LMFDB - Embedded newform 2736.2.bm.g.1855.1

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 304)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1855.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.1855
Dual form			2736.2.bm.g.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{5} +3.46410i q^{7} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(4.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-7.50000 - 4.33013i) q^{29} +4.00000 q^{31} +(-9.00000 + 5.19615i) q^{35} +(-7.50000 + 4.33013i) q^{41} +(10.5000 - 6.06218i) q^{43} +(-1.50000 - 0.866025i) q^{47} -5.00000 q^{49} +(-1.50000 - 0.866025i) q^{53} +(-9.00000 + 5.19615i) q^{55} +(-1.50000 - 2.59808i) q^{59} +(-3.50000 + 6.06218i) q^{61} -15.5885i q^{65} +(2.50000 - 4.33013i) q^{67} +(4.50000 + 7.79423i) q^{71} +(-3.50000 - 6.06218i) q^{73} -12.0000 q^{77} +(3.50000 + 6.06218i) q^{79} -3.46410i q^{83} +(-4.50000 + 7.79423i) q^{85} +(-7.50000 - 4.33013i) q^{89} +(9.00000 - 15.5885i) q^{91} +(-10.5000 - 7.79423i) q^{95} +(7.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 9 q^{13} + 3 q^{17} - 8 q^{19} + 9 q^{23} - 4 q^{25} - 15 q^{29} + 8 q^{31} - 18 q^{35} - 15 q^{41} + 21 q^{43} - 3 q^{47} - 10 q^{49} - 3 q^{53} - 18 q^{55} - 3 q^{59} - 7 q^{61} + 5 q^{67}+ \cdots + 15 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 9 * q^13 + 3 * q^17 - 8 * q^19 + 9 * q^23 - 4 * q^25 - 15 * q^29 + 8 * q^31 - 18 * q^35 - 15 * q^41 + 21 * q^43 - 3 * q^47 - 10 * q^49 - 3 * q^53 - 18 * q^55 - 3 * q^59 - 7 * q^61 + 5 * q^67 + 9 * q^71 - 7 * q^73 - 24 * q^77 + 7 * q^79 - 9 * q^85 - 15 * q^89 + 18 * q^91 - 21 * q^95 + 15 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−4.50000	−	2.59808i	−1.24808	−	0.720577i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.50000	+	2.59808i	0.938315	+	0.541736i	0.889432	−	0.457068i	$-0.151100\pi$
$23$	4.50000	+	2.59808i	0.938315	+	0.541736i	0.0488832	+	0.998805i	$0.484434\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.50000	−	4.33013i	−1.39272	−	0.804084i	−0.399100	−	0.916907i	$-0.630677\pi$
$29$	−7.50000	−	4.33013i	−1.39272	−	0.804084i	−0.993615	+	0.112823i	$0.964011\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.00000	+	5.19615i	−1.52128	+	0.878310i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.50000	+	4.33013i	−1.17130	+	0.676252i	−0.953987	−	0.299849i	$-0.903064\pi$
$41$	−7.50000	+	4.33013i	−1.17130	+	0.676252i	−0.217317	+	0.976101i	$0.569730\pi$
$42$		0			0
$43$	10.5000	−	6.06218i	1.60123	−	0.924473i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	10.5000	−	6.06218i	1.60123	−	0.924473i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.50000	−	0.866025i	−0.218797	−	0.126323i	0.386596	−	0.922249i	$-0.373651\pi$
$47$	−1.50000	−	0.866025i	−0.218797	−	0.126323i	−0.605393	+	0.795926i	$0.706984\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.50000	−	0.866025i	−0.206041	−	0.118958i	0.393429	−	0.919355i	$-0.371289\pi$
$53$	−1.50000	−	0.866025i	−0.206041	−	0.118958i	−0.599470	+	0.800397i	$0.704622\pi$
$54$		0			0
$55$	−9.00000	+	5.19615i	−1.21356	+	0.700649i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	15.5885i		−	1.93351i
$66$		0			0
$67$	2.50000	−	4.33013i	0.305424	−	0.529009i	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	2.50000	−	4.33013i	0.305424	−	0.529009i	0.977356	+	0.211604i	$0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.50000	+	7.79423i	0.534052	+	0.925005i	0.999209	+	0.0397765i	$0.0126646\pi$
$71$	4.50000	+	7.79423i	0.534052	+	0.925005i	−0.465157	+	0.885228i	$0.654002\pi$
$72$		0			0
$73$	−3.50000	−	6.06218i	−0.409644	−	0.709524i	0.585206	−	0.810885i	$-0.301014\pi$
$73$	−3.50000	−	6.06218i	−0.409644	−	0.709524i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	0.992945	−	0.118578i	$-0.0378336\pi$
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	−0.599164	+	0.800626i	$0.704500\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.46410i		−	0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$		−	3.46410i		−	0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.50000	−	4.33013i	−0.794998	−	0.458993i	0.0467209	−	0.998908i	$-0.485123\pi$
$89$	−7.50000	−	4.33013i	−0.794998	−	0.458993i	−0.841719	+	0.539915i	$0.818456\pi$
$90$		0			0
$91$	9.00000	−	15.5885i	0.943456	−	1.63411i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.5000	−	7.79423i	−1.07728	−	0.799671i
$96$		0			0
$97$	7.50000	−	4.33013i	0.761510	−	0.439658i	−0.0683279	−	0.997663i	$-0.521766\pi$
$97$	7.50000	−	4.33013i	0.761510	−	0.439658i	0.829837	+	0.558005i	$0.188433\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	13.5000	−	7.79423i	1.29307	−	0.746552i	0.313869	−	0.949466i	$-0.398375\pi$
$109$	13.5000	−	7.79423i	1.29307	−	0.746552i	0.979196	+	0.202915i	$0.0650414\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	6.92820i		−	0.651751i	−0.945413	−	0.325875i	$-0.894341\pi$
