LMFDB - Embedded newform 2736.2.bm.i.559.2

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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2736.559
Dual form			2736.2.bm.i.1855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{5} +1.09638i q^{7} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{5} +1.09638i q^{7} +5.65685i q^{11} +(-0.949490 + 0.548188i) q^{13} +(-1.00000 + 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(4.89898 - 2.82843i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-7.89898 + 4.56048i) q^{29} +1.89898 q^{31} +(-1.89898 - 1.09638i) q^{35} -4.56048i q^{37} +(-4.50000 - 2.59808i) q^{43} +(-1.89898 + 1.09638i) q^{47} +5.79796 q^{49} +(-1.10102 + 0.635674i) q^{53} +(-9.79796 - 5.65685i) q^{55} +(3.89898 - 6.75323i) q^{59} +(0.0505103 + 0.0874863i) q^{61} -2.19275i q^{65} +(-5.39898 - 9.35131i) q^{67} +(2.00000 - 3.46410i) q^{71} +(-6.39898 + 11.0834i) q^{73} -6.20204 q^{77} +(4.94949 - 8.57277i) q^{79} +3.46410i q^{83} +(-2.00000 - 3.46410i) q^{85} +(-7.89898 + 4.56048i) q^{89} +(-0.601021 - 1.04100i) q^{91} +(7.00000 - 5.19615i) q^{95} +(-15.7980 - 9.12096i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} - 16 q^{19} + 2 q^{25} - 12 q^{29} - 12 q^{31} + 12 q^{35} - 18 q^{43} + 12 q^{47} - 16 q^{49} - 24 q^{53} - 4 q^{59} + 10 q^{61} - 2 q^{67} + 8 q^{71} - 6 q^{73} - 64 q^{77} + 10 q^{79} - 8 q^{85} - 12 q^{89} - 22 q^{91} + 28 q^{95} - 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 6 * q^13 - 4 * q^17 - 16 * q^19 + 2 * q^25 - 12 * q^29 - 12 * q^31 + 12 * q^35 - 18 * q^43 + 12 * q^47 - 16 * q^49 - 24 * q^53 - 4 * q^59 + 10 * q^61 - 2 * q^67 + 8 * q^71 - 6 * q^73 - 64 * q^77 + 10 * q^79 - 8 * q^85 - 12 * q^89 - 22 * q^91 + 28 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	−0.998203	−	0.0599153i	$-0.980917\pi$
$5$	−1.00000	+	1.73205i	−0.447214	+	0.774597i	0.550990	+	0.834512i	$0.314250\pi$
$6$		0			0
$7$			1.09638i			0.414391i	0.978300	+	0.207196i	$0.0664337\pi$
$7$			1.09638i			0.414391i	−0.978300	+	0.207196i	$0.933566\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$	−0.949490	+	0.548188i	−0.263341	+	0.152040i	−0.625858	−	0.779937i	$-0.715251\pi$
$13$	−0.949490	+	0.548188i	−0.263341	+	0.152040i	0.362517	+	0.931977i	$0.381918\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		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
$23$	4.89898	−	2.82843i	1.02151	−	0.589768i	0.106967	−	0.994263i	$-0.465886\pi$
$23$	4.89898	−	2.82843i	1.02151	−	0.589768i	0.914540	+	0.404495i	$0.132553\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.89898	+	4.56048i	−1.46680	+	0.846859i	−0.999310	−	0.0371370i	$-0.988176\pi$
$29$	−7.89898	+	4.56048i	−1.46680	+	0.846859i	−0.467493	+	0.883997i	$0.654843\pi$
$30$		0			0
$31$	1.89898			0.341067			0.170533	−	0.985352i	$-0.445451\pi$
$31$	1.89898			0.341067			0.170533	+	0.985352i	$0.445451\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.89898	−	1.09638i	−0.320986	−	0.185321i
$36$		0			0
$37$		−	4.56048i		−	0.749738i	−0.927078	−	0.374869i	$-0.877688\pi$
$37$		−	4.56048i		−	0.749738i	0.927078	−	0.374869i	$-0.122312\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	0.115960	−	0.993254i	$-0.463006\pi$
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	−0.802203	+	0.597051i	$0.796339\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.89898	+	1.09638i	−0.276995	+	0.159923i	−0.632062	−	0.774918i	$-0.717791\pi$
$47$	−1.89898	+	1.09638i	−0.276995	+	0.159923i	0.355067	+	0.934841i	$0.384458\pi$
$48$		0			0
$49$	5.79796			0.828280
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.10102	+	0.635674i	−0.151237	+	0.0873166i	−0.573709	−	0.819059i	$-0.694496\pi$
$53$	−1.10102	+	0.635674i	−0.151237	+	0.0873166i	0.422472	+	0.906376i	$0.361163\pi$
$54$		0			0
$55$	−9.79796	−	5.65685i	−1.32116	−	0.762770i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.89898	−	6.75323i	0.507604	−	0.879196i	−0.492357	−	0.870393i	$-0.663865\pi$
$59$	3.89898	−	6.75323i	0.507604	−	0.879196i	0.999961	−	0.00880259i	$-0.00280199\pi$
$60$		0			0
$61$	0.0505103	+	0.0874863i	0.00646718	+	0.0112015i	0.869241	−	0.494389i	$-0.164608\pi$
$61$	0.0505103	+	0.0874863i	0.00646718	+	0.0112015i	−0.862774	+	0.505590i	$0.831275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.19275i		−	0.271977i
$66$		0			0
$67$	−5.39898	−	9.35131i	−0.659590	−	1.14244i	−0.980722	−	0.195409i	$-0.937396\pi$
$67$	−5.39898	−	9.35131i	−0.659590	−	1.14244i	0.321131	−	0.947035i	$-0.395937\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.00000	−	3.46410i	0.237356	−	0.411113i	−0.722599	−	0.691268i	$-0.757052\pi$
$71$	2.00000	−	3.46410i	0.237356	−	0.411113i	0.959955	+	0.280155i	$0.0903858\pi$
$72$		0			0
$73$	−6.39898	+	11.0834i	−0.748944	+	1.29721i	0.199385	+	0.979921i	$0.436105\pi$
$73$	−6.39898	+	11.0834i	−0.748944	+	1.29721i	−0.948329	+	0.317288i	$0.897228\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.20204			−0.706788
$78$		0			0
$79$	4.94949	−	8.57277i	0.556861	−	0.964512i	−0.440895	−	0.897559i	$-0.645339\pi$
$79$	4.94949	−	8.57277i	0.556861	−	0.964512i	0.997756	−	0.0669530i	$-0.0213278\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	−2.00000	−	3.46410i	−0.216930	−	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.89898	+	4.56048i	−0.837290	+	0.483410i	−0.856342	−	0.516409i	$-0.827269\pi$
$89$	−7.89898	+	4.56048i	−0.837290	+	0.483410i	0.0190520	+	0.999818i	$0.493935\pi$
$90$		0			0
