LMFDB - Embedded newform 2736.2.bm.t.1855.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 28x^{14} + 542x^{12} + 5488x^{10} + 40451x^{8} + 151312x^{6} + 395134x^{4} + 52164x^{2} + 6561 $ x^16 + 28x^14 + 542x^12 + 5488x^10 + 40451x^8 + 151312x^6 + 395134x^4 + 52164*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1855.4
Root			$-1.09560 - 1.89764i$ of defining polynomial
Character	$\chi$	$=$	2736.1855
Dual form			2736.2.bm.t.559.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.767178 - 1.32879i) q^{5} +2.31392i q^{7} +O(q^{10})$	$q+(-0.767178 - 1.32879i) q^{5} +2.31392i q^{7} +5.49807i q^{11} +(-3.96863 - 2.29129i) q^{13} +(-2.79694 - 4.84444i) q^{17} +(1.09932 - 4.21800i) q^{19} +(-3.07472 - 1.77519i) q^{23} +(1.32288 - 2.29129i) q^{25} +(7.12824 + 4.11549i) q^{29} +6.20647 q^{31} +(3.07472 - 1.77519i) q^{35} -1.73205i q^{37} +(-5.86565 + 3.38654i) q^{41} +(9.30970 - 5.37496i) q^{43} +(1.68674 + 0.973842i) q^{47} +1.64575 q^{49} +(8.16719 + 4.71533i) q^{53} +(7.30579 - 4.21800i) q^{55} +(-4.76147 - 8.24710i) q^{59} +(-2.32288 + 4.02334i) q^{61} +7.03130i q^{65} +(4.20255 - 7.27903i) q^{67} +(7.83619 + 13.5727i) q^{71} +(6.79150 + 11.7632i) q^{73} -12.7221 q^{77} +(7.11107 + 12.3167i) q^{79} -1.94768i q^{83} +(-4.29150 + 7.43310i) q^{85} +(4.82670 + 2.78670i) q^{89} +(5.30187 - 9.18310i) q^{91} +(-6.44821 + 1.77519i) q^{95} +(7.40588 - 4.27579i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100}) $ 16 * q - 16 * q^49 - 16 * q^61 + 24 * q^73 + 16 * q^85 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.767178	−	1.32879i	−0.343092	−	0.594254i	0.641913	−	0.766778i	$-0.278141\pi$
$5$	−0.767178	−	1.32879i	−0.343092	−	0.594254i	−0.985005	+	0.172524i	$0.944808\pi$
$6$		0			0
$7$			2.31392i			0.874581i	0.899320	+	0.437291i	$0.144062\pi$
$7$			2.31392i			0.874581i	−0.899320	+	0.437291i	$0.855938\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.49807i			1.65773i	0.559449	+	0.828865i	$0.311013\pi$
$11$			5.49807i			1.65773i	−0.559449	+	0.828865i	$0.688987\pi$
$12$		0			0
$13$	−3.96863	−	2.29129i	−1.10070	−	0.635489i	−0.164295	−	0.986411i	$-0.552535\pi$
$13$	−3.96863	−	2.29129i	−1.10070	−	0.635489i	−0.936405	+	0.350922i	$0.885868\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.79694	−	4.84444i	−0.678358	−	1.17495i	−0.975475	−	0.220109i	$-0.929359\pi$
$17$	−2.79694	−	4.84444i	−0.678358	−	1.17495i	0.297118	−	0.954841i	$-0.403975\pi$
$18$		0			0
$19$	1.09932	−	4.21800i	0.252201	−	0.967675i
$19$	1.09932	−	4.21800i	0.252201	−	0.967675i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.07472	−	1.77519i	−0.641124	−	0.370153i	0.143923	−	0.989589i	$-0.454028\pi$
$23$	−3.07472	−	1.77519i	−0.641124	−	0.370153i	−0.785047	+	0.619436i	$0.787361\pi$
$24$		0			0
$25$	1.32288	−	2.29129i	0.264575	−	0.458258i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.12824	+	4.11549i	1.32368	+	0.764227i	0.984314	−	0.176427i	$-0.0564539\pi$
$29$	7.12824	+	4.11549i	1.32368	+	0.764227i	0.339367	+	0.940654i	$0.389787\pi$
$30$		0			0
$31$	6.20647			1.11471			0.557357	−	0.830273i	$-0.311815\pi$