$113$		−	6.92820i		−	0.651751i	0.945413	−	0.325875i	$-0.105659\pi$
$114$		0			0
$115$			15.5885i			1.45363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.00000	+	5.19615i	−0.825029	+	0.476331i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	−3.50000	+	6.06218i	−0.310575	+	0.537931i	−0.978487	−	0.206309i	$-0.933855\pi$
$127$	−3.50000	+	6.06218i	−0.310575	+	0.537931i	0.667912	+	0.744240i	$0.267188\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.50000	+	2.59808i	−0.393167	+	0.226995i	−0.683531	−	0.729921i	$-0.739557\pi$
$131$	−4.50000	+	2.59808i	−0.393167	+	0.226995i	0.290365	+	0.956916i	$0.406223\pi$
$132$		0			0
$133$	−6.00000	−	13.8564i	−0.520266	−	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	0.922961	+	0.384893i	$0.125762\pi$
$138$		0			0
$139$	−16.5000	−	9.52628i	−1.39951	−	0.808008i	−0.405170	−	0.914241i	$-0.632788\pi$
$139$	−16.5000	−	9.52628i	−1.39951	−	0.808008i	−0.994341	+	0.106233i	$0.966121\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.00000	−	15.5885i	0.752618	−	1.30357i
$144$		0			0
$145$		−	25.9808i		−	2.15758i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.50000	+	12.9904i	0.614424	+	1.06421i	0.990485	+	0.137619i	$0.0439449\pi$
$149$	7.50000	+	12.9904i	0.614424	+	1.06421i	−0.376061	+	0.926595i	$0.622722\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.00000	+	10.3923i	0.481932	+	0.834730i
$156$		0			0
$157$	2.50000	+	4.33013i	0.199522	+	0.345582i	0.948373	−	0.317156i	$-0.102728\pi$
$157$	2.50000	+	4.33013i	0.199522	+	0.345582i	−0.748852	+	0.662738i	$0.769394\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.00000	+	15.5885i	−0.709299	+	1.22854i
$162$		0			0
$163$			10.3923i			0.813988i	0.913431	+	0.406994i	$0.133423\pi$
$163$			10.3923i			0.813988i	−0.913431	+	0.406994i	$0.866577\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.5000	+	18.1865i	−0.812514	+	1.40732i	0.0985846	+	0.995129i	$0.468568\pi$
$167$	−10.5000	+	18.1865i	−0.812514	+	1.40732i	−0.911099	+	0.412188i	$0.864765\pi$
$168$		0			0
$169$	7.00000	+	12.1244i	0.538462	+	0.932643i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.50000	+	4.33013i	−0.570214	+	0.329213i	−0.757235	−	0.653143i	$-0.773450\pi$
$173$	−7.50000	+	4.33013i	−0.570214	+	0.329213i	0.187021	+	0.982356i	$0.440117\pi$
$174$		0			0
$175$	−12.0000	−	6.92820i	−0.907115	−	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	1.50000	+	0.866025i	0.111494	+	0.0643712i	0.554710	−	0.832044i	$-0.312829\pi$
$181$	1.50000	+	0.866025i	0.111494	+	0.0643712i	−0.443216	+	0.896415i	$0.646163\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−9.00000	+	5.19615i	−0.658145	+	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			24.2487i			1.75458i	0.479965	+	0.877288i	$0.340649\pi$
$191$			24.2487i			1.75458i	−0.479965	+	0.877288i	$0.659351\pi$
$192$		0			0
$193$	−10.5000	+	6.06218i	−0.755807	+	0.436365i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−10.5000	+	6.06218i	−0.755807	+	0.436365i	0.0719816	+	0.997406i	$0.477068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	−10.5000	−	6.06218i	−0.744325	−	0.429736i	0.0793146	−	0.996850i	$-0.474727\pi$
$199$	−10.5000	−	6.06218i	−0.744325	−	0.429736i	−0.823640	+	0.567113i	$0.808060\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.0000	−	25.9808i	1.05279	−	1.82349i
$204$		0			0
$205$	−22.5000	−	12.9904i	−1.57147	−	0.907288i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000	−	13.8564i	−0.415029	−	0.958468i
$210$		0			0
$211$	11.5000	+	19.9186i	0.791693	+	1.37125i	0.924918	+	0.380166i	$0.124133\pi$
$211$	11.5000	+	19.9186i	0.791693	+	1.37125i	−0.133226	+	0.991086i	$0.542533\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	31.5000	+	18.1865i	2.14828	+	1.24031i
$216$		0			0
$217$			13.8564i			0.940634i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	15.5885i		−	1.04859i
$222$		0			0
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	0.986267	+	0.165161i	$0.0528144\pi$
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	−0.350100	+	0.936713i	$0.613852\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.5000	+	23.3827i	0.884414	+	1.53185i	0.846383	+	0.532574i	$0.178775\pi$
$233$	13.5000	+	23.3827i	0.884414	+	1.53185i	0.0380310	+	0.999277i	$0.487891\pi$
$234$		0			0
$235$		−	5.19615i		−	0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			17.3205i			1.12037i	0.828367	+	0.560185i	$0.189270\pi$
$239$			17.3205i			1.12037i	−0.828367	+	0.560185i	$0.810730\pi$
$240$		0			0
$241$	19.5000	+	11.2583i	1.25611	+	0.725213i	0.972315	−	0.233674i	$-0.0750747\pi$
$241$	19.5000	+	11.2583i	1.25611	+	0.725213i	0.283790	+	0.958886i	$0.408408\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.50000	−	12.9904i	−0.479157	−	0.829925i
$246$		0			0
$247$	22.5000	+	2.59808i	1.43164	+	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.50000	+	2.59808i	0.284037	+	0.163989i	0.635250	−	0.772307i	$-0.280897\pi$