$91$	−0.601021	−	1.04100i	−0.0630041	−	0.109126i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	7.00000	−	5.19615i	0.718185	−	0.533114i
$96$		0			0
$97$	−15.7980	−	9.12096i	−1.60404	−	0.926093i	−0.990668	−	0.136299i	$-0.956479\pi$
$97$	−15.7980	−	9.12096i	−1.60404	−	0.926093i	−0.613372	−	0.789794i	$-0.710187\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	11.8990			1.17244			0.586221	−	0.810151i	$-0.300615\pi$
$103$	11.8990			1.17244			0.586221	+	0.810151i	$0.300615\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$	14.6969	+	8.48528i	1.40771	+	0.812743i	0.995167	−	0.0981950i	$-0.0313069\pi$
$109$	14.6969	+	8.48528i	1.40771	+	0.812743i	0.412544	+	0.910938i	$0.364640\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.0492i			1.50978i	0.655853	+	0.754889i	$0.272309\pi$
$113$			16.0492i			1.50978i	−0.655853	+	0.754889i	$0.727691\pi$
$114$		0			0
$115$			11.3137i			1.05501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.89898	−	1.09638i	−0.174079	−	0.100505i
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	6.89898	+	11.9494i	0.612185	+	1.06034i	0.990871	+	0.134811i	$0.0430427\pi$
$127$	6.89898	+	11.9494i	0.612185	+	1.06034i	−0.378686	+	0.925525i	$0.623624\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000	−	6.92820i	−1.04844	−	0.605320i	−0.126231	−	0.992001i	$-0.540288\pi$
$131$	−12.0000	−	6.92820i	−1.04844	−	0.605320i	−0.922214	+	0.386681i	$0.873621\pi$
$132$		0			0
$133$	1.89898	−	4.38551i	0.164662	−	0.380272i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	6.39898	−	3.69445i	0.542754	−	0.313359i	−0.203440	−	0.979087i	$-0.565212\pi$
$139$	6.39898	−	3.69445i	0.542754	−	0.313359i	0.746194	+	0.665728i	$0.231879\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.10102	−	5.37113i	−0.259320	−	0.449156i
$144$		0			0
$145$		−	18.2419i		−	1.51491i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.89898	+	5.02118i	−0.237494	+	0.411351i	−0.959994	−	0.280019i	$-0.909659\pi$
$149$	−2.89898	+	5.02118i	−0.237494	+	0.411351i	0.722501	+	0.691370i	$0.242993\pi$
$150$		0			0
$151$	2.20204			0.179200			0.0895998	−	0.995978i	$-0.471441\pi$
$151$	2.20204			0.179200			0.0895998	+	0.995978i	$0.471441\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.89898	+	3.28913i	−0.152530	+	0.264189i
$156$		0			0
$157$	−6.84847	+	11.8619i	−0.546567	+	0.946682i	0.451939	+	0.892049i	$0.350732\pi$
$157$	−6.84847	+	11.8619i	−0.546567	+	0.946682i	−0.998506	+	0.0546336i	$0.982601\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.10102	+	5.37113i	0.244395	+	0.423304i
$162$		0			0
$163$			15.5885i			1.22098i	0.792023	+	0.610491i	$0.209028\pi$
$163$			15.5885i			1.22098i	−0.792023	+	0.610491i	$0.790972\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	−5.89898	+	10.2173i	−0.453768	+	0.785949i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.89898	−	4.56048i	−0.600548	−	0.346727i	0.168709	−	0.985666i	$-0.446040\pi$
$173$	−7.89898	−	4.56048i	−0.600548	−	0.346727i	−0.769257	+	0.638939i	$0.779374\pi$
$174$		0			0
$175$	−0.949490	+	0.548188i	−0.0717747	+	0.0414391i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−6.20204			−0.463562			−0.231781	−	0.972768i	$-0.574455\pi$
$179$	−6.20204			−0.463562			−0.231781	+	0.972768i	$0.574455\pi$
$180$		0			0
$181$	−14.6969	+	8.48528i	−1.09241	+	0.630706i	−0.934218	−	0.356702i	$-0.883901\pi$
$181$	−14.6969	+	8.48528i	−1.09241	+	0.630706i	−0.158196	+	0.987408i	$0.550568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.89898	+	4.56048i	0.580745	+	0.335293i
$186$		0			0
$187$	−9.79796	−	5.65685i	−0.716498	−	0.413670i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	24.2487i		−	1.75458i	−0.479965	−	0.877288i	$-0.659351\pi$
$191$		−	24.2487i		−	1.75458i	0.479965	−	0.877288i	$-0.340649\pi$
$192$		0			0
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	15.9495	−	9.20844i	1.13063	−	0.652769i	0.186536	−	0.982448i	$-0.440274\pi$
$199$	15.9495	−	9.20844i	1.13063	−	0.652769i	0.944093	+	0.329679i	$0.106940\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.00000	−	8.66025i	−0.350931	−	0.607831i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.79796	−	22.6274i	0.677739	−	1.56517i
$210$		0			0
$211$	−0.601021	+	1.04100i	−0.0413760	+	0.0716653i	−0.885972	−	0.463739i	$-0.846507\pi$
$211$	−0.601021	+	1.04100i	−0.0413760	+	0.0716653i	0.844596	+	0.535404i	$0.179841\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	9.00000	−	5.19615i	0.613795	−	0.354375i
$216$		0			0
$217$			2.08200i			0.141335i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.19275i		−	0.147501i
$222$		0			0
$223$	−1.94949	+	3.37662i	−0.130547	+	0.226115i	−0.923888	−	0.382664i	$-0.875007\pi$
$223$	−1.94949	+	3.37662i	−0.130547	+	0.226115i	0.793340	+	0.608778i	$0.208340\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.79796			0.252079			0.126040	−	0.992025i	$-0.459773\pi$
$227$	3.79796			0.252079			0.126040	+	0.992025i	$0.459773\pi$
$228$		0			0
$229$	21.6969			1.43377			0.716887	−	0.697189i	$-0.245566\pi$
$229$	21.6969			1.43377			0.716887	+	0.697189i	$0.245566\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.79796	+	16.9706i	−0.641886	+	1.11178i	0.343126	+	0.939289i	$0.388514\pi$
$233$	−9.79796	+	16.9706i	−0.641886	+	1.11178i	−0.985011	+	0.172489i	$0.944819\pi$
$234$		0			0
$235$		−	4.38551i		−	0.286079i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	0.921404i		−	0.0596006i	−0.999556	−	0.0298003i	$-0.990513\pi$