$31$	6.20647			1.11471			0.557357	+	0.830273i	$0.311815\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.07472	−	1.77519i	0.519723	−	0.300062i
$36$		0			0
$37$		−	1.73205i		−	0.284747i	−0.989813	−	0.142374i	$-0.954527\pi$
$37$		−	1.73205i		−	0.284747i	0.989813	−	0.142374i	$-0.0454735\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.86565	+	3.38654i	−0.916061	+	0.528888i	−0.882376	−	0.470544i	$-0.844058\pi$
$41$	−5.86565	+	3.38654i	−0.916061	+	0.528888i	−0.0336847	+	0.999433i	$0.510724\pi$
$42$		0			0
$43$	9.30970	−	5.37496i	1.41972	−	0.819674i	0.423443	−	0.905923i	$-0.360821\pi$
$43$	9.30970	−	5.37496i	1.41972	−	0.819674i	0.996274	+	0.0862491i	$0.0274881\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.68674	+	0.973842i	0.246037	+	0.142049i	0.617948	−	0.786219i	$-0.287964\pi$
$47$	1.68674	+	0.973842i	0.246037	+	0.142049i	−0.371911	+	0.928268i	$0.621297\pi$
$48$		0			0
$49$	1.64575			0.235107
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.16719	+	4.71533i	1.12185	+	0.647700i	0.941872	−	0.335971i	$-0.109064\pi$
$53$	8.16719	+	4.71533i	1.12185	+	0.647700i	0.179977	+	0.983671i	$0.442398\pi$
$54$		0			0
$55$	7.30579	−	4.21800i	0.985112	−	0.568755i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.76147	−	8.24710i	−0.619890	−	1.07368i	−0.989505	−	0.144496i	$-0.953844\pi$
$59$	−4.76147	−	8.24710i	−0.619890	−	1.07368i	0.369615	−	0.929185i	$-0.379490\pi$
$60$		0			0
$61$	−2.32288	+	4.02334i	−0.297414	+	0.515136i	−0.975543	−	0.219806i	$-0.929457\pi$
$61$	−2.32288	+	4.02334i	−0.297414	+	0.515136i	0.678130	+	0.734942i	$0.262791\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			7.03130i			0.872126i
$66$		0			0
$67$	4.20255	−	7.27903i	0.513423	−	0.889275i	−0.486455	−	0.873705i	$-0.661710\pi$
$67$	4.20255	−	7.27903i	0.513423	−	0.889275i	0.999879	−	0.0155700i	$-0.00495627\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.83619	+	13.5727i	0.929985	+	1.61078i	0.783342	+	0.621591i	$0.213514\pi$
$71$	7.83619	+	13.5727i	0.929985	+	1.61078i	0.146643	+	0.989189i	$0.453153\pi$
$72$		0			0
$73$	6.79150	+	11.7632i	0.794885	+	1.37678i	0.922912	+	0.385011i	$0.125802\pi$
$73$	6.79150	+	11.7632i	0.794885	+	1.37678i	−0.128027	+	0.991771i	$0.540864\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.7221			−1.44982
$78$		0			0
$79$	7.11107	+	12.3167i	0.800058	+	1.38574i	0.919577	+	0.392909i	$0.128531\pi$
$79$	7.11107	+	12.3167i	0.800058	+	1.38574i	−0.119519	+	0.992832i	$0.538135\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1.94768i		−	0.213786i	−0.994271	−	0.106893i	$-0.965910\pi$
$83$		−	1.94768i		−	0.213786i	0.994271	−	0.106893i	$-0.0340902\pi$
$84$		0			0
$85$	−4.29150	+	7.43310i	−0.465479	+	0.806233i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.82670	+	2.78670i	0.511629	+	0.295389i	0.733503	−	0.679686i	$-0.237884\pi$
$89$	4.82670	+	2.78670i	0.511629	+	0.295389i	−0.221874	+	0.975075i	$0.571217\pi$
$90$		0			0
$91$	5.30187	−	9.18310i	0.555787	−	0.962651i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.44821	+	1.77519i	−0.661572	+	0.182131i
$96$		0			0
$97$	7.40588	−	4.27579i	0.751953	−	0.434140i	−0.0744460	−	0.997225i	$-0.523719\pi$