$251$	4.50000	+	2.59808i	0.284037	+	0.163989i	−0.351212	+	0.936296i	$0.614230\pi$
$252$		0			0
$253$	−9.00000	+	15.5885i	−0.565825	+	0.980038i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.5000	−	7.79423i	−0.842107	−	0.486191i	0.0158730	−	0.999874i	$-0.494947\pi$
$257$	−13.5000	−	7.79423i	−0.842107	−	0.486191i	−0.857980	+	0.513683i	$0.828281\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.5000	+	6.06218i	−0.647458	+	0.373810i	−0.787482	−	0.616338i	$-0.788615\pi$
$263$	−10.5000	+	6.06218i	−0.647458	+	0.373810i	0.140024	+	0.990148i	$0.455282\pi$
$264$		0			0
$265$		−	5.19615i		−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.5000	−	12.9904i	1.37185	−	0.792038i	0.380688	−	0.924703i	$-0.375687\pi$
$269$	22.5000	−	12.9904i	1.37185	−	0.792038i	0.991161	+	0.132666i	$0.0423537\pi$
$270$		0			0
$271$	10.5000	−	6.06218i	0.637830	−	0.368251i	−0.145948	−	0.989292i	$-0.546623\pi$
$271$	10.5000	−	6.06218i	0.637830	−	0.368251i	0.783778	+	0.621041i	$0.213290\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.0000	−	6.92820i	−0.723627	−	0.417786i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.5000	−	7.79423i	−0.805342	−	0.464965i	0.0399934	−	0.999200i	$-0.487266\pi$
$281$	−13.5000	−	7.79423i	−0.805342	−	0.464965i	−0.845336	+	0.534235i	$0.820600\pi$
$282$		0			0
$283$	4.50000	−	2.59808i	0.267497	−	0.154440i	−0.360252	−	0.932855i	$-0.617309\pi$
$283$	4.50000	−	2.59808i	0.267497	−	0.154440i	0.627750	+	0.778415i	$0.283976\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−15.0000	−	25.9808i	−0.885422	−	1.53360i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$	4.50000	−	7.79423i	0.262000	−	0.453798i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.5000	−	23.3827i	−0.780725	−	1.35226i
$300$		0			0
$301$	21.0000	+	36.3731i	1.21042	+	2.09651i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−21.0000			−1.20246
$306$		0			0
$307$	−2.50000	−	4.33013i	−0.142683	−	0.247133i	0.785823	−	0.618451i	$-0.212239\pi$
$307$	−2.50000	−	4.33013i	−0.142683	−	0.247133i	−0.928506	+	0.371318i	$0.878906\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			3.46410i			0.196431i	0.995165	+	0.0982156i	$0.0313135\pi$
$311$			3.46410i			0.196431i	−0.995165	+	0.0982156i	$0.968687\pi$
$312$		0			0
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	0.462093	+	0.886831i	$0.347098\pi$
$313$	−9.50000	+	16.4545i	−0.536972	+	0.930062i	−0.999065	+	0.0432311i	$0.986235\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.50000	+	2.59808i	0.252745	+	0.145922i	0.621021	−	0.783794i	$-0.286718\pi$
$317$	4.50000	+	2.59808i	0.252745	+	0.145922i	−0.368275	+	0.929717i	$0.620052\pi$
$318$		0			0
$319$	15.0000	−	25.9808i	0.839839	−	1.45464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.5000	−	7.79423i	−0.584236	−	0.433682i
$324$		0			0
$325$	18.0000	−	10.3923i	0.998460	−	0.576461i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.00000	−	5.19615i	0.165395	−	0.286473i
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.0000			0.819538
$336$		0			0
$337$	−28.5000	+	16.4545i	−1.55249	+	0.896333i	−0.554556	+	0.832146i	$0.687112\pi$
$337$	−28.5000	+	16.4545i	−1.55249	+	0.896333i	−0.997938	+	0.0641864i	$0.979555\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.8564i			0.750366i
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.5000	−	7.79423i	0.724718	−	0.418416i	−0.0917687	−	0.995780i	$-0.529252\pi$
$347$	13.5000	−	7.79423i	0.724718	−	0.418416i	0.816487	+	0.577364i	$0.195919\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$	−13.5000	+	23.3827i	−0.716506	+	1.24102i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	13.5000	−	7.79423i	0.712503	−	0.411364i	−0.0994843	−	0.995039i	$-0.531719\pi$
$359$	13.5000	−	7.79423i	0.712503	−	0.411364i	0.811987	+	0.583675i	$0.198386\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.5000	−	18.1865i	0.549595	−	0.951927i
$366$		0			0
$367$	13.5000	+	7.79423i	0.704694	+	0.406855i	0.809093	−	0.587680i	$-0.199959\pi$
$367$	13.5000	+	7.79423i	0.704694	+	0.406855i	−0.104399	+	0.994535i	$0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	−	5.19615i	0.155752	−	0.269771i
$372$		0			0
$373$			6.92820i			0.358729i	0.983783	+	0.179364i	$0.0574041\pi$
$373$			6.92820i			0.358729i	−0.983783	+	0.179364i	$0.942596\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	22.5000	+	38.9711i	1.15881	+	2.00712i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	0.887252	+	0.461285i	$0.152611\pi$
$383$	16.5000	+	28.5788i	0.843111	+	1.46031i	−0.0441413	+	0.999025i	$0.514055\pi$
$384$		0			0
$385$	−18.0000	−	31.1769i	−0.917365	−	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	−0.610835	−	0.791758i	$-0.709166\pi$
$389$	7.50000	−	12.9904i	0.380265	−	0.658638i	0.991100	+	0.133120i	$0.0424994\pi$
$390$		0			0