$239$		−	0.921404i		−	0.0596006i	0.999556	−	0.0298003i	$-0.00948714\pi$
$240$		0			0
$241$	5.29796	−	3.05878i	0.341272	−	0.197033i	−0.319563	−	0.947565i	$-0.603536\pi$
$241$	5.29796	−	3.05878i	0.341272	−	0.197033i	0.660834	+	0.750532i	$0.270203\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.79796	+	10.0424i	−0.370418	+	0.641583i
$246$		0			0
$247$	4.74745	−	0.548188i	0.302073	−	0.0348804i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.7980	+	10.8530i	−1.18652	+	0.685036i	−0.957513	−	0.288390i	$-0.906880\pi$
$251$	−18.7980	+	10.8530i	−1.18652	+	0.685036i	−0.229004	+	0.973426i	$0.573547\pi$
$252$		0			0
$253$	16.0000	+	27.7128i	1.00591	+	1.74229i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.10102	−	0.635674i	0.0686798	−	0.0396523i	−0.465267	−	0.885170i	$-0.654042\pi$
$257$	1.10102	−	0.635674i	0.0686798	−	0.0396523i	0.533947	+	0.845518i	$0.320708\pi$
$258$		0			0
$259$	5.00000			0.310685
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.79796	−	2.19275i	−0.234192	−	0.135211i	0.378312	−	0.925678i	$-0.376505\pi$
$263$	−3.79796	−	2.19275i	−0.234192	−	0.135211i	−0.612505	+	0.790467i	$0.709838\pi$
$264$		0			0
$265$		−	2.54270i		−	0.156197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.797959	+	0.460702i	0.0486524	+	0.0280895i	0.524129	−	0.851639i	$-0.324391\pi$
$269$	0.797959	+	0.460702i	0.0486524	+	0.0280895i	−0.475476	+	0.879728i	$0.657724\pi$
$270$		0			0
$271$	−12.7980	−	7.38891i	−0.777421	−	0.448844i	0.0580947	−	0.998311i	$-0.481497\pi$
$271$	−12.7980	−	7.38891i	−0.777421	−	0.448844i	−0.835515	+	0.549467i	$0.814831\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.89898	+	2.82843i	−0.295420	+	0.170561i
$276$		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
$281$	16.1010	−	9.29593i	0.960506	−	0.554549i	0.0641775	−	0.997938i	$-0.479558\pi$
$281$	16.1010	−	9.29593i	0.960506	−	0.554549i	0.896329	+	0.443390i	$0.146224\pi$
$282$		0			0
$283$	12.7980	+	7.38891i	0.760760	+	0.439225i	0.829568	−	0.558405i	$-0.188586\pi$
$283$	12.7980	+	7.38891i	0.760760	+	0.439225i	−0.0688087	+	0.997630i	$0.521920\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.57826i		−	0.384306i	−0.981365	−	0.192153i	$-0.938453\pi$
$293$		−	6.57826i		−	0.384306i	0.981365	−	0.192153i	$-0.0615470\pi$
$294$		0			0
$295$	7.79796	+	13.5065i	0.454015	+	0.786377i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.10102	+	5.37113i	−0.179337	+	0.310620i
$300$		0			0
$301$	2.84847	−	4.93369i	0.164183	−	0.284373i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.202041			−0.0115688
$306$		0			0
$307$	−15.7980	+	27.3629i	−0.901637	+	1.56168i	−0.0762684	+	0.997087i	$0.524301\pi$
$307$	−15.7980	+	27.3629i	−0.901637	+	1.56168i	−0.825369	+	0.564594i	$0.809033\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			33.0197i			1.87238i	0.351499	+	0.936188i	$0.385672\pi$
$311$			33.0197i			1.87238i	−0.351499	+	0.936188i	$0.614328\pi$
$312$		0			0
$313$	8.79796	+	15.2385i	0.497290	+	0.861332i	0.999995	−	0.00312637i	$-0.000995155\pi$
$313$	8.79796	+	15.2385i	0.497290	+	0.861332i	−0.502705	+	0.864458i	$0.667662\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.7980	−	7.38891i	0.718805	−	0.415002i	−0.0955077	−	0.995429i	$-0.530447\pi$
$317$	12.7980	−	7.38891i	0.718805	−	0.415002i	0.814313	+	0.580426i	$0.197114\pi$
$318$		0			0
$319$	−25.7980	−	44.6834i	−1.44441	−	2.50179i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	7.00000	−	5.19615i	0.389490	−	0.289122i
$324$		0			0
$325$	−0.949490	−	0.548188i	−0.0526682	−	0.0304080i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.20204	−	2.08200i	−0.0662707	−	0.114784i
$330$		0			0
$331$	1.20204			0.0660702			0.0330351	−	0.999454i	$-0.489483\pi$
$331$	1.20204			0.0660702			0.0330351	+	0.999454i	$0.489483\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	21.5959			1.17991
$336$		0			0
$337$	−22.1969	−	12.8154i	−1.20914	−	0.698100i	−0.246572	−	0.969125i	$-0.579304\pi$
$337$	−22.1969	−	12.8154i	−1.20914	−	0.698100i	−0.962572	+	0.271025i	$0.912637\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10.7423i			0.581725i
$342$		0			0
$343$			14.0314i			0.757623i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.89898	−	1.09638i	−0.101943	−	0.0588566i	0.448162	−	0.893952i	$-0.352079\pi$
$347$	−1.89898	−	1.09638i	−0.101943	−	0.0588566i	−0.550105	+	0.835096i	$0.685412\pi$
$348$		0			0
$349$	−21.6969			−1.16141			−0.580705	−	0.814114i	$-0.697223\pi$
$349$	−21.6969			−1.16141			−0.580705	+	0.814114i	$0.697223\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.79796			0.202145			0.101072	−	0.994879i	$-0.467773\pi$
$353$	3.79796			0.202145			0.101072	+	0.994879i	$0.467773\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.79796	−	5.65685i	−0.517116	−	0.298557i	0.218638	−	0.975806i	$-0.429839\pi$
$359$	−9.79796	−	5.65685i	−0.517116	−	0.298557i	−0.735754	+	0.677249i	$0.763172\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.7980	−	22.1667i	−0.669876	−	1.16026i
$366$		0			0
$367$	−5.84847	+	3.37662i	−0.305288	+	0.176258i	−0.644816	−	0.764338i	$-0.723066\pi$
$367$	−5.84847	+	3.37662i	−0.305288	+	0.176258i	0.339528	+	0.940596i	$0.389733\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.696938	−	1.20713i	−0.0361832	−	0.0626712i
$372$		0			0
$373$		−	33.3697i		−	1.72782i	−0.503650	−	0.863908i	$-0.668010\pi$
$373$		−	33.3697i		−	1.72782i	0.503650	−	0.863908i	$-0.331990\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.00000	−	8.66025i	0.257513	−	0.446026i