$97$	7.40588	−	4.27579i	0.751953	−	0.434140i	0.826399	+	0.563085i	$0.190385\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.92518	−	17.1909i	0.987592	−	1.71056i	0.357794	−	0.933800i	$-0.383529\pi$
$101$	9.92518	−	17.1909i	0.987592	−	1.71056i	0.629798	−	0.776759i	$-0.283138\pi$
$102$		0			0
$103$	0.389432			0.0383718			0.0191859	−	0.999816i	$-0.493893\pi$
$103$	0.389432			0.0383718			0.0191859	+	0.999816i	$0.493893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−19.0459			−1.84123			−0.920617	−	0.390467i	$-0.872313\pi$
$107$	−19.0459			−1.84123			−0.920617	+	0.390467i	$0.872313\pi$
$108$		0			0
$109$	5.46863	−	3.15731i	0.523799	−	0.302416i	−0.214688	−	0.976683i	$-0.568874\pi$
$109$	5.46863	−	3.15731i	0.523799	−	0.302416i	0.738488	+	0.674267i	$0.235540\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	15.2623i		−	1.43575i	−0.696170	−	0.717877i	$-0.745114\pi$
$113$		−	15.2623i		−	1.43575i	0.696170	−	0.717877i	$-0.254886\pi$
$114$		0			0
$115$			5.44755i			0.507987i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.2097	−	6.47191i	1.02759	−	0.593279i
$120$		0			0
$121$	−19.2288			−1.74807
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.7313			−1.04928
$126$		0			0
$127$	6.20647	−	10.7499i	0.550735	−	0.953901i	−0.447487	−	0.894291i	$-0.647681\pi$
$127$	6.20647	−	10.7499i	0.550735	−	0.953901i	0.998222	−	0.0596105i	$-0.0189859\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	17.3591	−	10.0223i	1.51667	−	0.875652i	0.516866	−	0.856067i	$-0.327099\pi$
$131$	17.3591	−	10.0223i	1.51667	−	0.875652i	0.999808	−	0.0195854i	$-0.00623463\pi$
$132$		0			0
$133$	9.76013	+	2.54374i	0.846311	+	0.220570i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.06871	+	5.31517i	−0.262178	+	0.454105i	−0.966820	−	0.255457i	$-0.917774\pi$
$137$	−3.06871	+	5.31517i	−0.262178	+	0.454105i	0.704643	+	0.709562i	$0.251107\pi$
$138$		0			0
$139$	17.3254	+	10.0028i	1.46952	+	0.848427i	0.999416	−	0.0341840i	$-0.0108832\pi$
$139$	17.3254	+	10.0028i	1.46952	+	0.848427i	0.470104	+	0.882611i	$0.344217\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.5977	−	21.8198i	1.05347	−	1.82466i
$144$		0			0
$145$		−	12.6293i		−	1.04880i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.495406	+	0.858069i	0.0405853	+	0.0702957i	0.885604	−	0.464440i	$-0.153744\pi$
$149$	0.495406	+	0.858069i	0.0405853	+	0.0702957i	−0.845019	+	0.534736i	$0.820411\pi$
$150$		0			0
$151$	4.39727			0.357845			0.178922	−	0.983863i	$-0.442739\pi$
$151$	4.39727			0.357845			0.178922	+	0.983863i	$0.442739\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.76147	−	8.24710i	−0.382450	−	0.662423i
$156$		0			0
$157$	−4.79150	−	8.29913i	−0.382404	−	0.662342i	0.609002	−	0.793169i	$-0.291570\pi$
$157$	−4.79150	−	8.29913i	−0.382404	−	0.662342i	−0.991405	+	0.130827i	$0.958237\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.10766	−	7.11468i	0.323729	−	0.560715i
$162$		0			0
$163$		−	19.1859i		−	1.50276i	−0.659872	−	0.751378i	$-0.729389\pi$
$163$		−	19.1859i		−	1.50276i	0.659872	−	0.751378i	$-0.270611\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.07472	+	5.32558i	−0.237929	+	0.412106i	−0.960120	−	0.279588i	$-0.909802\pi$