$391$			15.5885i			0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10.5000	+	18.1865i	−0.528312	+	0.915064i
$396$		0			0
$397$	−5.50000	−	9.52628i	−0.276037	−	0.478110i	0.694359	−	0.719629i	$-0.255688\pi$
$397$	−5.50000	−	9.52628i	−0.276037	−	0.478110i	−0.970396	+	0.241518i	$0.922355\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.50000	−	2.59808i	0.224719	−	0.129742i	−0.383414	−	0.923576i	$-0.625252\pi$
$401$	4.50000	−	2.59808i	0.224719	−	0.129742i	0.608134	+	0.793835i	$0.291919\pi$
$402$		0			0
$403$	−18.0000	−	10.3923i	−0.896644	−	0.517678i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.50000	+	0.866025i	0.0741702	+	0.0428222i	0.536626	−	0.843820i	$-0.319698\pi$
$409$	1.50000	+	0.866025i	0.0741702	+	0.0428222i	−0.462456	+	0.886642i	$0.653032\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.00000	−	5.19615i	0.442861	−	0.255686i
$414$		0			0
$415$	9.00000	−	5.19615i	0.441793	−	0.255069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.2487i		−	1.18463i	−0.805708	−	0.592314i	$-0.798215\pi$
$419$		−	24.2487i		−	1.18463i	0.805708	−	0.592314i	$-0.201785\pi$
$420$		0			0
$421$	19.5000	−	11.2583i	0.950372	−	0.548697i	0.0571754	−	0.998364i	$-0.481791\pi$
$421$	19.5000	−	11.2583i	0.950372	−	0.548697i	0.893196	+	0.449667i	$0.148457\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$	−21.0000	−	12.1244i	−1.01626	−	0.586739i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7.50000	−	12.9904i	0.361262	−	0.625725i	−0.626907	−	0.779094i	$-0.715679\pi$
$431$	7.50000	−	12.9904i	0.361262	−	0.625725i	0.988169	+	0.153370i	$0.0490126\pi$
$432$		0			0
$433$	1.50000	+	0.866025i	0.0720854	+	0.0416185i	0.535609	−	0.844466i	$-0.320082\pi$
$433$	1.50000	+	0.866025i	0.0720854	+	0.0416185i	−0.463524	+	0.886084i	$0.653415\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.5000	−	2.59808i	−1.07632	−	0.124283i
$438$		0			0
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	0.588718	−	0.808339i	$-0.299633\pi$
$439$	−8.50000	−	14.7224i	−0.405683	−	0.702663i	−0.994401	+	0.105675i	$0.966300\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−19.5000	−	11.2583i	−0.926473	−	0.534899i	−0.0407786	−	0.999168i	$-0.512984\pi$
$443$	−19.5000	−	11.2583i	−0.926473	−	0.534899i	−0.885694	+	0.464269i	$0.846317\pi$
$444$		0			0
$445$		−	25.9808i		−	1.23161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			6.92820i			0.326962i	0.986546	+	0.163481i	$0.0522723\pi$
$449$			6.92820i			0.326962i	−0.986546	+	0.163481i	$0.947728\pi$
$450$		0			0
$451$	−15.0000	−	25.9808i	−0.706322	−	1.22339i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	54.0000			2.53156
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.50000	+	12.9904i	0.349310	+	0.605022i	0.986127	−	0.165992i	$-0.0530827\pi$
$461$	7.50000	+	12.9904i	0.349310	+	0.605022i	−0.636817	+	0.771015i	$0.719749\pi$
$462$		0			0
$463$			10.3923i			0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$			10.3923i			0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			17.3205i			0.801498i	0.916188	+	0.400749i	$0.131250\pi$
$467$			17.3205i			0.801498i	−0.916188	+	0.400749i	$0.868750\pi$
$468$		0			0
$469$	15.0000	+	8.66025i	0.692636	+	0.399893i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.0000	+	36.3731i	0.965581	+	1.67244i
$474$		0			0
$475$	2.00000	−	17.3205i	0.0917663	−	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.5000	−	7.79423i	−0.616831	−	0.356127i	0.158803	−	0.987310i	$-0.449236\pi$
$479$	−13.5000	−	7.79423i	−0.616831	−	0.356127i	−0.775634	+	0.631183i	$0.782570\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.5000	+	12.9904i	1.02167	+	0.589863i
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16.5000	+	9.52628i	−0.744635	+	0.429915i	−0.823752	−	0.566950i	$-0.808123\pi$
$491$	−16.5000	+	9.52628i	−0.744635	+	0.429915i	0.0791174	+	0.996865i	$0.474790\pi$
$492$		0			0
$493$		−	25.9808i		−	1.17011i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.0000	+	15.5885i	−1.21112	+	0.699238i
$498$		0			0
$499$	10.5000	−	6.06218i	0.470045	−	0.271380i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	10.5000	−	6.06218i	0.470045	−	0.271380i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.50000	−	4.33013i	−0.334408	−	0.193071i	0.323388	−	0.946266i	$-0.395178\pi$
$503$	−7.50000	−	4.33013i	−0.334408	−	0.193071i	−0.657797	+	0.753196i	$0.728511\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.5000	+	9.52628i	0.731350	+	0.422245i	0.818916	−	0.573914i	$-0.194576\pi$
$509$	16.5000	+	9.52628i	0.731350	+	0.422245i	−0.0875661	+	0.996159i	$0.527909\pi$
$510$		0			0
$511$	21.0000	−	12.1244i	0.928985	−	0.536350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000	−	41.5692i	−1.05757	−	1.83176i
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	41.5692i		−	1.82118i	−0.413310	−	0.910590i	$-0.635627\pi$
$521$		−	41.5692i		−	1.82118i	0.413310	−	0.910590i	$-0.364373\pi$
$522$		0			0