$378$		0			0
$379$	6.59592			0.338810			0.169405	−	0.985547i	$-0.445815\pi$
$379$	6.59592			0.338810			0.169405	+	0.985547i	$0.445815\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.89898	−	11.9494i	0.352521	−	0.610585i	−0.634169	−	0.773194i	$-0.718658\pi$
$383$	6.89898	−	11.9494i	0.352521	−	0.610585i	0.986691	+	0.162609i	$0.0519910\pi$
$384$		0			0
$385$	6.20204	−	10.7423i	0.316085	−	0.547476i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.6969	+	32.3840i	0.947972	+	1.64194i	0.749688	+	0.661792i	$0.230204\pi$
$389$	18.6969	+	32.3840i	0.947972	+	1.64194i	0.198285	+	0.980144i	$0.436463\pi$
$390$		0			0
$391$			11.3137i			0.572159i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.89898	+	17.1455i	0.498072	+	0.862686i
$396$		0			0
$397$	3.05051	−	5.28364i	0.153101	−	0.265178i	−0.779265	−	0.626694i	$-0.784407\pi$
$397$	3.05051	−	5.28364i	0.153101	−	0.265178i	0.932366	+	0.361516i	$0.117741\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.10102	+	2.36773i	0.204795	+	0.118239i	0.598890	−	0.800831i	$-0.295609\pi$
$401$	4.10102	+	2.36773i	0.204795	+	0.118239i	−0.394095	+	0.919070i	$0.628942\pi$
$402$		0			0
$403$	−1.80306	+	1.04100i	−0.0898169	+	0.0518558i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.7980			1.27876
$408$		0			0
$409$	−6.00000	+	3.46410i	−0.296681	+	0.171289i	−0.640951	−	0.767582i	$-0.721460\pi$
$409$	−6.00000	+	3.46410i	−0.296681	+	0.171289i	0.344270	+	0.938871i	$0.388126\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7.40408	+	4.27475i	0.364331	+	0.210347i
$414$		0			0
$415$	−6.00000	−	3.46410i	−0.294528	−	0.170046i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			35.2125i			1.72024i	0.510090	+	0.860121i	$0.329612\pi$
$419$			35.2125i			1.72024i	−0.510090	+	0.860121i	$0.670388\pi$
$420$		0			0
$421$	4.89898	+	2.82843i	0.238762	+	0.137849i	0.614607	−	0.788833i	$-0.289314\pi$
$421$	4.89898	+	2.82843i	0.238762	+	0.137849i	−0.375846	+	0.926682i	$0.622648\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$	−0.0959179	+	0.0553782i	−0.00464179	+	0.00267994i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.89898	−	13.6814i	−0.380480	−	0.659011i	0.610651	−	0.791900i	$-0.290908\pi$
$431$	−7.89898	−	13.6814i	−0.380480	−	0.659011i	−0.991131	+	0.132889i	$0.957575\pi$
$432$		0			0
$433$	10.1969	−	5.88721i	0.490034	−	0.282921i	−0.234555	−	0.972103i	$-0.575363\pi$
$433$	10.1969	−	5.88721i	0.490034	−	0.282921i	0.724588	+	0.689182i	$0.242030\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−24.4949	+	2.82843i	−1.17175	+	0.135302i
$438$		0			0
$439$	10.8485	−	18.7901i	0.517769	−	0.896803i	−0.482018	−	0.876161i	$-0.660096\pi$
$439$	10.8485	−	18.7901i	0.517769	−	0.896803i	0.999787	−	0.0206411i	$-0.00657073\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.7980	−	10.8530i	0.893118	−	0.515642i	0.0181569	−	0.999835i	$-0.494220\pi$
$443$	18.7980	−	10.8530i	0.893118	−	0.515642i	0.874961	+	0.484193i	$0.160887\pi$
$444$		0			0
$445$		−	18.2419i		−	0.864750i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			28.6342i			1.35133i	0.737208	+	0.675666i	$0.236144\pi$
$449$			28.6342i			1.35133i	−0.737208	+	0.675666i	$0.763856\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.40408			0.112705
$456$		0			0
$457$	20.7980			0.972887			0.486444	−	0.873712i	$-0.338294\pi$
$457$	20.7980			0.972887			0.486444	+	0.873712i	$0.338294\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.7980	+	18.7026i	−0.502911	+	0.871068i	0.497083	+	0.867703i	$0.334404\pi$
$461$	−10.7980	+	18.7026i	−0.502911	+	0.871068i	−0.999994	+	0.00336470i	$0.998929\pi$
$462$		0			0
$463$			3.28913i			0.152859i	0.997075	+	0.0764294i	$0.0243520\pi$
$463$			3.28913i			0.152859i	−0.997075	+	0.0764294i	$0.975648\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			16.3991i			0.758860i	0.925220	+	0.379430i	$0.123880\pi$
$467$			16.3991i			0.758860i	−0.925220	+	0.379430i	$0.876120\pi$
$468$		0			0
$469$	10.2526	−	5.91931i	0.473419	−	0.273328i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.6969	−	25.4558i	0.675766	−	1.17046i
$474$		0			0
$475$	−0.500000	−	4.33013i	−0.0229416	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.7980	+	12.5851i	−0.995974	+	0.575026i	−0.907055	−	0.421013i	$-0.861675\pi$
$479$	−21.7980	+	12.5851i	−0.995974	+	0.575026i	−0.0889195	+	0.996039i	$0.528341\pi$
$480$		0			0
$481$	2.50000	+	4.33013i	0.113990	+	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	31.5959	−	18.2419i	1.43470	−	0.828323i
$486$		0			0
$487$	1.79796			0.0814733			0.0407366	−	0.999170i	$-0.487030\pi$
$487$	1.79796			0.0814733			0.0407366	+	0.999170i	$0.487030\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	21.7980	+	12.5851i	0.983728	+	0.567956i	0.903394	−	0.428812i	$-0.141068\pi$
$491$	21.7980	+	12.5851i	0.983728	+	0.567956i	0.0803345	+	0.996768i	$0.474401\pi$
$492$		0			0
$493$		−	18.2419i		−	0.821574i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.79796	+	2.19275i	0.170362	+	0.0983584i
$498$		0			0
$499$	1.80306	+	1.04100i	0.0807161	+	0.0466015i	0.539815	−	0.841784i	$-0.318494\pi$
$499$	1.80306	+	1.04100i	0.0807161	+	0.0466015i	−0.459099	+	0.888385i	$0.651828\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	23.6969	−	13.6814i	1.05659	−	0.610025i	0.132106	−	0.991236i	$-0.457826\pi$
