LMFDB - Embedded newform 2736.2.s.bb.1873.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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, 0, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.764411904.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 $ x^8 - 6x^6 + 21x^4 - 54*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 171)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.3
Root			$1.69185 - 0.370982i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.bb.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.524648 - 0.908716i) q^{5} +3.44949 q^{7} +O(q^{10})$	$q+(0.524648 - 0.908716i) q^{5} +3.44949 q^{7} -5.71812 q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.04930 + 1.81743i) q^{17} +(-1.00000 - 4.24264i) q^{19} +(1.80977 + 3.13461i) q^{23} +(1.94949 + 3.37662i) q^{25} +(3.61953 + 6.26922i) q^{29} +9.44949 q^{31} +(1.80977 - 3.13461i) q^{35} +3.89898 q^{37} +(4.66883 - 8.08665i) q^{41} +(3.17423 - 5.49794i) q^{43} +(4.66883 + 8.08665i) q^{47} +4.89898 q^{49} +(-0.524648 - 0.908716i) q^{53} +(-3.00000 + 5.19615i) q^{55} +(3.90836 - 6.76947i) q^{59} +(-2.50000 - 4.33013i) q^{61} +1.04930 q^{65} +(0.174235 + 0.301783i) q^{67} +(3.61953 - 6.26922i) q^{71} +(-2.50000 + 4.33013i) q^{73} -19.7246 q^{77} +(0.174235 - 0.301783i) q^{79} +11.4362 q^{83} +(1.10102 + 1.90702i) q^{85} +(2.62324 + 4.54358i) q^{89} +(1.72474 + 2.98735i) q^{91} +(-4.38000 - 1.31718i) q^{95} +(-1.55051 + 2.68556i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 4 q^{13} - 8 q^{19} - 4 q^{25} + 56 q^{31} - 8 q^{37} - 4 q^{43} - 24 q^{55} - 20 q^{61} - 28 q^{67} - 20 q^{73} - 28 q^{79} + 48 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 4 * q^13 - 8 * q^19 - 4 * q^25 + 56 * q^31 - 8 * q^37 - 4 * q^43 - 24 * q^55 - 20 * q^61 - 28 * q^67 - 20 * q^73 - 28 * q^79 + 48 * q^85 + 4 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.524648	−	0.908716i	0.234630	−	0.406390i	−0.724535	−	0.689238i	$-0.757946\pi$
$5$	0.524648	−	0.908716i	0.234630	−	0.406390i	0.959165	+	0.282847i	$0.0912790\pi$
$6$		0			0
$7$	3.44949			1.30378			0.651892	−	0.758312i	$-0.273975\pi$
$7$	3.44949			1.30378			0.651892	+	0.758312i	$0.273975\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.71812			−1.72408			−0.862040	−	0.506841i	$-0.830813\pi$
$11$	−5.71812			−1.72408			−0.862040	+	0.506841i	$0.830813\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.04930	+	1.81743i	−0.254491	+	0.440792i	−0.964757	−	0.263141i	$-0.915241\pi$
$17$	−1.04930	+	1.81743i	−0.254491	+	0.440792i	0.710266	+	0.703934i	$0.248575\pi$
$18$		0			0
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.80977	+	3.13461i	0.377362	+	0.653611i	0.990678	−	0.136227i	$-0.0434978\pi$
$23$	1.80977	+	3.13461i	0.377362	+	0.653611i	−0.613315	+	0.789838i	$0.710164\pi$
$24$		0			0
$25$	1.94949	+	3.37662i	0.389898	+	0.675323i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.61953	+	6.26922i	0.672130	+	1.16416i	0.977299	+	0.211866i	$0.0679539\pi$
$29$	3.61953	+	6.26922i	0.672130	+	1.16416i	−0.305168	+	0.952298i	$0.598713\pi$
$30$		0			0
$31$	9.44949			1.69718			0.848589	−	0.529052i	$-0.177452\pi$
$31$	9.44949			1.69718			0.848589	+	0.529052i	$0.177452\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.80977	−	3.13461i	0.305906	−	0.529845i
$36$		0			0
$37$	3.89898			0.640988			0.320494	−	0.947250i	$-0.396151\pi$
$37$	3.89898			0.640988			0.320494	+	0.947250i	$0.396151\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.66883	−	8.08665i	0.729149	−	1.26292i	−0.228095	−	0.973639i	$-0.573250\pi$
$41$	4.66883	−	8.08665i	0.729149	−	1.26292i	0.957244	−	0.289283i	$-0.0934170\pi$
$42$		0			0
$43$	3.17423	−	5.49794i	0.484066	−	0.838427i	−0.515766	−	0.856729i	$-0.672493\pi$
$43$	3.17423	−	5.49794i	0.484066	−	0.838427i	0.999833	+	0.0183020i	$0.00582604\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.66883	+	8.08665i	0.681019	+	1.17956i	0.974670	+	0.223646i	$0.0717961\pi$
$47$	4.66883	+	8.08665i	0.681019	+	1.17956i	−0.293652	+	0.955912i	$0.594871\pi$
$48$		0			0
$49$	4.89898			0.699854
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.524648	−	0.908716i	−0.0720659	−	0.124822i	0.827741	−	0.561111i	$-0.189626\pi$
$53$	−0.524648	−	0.908716i	−0.0720659	−	0.124822i	−0.899807	+	0.436289i	$0.856293\pi$
$54$		0			0
$55$	−3.00000	+	5.19615i	−0.404520	+	0.700649i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.90836	−	6.76947i	0.508825	−	0.881310i	−0.491123	−	0.871090i	$-0.663413\pi$
$59$	3.90836	−	6.76947i	0.508825	−	0.881310i	0.999948	−	0.0102201i	$-0.00325322\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.04930			0.130149
$66$		0			0
$67$	0.174235	+	0.301783i	0.0212861	+	0.0368687i	0.876472	−	0.481452i	$-0.159891\pi$
$67$	0.174235	+	0.301783i	0.0212861	+	0.0368687i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.61953	−	6.26922i	0.429560	−	0.744019i	−0.567275	−	0.823529i	$-0.692002\pi$
$71$	3.61953	−	6.26922i	0.429560	−	0.744019i	0.996834	+	0.0795098i	$0.0253355\pi$
$72$		0			0
$73$	−2.50000	+	4.33013i	−0.292603	+	0.506803i	−0.974424	−	0.224716i	$-0.927855\pi$
$73$	−2.50000	+	4.33013i	−0.292603	+	0.506803i	0.681822	+	0.731519i	$0.261188\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19.7246			−2.24783
$78$		0			0
$79$	0.174235	−	0.301783i	0.0196029	−	0.0339533i	−0.856058	−	0.516881i	$-0.827093\pi$
$79$	0.174235	−	0.301783i	0.0196029	−	0.0339533i	0.875660	+	0.482927i	$0.160426\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.4362			1.25529			0.627646	−	0.778499i	$-0.284019\pi$
$83$	11.4362			1.25529			0.627646	+	0.778499i	$0.284019\pi$
$84$		0			0