$523$	2.50000	−	4.33013i	0.109317	−	0.189343i	−0.806177	−	0.591675i	$-0.798467\pi$
$523$	2.50000	−	4.33013i	0.109317	−	0.189343i	0.915494	+	0.402332i	$0.131800\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	2.00000	+	3.46410i	0.0869565	+	0.150613i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	45.0000			1.94917
$534$		0			0
$535$	18.0000	+	31.1769i	0.778208	+	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	17.3205i		−	0.746047i
$540$		0			0
$541$	2.50000	−	4.33013i	0.107483	−	0.186167i	−0.807267	−	0.590187i	$-0.799054\pi$
$541$	2.50000	−	4.33013i	0.107483	−	0.186167i	0.914750	+	0.404020i	$0.132387\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	40.5000	+	23.3827i	1.73483	+	1.00160i
$546$		0			0
$547$	0.500000	−	0.866025i	0.0213785	−	0.0370286i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	0.500000	−	0.866025i	0.0213785	−	0.0370286i	0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	37.5000	+	4.33013i	1.59755	+	0.184470i
$552$		0			0
$553$	−21.0000	+	12.1244i	−0.893011	+	0.515580i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	−0.945473	−	0.325701i	$-0.894400\pi$
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	0.754802	+	0.655953i	$0.227733\pi$
$558$		0			0
$559$	−63.0000			−2.66462
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$	18.0000	−	10.3923i	0.757266	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	20.7846i		−	0.871336i	−0.900107	−	0.435668i	$-0.856512\pi$
$569$		−	20.7846i		−	0.871336i	0.900107	−	0.435668i	$-0.143488\pi$
$570$		0			0
$571$			10.3923i			0.434904i	0.976071	+	0.217452i	$0.0697746\pi$
$571$			10.3923i			0.434904i	−0.976071	+	0.217452i	$0.930225\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−18.0000	+	10.3923i	−0.750652	+	0.433389i
$576$		0			0
$577$	46.0000			1.91501			0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000			1.91501			0.957503	+	0.288425i	$0.0931316\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000			0.497844
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.50000	+	2.59808i	−0.185735	+	0.107234i	−0.589984	−	0.807415i	$-0.700866\pi$
$587$	−4.50000	+	2.59808i	−0.185735	+	0.107234i	0.404249	+	0.914649i	$0.367533\pi$
$588$		0			0
$589$	−16.0000	+	6.92820i	−0.659269	+	0.285472i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.50000	−	2.59808i	0.0615976	−	0.106690i	−0.833582	−	0.552396i	$-0.813714\pi$
$593$	1.50000	−	2.59808i	0.0615976	−	0.106690i	0.895180	+	0.445705i	$0.147047\pi$
$594$		0			0
$595$	−27.0000	−	15.5885i	−1.10689	−	0.639064i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.50000	+	7.79423i	−0.183865	+	0.318464i	−0.943193	−	0.332244i	$-0.892194\pi$
$599$	−4.50000	+	7.79423i	−0.183865	+	0.318464i	0.759328	+	0.650708i	$0.225528\pi$
$600$		0			0
$601$		−	6.92820i		−	0.282607i	−0.989966	−	0.141304i	$-0.954871\pi$
$601$		−	6.92820i		−	0.282607i	0.989966	−	0.141304i	$-0.0451294\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.50000	−	2.59808i	−0.0609837	−	0.105627i
$606$		0			0
$607$	40.0000			1.62355			0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000			1.62355			0.811775	+	0.583970i	$0.198502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.50000	+	7.79423i	0.182051	+	0.315321i
$612$		0			0
$613$	8.50000	+	14.7224i	0.343312	+	0.594633i	0.985046	−	0.172294i	$-0.0551179\pi$
$613$	8.50000	+	14.7224i	0.343312	+	0.594633i	−0.641734	+	0.766927i	$0.721785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	13.5000	−	23.3827i	0.543490	−	0.941351i	−0.455211	−	0.890384i	$-0.650436\pi$
$617$	13.5000	−	23.3827i	0.543490	−	0.941351i	0.998700	−	0.0509678i	$-0.0162306\pi$
$618$		0			0
$619$		−	31.1769i		−	1.25311i	−0.779379	−	0.626553i	$-0.784465\pi$
$619$		−	31.1769i		−	1.25311i	0.779379	−	0.626553i	$-0.215535\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	15.0000	−	25.9808i	0.600962	−	1.04090i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	13.5000	+	7.79423i	0.537427	+	0.310283i	0.744035	−	0.668140i	$-0.232909\pi$
$631$	13.5000	+	7.79423i	0.537427	+	0.310283i	−0.206609	+	0.978424i	$0.566243\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−21.0000			−0.833360
$636$		0			0
$637$	22.5000	+	12.9904i	0.891482	+	0.514698i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	34.5000	−	19.9186i	1.36267	−	0.786737i	0.372690	−	0.927956i	$-0.378436\pi$
$641$	34.5000	−	19.9186i	1.36267	−	0.786737i	0.989978	+	0.141219i	$0.0451022\pi$
$642$		0			0
$643$	−1.50000	+	0.866025i	−0.0591542	+	0.0341527i	−0.529285	−	0.848444i	$-0.677540\pi$
$643$	−1.50000	+	0.866025i	−0.0591542	+	0.0341527i	0.470131	+	0.882597i	$0.344207\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	3.46410i		−	0.136188i	−0.997679	−	0.0680939i	$-0.978308\pi$
$647$		−	3.46410i		−	0.136188i	0.997679	−	0.0680939i	$-0.0216918\pi$
$648$		0			0
$649$	9.00000	−	5.19615i	0.353281	−	0.203967i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$		0			0