$503$	23.6969	−	13.6814i	1.05659	−	0.610025i	0.924488	+	0.381211i	$0.124493\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.8990	−	18.4169i	1.41390	−	0.816314i	0.418145	−	0.908381i	$-0.362681\pi$
$509$	31.8990	−	18.4169i	1.41390	−	0.816314i	0.995753	+	0.0920665i	$0.0293472\pi$
$510$		0			0
$511$	−12.1515	−	7.01569i	−0.537552	−	0.310356i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.8990	+	20.6096i	−0.524332	+	0.908169i
$516$		0			0
$517$	−6.20204	−	10.7423i	−0.272765	−	0.472444i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1.27135i		−	0.0556988i	−0.999612	−	0.0278494i	$-0.991134\pi$
$521$		−	1.27135i		−	0.0556988i	0.999612	−	0.0278494i	$-0.00886589\pi$
$522$		0			0
$523$	−13.2980	−	23.0327i	−0.581479	−	1.00715i	−0.995304	−	0.0967951i	$-0.969141\pi$
$523$	−13.2980	−	23.0327i	−0.581479	−	1.00715i	0.413825	−	0.910356i	$-0.364192\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.89898	+	3.28913i	−0.0827208	+	0.143277i
$528$		0			0
$529$	4.50000	−	7.79423i	0.195652	−	0.338880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	8.00000	−	13.8564i	0.345870	−	0.599065i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			32.7982i			1.41272i
$540$		0			0
$541$	11.0505	+	19.1400i	0.475099	+	0.822895i	0.999593	−	0.0285189i	$-0.00907908\pi$
$541$	11.0505	+	19.1400i	0.475099	+	0.822895i	−0.524495	+	0.851414i	$0.675746\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−29.3939	+	16.9706i	−1.25910	+	0.726939i
$546$		0			0
$547$	−19.5000	−	33.7750i	−0.833760	−	1.44411i	−0.895036	−	0.445993i	$-0.852850\pi$
$547$	−19.5000	−	33.7750i	−0.833760	−	1.44411i	0.0612764	−	0.998121i	$-0.480483\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	39.4949	−	4.56048i	1.68254	−	0.194283i
$552$		0			0
$553$	9.39898	+	5.42650i	0.399685	+	0.230758i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	−0.999952	−	0.00979220i	$-0.996883\pi$
$557$	−12.0000	−	20.7846i	−0.508456	−	0.880672i	0.491496	−	0.870880i	$-0.336450\pi$
$558$		0			0
$559$	5.69694			0.240955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−43.5959			−1.83735			−0.918674	−	0.395016i	$-0.870739\pi$
$563$	−43.5959			−1.83735			−0.918674	+	0.395016i	$0.870739\pi$
$564$		0			0
$565$	−27.7980	−	16.0492i	−1.16947	−	0.675193i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			2.54270i			0.106595i	0.998579	+	0.0532977i	$0.0169732\pi$
$569$			2.54270i			0.106595i	−0.998579	+	0.0532977i	$0.983027\pi$
$570$		0			0
$571$		−	9.93160i		−	0.415625i	−0.978169	−	0.207812i	$-0.933366\pi$
$571$		−	9.93160i		−	0.415625i	0.978169	−	0.207812i	$-0.0666343\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.89898	+	2.82843i	0.204302	+	0.117954i
$576$		0			0
$577$	−2.40408			−0.100083			−0.0500416	−	0.998747i	$-0.515935\pi$
$577$	−2.40408			−0.100083			−0.0500416	+	0.998747i	$0.515935\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.79796			−0.157566
$582$		0			0
$583$	−3.59592	−	6.22831i	−0.148928	−	0.257950i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.10102	−	2.36773i	−0.169267	−	0.0977265i	0.412973	−	0.910743i	$-0.364490\pi$
$587$	−4.10102	−	2.36773i	−0.169267	−	0.0977265i	−0.582240	+	0.813017i	$0.697824\pi$
$588$		0			0
$589$	−7.59592	−	3.28913i	−0.312984	−	0.135526i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.89898	+	3.28913i	0.0779817	+	0.135068i	0.902379	−	0.430943i	$-0.141819\pi$
$593$	1.89898	+	3.28913i	0.0779817	+	0.135068i	−0.824397	+	0.566012i	$0.808486\pi$
$594$		0			0
$595$	3.79796	−	2.19275i	0.155701	−	0.0898941i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.8990	+	36.1981i	0.853909	+	1.47901i	0.877653	+	0.479296i	$0.159108\pi$
$599$	20.8990	+	36.1981i	0.853909	+	1.47901i	−0.0237442	+	0.999718i	$0.507559\pi$
$600$		0			0
$601$		−	27.2521i		−	1.11164i	−0.831304	−	0.555818i	$-0.812405\pi$
$601$		−	27.2521i		−	1.11164i	0.831304	−	0.555818i	$-0.187595\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	21.0000	−	36.3731i	0.853771	−	1.47878i
$606$		0			0
$607$	12.1010			0.491165			0.245583	−	0.969376i	$-0.421021\pi$
$607$	12.1010			0.491165			0.245583	+	0.969376i	$0.421021\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.20204	−	2.08200i	0.0486294	−	0.0842285i
$612$		0			0
$613$	16.7980	−	29.0949i	0.678463	−	1.17513i	−0.296980	−	0.954884i	$-0.595980\pi$
$613$	16.7980	−	29.0949i	0.678463	−	1.17513i	0.975444	−	0.220249i	$-0.0706870\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.89898	−	11.9494i	−0.277742	−	0.481064i	0.693081	−	0.720860i	$-0.256253\pi$
$617$	−6.89898	−	11.9494i	−0.277742	−	0.481064i	−0.970823	+	0.239796i	$0.922920\pi$
$618$		0			0
$619$			32.5590i			1.30866i	0.756210	+	0.654329i	$0.227049\pi$
$619$			32.5590i			1.30866i	−0.756210	+	0.654329i	$0.772951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00000	−	8.66025i	−0.200321	−	0.346966i
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.89898	+	4.56048i	0.314953	+	0.181838i
$630$		0			0
$631$	−36.6464	+	21.1578i	−1.45887	+	0.842280i	−0.998956	−	0.0456831i	$-0.985454\pi$
$631$	−36.6464	+	21.1578i	−1.45887	+	0.842280i	−0.459915	+	0.887963i	$0.652120\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−27.5959			−1.09511
$636$		0			0
$637$	−5.50510	+	3.17837i	−0.218120	+	0.125932i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.2020	+	6.46750i	0.442454	+	0.255451i	0.704638	−	0.709567i	$-0.251109\pi$
$641$	11.2020	+	6.46750i	0.442454	+	0.255451i	−0.262184	+	0.965018i	$0.584443\pi$
$642$		0			0
$643$	15.0959	+	8.71563i	0.595325	+	0.343711i	0.767200	−	0.641408i	$-0.221649\pi$
