−2.24888
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.09809	+	10.5622i	0.196304	+	0.340009i
$966$		0			0
$967$	−6.41811	+	11.1165i	−0.206392	+	0.357482i	−0.950575	−	0.310494i	$-0.899506\pi$
$967$	−6.41811	+	11.1165i	−0.206392	+	0.357482i	0.744183	+	0.667976i	$0.232839\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.87774	+	10.1805i	−0.188626	+	0.326709i	−0.944792	−	0.327670i	$-0.893737\pi$
$971$	−5.87774	+	10.1805i	−0.188626	+	0.326709i	0.756167	+	0.654379i	$0.227070\pi$
$972$		0			0
$973$	−0.456231	−	0.790216i	−0.0146261	−	0.0253332i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	33.6204			1.07561			0.537806	−	0.843069i	$-0.319253\pi$
$977$	33.6204			1.07561			0.537806	+	0.843069i	$0.319253\pi$
$978$		0			0
$979$	−48.2894	+	27.8799i	−1.54334	+	0.891045i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.2420	−	21.2037i	0.390458	−	0.676294i	−0.602052	−	0.798457i	$-0.705650\pi$
$983$	12.2420	−	21.2037i	0.390458	−	0.676294i	0.992510	+	0.122163i	$0.0389832\pi$
$984$		0			0
$985$	30.7519	−	53.2638i	0.979836	−	1.69713i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	15.2928i		−	0.486282i
$990$		0			0
$991$	−4.56804	−	2.63736i	−0.145108	−	0.0837784i	0.425688	−	0.904870i	$-0.360032\pi$
$991$	−4.56804	−	2.63736i	−0.145108	−	0.0837784i	−0.570796	+	0.821092i	$0.693365\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			48.5635i			1.53957i
$996$		0			0
$997$	−0.422417	−	0.731648i	−0.0133781	−	0.0231715i	0.859259	−	0.511541i	$-0.170925\pi$
$997$	−0.422417	−	0.731648i	−0.0133781	−	0.0231715i	−0.872637	+	0.488369i	$0.837592\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.d.1889.8		16
3.2	odd	2	inner	2736.2.dc.d.1889.1		16
4.3	odd	2		684.2.bk.a.521.8	yes	16
12.11	even	2		684.2.bk.a.521.1	yes	16
19.12	odd	6	inner	2736.2.dc.d.449.1		16
57.50	even	6	inner	2736.2.dc.d.449.8		16
76.31	even	6		684.2.bk.a.449.1	✓	16
228.107	odd	6		684.2.bk.a.449.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.bk.a.449.1	✓	16	76.31	even	6
684.2.bk.a.449.8	yes	16	228.107	odd	6
684.2.bk.a.521.1	yes	16	12.11	even	2
684.2.bk.a.521.8	yes	16	4.3	odd	2
2736.2.dc.d.449.1		16	19.12	odd	6	inner
2736.2.dc.d.449.8		16	57.50	even	6	inner
2736.2.dc.d.1889.1		16	3.2	odd	2	inner
2736.2.dc.d.1889.8		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(2.99029 + 1.72644i) q^{5} -0.128302 q^{7} +O(q^{10})\)	\(q+(2.99029 + 1.72644i) q^{5} -0.128302 q^{7} +6.15544i q^{11} +(2.67404 - 1.54386i) q^{13} +(4.91297 + 2.83650i) q^{17} +(3.96121 - 1.81902i) q^{19} +(-2.18882 + 1.26372i) q^{23} +(3.46121 + 5.99499i) q^{25} +(-3.40809 - 5.90299i) q^{29} -10.0357i q^{31} +(-0.383660 - 0.221506i) q^{35} +10.1899i q^{37} +(2.34048 - 4.05384i) q^{41} +(-3.02536 + 5.24008i) q^{43} +(-0.266142 + 0.153657i) q^{47} -6.98354 q^{49} +(4.05789 + 7.02848i) q^{53} +(-10.6270 + 18.4065i) q^{55} +(-4.05789 + 7.02848i) q^{59} +(-2.20460 - 3.81849i) q^{61} +10.6615 q^{65} +(4.51713 - 2.60797i) q^{67} +(3.67423 - 6.36396i) q^{71} +(3.15886 - 5.47131i) q^{73} -0.789755i q^{77} +(-2.95818 - 1.70791i) q^{79} +0.267885i q^{83} +(9.79411 + 16.9639i) q^{85} +(4.52931 + 7.84499i) q^{89} +(-0.343085 + 0.198080i) q^{91} +(14.9856 + 1.39941i) q^{95} +(-16.2083 - 9.35787i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) \) 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

Introduction

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