$85$	1.10102	+	1.90702i	0.119422	+	0.206846i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.62324	+	4.54358i	0.278063	+	0.481619i	0.970903	−	0.239472i	$-0.0769744\pi$
$89$	2.62324	+	4.54358i	0.278063	+	0.481619i	−0.692841	+	0.721091i	$0.743641\pi$
$90$		0			0
$91$	1.72474	+	2.98735i	0.180802	+	0.313159i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.38000	−	1.31718i	−0.449379	−	0.135139i
$96$		0			0
$97$	−1.55051	+	2.68556i	−0.157430	+	0.272678i	−0.933941	−	0.357426i	$-0.883654\pi$
$97$	−1.55051	+	2.68556i	−0.157430	+	0.272678i	0.776511	+	0.630104i	$0.216988\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.04930	+	1.81743i	0.104409	+	0.180841i	0.913497	−	0.406847i	$-0.133372\pi$
$101$	1.04930	+	1.81743i	0.104409	+	0.180841i	−0.809088	+	0.587688i	$0.800038\pi$
$102$		0			0
$103$	3.44949			0.339888			0.169944	−	0.985454i	$-0.445641\pi$
$103$	3.44949			0.339888			0.169944	+	0.985454i	$0.445641\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.5767			−1.60253			−0.801266	−	0.598308i	$-0.795840\pi$
$107$	−16.5767			−1.60253			−0.801266	+	0.598308i	$0.795840\pi$
$108$		0			0
$109$	4.44949	−	7.70674i	0.426184	−	0.738172i	−0.570346	−	0.821404i	$-0.693191\pi$
$109$	4.44949	−	7.70674i	0.426184	−	0.738172i	0.996530	+	0.0832323i	$0.0265243\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.28836			−0.779703			−0.389852	−	0.920878i	$-0.627474\pi$
$113$	−8.28836			−0.779703			−0.389852	+	0.920878i	$0.627474\pi$
$114$		0			0
$115$	3.79796			0.354162
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.61953	+	6.26922i	−0.331802	+	0.574698i
$120$		0			0
$121$	21.6969			1.97245
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.33766			0.835185
$126$		0			0
$127$	2.89898	+	5.02118i	0.257243	+	0.445558i	0.965502	−	0.260395i	$-0.0838526\pi$
$127$	2.89898	+	5.02118i	0.257243	+	0.445558i	−0.708259	+	0.705952i	$0.750519\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.81671	+	13.5389i	−0.682949	+	1.18290i	0.291127	+	0.956684i	$0.405970\pi$
$131$	−7.81671	+	13.5389i	−0.682949	+	1.18290i	−0.974077	+	0.226219i	$0.927364\pi$
$132$		0			0
$133$	−3.44949	−	14.6349i	−0.299109	−	1.26901i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.81671	−	13.5389i	−0.667827	−	1.15671i	−0.978511	−	0.206197i	$-0.933891\pi$
$137$	−7.81671	−	13.5389i	−0.667827	−	1.15671i	0.310684	−	0.950513i	$-0.399442\pi$
$138$		0			0
$139$	−1.17423	−	2.03383i	−0.0995973	−	0.172508i	0.811921	−	0.583768i	$-0.198422\pi$
$139$	−1.17423	−	2.03383i	−0.0995973	−	0.172508i	−0.911518	+	0.411260i	$0.865089\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.85906	−	4.95204i	−0.239087	−	0.414110i
$144$		0			0
$145$	7.59592			0.630807
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.24277	−	10.8128i	0.511428	−	0.885819i	−0.488485	−	0.872573i	$-0.662450\pi$
$149$	6.24277	−	10.8128i	0.511428	−	0.885819i	0.999912	−	0.0132463i	$-0.00421655\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.95765	−	8.58691i	0.398208	−	0.689717i
$156$		0			0
$157$	7.84847	−	13.5939i	0.626376	−	1.08492i	−0.361897	−	0.932218i	$-0.617871\pi$
$157$	7.84847	−	13.5939i	0.626376	−	1.08492i	0.988273	−	0.152697i	$-0.0487959\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.24277	+	10.8128i	0.491999	+	0.852168i
$162$		0			0
$163$	9.44949			0.740141			0.370071	−	0.929004i	$-0.379333\pi$
$163$	9.44949			0.740141			0.370071	+	0.929004i	$0.379333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.62648	−	16.6736i	−0.744919	−	1.29024i	−0.950232	−	0.311542i	$-0.899155\pi$
$167$	−9.62648	−	16.6736i	−0.744919	−	1.29024i	0.205313	−	0.978696i	$-0.434179\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.61953	+	6.26922i	−0.275188	+	0.476640i	−0.970183	−	0.242375i	$-0.922073\pi$
$173$	−3.61953	+	6.26922i	−0.275188	+	0.476640i	0.694995	+	0.719015i	$0.255407\pi$
$174$		0			0
$175$	6.72474	+	11.6476i	0.508343	+	0.880476i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.577648			−0.0431754			−0.0215877	−	0.999767i	$-0.506872\pi$
$179$	−0.577648			−0.0431754			−0.0215877	+	0.999767i	$0.506872\pi$
$180$		0			0
$181$	10.4495	+	18.0990i	0.776704	+	1.34529i	0.933832	+	0.357713i	$0.116443\pi$
$181$	10.4495	+	18.0990i	0.776704	+	1.34529i	−0.157127	+	0.987578i	$0.550223\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.04559	−	3.54307i	0.150395	−	0.260491i
$186$		0			0
$187$	6.00000	−	10.3923i	0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.71812			−0.413749			−0.206874	−	0.978367i	$-0.566329\pi$
$191$	−5.71812			−0.413749			−0.206874	+	0.978367i	$0.566329\pi$
$192$		0			0
$193$	−9.84847	+	17.0580i	−0.708908	+	1.22787i	0.256354	+	0.966583i	$0.417479\pi$
$193$	−9.84847	+	17.0580i	−0.708908	+	1.22787i	−0.965262	+	0.261282i	$0.915855\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	14.5841			1.03908			0.519538	−	0.854447i	$-0.326104\pi$
$197$	14.5841			1.03908			0.519538	+	0.854447i	$0.326104\pi$
$198$		0			0
$199$	0.174235	+	0.301783i	0.0123512	+	0.0213928i	0.872135	−	0.489265i	$-0.162735\pi$
$199$	0.174235	+	0.301783i	0.0123512	+	0.0213928i	−0.859784	+	0.510658i	$0.829402\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.4855	+	21.6256i	0.876313	+	1.51782i
$204$		0			0
$205$	−4.89898	−	8.48528i	−0.342160	−	0.592638i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.71812	+	24.2599i	0.395531	+	1.67810i
$210$		0			0
$211$	−7.17423	+	12.4261i	−0.493895	+	0.855451i	−0.999975	−	0.00703553i	$-0.997761\pi$
$211$	−7.17423	+	12.4261i	−0.493895	+	0.855451i	0.506081	+	0.862486i	$0.331094\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.33071	−	5.76896i	−0.227152	−	0.393440i
$216$		0			0
$217$	32.5959			2.21276
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.09859			−0.141166
$222$		0			0