$643$	15.0959	+	8.71563i	0.595325	+	0.343711i	−0.171875	+	0.985119i	$0.554983\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	39.3765i		−	1.54805i	−0.633156	−	0.774024i	$-0.718241\pi$
$647$		−	39.3765i		−	1.54805i	0.633156	−	0.774024i	$-0.281759\pi$
$648$		0			0
$649$	38.2020	+	22.0560i	1.49956	+	0.865772i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	43.5959			1.70604			0.853020	−	0.521878i	$-0.174768\pi$
$653$	43.5959			1.70604			0.853020	+	0.521878i	$0.174768\pi$
$654$		0			0
$655$	24.0000	−	13.8564i	0.937758	−	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.6969	+	32.3840i	0.728329	+	1.26150i	0.957589	+	0.288138i	$0.0930362\pi$
$659$	18.6969	+	32.3840i	0.728329	+	1.26150i	−0.229260	+	0.973365i	$0.573630\pi$
$660$		0			0
$661$	−42.4949	+	24.5344i	−1.65286	+	0.954279i	−0.676972	+	0.736009i	$0.736708\pi$
$661$	−42.4949	+	24.5344i	−1.65286	+	0.954279i	−0.975888	+	0.218270i	$0.929959\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.69694	+	7.67463i	0.220918	+	0.297610i
$666$		0			0
$667$	−25.7980	+	44.6834i	−0.998901	+	1.73015i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.494897	+	0.285729i	−0.0191053	+	0.0110305i
$672$		0			0
$673$			41.3300i			1.59316i	0.604536	+	0.796578i	$0.293359\pi$
$673$			41.3300i			1.59316i	−0.604536	+	0.796578i	$0.706641\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			36.8338i			1.41564i	0.706394	+	0.707818i	$0.250320\pi$
$677$			36.8338i			1.41564i	−0.706394	+	0.707818i	$0.749680\pi$
$678$		0			0
$679$	10.0000	−	17.3205i	0.383765	−	0.664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.7980			1.36977			0.684885	−	0.728651i	$-0.259852\pi$
$683$	35.7980			1.36977			0.684885	+	0.728651i	$0.259852\pi$
$684$		0			0
$685$	24.0000			0.916993
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.696938	−	1.20713i	0.0265512	−	0.0459881i
$690$		0			0
$691$			24.2487i			0.922464i	0.887279	+	0.461232i	$0.152592\pi$
$691$			24.2487i			0.922464i	−0.887279	+	0.461232i	$0.847408\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			14.7778i			0.560554i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.0000	−	36.3731i	0.793159	−	1.37379i	−0.130843	−	0.991403i	$-0.541768\pi$
$701$	21.0000	−	36.3731i	0.793159	−	1.37379i	0.924002	−	0.382389i	$-0.124898\pi$
$702$		0			0
$703$	−7.89898	+	18.2419i	−0.297916	+	0.688007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.3939	−	6.57826i	0.428511	−	0.247401i
$708$		0			0
$709$	18.8485	+	32.6465i	0.707869	+	1.22607i	0.965646	+	0.259861i	$0.0836767\pi$
$709$	18.8485	+	32.6465i	0.707869	+	1.22607i	−0.257777	+	0.966204i	$0.582990\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.30306	−	5.37113i	0.348402	−	0.201150i
$714$		0			0
$715$	12.4041			0.463886
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.1010	−	7.56388i	−0.488586	−	0.282085i	0.235402	−	0.971898i	$-0.424359\pi$
$719$	−13.1010	−	7.56388i	−0.488586	−	0.282085i	−0.723988	+	0.689813i	$0.757693\pi$
$720$		0			0
$721$			13.0458i			0.485849i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.89898	−	4.56048i	−0.293361	−	0.169372i
$726$		0			0
$727$	42.3434	+	24.4470i	1.57043	+	0.906687i	0.996116	+	0.0880513i	$0.0280639\pi$
$727$	42.3434	+	24.4470i	1.57043	+	0.906687i	0.574313	+	0.818636i	$0.305269\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.00000	−	5.19615i	0.332877	−	0.192187i
$732$		0			0
$733$	−41.5959			−1.53638			−0.768190	−	0.640222i	$-0.778842\pi$
$733$	−41.5959			−1.53638			−0.768190	+	0.640222i	$0.778842\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	52.8990	−	30.5412i	1.94856	−	1.12500i
$738$		0			0
$739$	−8.60102	−	4.96580i	−0.316394	−	0.182670i	0.333390	−	0.942789i	$-0.391807\pi$
$739$	−8.60102	−	4.96580i	−0.316394	−	0.182670i	−0.649784	+	0.760119i	$0.725141\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.10102	+	1.90702i	−0.0403925	+	0.0699619i	−0.885515	−	0.464611i	$-0.846194\pi$
$743$	−1.10102	+	1.90702i	−0.0403925	+	0.0699619i	0.845122	+	0.534573i	$0.179528\pi$
$744$		0			0
$745$	−5.79796	−	10.0424i	−0.212421	−	0.367924i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	8.77101i		−	0.320486i
$750$		0			0
$751$	20.7474	+	35.9356i	0.757085	+	1.31131i	0.944331	+	0.328997i	$0.106711\pi$
$751$	20.7474	+	35.9356i	0.757085	+	1.31131i	−0.187246	+	0.982313i	$0.559956\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.20204	+	3.81405i	−0.0801405	+	0.138807i
$756$		0			0
$757$	−15.7474	+	27.2754i	−0.572351	+	0.991341i	0.423973	+	0.905675i	$0.360635\pi$
$757$	−15.7474	+	27.2754i	−0.572351	+	0.991341i	−0.996324	+	0.0856658i	$0.972698\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.7980			0.572676			0.286338	−	0.958129i	$-0.407562\pi$
$761$	15.7980			0.572676			0.286338	+	0.958129i	$0.407562\pi$
$762$		0			0
$763$	−9.30306	+	16.1134i	−0.336793	+	0.583343i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			8.54950i			0.308704i
$768$		0			0
$769$	−16.0959	−	27.8789i	−0.580434	−	1.00534i	−0.995428	−	0.0955165i	$-0.969550\pi$
$769$	−16.0959	−	27.8789i	−0.580434	−	1.00534i	0.414994	−	0.909824i	$-0.363784\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.30306	−	3.63907i	0.226705	−	0.130888i	−0.382346	−	0.924019i	$-0.624884\pi$
$773$	6.30306	−	3.63907i	0.226705	−	0.130888i	0.609051	+	0.793131i	$0.291550\pi$
$774$		0			0