$655$	−13.5000	−	7.79423i	−0.527489	−	0.304546i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−22.5000	+	38.9711i	−0.876476	+	1.51810i	−0.0212930	+	0.999773i	$0.506778\pi$
$659$	−22.5000	+	38.9711i	−0.876476	+	1.51810i	−0.855183	+	0.518327i	$0.826555\pi$
$660$		0			0
$661$	19.5000	+	11.2583i	0.758462	+	0.437898i	0.828743	−	0.559629i	$-0.189056\pi$
$661$	19.5000	+	11.2583i	0.758462	+	0.437898i	−0.0702812	+	0.997527i	$0.522390\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	27.0000	−	36.3731i	1.04702	−	1.41049i
$666$		0			0
$667$	−22.5000	−	38.9711i	−0.871203	−	1.50897i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.0000	−	12.1244i	−0.810696	−	0.468056i
$672$		0			0
$673$			48.4974i			1.86944i	0.355387	+	0.934719i	$0.384349\pi$
$673$			48.4974i			1.86944i	−0.355387	+	0.934719i	$0.615651\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			41.5692i			1.59763i	0.601574	+	0.798817i	$0.294541\pi$
$677$			41.5692i			1.59763i	−0.601574	+	0.798817i	$0.705459\pi$
$678$		0			0
$679$	15.0000	+	25.9808i	0.575647	+	0.997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.50000	+	7.79423i	0.171436	+	0.296936i
$690$		0			0
$691$			10.3923i			0.395342i	0.980268	+	0.197671i	$0.0633378\pi$
$691$			10.3923i			0.395342i	−0.980268	+	0.197671i	$0.936662\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	57.1577i		−	2.16811i
$696$		0			0
$697$	−22.5000	−	12.9904i	−0.852248	−	0.492046i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16.5000	−	28.5788i	−0.623196	−	1.07941i	−0.988887	−	0.148671i	$-0.952500\pi$
$701$	−16.5000	−	28.5788i	−0.623196	−	1.07941i	0.365690	−	0.930737i	$-0.380833\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.00000	+	5.19615i	0.338480	+	0.195421i
$708$		0			0
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	0.413114	+	0.910679i	$0.364441\pi$
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.0000	+	10.3923i	0.674105	+	0.389195i
$714$		0			0
$715$	54.0000			2.01949
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.50000	−	0.866025i	0.0559406	−	0.0322973i	−0.471769	−	0.881722i	$-0.656384\pi$
$719$	1.50000	−	0.866025i	0.0559406	−	0.0322973i	0.527709	+	0.849425i	$0.323051\pi$
$720$		0			0
$721$		−	55.4256i		−	2.06416i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	30.0000	−	17.3205i	1.11417	−	0.643268i
$726$		0			0
$727$	−1.50000	+	0.866025i	−0.0556319	+	0.0321191i	−0.527558	−	0.849519i	$-0.676892\pi$
$727$	−1.50000	+	0.866025i	−0.0556319	+	0.0321191i	0.471926	+	0.881638i	$0.343559\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	31.5000	+	18.1865i	1.16507	+	0.672653i
$732$		0			0
$733$	10.0000			0.369358			0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000			0.369358			0.184679	+	0.982799i	$0.440875\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.0000	+	8.66025i	0.552532	+	0.319005i
$738$		0			0
$739$	−1.50000	+	0.866025i	−0.0551784	+	0.0318573i	−0.527335	−	0.849657i	$-0.676809\pi$
$739$	−1.50000	+	0.866025i	−0.0551784	+	0.0318573i	0.472157	+	0.881514i	$0.343476\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.50000	−	2.59808i	−0.0550297	−	0.0953142i	0.837198	−	0.546899i	$-0.184192\pi$
$743$	−1.50000	−	2.59808i	−0.0550297	−	0.0953142i	−0.892228	+	0.451585i	$0.850859\pi$
$744$		0			0
$745$	−22.5000	+	38.9711i	−0.824336	+	1.42779i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			41.5692i			1.51891i
$750$		0			0
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	0.576262	+	0.817265i	$0.304511\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000	+	20.7846i	0.436725	+	0.756429i
$756$		0			0
$757$	6.50000	+	11.2583i	0.236247	+	0.409191i	0.959634	−	0.281251i	$-0.0907494\pi$
$757$	6.50000	+	11.2583i	0.236247	+	0.409191i	−0.723388	+	0.690442i	$0.757416\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$	27.0000	+	46.7654i	0.977466	+	1.69302i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			15.5885i			0.562867i
$768$		0			0
$769$	20.5000	−	35.5070i	0.739249	−	1.28042i	−0.213585	−	0.976924i	$-0.568514\pi$
$769$	20.5000	−	35.5070i	0.739249	−	1.28042i	0.952834	−	0.303492i	$-0.0981526\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−37.5000	−	21.6506i	−1.34878	−	0.778719i	−0.360704	−	0.932680i	$-0.617464\pi$
$773$	−37.5000	−	21.6506i	−1.34878	−	0.778719i	−0.988077	+	0.153961i	$0.950797\pi$
$774$		0			0
$775$	−8.00000	+	13.8564i	−0.287368	+	0.497737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	22.5000	−	30.3109i	0.806146	−	1.08600i
$780$		0			0
$781$	−27.0000	+	15.5885i	−0.966136	+	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.50000	+	12.9904i	−0.267686	+	0.463647i
$786$		0			0
$787$	28.0000			0.998092			0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000			0.998092			0.499046	+	0.866575i	$0.333684\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	31.5000	−	18.1865i	1.11860	−	0.645823i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.8564i		−	0.490819i	−0.969419	−	0.245410i	$-0.921078\pi$