$223$	0.174235	−	0.301783i	0.0116676	−	0.0202089i	−0.860133	−	0.510070i	$-0.829619\pi$
$223$	0.174235	−	0.301783i	0.0116676	−	0.0202089i	0.871800	+	0.489862i	$0.162953\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.8586			0.720711			0.360355	−	0.932815i	$-0.382655\pi$
$227$	10.8586			0.720711			0.360355	+	0.932815i	$0.382655\pi$
$228$		0			0
$229$	−10.7980			−0.713549			−0.356775	−	0.934190i	$-0.616124\pi$
$229$	−10.7980			−0.713549			−0.356775	+	0.934190i	$0.616124\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.3870	−	17.9907i	0.680472	−	1.17861i	−0.294365	−	0.955693i	$-0.595108\pi$
$233$	10.3870	−	17.9907i	0.680472	−	1.17861i	0.974837	−	0.222919i	$-0.0715585\pi$
$234$		0			0
$235$	9.79796			0.639148
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.8586			0.702384			0.351192	−	0.936303i	$-0.385776\pi$
$239$	10.8586			0.702384			0.351192	+	0.936303i	$0.385776\pi$
$240$		0			0
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	0.990912	+	0.134515i	$0.0429475\pi$
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	−0.378963	+	0.925412i	$0.623719\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.57024	−	4.45178i	0.164206	−	0.284414i
$246$		0			0
$247$	3.17423	−	2.98735i	0.201972	−	0.190080i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.23907	+	12.5384i	0.456926	+	0.791419i	0.998797	−	0.0490430i	$-0.0156171\pi$
$251$	7.23907	+	12.5384i	0.456926	+	0.791419i	−0.541871	+	0.840462i	$0.682284\pi$
$252$		0			0
$253$	−10.3485	−	17.9241i	−0.650603	−	1.12688i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.24277	−	10.8128i	−0.389413	−	0.674484i	0.602957	−	0.797773i	$-0.293989\pi$
$257$	−6.24277	−	10.8128i	−0.389413	−	0.674484i	−0.992371	+	0.123290i	$0.960656\pi$
$258$		0			0
$259$	13.4495			0.835711
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.9572	+	22.4425i	−0.798975	+	1.38386i	0.121310	+	0.992615i	$0.461291\pi$
$263$	−12.9572	+	22.4425i	−0.798975	+	1.38386i	−0.920284	+	0.391250i	$0.872043\pi$
$264$		0			0
$265$	−1.10102			−0.0676352
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.76371	+	13.4471i	−0.473362	+	0.819887i	−0.999535	−	0.0304905i	$-0.990293\pi$
$269$	−7.76371	+	13.4471i	−0.473362	+	0.819887i	0.526173	+	0.850378i	$0.323626\pi$
$270$		0			0
$271$	8.89898	−	15.4135i	0.540575	−	0.936303i	−0.458297	−	0.888799i	$-0.651540\pi$
$271$	8.89898	−	15.4135i	0.540575	−	0.936303i	0.998871	−	0.0475032i	$-0.0151264\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.1474	−	19.3079i	−0.672215	−	1.16431i
$276$		0			0
$277$	−18.6969			−1.12339			−0.561695	−	0.827344i	$-0.689851\pi$
$277$	−18.6969			−1.12339			−0.561695	+	0.827344i	$0.689851\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.24277	−	10.8128i	−0.372413	−	0.645037i	0.617524	−	0.786552i	$-0.288136\pi$
$281$	−6.24277	−	10.8128i	−0.372413	−	0.645037i	−0.989936	+	0.141515i	$0.954803\pi$
$282$		0			0
$283$	−3.10102	+	5.37113i	−0.184337	+	0.319280i	−0.943353	−	0.331791i	$-0.892347\pi$
$283$	−3.10102	+	5.37113i	−0.184337	+	0.319280i	0.759016	+	0.651072i	$0.225680\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.1051	−	27.8948i	0.950653	−	1.64658i
$288$		0			0
$289$	6.29796	+	10.9084i	0.370468	+	0.641670i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.2810			0.600620			0.300310	−	0.953842i	$-0.402910\pi$
$293$	10.2810			0.600620			0.300310	+	0.953842i	$0.402910\pi$
$294$		0			0
$295$	−4.10102	−	7.10318i	−0.238771	−	0.413563i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.80977	+	3.13461i	−0.104662	+	0.181279i
$300$		0			0
$301$	10.9495	−	18.9651i	0.631118	−	1.09313i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.24648			−0.300412
$306$		0			0
$307$	−14.7980	+	25.6308i	−0.844564	+	1.46283i	0.0414351	+	0.999141i	$0.486807\pi$
$307$	−14.7980	+	25.6308i	−0.844564	+	1.46283i	−0.885999	+	0.463687i	$0.846526\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	23.4501			1.32974			0.664868	−	0.746961i	$-0.268488\pi$
$311$	23.4501			1.32974			0.664868	+	0.746961i	$0.268488\pi$
$312$		0			0
$313$	−1.55051	−	2.68556i	−0.0876400	−	0.151797i	0.818873	−	0.573975i	$-0.194599\pi$
$313$	−1.55051	−	2.68556i	−0.0876400	−	0.151797i	−0.906513	+	0.422178i	$0.861266\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.19348	+	8.99536i	0.291695	+	0.505230i	0.974211	−	0.225641i	$-0.0724475\pi$
$317$	5.19348	+	8.99536i	0.291695	+	0.505230i	−0.682516	+	0.730871i	$0.739114\pi$
$318$		0			0
$319$	−20.6969	−	35.8481i	−1.15881	−	2.00711i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.76001	+	2.63435i	0.487420	+	0.146579i
$324$		0			0
$325$	−1.94949	+	3.37662i	−0.108138	+	0.187301i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.1051	+	27.8948i	0.887902	+	1.53789i
$330$		0			0
$331$	3.44949			0.189601			0.0948006	−	0.995496i	$-0.469779\pi$
$331$	3.44949			0.189601			0.0948006	+	0.995496i	$0.469779\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.365647			0.0199774
$336$		0			0
$337$	−12.8485	+	22.2542i	−0.699901	+	1.21226i	0.268600	+	0.963252i	$0.413439\pi$
$337$	−12.8485	+	22.2542i	−0.699901	+	1.21226i	−0.968500	+	0.249012i	$0.919894\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−54.0334			−2.92607
$342$		0			0
$343$	−7.24745			−0.391325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.38000	+	7.58639i	−0.235131	+	0.407259i	−0.959311	−	0.282352i	$-0.908885\pi$
$347$	−4.38000	+	7.58639i	−0.235131	+	0.407259i	0.724180	+	0.689611i	$0.242219\pi$
$348$		0			0
$349$	−16.7980			−0.899174			−0.449587	−	0.893237i	$-0.648429\pi$
$349$	−16.7980			−0.899174			−0.449587	+	0.893237i	$0.648429\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.7246			−1.04984			−0.524918	−	0.851153i	$-0.675904\pi$