$167$	−3.07472	+	5.32558i	−0.237929	+	0.412106i	0.722191	+	0.691694i	$0.243135\pi$
$168$		0			0
$169$	4.00000	+	6.92820i	0.307692	+	0.532939i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.05034	+	2.91581i	−0.383970	+	0.221685i	−0.679544	−	0.733635i	$-0.737822\pi$
$173$	−5.05034	+	2.91581i	−0.383970	+	0.221685i	0.295574	+	0.955320i	$0.404489\pi$
$174$		0			0
$175$	5.30187	+	3.06103i	0.400784	+	0.231392i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.52293			−0.711777			−0.355889	−	0.934528i	$-0.615822\pi$
$179$	−9.52293			−0.711777			−0.355889	+	0.934528i	$0.615822\pi$
$180$		0			0
$181$	−16.4059	−	9.47194i	−1.21944	−	0.704044i	−0.254641	−	0.967036i	$-0.581957\pi$
$181$	−16.4059	−	9.47194i	−1.21944	−	0.704044i	−0.964798	+	0.262992i	$0.915291\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.30153	+	1.32879i	−0.169212	+	0.0976947i
$186$		0			0
$187$	26.6351	−	15.3778i	1.94775	−	1.12453i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.4419i			1.33441i	0.744875	+	0.667204i	$0.232509\pi$
$191$			18.4419i			1.33441i	−0.744875	+	0.667204i	$0.767491\pi$
$192$		0			0
$193$	5.03137	−	2.90486i	0.362166	−	0.209097i	−0.307864	−	0.951430i	$-0.599614\pi$
$193$	5.03137	−	2.90486i	0.362166	−	0.209097i	0.670031	+	0.742334i	$0.266281\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.7405			−0.765228			−0.382614	−	0.923908i	$-0.624976\pi$
$197$	−10.7405			−0.765228			−0.382614	+	0.923908i	$0.624976\pi$
$198$		0			0
$199$	9.30970	+	5.37496i	0.659947	+	0.381021i	0.792257	−	0.610188i	$-0.208906\pi$
$199$	9.30970	+	5.37496i	0.659947	+	0.381021i	−0.132310	+	0.991208i	$0.542239\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.52293	+	16.4942i	−0.668379	+	1.15767i
$204$		0			0
$205$	9.00000	+	5.19615i	0.628587	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	23.1908	+	6.04412i	1.60414	+	0.418080i
$210$		0			0
$211$	−3.10323	−	5.37496i	−0.213635	−	0.370028i	0.739214	−	0.673470i	$-0.235197\pi$
$211$	−3.10323	−	5.37496i	−0.213635	−	0.370028i	−0.952850	+	0.303443i	$0.901864\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.2844	−	8.24710i	−0.974188	−	0.562448i
$216$		0			0
$217$			14.3613i			0.974909i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			25.6344i			1.72436i
$222$		0			0
$223$	−4.91244	−	8.50859i	−0.328961	−	0.569777i	0.653345	−	0.757060i	$-0.273365\pi$
$223$	−4.91244	−	8.50859i	−0.328961	−	0.569777i	−0.982306	+	0.187283i	$0.940032\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.8964			0.855966			0.427983	−	0.903787i	$-0.359224\pi$
$227$	12.8964			0.855966			0.427983	+	0.903787i	$0.359224\pi$
$228$		0			0
$229$	1.35425			0.0894913			0.0447456	−	0.998998i	$-0.485752\pi$
$229$	1.35425			0.0894913			0.0447456	+	0.998998i	$0.485752\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.40001	+	12.8172i	0.484791	+	0.839682i	0.999847	−	0.0174740i	$-0.00556244\pi$
$233$	7.40001	+	12.8172i	0.484791	+	0.839682i	−0.515057	+	0.857156i	$0.672229\pi$
$234$		0			0
$235$		−	2.98844i		−	0.194944i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			27.4903i			1.77820i	0.457711	+	0.889101i	$0.348670\pi$