$797$		−	13.8564i		−	0.490819i	0.969419	−	0.245410i	$-0.0789224\pi$
$798$		0			0
$799$		−	5.19615i		−	0.183827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	21.0000	−	12.1244i	0.741074	−	0.427859i
$804$		0			0
$805$	−54.0000			−1.90325
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	−0.729042	−	0.684468i	$-0.760034\pi$
$811$	6.50000	−	11.2583i	0.228246	−	0.395333i	0.957288	+	0.289135i	$0.0933677\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.0000	+	15.5885i	−0.945769	+	0.546040i
$816$		0			0
$817$	−31.5000	+	42.4352i	−1.10205	+	1.48462i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.5000	+	38.9711i	−0.785255	+	1.36010i	0.143591	+	0.989637i	$0.454135\pi$
$821$	−22.5000	+	38.9711i	−0.785255	+	1.36010i	−0.928846	+	0.370465i	$0.879198\pi$
$822$		0			0
$823$	1.50000	+	0.866025i	0.0522867	+	0.0301877i	0.525915	−	0.850537i	$-0.323723\pi$
$823$	1.50000	+	0.866025i	0.0522867	+	0.0301877i	−0.473629	+	0.880725i	$0.657056\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.5000	−	33.7750i	0.678081	−	1.17447i	−0.297477	−	0.954729i	$-0.596145\pi$
$827$	19.5000	−	33.7750i	0.678081	−	1.17447i	0.975558	−	0.219742i	$-0.0705217\pi$
$828$		0			0
$829$			6.92820i			0.240626i	0.992736	+	0.120313i	$0.0383899\pi$
$829$			6.92820i			0.240626i	−0.992736	+	0.120313i	$0.961610\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7.50000	−	12.9904i	−0.259860	−	0.450090i
$834$		0			0
$835$	−63.0000			−2.18020
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.5000	+	18.1865i	0.362500	+	0.627869i	0.988372	−	0.152057i	$-0.0485899\pi$
$839$	10.5000	+	18.1865i	0.362500	+	0.627869i	−0.625871	+	0.779926i	$0.715257\pi$
$840$		0			0
$841$	23.0000	+	39.8372i	0.793103	+	1.37370i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−21.0000	+	36.3731i	−0.722422	+	1.25127i
$846$		0			0
$847$		−	3.46410i		−	0.119028i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−17.5000	−	30.3109i	−0.599189	−	1.03783i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−17.5000	−	30.3109i	−0.599189	−	1.03783i	0.393753	−	0.919216i	$-0.371177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−37.5000	+	21.6506i	−1.28098	+	0.739572i	−0.977027	−	0.213117i	$-0.931639\pi$
$857$	−37.5000	+	21.6506i	−1.28098	+	0.739572i	−0.303949	+	0.952688i	$0.598305\pi$
$858$		0			0
$859$	49.5000	+	28.5788i	1.68892	+	0.975097i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	49.5000	+	28.5788i	1.68892	+	0.975097i	0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$		0			0
$865$	−22.5000	−	12.9904i	−0.765023	−	0.441686i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.0000	+	12.1244i	−0.712376	+	0.411291i
$870$		0			0
$871$	−22.5000	+	12.9904i	−0.762383	+	0.440162i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			10.3923i			0.351324i
$876$		0			0
$877$	−22.5000	+	12.9904i	−0.759771	+	0.438654i	−0.829214	−	0.558932i	$-0.811211\pi$
$877$	−22.5000	+	12.9904i	−0.759771	+	0.438654i	0.0694425	+	0.997586i	$0.477878\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−4.50000	−	2.59808i	−0.151437	−	0.0874322i	0.422367	−	0.906425i	$-0.361199\pi$
$883$	−4.50000	−	2.59808i	−0.151437	−	0.0874322i	−0.573804	+	0.818993i	$0.694533\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	−0.839744	−	0.542983i	$-0.817295\pi$
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	0.890109	+	0.455748i	$0.150628\pi$
$888$		0			0
$889$	−21.0000	−	12.1244i	−0.704317	−	0.406638i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.50000	+	0.866025i	0.250978	+	0.0289804i
$894$		0			0
$895$	−18.0000	−	31.1769i	−0.601674	−	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30.0000	−	17.3205i	−1.00056	−	0.577671i
$900$		0			0
$901$		−	5.19615i		−	0.173109i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			5.19615i			0.172726i
$906$		0			0
$907$	−6.50000	−	11.2583i	−0.215829	−	0.373827i	0.737700	−	0.675129i	$-0.235912\pi$
$907$	−6.50000	−	11.2583i	−0.215829	−	0.373827i	−0.953529	+	0.301302i	$0.902579\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.00000	−	15.5885i	−0.297206	−	0.514776i
$918$		0			0
$919$		−	38.1051i		−	1.25697i	−0.777821	−	0.628486i	$-0.783675\pi$
$919$		−	38.1051i		−	1.25697i	0.777821	−	0.628486i	$-0.216325\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	46.7654i		−	1.53930i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.5000	+	33.7750i	0.639774	+	1.10812i	0.985482	+	0.169779i	$0.0543055\pi$
$929$	19.5000	+	33.7750i	0.639774	+	1.10812i	−0.345708	+	0.938342i	$0.612361\pi$
$930$		0			0
$931$	20.0000	−	8.66025i	0.655474	−	0.283828i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−27.0000	−	15.5885i	−0.882994	−	0.509797i
$936$		0			0
$937$	24.5000	−	42.4352i	0.800380	−	1.38630i	−0.118986	−	0.992896i	$-0.537964\pi$