$353$	−19.7246			−1.04984			−0.524918	+	0.851153i	$0.675904\pi$
$354$		0			0
$355$	−3.79796	−	6.57826i	−0.201575	−	0.349138i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.23907	+	12.5384i	−0.382063	+	0.661753i	−0.991357	−	0.131192i	$-0.958120\pi$
$359$	−7.23907	+	12.5384i	−0.382063	+	0.661753i	0.609294	+	0.792945i	$0.291453\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.62324	+	4.54358i	0.137307	+	0.237822i
$366$		0			0
$367$	−10.1742	−	17.6223i	−0.531091	−	0.919876i	−0.999342	−	0.0362806i	$-0.988449\pi$
$367$	−10.1742	−	17.6223i	−0.531091	−	0.919876i	0.468251	−	0.883596i	$-0.344884\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.80977	−	3.13461i	−0.0939584	−	0.162741i
$372$		0			0
$373$	−30.6969			−1.58943			−0.794714	−	0.606985i	$-0.792379\pi$
$373$	−30.6969			−1.58943			−0.794714	+	0.606985i	$0.792379\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3.61953	+	6.26922i	−0.186415	+	0.322881i
$378$		0			0
$379$	3.44949			0.177188			0.0885942	−	0.996068i	$-0.471763\pi$
$379$	3.44949			0.177188			0.0885942	+	0.996068i	$0.471763\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.95765	+	8.58691i	−0.253324	+	0.438770i	−0.964439	−	0.264306i	$-0.914857\pi$
$383$	−4.95765	+	8.58691i	−0.253324	+	0.438770i	0.711115	+	0.703076i	$0.248191\pi$
$384$		0			0
$385$	−10.3485	+	17.9241i	−0.527407	+	0.913495i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.24277	−	10.8128i	−0.316521	−	0.548231i	0.663239	−	0.748408i	$-0.269181\pi$
$389$	−6.24277	−	10.8128i	−0.316521	−	0.548231i	−0.979760	+	0.200177i	$0.935848\pi$
$390$		0			0
$391$	−7.59592			−0.384142
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.182824	−	0.316660i	−0.00919885	−	0.0159329i
$396$		0			0
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	−0.764730	−	0.644351i	$-0.777127\pi$
$397$	3.50000	−	6.06218i	0.175660	−	0.304252i	0.940389	+	0.340099i	$0.110461\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.67253	−	6.36101i	0.183398	−	0.317654i	−0.759638	−	0.650346i	$-0.774624\pi$
$401$	3.67253	−	6.36101i	0.183398	−	0.317654i	0.943035	+	0.332692i	$0.107957\pi$
$402$		0			0
$403$	4.72474	+	8.18350i	0.235356	+	0.407649i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−22.2948			−1.10511
$408$		0			0
$409$	−1.55051	−	2.68556i	−0.0766678	−	0.132793i	0.825143	−	0.564925i	$-0.191095\pi$
$409$	−1.55051	−	2.68556i	−0.0766678	−	0.132793i	−0.901810	+	0.432132i	$0.857761\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.4818	−	23.3512i	0.663398	−	1.14904i
$414$		0			0
$415$	6.00000	−	10.3923i	0.294528	−	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.577648			−0.0282199			−0.0141100	−	0.999900i	$-0.504491\pi$
$419$	−0.577648			−0.0282199			−0.0141100	+	0.999900i	$0.504491\pi$
$420$		0			0
$421$	−1.55051	+	2.68556i	−0.0755672	+	0.130886i	−0.901333	−	0.433127i	$-0.857410\pi$
$421$	−1.55051	+	2.68556i	−0.0755672	+	0.130886i	0.825766	+	0.564014i	$0.190743\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.18236			−0.396903
$426$		0			0
$427$	−8.62372	−	14.9367i	−0.417331	−	0.722839i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.66883	+	8.08665i	0.224890	+	0.389520i	0.956286	−	0.292432i	$-0.0944645\pi$
$431$	4.66883	+	8.08665i	0.224890	+	0.389520i	−0.731397	+	0.681952i	$0.761131\pi$
$432$		0			0
$433$	10.8485	+	18.7901i	0.521344	+	0.902995i	0.999692	+	0.0248240i	$0.00790255\pi$
$433$	10.8485	+	18.7901i	0.521344	+	0.902995i	−0.478348	+	0.878171i	$0.658764\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.4892	−	10.8128i	0.549605	−	0.517246i
$438$		0			0
$439$	−19.1742	+	33.2107i	−0.915136	+	1.58506i	−0.108435	+	0.994104i	$0.534584\pi$
$439$	−19.1742	+	33.2107i	−0.915136	+	1.58506i	−0.806702	+	0.590959i	$0.798750\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−17.6260	−	30.5292i	−0.837437	−	1.45048i	−0.892030	−	0.451975i	$-0.850719\pi$
$443$	−17.6260	−	30.5292i	−0.837437	−	1.45048i	0.0545930	−	0.998509i	$-0.482614\pi$
$444$		0			0
$445$	5.50510			0.260967
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.7172			1.02490			0.512449	−	0.858718i	$-0.328738\pi$
$449$	21.7172			1.02490			0.512449	+	0.858718i	$0.328738\pi$
$450$		0			0
$451$	−26.6969	+	46.2405i	−1.25711	+	2.17738i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.61953			0.169686
$456$		0			0
$457$	−20.1010			−0.940286			−0.470143	−	0.882590i	$-0.655798\pi$
$457$	−20.1010			−0.940286			−0.470143	+	0.882590i	$0.655798\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.81301	−	15.2646i	0.410463	−	0.710942i	−0.584478	−	0.811410i	$-0.698701\pi$
$461$	8.81301	−	15.2646i	0.410463	−	0.710942i	0.994940	+	0.100468i	$0.0320338\pi$
$462$		0			0
$463$	0.146428			0.00680510			0.00340255	−	0.999994i	$-0.498917\pi$
$463$	0.146428			0.00680510			0.00340255	+	0.999994i	$0.498917\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.14048			−0.237873			−0.118936	−	0.992902i	$-0.537948\pi$
$467$	−5.14048			−0.237873			−0.118936	+	0.992902i	$0.537948\pi$
$468$		0			0
$469$	0.601021	+	1.04100i	0.0277525	+	0.0480688i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−18.1507	+	31.4379i	−0.834569	+	1.44552i
$474$		0			0
$475$	12.3763	−	11.6476i	0.567862	−	0.534429i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.61953	−	6.26922i	−0.165381	−	0.286448i	0.771410	−	0.636339i	$-0.219552\pi$
$479$	−3.61953	−	6.26922i	−0.165381	−	0.286448i	−0.936790	+	0.349891i	$0.886219\pi$
$480$		0			0
$481$	1.94949	+	3.37662i	0.0888891	+	0.153960i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.62694	+	2.81795i	0.0738757	+	0.127956i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.91530	−	17.1738i	0.447471	−	0.775043i	−0.550749	−	0.834671i	$-0.685658\pi$