$239$			27.4903i			1.77820i	−0.457711	+	0.889101i	$0.651330\pi$
$240$		0			0
$241$	−14.3745	−	8.29913i	−0.925943	−	0.534594i	−0.0404171	−	0.999183i	$-0.512869\pi$
$241$	−14.3745	−	8.29913i	−0.925943	−	0.534594i	−0.885526	+	0.464589i	$0.846202\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.26258	−	2.18686i	−0.0806636	−	0.139713i
$246$		0			0
$247$	−14.0274	+	14.2208i	−0.892544	+	0.904848i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.14945	+	3.55038i	0.388150	+	0.224098i	0.681358	−	0.731950i	$-0.261390\pi$
$251$	6.14945	+	3.55038i	0.388150	+	0.224098i	−0.293208	+	0.956049i	$0.594723\pi$
$252$		0			0
$253$	9.76013	−	16.9050i	0.613614	−	1.06281i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.08929	+	3.51565i	0.379839	+	0.219300i	0.677748	−	0.735294i	$-0.262956\pi$
$257$	6.08929	+	3.51565i	0.379839	+	0.219300i	−0.297909	+	0.954594i	$0.596289\pi$
$258$		0			0
$259$	4.00784			0.249035
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.68674	−	0.973842i	0.104009	−	0.0600497i	−0.447093	−	0.894487i	$-0.647541\pi$
$263$	1.68674	−	0.973842i	0.104009	−	0.0600497i	0.551102	+	0.834438i	$0.314207\pi$
$264$		0			0
$265$		−	14.4700i		−	0.888884i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.08929	−	3.51565i	0.371270	−	0.214353i	−0.302743	−	0.953072i	$-0.597902\pi$
$269$	6.08929	−	3.51565i	0.371270	−	0.214353i	0.674013	+	0.738719i	$0.264569\pi$
$270$		0			0
$271$	12.0235	−	6.94177i	0.730376	−	0.421683i	−0.0881837	−	0.996104i	$-0.528106\pi$
$271$	12.0235	−	6.94177i	0.730376	−	0.421683i	0.818560	+	0.574421i	$0.194773\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.5977	+	7.27326i	0.759667	+	0.438594i
$276$		0			0
$277$	2.22876			0.133913			0.0669565	−	0.997756i	$-0.478671\pi$
$277$	2.22876			0.133913			0.0669565	+	0.997756i	$0.478671\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.87704	−	5.70251i	−0.589215	−	0.340183i	0.175572	−	0.984467i	$-0.443822\pi$
$281$	−9.87704	−	5.70251i	−0.589215	−	0.340183i	−0.764787	+	0.644283i	$0.777156\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.83619	−	13.5727i	−0.462556	−	0.801170i
$288$		0			0
$289$	−7.14575	+	12.3768i	−0.420338	+	0.728047i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	24.1765i		−	1.41240i	−0.708010	−	0.706202i	$-0.750407\pi$
$293$		−	24.1765i		−	1.41240i	0.708010	−	0.706202i	$-0.249593\pi$
$294$		0			0
$295$	−7.30579	+	12.6540i	−0.425359	+	0.736744i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.13495	+	14.0902i	0.470457	+	0.814855i
$300$		0			0
$301$	12.4373	+	21.5420i	0.716871	+	1.24166i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.12824			0.408162
$306$		0			0
$307$	−5.49658	−	9.52036i	−0.313707	−	0.543356i	0.665455	−	0.746438i	$-0.268238\pi$
$307$	−5.49658	−	9.52036i	−0.313707	−	0.543356i	−0.979162	+	0.203082i	$0.934904\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	21.6473i		−	1.22751i	−0.789498	−	0.613753i	$-0.789659\pi$
$311$		−	21.6473i		−	1.22751i	0.789498	−	0.613753i	$-0.210341\pi$
$312$		0			0
$313$	−0.822876	+	1.42526i	−0.0465117	+	0.0805606i	−0.888344	−	0.459179i	$-0.848144\pi$