$775$	0.949490	+	1.64456i	0.0341067	+	0.0590745i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	19.5959	+	11.3137i	0.701197	+	0.404836i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.6969	−	23.7238i	−0.488865	−	0.846738i
$786$		0			0
$787$	34.5959			1.23321			0.616606	−	0.787272i	$-0.288507\pi$
$787$	34.5959			1.23321			0.616606	+	0.787272i	$0.288507\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17.5959			−0.625639
$792$		0			0
$793$	−0.0959179	−	0.0553782i	−0.00340615	−	0.00196654i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			52.1830i			1.84842i	0.381887	+	0.924209i	$0.375274\pi$
$797$			52.1830i			1.84842i	−0.381887	+	0.924209i	$0.624726\pi$
$798$		0			0
$799$		−	4.38551i		−	0.155148i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−62.6969	−	36.1981i	−2.21253	−	1.27740i
$804$		0			0
$805$	−12.4041			−0.437186
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−43.3939			−1.52565			−0.762824	−	0.646606i	$-0.776188\pi$
$809$	−43.3939			−1.52565			−0.762824	+	0.646606i	$0.776188\pi$
$810$		0			0
$811$	4.00000	+	6.92820i	0.140459	+	0.243282i	0.927670	−	0.373402i	$-0.121809\pi$
$811$	4.00000	+	6.92820i	0.140459	+	0.243282i	−0.787211	+	0.616684i	$0.788476\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.0000	−	15.5885i	−0.945769	−	0.546040i
$816$		0			0
$817$	13.5000	+	18.1865i	0.472305	+	0.636266i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	23.6969	+	41.0443i	0.827029	+	1.43246i	0.900359	+	0.435148i	$0.143304\pi$
$821$	23.6969	+	41.0443i	0.827029	+	1.43246i	−0.0733300	+	0.997308i	$0.523363\pi$
$822$		0			0
$823$	−38.3939	+	22.1667i	−1.33833	+	0.772683i	−0.986559	−	0.163404i	$-0.947752\pi$
$823$	−38.3939	+	22.1667i	−1.33833	+	0.772683i	−0.351767	+	0.936088i	$0.614419\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.7980	+	39.4872i	0.792763	+	1.37311i	0.924250	+	0.381787i	$0.124691\pi$
$827$	22.7980	+	39.4872i	0.792763	+	1.37311i	−0.131487	+	0.991318i	$0.541975\pi$
$828$		0			0
$829$			42.3157i			1.46968i	0.678239	+	0.734842i	$0.262744\pi$
$829$			42.3157i			1.46968i	−0.678239	+	0.734842i	$0.737256\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.79796	+	10.0424i	−0.200887	+	0.347947i
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.00000	−	5.19615i	0.103572	−	0.179391i	−0.809582	−	0.587007i	$-0.800306\pi$
$839$	3.00000	−	5.19615i	0.103572	−	0.179391i	0.913154	+	0.407615i	$0.133640\pi$
$840$		0			0
$841$	27.0959	−	46.9315i	0.934342	−	1.61833i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.7980	−	20.4347i	−0.405862	−	0.702974i
$846$		0			0
$847$		−	23.0239i		−	0.791111i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.8990	−	22.3417i	−0.442171	−	0.765863i
$852$		0			0
$853$	22.6464	−	39.2248i	0.775399	−	1.34303i	−0.159171	−	0.987251i	$-0.550882\pi$
$853$	22.6464	−	39.2248i	0.775399	−	1.34303i	0.934570	−	0.355779i	$-0.115785\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−46.5959	−	26.9022i	−1.59169	−	0.918960i	−0.993018	−	0.117963i	$-0.962363\pi$
$857$	−46.5959	−	26.9022i	−1.59169	−	0.918960i	−0.598668	−	0.800997i	$-0.704303\pi$
$858$		0			0
$859$	3.70204	−	2.13737i	0.126312	−	0.0729263i	−0.435513	−	0.900183i	$-0.643433\pi$
$859$	3.70204	−	2.13737i	0.126312	−	0.0729263i	0.561825	+	0.827256i	$0.310100\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.79796			−0.265446			−0.132723	−	0.991153i	$-0.542372\pi$
$863$	−7.79796			−0.265446			−0.132723	+	0.991153i	$0.542372\pi$
$864$		0			0
$865$	15.7980	−	9.12096i	0.537147	−	0.310122i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.4949	+	27.9985i	1.64508	+	0.949785i
$870$		0			0
$871$	10.2526	+	5.91931i	0.347394	+	0.200568i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	13.1565i		−	0.444771i
$876$		0			0
$877$	20.5454	+	11.8619i	0.693769	+	0.400548i	0.805022	−	0.593244i	$-0.202153\pi$
$877$	20.5454	+	11.8619i	0.693769	+	0.400548i	−0.111253	+	0.993792i	$0.535487\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.79796			0.127956			0.0639782	−	0.997951i	$-0.479621\pi$
$881$	3.79796			0.127956			0.0639782	+	0.997951i	$0.479621\pi$
$882$		0			0
$883$	−18.7020	+	10.7976i	−0.629374	+	0.363369i	−0.780510	−	0.625144i	$-0.785040\pi$
$883$	−18.7020	+	10.7976i	−0.629374	+	0.363369i	0.151136	+	0.988513i	$0.451707\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.5959	−	21.8168i	−0.422930	−	0.732535i	0.573295	−	0.819349i	$-0.305665\pi$
$887$	−12.5959	−	21.8168i	−0.422930	−	0.732535i	−0.996225	+	0.0868136i	$0.972332\pi$
$888$		0			0
$889$	−13.1010	+	7.56388i	−0.439394	+	0.253684i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.49490	−	1.09638i	0.317735	−	0.0366888i
$894$		0			0
$895$	6.20204	−	10.7423i	0.207311	−	0.359074i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−15.0000	+	8.66025i	−0.500278	+	0.288836i
$900$		0			0
$901$		−	2.54270i		−	0.0847096i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	33.9411i		−	1.12824i
$906$		0			0
$907$	27.5959	−	47.7975i	0.916307	−	1.58709i	0.111332	−	0.993783i	$-0.464488\pi$
$907$	27.5959	−	47.7975i	0.916307	−	1.58709i	0.804976	−	0.593308i	$-0.202178\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−23.5959			−0.781768			−0.390884	−	0.920440i	$-0.627831\pi$
$911$	−23.5959			−0.781768			−0.390884	+	0.920440i	$0.627831\pi$
$912$		0			0
$913$	−19.5959			−0.648530
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.59592	−	13.1565i	0.250839	−	0.434466i
$918$		0			0
$919$			36.3089i			1.19772i	0.800854	+	0.598859i	$0.204379\pi$