$491$	9.91530	−	17.1738i	0.447471	−	0.775043i	0.998221	+	0.0596275i	$0.0189913\pi$
$492$		0			0
$493$	−15.1918			−0.684206
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.4855	−	21.6256i	0.560053	−	0.970040i
$498$		0			0
$499$	13.8258	−	23.9469i	0.618926	−	1.07201i	−0.370756	−	0.928730i	$-0.620901\pi$
$499$	13.8258	−	23.9469i	0.618926	−	1.07201i	0.989682	−	0.143281i	$-0.0457654\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.3870	+	17.9907i	0.463131	+	0.802167i	0.999115	−	0.0420614i	$-0.0133925\pi$
$503$	10.3870	+	17.9907i	0.463131	+	0.802167i	−0.535984	+	0.844228i	$0.680059\pi$
$504$		0			0
$505$	2.20204			0.0979895
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.6334	+	27.0779i	0.692940	+	1.20021i	0.970870	+	0.239605i	$0.0770181\pi$
$509$	15.6334	+	27.0779i	0.692940	+	1.20021i	−0.277931	+	0.960601i	$0.589649\pi$
$510$		0			0
$511$	−8.62372	+	14.9367i	−0.381491	+	0.660762i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.80977	−	3.13461i	0.0797478	−	0.138127i
$516$		0			0
$517$	−26.6969	−	46.2405i	−1.17413	−	2.03365i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.14789			−0.137911			−0.0689557	−	0.997620i	$-0.521967\pi$
$521$	−3.14789			−0.137911			−0.0689557	+	0.997620i	$0.521967\pi$
$522$		0			0
$523$	−10.1742	−	17.6223i	−0.444888	−	0.770569i	0.553156	−	0.833078i	$-0.313423\pi$
$523$	−10.1742	−	17.6223i	−0.444888	−	0.770569i	−0.998044	+	0.0625086i	$0.980090\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.91530	+	17.1738i	−0.431917	+	0.748103i
$528$		0			0
$529$	4.94949	−	8.57277i	0.215195	−	0.372729i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	9.33766			0.404459
$534$		0			0
$535$	−8.69694	+	15.0635i	−0.376001	+	0.651254i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−28.0130			−1.20660
$540$		0			0
$541$	4.84847	+	8.39780i	0.208452	+	0.361049i	0.951227	−	0.308492i	$-0.0998242\pi$
$541$	4.84847	+	8.39780i	0.208452	+	0.361049i	−0.742775	+	0.669541i	$0.766491\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.66883	−	8.08665i	−0.199991	−	0.346394i
$546$		0			0
$547$	−8.82577	−	15.2867i	−0.377362	−	0.653611i	0.613315	−	0.789838i	$-0.289836\pi$
$547$	−8.82577	−	15.2867i	−0.377362	−	0.653611i	−0.990678	+	0.136227i	$0.956502\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.9785	−	21.6256i	0.978917	−	0.921281i
$552$		0			0
$553$	0.601021	−	1.04100i	0.0255580	−	0.0442677i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.52094	−	2.63435i	−0.0644444	−	0.111621i	0.832003	−	0.554771i	$-0.187194\pi$
$557$	−1.52094	−	2.63435i	−0.0644444	−	0.111621i	−0.896447	+	0.443150i	$0.853861\pi$
$558$		0			0
$559$	6.34847			0.268512
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.29577			0.265335			0.132668	−	0.991161i	$-0.457646\pi$
$563$	6.29577			0.265335			0.132668	+	0.991161i	$0.457646\pi$
$564$		0			0
$565$	−4.34847	+	7.53177i	−0.182941	+	0.316864i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.0130			1.17436			0.587182	−	0.809455i	$-0.300237\pi$
$569$	28.0130			1.17436			0.587182	+	0.809455i	$0.300237\pi$
$570$		0			0
$571$	−11.2474			−0.470691			−0.235346	−	0.971912i	$-0.575622\pi$
$571$	−11.2474			−0.470691			−0.235346	+	0.971912i	$0.575622\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.05624	+	12.2218i	−0.294266	+	0.509683i
$576$		0			0
$577$	22.6969			0.944886			0.472443	−	0.881361i	$-0.343372\pi$
$577$	22.6969			0.944886			0.472443	+	0.881361i	$0.343372\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	39.4492			1.63663
$582$		0			0
$583$	3.00000	+	5.19615i	0.124247	+	0.215203i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.760471	−	1.31718i	0.0313880	−	0.0543656i	−0.849905	−	0.526936i	$-0.823341\pi$
$587$	0.760471	−	1.31718i	0.0313880	−	0.0543656i	0.881293	+	0.472571i	$0.156674\pi$
$588$		0			0
$589$	−9.44949	−	40.0908i	−0.389359	−	1.65191i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.9609	−	20.7169i	−0.491175	−	0.850740i	0.508773	−	0.860901i	$-0.330099\pi$
$593$	−11.9609	−	20.7169i	−0.491175	−	0.850740i	−0.999948	+	0.0101603i	$0.996766\pi$
$594$		0			0
$595$	3.79796	+	6.57826i	0.155701	+	0.269682i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.7744	−	22.1259i	−0.521946	−	0.904038i	−0.999674	−	0.0255298i	$-0.991873\pi$
$599$	−12.7744	−	22.1259i	−0.521946	−	0.904038i	0.477728	−	0.878508i	$-0.341461\pi$
$600$		0			0
$601$	21.8990			0.893278			0.446639	−	0.894714i	$-0.352621\pi$
$601$	21.8990			0.893278			0.446639	+	0.894714i	$0.352621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.3832	−	19.7164i	0.462795	−	0.801584i
$606$		0			0
$607$	−7.94439			−0.322453			−0.161226	−	0.986917i	$-0.551545\pi$
$607$	−7.94439			−0.322453			−0.161226	+	0.986917i	$0.551545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.66883	+	8.08665i	−0.188881	+	0.327151i
$612$		0			0
$613$	−1.55051	+	2.68556i	−0.0626245	+	0.108469i	−0.895638	−	0.444784i	$-0.853280\pi$
$613$	−1.55051	+	2.68556i	−0.0626245	+	0.108469i	0.833013	+	0.553253i	$0.186614\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.67253	−	6.36101i	−0.147851	−	0.256085i	0.782582	−	0.622547i	$-0.213902\pi$
$617$	−3.67253	−	6.36101i	−0.147851	−	0.256085i	−0.930433	+	0.366462i	$0.880569\pi$
$618$		0			0
$619$	0.752551			0.0302476			0.0151238	−	0.999886i	$-0.495186\pi$
$619$	0.752551			0.0302476			0.0151238	+	0.999886i	$0.495186\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.04883	+	15.6730i	0.362534	+	0.627927i
$624$		0			0
$625$	−4.84847	+	8.39780i	−0.193939	+	0.335912i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.09118	+	7.08613i	−0.163126	+	0.282543i
$630$		0			0
$631$	3.17423	+	5.49794i	0.126364	+	0.218869i	0.922265	−	0.386557i	$-0.126336\pi$