$937$	24.5000	−	42.4352i	0.800380	−	1.38630i	0.919366	−	0.393403i	$-0.128702\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	0.263547	−	0.964646i	$-0.415107\pi$
$941$	−13.5000	−	7.79423i	−0.440087	−	0.254085i	−0.703635	+	0.710562i	$0.748441\pi$
$942$		0			0
$943$	−45.0000			−1.46540
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.5000	−	14.7224i	0.828639	−	0.478415i	−0.0247477	−	0.999694i	$-0.507878\pi$
$947$	25.5000	−	14.7224i	0.828639	−	0.478415i	0.853386	+	0.521279i	$0.174545\pi$
$948$		0			0
$949$			36.3731i			1.18072i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.5000	−	6.06218i	0.340128	−	0.196373i	−0.320200	−	0.947350i	$-0.603750\pi$
$953$	10.5000	−	6.06218i	0.340128	−	0.196373i	0.660329	+	0.750977i	$0.270417\pi$
$954$		0			0
$955$	−63.0000	+	36.3731i	−2.03863	+	1.17700i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.00000	+	5.19615i	0.290625	+	0.167793i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−31.5000	−	18.1865i	−1.01402	−	0.585445i
$966$		0			0
$967$	−43.5000	+	25.1147i	−1.39887	+	0.807635i	−0.994274	−	0.106862i	$-0.965920\pi$
$967$	−43.5000	+	25.1147i	−1.39887	+	0.807635i	−0.404592	+	0.914497i	$0.632586\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	0.563914	−	0.825833i	$-0.309295\pi$
$971$	−13.5000	−	23.3827i	−0.433236	−	0.750386i	−0.997150	+	0.0754473i	$0.975962\pi$
$972$		0			0
$973$	33.0000	−	57.1577i	1.05793	−	1.83239i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.92820i			0.221653i	0.993840	+	0.110826i	$0.0353498\pi$
$977$			6.92820i			0.221653i	−0.993840	+	0.110826i	$0.964650\pi$
$978$		0			0
$979$	15.0000	−	25.9808i	0.479402	−	0.830349i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.50000	+	7.79423i	0.143528	+	0.248597i	0.928823	−	0.370525i	$-0.120822\pi$
$983$	4.50000	+	7.79423i	0.143528	+	0.248597i	−0.785295	+	0.619122i	$0.787489\pi$
$984$		0			0
$985$	−27.0000	−	46.7654i	−0.860292	−	1.49007i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	63.0000			2.00328
$990$		0			0
$991$	−12.5000	−	21.6506i	−0.397076	−	0.687755i	0.596288	−	0.802771i	$-0.296642\pi$
$991$	−12.5000	−	21.6506i	−0.397076	−	0.687755i	−0.993364	+	0.115015i	$0.963308\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	36.3731i		−	1.15310i
$996$		0			0
$997$	−15.5000	+	26.8468i	−0.490890	+	0.850246i	−0.999945	−	0.0104877i	$-0.996662\pi$
$997$	−15.5000	+	26.8468i	−0.490890	+	0.850246i	0.509055	+	0.860734i	$0.329995\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.g.1855.1		2
3.2	odd	2		304.2.n.a.31.1	✓	2
4.3	odd	2		2736.2.bm.h.1855.1		2
12.11	even	2		304.2.n.b.31.1	yes	2
19.8	odd	6		2736.2.bm.h.559.1		2
24.5	odd	2		1216.2.n.b.639.1		2
24.11	even	2		1216.2.n.a.639.1		2
57.8	even	6		304.2.n.b.255.1	yes	2
76.27	even	6	inner	2736.2.bm.g.559.1		2
228.179	odd	6		304.2.n.a.255.1	yes	2
456.179	odd	6		1216.2.n.b.255.1		2
456.293	even	6		1216.2.n.a.255.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.n.a.31.1	✓	2	3.2	odd	2
304.2.n.a.255.1	yes	2	228.179	odd	6
304.2.n.b.31.1	yes	2	12.11	even	2
304.2.n.b.255.1	yes	2	57.8	even	6
1216.2.n.a.255.1		2	456.293	even	6
1216.2.n.a.639.1		2	24.11	even	2
1216.2.n.b.255.1		2	456.179	odd	6
1216.2.n.b.639.1		2	24.5	odd	2
2736.2.bm.g.559.1		2	76.27	even	6	inner
2736.2.bm.g.1855.1		2	1.1	even	1	trivial
2736.2.bm.h.559.1		2	19.8	odd	6
2736.2.bm.h.1855.1		2	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{5} +3.46410i q^{7} +3.46410i q^{11} +(-4.50000 - 2.59808i) q^{13} +(1.50000 + 2.59808i) q^{17} +(-4.00000 + 1.73205i) q^{19} +(4.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-7.50000 - 4.33013i) q^{29} +4.00000 q^{31} +(-9.00000 + 5.19615i) q^{35} +(-7.50000 + 4.33013i) q^{41} +(10.5000 - 6.06218i) q^{43} +(-1.50000 - 0.866025i) q^{47} -5.00000 q^{49} +(-1.50000 - 0.866025i) q^{53} +(-9.00000 + 5.19615i) q^{55} +(-1.50000 - 2.59808i) q^{59} +(-3.50000 + 6.06218i) q^{61} -15.5885i q^{65} +(2.50000 - 4.33013i) q^{67} +(4.50000 + 7.79423i) q^{71} +(-3.50000 - 6.06218i) q^{73} -12.0000 q^{77} +(3.50000 + 6.06218i) q^{79} -3.46410i q^{83} +(-4.50000 + 7.79423i) q^{85} +(-7.50000 - 4.33013i) q^{89} +(9.00000 - 15.5885i) q^{91} +(-10.5000 - 7.79423i) q^{95} +(7.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 9 q^{13} + 3 q^{17} - 8 q^{19} + 9 q^{23} - 4 q^{25} - 15 q^{29} + 8 q^{31} - 18 q^{35} - 15 q^{41} + 21 q^{43} - 3 q^{47} - 10 q^{49} - 3 q^{53} - 18 q^{55} - 3 q^{59} - 7 q^{61} + 5 q^{67}+ \cdots + 15 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 9 * q^13 + 3 * q^17 - 8 * q^19 + 9 * q^23 - 4 * q^25 - 15 * q^29 + 8 * q^31 - 18 * q^35 - 15 * q^41 + 21 * q^43 - 3 * q^47 - 10 * q^49 - 3 * q^53 - 18 * q^55 - 3 * q^59 - 7 * q^61 + 5 * q^67 + 9 * q^71 - 7 * q^73 - 24 * q^77 + 7 * q^79 - 9 * q^85 - 15 * q^89 + 18 * q^91 - 21 * q^95 + 15 * q^97

Introduction

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