$313$	−0.822876	+	1.42526i	−0.0465117	+	0.0805606i	0.841832	+	0.539739i	$0.181477\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.2954	−	8.83082i	−0.859077	−	0.495988i	0.00462637	−	0.999989i	$-0.498527\pi$
$317$	−15.2954	−	8.83082i	−0.859077	−	0.495988i	−0.863703	+	0.504001i	$0.831861\pi$
$318$		0			0
$319$	−22.6272	+	39.1915i	−1.26688	+	2.19430i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.5086	+	6.47191i	−1.30805	+	0.360107i
$324$		0			0
$325$	−10.5000	+	6.06218i	−0.582435	+	0.336269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.25340	+	3.90300i	−0.124234	+	0.215179i
$330$		0			0
$331$	−22.2378			−1.22230			−0.611150	−	0.791515i	$-0.709293\pi$
$331$	−22.2378			−1.22230			−0.611150	+	0.791515i	$0.709293\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.8964			−0.704607
$336$		0			0
$337$	17.9059	−	10.3380i	0.975395	−	0.563145i	0.0745186	−	0.997220i	$-0.476258\pi$
$337$	17.9059	−	10.3380i	0.975395	−	0.563145i	0.900877	+	0.434075i	$0.142925\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			34.1236i			1.84790i
$342$		0			0
$343$			20.0056i			1.08020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.4338	−	11.7975i	1.09695	−	0.633322i	0.161529	−	0.986868i	$-0.448357\pi$
$347$	20.4338	−	11.7975i	1.09695	−	0.633322i	0.935417	+	0.353546i	$0.115024\pi$
$348$		0			0
$349$	−14.0627			−0.752762			−0.376381	−	0.926465i	$-0.622832\pi$
$349$	−14.0627			−0.752762			−0.376381	+	0.926465i	$0.622832\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.4718			−1.19605			−0.598027	−	0.801476i	$-0.704048\pi$
$353$	−22.4718			−1.19605			−0.598027	+	0.801476i	$0.704048\pi$
$354$		0			0
$355$	12.0235	−	20.8253i	0.638141	−	1.10529i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.14945	+	3.55038i	−0.324555	+	0.187382i	−0.653421	−	0.756995i	$-0.726667\pi$
$359$	−6.14945	+	3.55038i	−0.324555	+	0.187382i	0.328866	+	0.944377i	$0.393334\pi$
$360$		0			0
$361$	−16.5830	−	9.27383i	−0.872790	−	0.488096i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.4206	−	18.0490i	0.545438	−	0.944727i
$366$		0			0
$367$	−9.30970	−	5.37496i	−0.485963	−	0.280571i	0.236935	−	0.971525i	$-0.423857\pi$
$367$	−9.30970	−	5.37496i	−0.485963	−	0.280571i	−0.722898	+	0.690955i	$0.757190\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.9109	+	18.8983i	−0.566466	+	0.981149i
$372$		0			0
$373$			6.31463i			0.326959i	0.986547	+	0.163479i	$0.0522718\pi$
$373$			6.31463i			0.326959i	−0.986547	+	0.163479i	$0.947728\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.8595	−	32.6657i	−0.971316	−	1.68237i
$378$		0			0
$379$	1.80920			0.0929325			0.0464662	−	0.998920i	$-0.485204\pi$
$379$	1.80920			0.0929325			0.0464662	+	0.998920i	$0.485204\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.07472	+	5.32558i	0.157111	+	0.272124i	0.933826	−	0.357728i	$-0.116449\pi$
$383$	3.07472	+	5.32558i	0.157111	+	0.272124i	−0.776715	+	0.629853i	$0.783115\pi$
$384$		0			0
$385$	9.76013	+	16.9050i	0.497422	+	0.861561i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.83589	−	6.64396i	0.194487	−	0.336862i	−0.752245	−	0.658884i	$-0.771029\pi$