$631$	3.17423	+	5.49794i	0.126364	+	0.218869i	−0.795901	+	0.605427i	$0.793003\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.08377			0.241427
$636$		0			0
$637$	2.44949	+	4.24264i	0.0970523	+	0.168100i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.3441	+	40.4332i	−0.922038	+	1.59702i	−0.125782	+	0.992058i	$0.540144\pi$
$641$	−23.3441	+	40.4332i	−0.922038	+	1.59702i	−0.796257	+	0.604959i	$0.793189\pi$
$642$		0			0
$643$	1.82577	−	3.16232i	0.0720012	−	0.124710i	−0.827777	−	0.561057i	$-0.810395\pi$
$643$	1.82577	−	3.16232i	0.0720012	−	0.124710i	0.899778	+	0.436347i	$0.143728\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.5767			0.651698			0.325849	−	0.945422i	$-0.394350\pi$
$647$	16.5767			0.651698			0.325849	+	0.945422i	$0.394350\pi$
$648$		0			0
$649$	−22.3485	+	38.7087i	−0.877254	+	1.51945i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−39.4492			−1.54377			−0.771884	−	0.635764i	$-0.780685\pi$
$653$	−39.4492			−1.54377			−0.771884	+	0.635764i	$0.780685\pi$
$654$		0			0
$655$	8.20204	+	14.2064i	0.320480	+	0.555088i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.90095	−	10.2207i	−0.229868	−	0.398144i	0.727901	−	0.685683i	$-0.240496\pi$
$659$	−5.90095	−	10.2207i	−0.229868	−	0.398144i	−0.957769	+	0.287539i	$0.907163\pi$
$660$		0			0
$661$	−20.5959	−	35.6732i	−0.801088	−	1.38753i	−0.918900	−	0.394490i	$-0.870921\pi$
$661$	−20.5959	−	35.6732i	−0.801088	−	1.38753i	0.117812	−	0.993036i	$-0.462412\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.1088	−	4.54358i	−0.585893	−	0.176193i
$666$		0			0
$667$	−13.1010	+	22.6916i	−0.507274	+	0.878624i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.2953	+	24.7602i	0.551864	+	0.955857i
$672$		0			0
$673$	21.8990			0.844144			0.422072	−	0.906562i	$-0.361303\pi$
$673$	21.8990			0.844144			0.422072	+	0.906562i	$0.361303\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.8725			−0.879061			−0.439531	−	0.898228i	$-0.644855\pi$
$677$	−22.8725			−0.879061			−0.439531	+	0.898228i	$0.644855\pi$
$678$		0			0
$679$	−5.34847	+	9.26382i	−0.205255	+	0.355513i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.577648			−0.0221031			−0.0110515	−	0.999939i	$-0.503518\pi$
$683$	−0.577648			−0.0221031			−0.0110515	+	0.999939i	$0.503518\pi$
$684$		0			0
$685$	−16.4041			−0.626768
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.524648	−	0.908716i	0.0199875	−	0.0346193i
$690$		0			0
$691$	−25.3939			−0.966029			−0.483014	−	0.875612i	$-0.660458\pi$
$691$	−25.3939			−0.966029			−0.483014	+	0.875612i	$0.660458\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.46424			−0.0934739
$696$		0			0
$697$	9.79796	+	16.9706i	0.371124	+	0.642806i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.1088	−	26.1692i	0.570651	−	0.988396i	−0.425848	−	0.904794i	$-0.640024\pi$
$701$	15.1088	−	26.1692i	0.570651	−	0.988396i	0.996499	−	0.0836016i	$-0.0266423\pi$
$702$		0			0
$703$	−3.89898	−	16.5420i	−0.147053	−	0.623892i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.61953	+	6.26922i	0.136127	+	0.235778i
$708$		0			0
$709$	−14.1969	−	24.5898i	−0.533177	−	0.923490i	−0.999249	−	0.0387432i	$-0.987665\pi$
$709$	−14.1969	−	24.5898i	−0.533177	−	0.923490i	0.466072	−	0.884747i	$-0.345669\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.1014	+	29.6204i	0.640451	+	1.10929i
$714$		0			0
$715$	−6.00000			−0.224387
$716$		0			0
$717$		0			0
$718$		0			0
$719$	23.6330	−	40.9335i	0.881361	−	1.52656i	0.0315323	−	0.999503i	$-0.489961\pi$
$719$	23.6330	−	40.9335i	0.881361	−	1.52656i	0.849829	−	0.527059i	$-0.176705\pi$
$720$		0			0
$721$	11.8990			0.443141
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.1125	+	24.4435i	−0.524125	+	0.907810i
$726$		0			0
$727$	−20.5227	+	35.5464i	−0.761145	+	1.31834i	0.181116	+	0.983462i	$0.442029\pi$
$727$	−20.5227	+	35.5464i	−0.761145	+	1.31834i	−0.942261	+	0.334880i	$0.891304\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.66142	+	11.5379i	0.246381	+	0.426745i
$732$		0			0
$733$	5.30306			0.195873			0.0979365	−	0.995193i	$-0.468776\pi$
$733$	5.30306			0.195873			0.0979365	+	0.995193i	$0.468776\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.996295	−	1.72563i	−0.0366990	−	0.0635645i
$738$		0			0
$739$	−14.5227	+	25.1541i	−0.534226	+	0.925307i	0.464974	+	0.885324i	$0.346064\pi$
$739$	−14.5227	+	25.1541i	−0.534226	+	0.925307i	−0.999200	+	0.0399828i	$0.987270\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.8163	+	27.3946i	−0.580242	+	1.00501i	0.415208	+	0.909726i	$0.363709\pi$
$743$	−15.8163	+	27.3946i	−0.580242	+	1.00501i	−0.995450	+	0.0952823i	$0.969625\pi$
$744$		0			0
$745$	−6.55051	−	11.3458i	−0.239992	−	0.415679i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−57.1812			−2.08936
$750$		0			0
$751$	−17.5227	−	30.3502i	−0.639413	−	1.10750i	−0.985562	−	0.169316i	$-0.945844\pi$
$751$	−17.5227	−	30.3502i	−0.639413	−	1.10750i	0.346149	−	0.938179i	$-0.387489\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.09859	−	3.63487i	0.0763755	−	0.132286i
$756$		0			0
$757$	−8.19694	+	14.1975i	−0.297923	+	0.516017i	−0.975661	−	0.219286i	$-0.929627\pi$
$757$	−8.19694	+	14.1975i	−0.297923	+	0.516017i	0.677738	+	0.735304i	$0.262960\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.99259			−0.0722313			−0.0361157	−	0.999348i	$-0.511498\pi$
$761$	−1.99259			−0.0722313			−0.0361157	+	0.999348i	$0.511498\pi$
$762$		0			0
$763$	15.3485	−	26.5843i	0.555652	−	0.962417i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.81671			0.282245
$768$		0			0
$769$	−14.1969	−	24.5898i	−0.511955	−	0.886732i	−0.999904	−	0.0138595i	$-0.995588\pi$