$919$			36.3089i			1.19772i	−0.800854	+	0.598859i	$0.795621\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			4.38551i			0.144351i
$924$		0			0
$925$	3.94949	−	2.28024i	0.129858	−	0.0749738i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.89898	+	10.2173i	−0.193539	+	0.335220i	−0.946421	−	0.322936i	$-0.895330\pi$
$929$	−5.89898	+	10.2173i	−0.193539	+	0.335220i	0.752881	+	0.658156i	$0.228663\pi$
$930$		0			0
$931$	−23.1918	−	10.0424i	−0.760082	−	0.329125i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.5959	−	11.3137i	0.640855	−	0.369998i
$936$		0			0
$937$	14.5000	+	25.1147i	0.473694	+	0.820463i	0.999546	−	0.0301133i	$-0.00958681\pi$
$937$	14.5000	+	25.1147i	0.473694	+	0.820463i	−0.525852	+	0.850576i	$0.676253\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.79796	−	5.65685i	0.319404	−	0.184408i	−0.331723	−	0.943377i	$-0.607630\pi$
$941$	9.79796	−	5.65685i	0.319404	−	0.184408i	0.651127	+	0.758969i	$0.274297\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.374556	−	0.927204i	$-0.622205\pi$
$947$	−42.0000	−	24.2487i	−1.36482	−	0.787977i	−0.990260	+	0.139227i	$0.955538\pi$
$948$		0			0
$949$		−	14.0314i		−	0.455478i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.8990	+	6.29253i	0.353053	+	0.203835i	0.666029	−	0.745926i	$-0.267993\pi$
$953$	10.8990	+	6.29253i	0.353053	+	0.203835i	−0.312976	+	0.949761i	$0.601326\pi$
$954$		0			0
$955$	42.0000	+	24.2487i	1.35909	+	0.784670i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.3939	−	6.57826i	0.367927	−	0.212423i
$960$		0			0
$961$	−27.3939			−0.883673
$962$		0			0
$963$		0			0
$964$		0			0
$965$	21.0000	−	12.1244i	0.676014	−	0.390297i
$966$		0			0
$967$	−42.3434	−	24.4470i	−1.36167	−	0.786161i	−0.371825	−	0.928303i	$-0.621268\pi$
$967$	−42.3434	−	24.4470i	−1.36167	−	0.786161i	−0.989846	+	0.142142i	$0.954601\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12.5959	+	21.8168i	−0.404222	+	0.700133i	−0.994231	−	0.107264i	$-0.965791\pi$
$971$	−12.5959	+	21.8168i	−0.404222	+	0.700133i	0.590008	+	0.807397i	$0.299124\pi$
$972$		0			0
$973$	4.05051	+	7.01569i	0.129853	+	0.224913i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			12.2351i			0.391436i	0.980660	+	0.195718i	$0.0627037\pi$
$977$			12.2351i			0.391436i	−0.980660	+	0.195718i	$0.937296\pi$
$978$		0			0
$979$	−25.7980	−	44.6834i	−0.824506	−	1.42809i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.10102	−	15.7634i	0.290277	−	0.502775i	−0.683598	−	0.729859i	$-0.739586\pi$
$983$	9.10102	−	15.7634i	0.290277	−	0.502775i	0.973875	+	0.227084i	$0.0729191\pi$
$984$		0			0
$985$	−2.00000	+	3.46410i	−0.0637253	+	0.110375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29.3939			−0.934671
$990$		0			0
$991$	11.8485	−	20.5222i	0.376379	−	0.651908i	−0.614153	−	0.789187i	$-0.710502\pi$
$991$	11.8485	−	20.5222i	0.376379	−	0.651908i	0.990532	+	0.137279i	$0.0438357\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			36.8338i			1.16771i
$996$		0			0
$997$	−11.9495	−	20.6971i	−0.378444	−	0.655484i	0.612392	−	0.790554i	$-0.290207\pi$
$997$	−11.9495	−	20.6971i	−0.378444	−	0.655484i	−0.990836	+	0.135070i	$0.956874\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.i.559.2		4
3.2	odd	2		912.2.bb.d.559.2	yes	4
4.3	odd	2		2736.2.bm.j.559.1		4
12.11	even	2		912.2.bb.c.559.1	yes	4
19.12	odd	6		2736.2.bm.j.1855.2		4
57.50	even	6		912.2.bb.c.31.2	✓	4
76.31	even	6	inner	2736.2.bm.i.1855.1		4
228.107	odd	6		912.2.bb.d.31.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bb.c.31.2	✓	4	57.50	even	6
912.2.bb.c.559.1	yes	4	12.11	even	2
912.2.bb.d.31.1	yes	4	228.107	odd	6
912.2.bb.d.559.2	yes	4	3.2	odd	2
2736.2.bm.i.559.2		4	1.1	even	1	trivial
2736.2.bm.i.1855.1		4	76.31	even	6	inner
2736.2.bm.j.559.1		4	4.3	odd	2
2736.2.bm.j.1855.2		4	19.12	odd	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.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{5} +1.09638i q^{7} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{5} +1.09638i q^{7} +5.65685i q^{11} +(-0.949490 + 0.548188i) q^{13} +(-1.00000 + 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(4.89898 - 2.82843i) q^{23} +(0.500000 + 0.866025i) q^{25} +(-7.89898 + 4.56048i) q^{29} +1.89898 q^{31} +(-1.89898 - 1.09638i) q^{35} -4.56048i q^{37} +(-4.50000 - 2.59808i) q^{43} +(-1.89898 + 1.09638i) q^{47} +5.79796 q^{49} +(-1.10102 + 0.635674i) q^{53} +(-9.79796 - 5.65685i) q^{55} +(3.89898 - 6.75323i) q^{59} +(0.0505103 + 0.0874863i) q^{61} -2.19275i q^{65} +(-5.39898 - 9.35131i) q^{67} +(2.00000 - 3.46410i) q^{71} +(-6.39898 + 11.0834i) q^{73} -6.20204 q^{77} +(4.94949 - 8.57277i) q^{79} +3.46410i q^{83} +(-2.00000 - 3.46410i) q^{85} +(-7.89898 + 4.56048i) q^{89} +(-0.601021 - 1.04100i) q^{91} +(7.00000 - 5.19615i) q^{95} +(-15.7980 - 9.12096i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} - 16 q^{19} + 2 q^{25} - 12 q^{29} - 12 q^{31} + 12 q^{35} - 18 q^{43} + 12 q^{47} - 16 q^{49} - 24 q^{53} - 4 q^{59} + 10 q^{61} - 2 q^{67} + 8 q^{71} - 6 q^{73} - 64 q^{77} + 10 q^{79} - 8 q^{85} - 12 q^{89} - 22 q^{91} + 28 q^{95} - 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 6 * q^13 - 4 * q^17 - 16 * q^19 + 2 * q^25 - 12 * q^29 - 12 * q^31 + 12 * q^35 - 18 * q^43 + 12 * q^47 - 16 * q^49 - 24 * q^53 - 4 * q^59 + 10 * q^61 - 2 * q^67 + 8 * q^71 - 6 * q^73 - 64 * q^77 + 10 * q^79 - 8 * q^85 - 12 * q^89 - 22 * q^91 + 28 * q^95 - 24 * q^97

Introduction

L-functions

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