$769$	−14.1969	−	24.5898i	−0.511955	−	0.886732i	0.487949	−	0.872872i	$-0.337745\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−4.66883	−	8.08665i	−0.167926	−	0.290857i	0.769764	−	0.638328i	$-0.220374\pi$
$773$	−4.66883	−	8.08665i	−0.167926	−	0.290857i	−0.937691	+	0.347472i	$0.887040\pi$
$774$		0			0
$775$	18.4217	+	31.9073i	0.661726	+	1.14614i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−38.9776	−	11.7215i	−1.39652	−	0.419967i
$780$		0			0
$781$	−20.6969	+	35.8481i	−0.740595	+	1.28275i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.23536	−	14.2641i	−0.293933	−	0.509106i
$786$		0			0
$787$	−47.2474			−1.68419			−0.842095	−	0.539329i	$-0.818678\pi$
$787$	−47.2474			−1.68419			−0.842095	+	0.539329i	$0.818678\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−28.5906			−1.01657
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26.8577			0.951348			0.475674	−	0.879622i	$-0.342204\pi$
$797$	26.8577			0.951348			0.475674	+	0.879622i	$0.342204\pi$
$798$		0			0
$799$	−19.5959			−0.693254
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.2953	−	24.7602i	0.504471	−	0.873769i
$804$		0			0
$805$	13.1010			0.461750
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−37.4566			−1.31690			−0.658452	−	0.752622i	$-0.728789\pi$
$809$	−37.4566			−1.31690			−0.658452	+	0.752622i	$0.728789\pi$
$810$		0			0
$811$	8.89898	+	15.4135i	0.312485	+	0.541241i	0.978900	−	0.204341i	$-0.0655051\pi$
$811$	8.89898	+	15.4135i	0.312485	+	0.541241i	−0.666414	+	0.745582i	$0.732172\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.95765	−	8.58691i	0.173659	−	0.300786i
$816$		0			0
$817$	−26.5000	−	7.96920i	−0.927118	−	0.278807i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0558	+	26.0774i	0.525450	+	0.910107i	0.999561	+	0.0296412i	$0.00943648\pi$
$821$	15.0558	+	26.0774i	0.525450	+	0.910107i	−0.474110	+	0.880465i	$0.657230\pi$
$822$		0			0
$823$	−15.1010	−	26.1557i	−0.526388	−	0.911732i	−0.999527	−	0.0307437i	$-0.990212\pi$
$823$	−15.1010	−	26.1557i	−0.526388	−	0.911732i	0.473139	−	0.880988i	$-0.343121\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.6827	+	28.8953i	0.580115	+	1.00479i	0.995465	+	0.0951272i	$0.0303258\pi$
$827$	16.6827	+	28.8953i	0.580115	+	1.00479i	−0.415350	+	0.909662i	$0.636341\pi$
$828$		0			0
$829$	−52.1918			−1.81270			−0.906349	−	0.422531i	$-0.861142\pi$
$829$	−52.1918			−1.81270			−0.906349	+	0.422531i	$0.861142\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.14048	+	8.90357i	−0.178107	+	0.308490i
$834$		0			0
$835$	−20.2020			−0.699120
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.53530	+	9.58742i	−0.191100	+	0.330995i	−0.945615	−	0.325288i	$-0.894539\pi$
$839$	−5.53530	+	9.58742i	−0.191100	+	0.330995i	0.754515	+	0.656283i	$0.227872\pi$
$840$		0			0
$841$	−11.7020	+	20.2685i	−0.403519	+	0.698915i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.29577	−	10.9046i	−0.216581	−	0.375130i
$846$		0			0
$847$	74.8434			2.57165
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.05624	+	12.2218i	0.241885	+	0.418957i
$852$		0			0
$853$	19.8485	−	34.3786i	0.679599	−	1.17710i	−0.295503	−	0.955342i	$-0.595487\pi$
$853$	19.8485	−	34.3786i	0.679599	−	1.17710i	0.975102	−	0.221758i	$-0.0711794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.19718	+	7.26973i	−0.143373	+	0.248329i	−0.928765	−	0.370670i	$-0.879128\pi$
$857$	−4.19718	+	7.26973i	−0.143373	+	0.248329i	0.785392	+	0.618999i	$0.212462\pi$
$858$		0			0
$859$	−26.5227	−	45.9387i	−0.904943	−	1.56741i	−0.820994	−	0.570937i	$-0.806580\pi$
$859$	−26.5227	−	45.9387i	−0.904943	−	1.56741i	−0.0839492	−	0.996470i	$-0.526753\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.7320			0.603605			0.301802	−	0.953370i	$-0.402412\pi$
$863$	17.7320			0.603605			0.301802	+	0.953370i	$0.402412\pi$
$864$		0			0
$865$	3.79796	+	6.57826i	0.129134	+	0.223667i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.996295	+	1.72563i	−0.0337970	+	0.0585381i
$870$		0			0
$871$	−0.174235	+	0.301783i	−0.00590371	+	0.0102255i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	32.2102			1.08890
$876$		0			0
$877$	1.84847	−	3.20164i	0.0624184	−	0.108112i	−0.833128	−	0.553081i	$-0.813452\pi$
$877$	1.84847	−	3.20164i	0.0624184	−	0.108112i	0.895546	+	0.444969i	$0.146785\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	58.0185			1.95469			0.977347	−	0.211643i	$-0.0678815\pi$
$881$	58.0185			1.95469			0.977347	+	0.211643i	$0.0678815\pi$
$882$		0			0
$883$	15.1742	+	26.2825i	0.510654	+	0.884478i	0.999924	+	0.0123458i	$0.00392988\pi$
$883$	15.1742	+	26.2825i	0.510654	+	0.884478i	−0.489270	+	0.872132i	$0.662737\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.2106	−	41.9340i	−0.812913	−	1.40801i	−0.910817	−	0.412811i	$-0.864547\pi$
$887$	−24.2106	−	41.9340i	−0.812913	−	1.40801i	0.0979041	−	0.995196i	$-0.468786\pi$
$888$		0			0
$889$	10.0000	+	17.3205i	0.335389	+	0.580911i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	29.6399	−	27.8948i	0.991862	−	0.933464i
$894$		0			0
$895$	−0.303062	+	0.524918i	−0.0101302	+	0.0175461i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	34.2027	+	59.2409i	1.14073	+	1.97579i
$900$		0			0
$901$	2.20204			0.0733606
$902$		0			0
$903$		0			0
$904$		0			0
$905$	21.9292			0.728951
$906$		0			0
$907$	17.5959	−	30.4770i	0.584263	−	1.01197i	−0.410704	−	0.911769i	$-0.634717\pi$
$907$	17.5959	−	30.4770i	0.584263	−	1.01197i	0.994967	−	0.100204i	$-0.0319496\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−17.7320			−0.587488			−0.293744	−	0.955884i	$-0.594901\pi$
$911$	−17.7320			−0.587488			−0.293744	+	0.955884i	$0.594901\pi$
$912$		0			0
$913$	−65.3939			−2.16422
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−26.9637	+	46.7025i	−0.890419	+	1.54225i
$918$		0			0
$919$	−29.2474			−0.964784			−0.482392	−	0.875955i	$-0.660232\pi$
$919$	−29.2474			−0.964784			−0.482392	+	0.875955i	$0.660232\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.23907			0.238277
$924$		0			0
$925$	7.60102	+	13.1654i	0.249920	+	0.432874i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.9600	−	48.4281i	0.917337	−	1.58887i	0.113893	−	0.993493i	$-0.463668\pi$
$929$	27.9600	−	48.4281i	0.917337	−	1.58887i	0.803443	−	0.595381i	$-0.202999\pi$
$930$		0			0
$931$	−4.89898	−	20.7846i	−0.160558	−	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.29577	−	10.9046i	−0.205894	−	0.356618i
$936$		0			0
$937$	19.5454	+	33.8536i	0.638521	+	1.10595i	0.985758	+	0.168173i	$0.0537866\pi$
$937$	19.5454	+	33.8536i	0.638521	+	1.10595i	−0.347237	+	0.937777i	$0.612880\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.04189	+	5.26870i	0.0991626	+	0.171755i	0.911338	−	0.411658i	$-0.135050\pi$
$941$	3.04189	+	5.26870i	0.0991626	+	0.171755i	−0.812176	+	0.583413i	$0.801717\pi$
$942$		0			0
$943$	33.7980			1.10061
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.2530	+	33.3471i	−0.625637	+	1.08364i	0.362780	+	0.931875i	$0.381828\pi$
$947$	−19.2530	+	33.3471i	−0.625637	+	1.08364i	−0.988417	+	0.151761i	$0.951506\pi$
$948$		0			0
$949$	−5.00000			−0.162307
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.3832	−	19.7164i	0.368740	−	0.638676i	−0.620629	−	0.784104i	$-0.713123\pi$
$953$	11.3832	−	19.7164i	0.368740	−	0.638676i	0.989369	+	0.145429i	$0.0464561\pi$
$954$		0			0
$955$	−3.00000	+	5.19615i	−0.0970777	+	0.168144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−26.9637	−	46.7025i	−0.870702	−	1.50810i
$960$		0			0
$961$	58.2929			1.88041
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.3340	+	17.8989i	0.332662	+	0.576187i
$966$		0			0
$967$	−10.1742	+	17.6223i	−0.327181	+	0.566695i	−0.981951	−	0.189133i	$-0.939432\pi$
$967$	−10.1742	+	17.6223i	−0.327181	+	0.566695i	0.654770	+	0.755828i	$0.272765\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.80930	+	16.9902i	−0.314796	+	0.545242i	−0.979394	−	0.201959i	$-0.935269\pi$
$971$	−9.80930	+	16.9902i	−0.314796	+	0.545242i	0.664598	+	0.747201i	$0.268603\pi$
$972$		0			0
$973$	−4.05051	−	7.01569i	−0.129853	−	0.224913i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.14048			0.164458			0.0822292	−	0.996613i	$-0.473796\pi$
$977$	5.14048			0.164458			0.0822292	+	0.996613i	$0.473796\pi$
$978$		0			0
$979$	−15.0000	−	25.9808i	−0.479402	−	0.830349i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.33071	−	5.76896i	0.106233	−	0.184001i	−0.808008	−	0.589171i	$-0.799454\pi$
$983$	3.33071	−	5.76896i	0.106233	−	0.184001i	0.914241	+	0.405170i	$0.132788\pi$
$984$		0			0
$985$	7.65153	−	13.2528i	0.243798	−	0.422271i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22.9785			0.730674
$990$		0			0
$991$	−1.47730	+	2.55875i	−0.0469279	+	0.0812814i	−0.888535	−	0.458808i	$-0.848276\pi$
$991$	−1.47730	+	2.55875i	−0.0469279	+	0.0812814i	0.841607	+	0.540090i	$0.181610\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.365647			0.0115918
$996$		0			0
$997$	22.5454	+	39.0498i	0.714020	+	1.23672i	0.963336	+	0.268297i	$0.0864609\pi$
$997$	22.5454	+	39.0498i	0.714020	+	1.23672i	−0.249316	+	0.968422i	$0.580206\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.bb.1873.3		8
3.2	odd	2	inner	2736.2.s.bb.1873.2		8
4.3	odd	2		171.2.f.c.163.4	yes	8
12.11	even	2		171.2.f.c.163.1	yes	8
19.7	even	3	inner	2736.2.s.bb.577.3		8
57.26	odd	6	inner	2736.2.s.bb.577.2		8
76.7	odd	6		171.2.f.c.64.4	yes	8
76.11	odd	6		3249.2.a.bd.1.1		4
76.27	even	6		3249.2.a.be.1.4		4
228.11	even	6		3249.2.a.bd.1.4		4
228.83	even	6		171.2.f.c.64.1	✓	8
228.179	odd	6		3249.2.a.be.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.f.c.64.1	✓	8	228.83	even	6
171.2.f.c.64.4	yes	8	76.7	odd	6
171.2.f.c.163.1	yes	8	12.11	even	2
171.2.f.c.163.4	yes	8	4.3	odd	2
2736.2.s.bb.577.2		8	57.26	odd	6	inner
2736.2.s.bb.577.3		8	19.7	even	3	inner
2736.2.s.bb.1873.2		8	3.2	odd	2	inner
2736.2.s.bb.1873.3		8	1.1	even	1	trivial
3249.2.a.bd.1.1		4	76.11	odd	6
3249.2.a.bd.1.4		4	228.11	even	6
3249.2.a.be.1.1		4	228.179	odd	6
3249.2.a.be.1.4		4	76.27	even	6

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.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.524648 - 0.908716i) q^{5} +3.44949 q^{7} +O(q^{10})\)	\(q+(0.524648 - 0.908716i) q^{5} +3.44949 q^{7} -5.71812 q^{11} +(0.500000 + 0.866025i) q^{13} +(-1.04930 + 1.81743i) q^{17} +(-1.00000 - 4.24264i) q^{19} +(1.80977 + 3.13461i) q^{23} +(1.94949 + 3.37662i) q^{25} +(3.61953 + 6.26922i) q^{29} +9.44949 q^{31} +(1.80977 - 3.13461i) q^{35} +3.89898 q^{37} +(4.66883 - 8.08665i) q^{41} +(3.17423 - 5.49794i) q^{43} +(4.66883 + 8.08665i) q^{47} +4.89898 q^{49} +(-0.524648 - 0.908716i) q^{53} +(-3.00000 + 5.19615i) q^{55} +(3.90836 - 6.76947i) q^{59} +(-2.50000 - 4.33013i) q^{61} +1.04930 q^{65} +(0.174235 + 0.301783i) q^{67} +(3.61953 - 6.26922i) q^{71} +(-2.50000 + 4.33013i) q^{73} -19.7246 q^{77} +(0.174235 - 0.301783i) q^{79} +11.4362 q^{83} +(1.10102 + 1.90702i) q^{85} +(2.62324 + 4.54358i) q^{89} +(1.72474 + 2.98735i) q^{91} +(-4.38000 - 1.31718i) q^{95} +(-1.55051 + 2.68556i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 4 q^{13} - 8 q^{19} - 4 q^{25} + 56 q^{31} - 8 q^{37} - 4 q^{43} - 24 q^{55} - 20 q^{61} - 28 q^{67} - 20 q^{73} - 28 q^{79} + 48 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 4 * q^13 - 8 * q^19 - 4 * q^25 + 56 * q^31 - 8 * q^37 - 4 * q^43 - 24 * q^55 - 20 * q^61 - 28 * q^67 - 20 * q^73 - 28 * q^79 + 48 * q^85 + 4 * q^91 - 32 * q^97

Introduction

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