$389$	3.83589	−	6.64396i	0.194487	−	0.336862i	0.946732	+	0.322022i	$0.104362\pi$
$390$		0			0
$391$			19.8604i			1.00439i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.9109	−	18.8983i	0.548988	−	0.950875i
$396$		0			0
$397$	18.6144	+	32.2410i	0.934229	+	1.61813i	0.776003	+	0.630729i	$0.217244\pi$
$397$	18.6144	+	32.2410i	0.934229	+	1.61813i	0.158226	+	0.987403i	$0.449423\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	19.0832	−	11.0177i	0.952968	−	0.550197i	0.0589666	−	0.998260i	$-0.481219\pi$
$401$	19.0832	−	11.0177i	0.952968	−	0.550197i	0.894002	+	0.448063i	$0.147886\pi$
$402$		0			0
$403$	−24.6312	−	14.2208i	−1.22697	−	0.708389i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.52293			0.472034
$408$		0			0
$409$	−3.53137	−	2.03884i	−0.174615	−	0.100814i	0.410145	−	0.912020i	$-0.365478\pi$
$409$	−3.53137	−	2.03884i	−0.174615	−	0.100814i	−0.584760	+	0.811206i	$0.698811\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	19.0832	−	11.0177i	0.939022	−	0.542144i
$414$		0			0
$415$	−2.58807	+	1.49422i	−0.127043	+	0.0733484i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	18.4419i		−	0.900945i	−0.892790	−	0.450473i	$-0.851255\pi$
$419$		−	18.4419i		−	0.900945i	0.892790	−	0.450473i	$-0.148745\pi$
$420$		0			0
$421$	1.59412	−	0.920365i	0.0776926	−	0.0448558i	−0.460650	−	0.887582i	$-0.652384\pi$
$421$	1.59412	−	0.920365i	0.0776926	−	0.0448558i	0.538343	+	0.842726i	$0.319050\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14.8000			−0.717906
$426$		0			0
$427$	−9.30970	−	5.37496i	−0.450528	−	0.260113i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.68674	−	2.92153i	0.0812476	−	0.140725i	−0.822538	−	0.568709i	$-0.807443\pi$
$431$	1.68674	−	2.92153i	0.0812476	−	0.140725i	0.903786	+	0.427985i	$0.140776\pi$
$432$		0			0
$433$	5.03137	+	2.90486i	0.241792	+	0.139599i	0.616000	−	0.787746i	$-0.288752\pi$
$433$	5.03137	+	2.90486i	0.241792	+	0.139599i	−0.374208	+	0.927345i	$0.622085\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.8679	+	11.0177i	−0.519880	+	0.527047i
$438$		0			0
$439$	2.39335	+	4.14540i	0.114228	+	0.197849i	0.917471	−	0.397803i	$-0.130227\pi$
$439$	2.39335	+	4.14540i	0.114228	+	0.197849i	−0.803243	+	0.595652i	$0.796894\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.0459	+	10.9961i	0.904896	+	0.522442i	0.878786	−	0.477217i	$-0.158354\pi$
$443$	19.0459	+	10.9961i	0.904896	+	0.522442i	0.0261109	+	0.999659i	$0.491688\pi$
$444$		0			0
$445$		−	8.55157i		−	0.405384i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.83163i		−	0.275211i	−0.990487	−	0.137606i	$-0.956059\pi$
$449$		−	5.83163i		−	0.275211i	0.990487	−	0.137606i	$-0.0439407\pi$
$450$		0			0
$451$	−18.6194	−	32.2498i	−0.876754	−	1.51858i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−16.2699			−0.762745
$456$		0			0
$457$	17.9373			0.839069			0.419535	−	0.907739i	$-0.362193\pi$
$457$	17.9373			0.839069			0.419535	+	0.907739i	$0.362193\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.10766	−	7.11468i	−0.191313	−	0.331364i	0.754373	−	0.656446i	$-0.227941\pi$
$461$	−4.10766	−	7.11468i	−0.191313	−	0.331364i	−0.945686	+	0.325083i	$0.894608\pi$
$462$		0			0
$463$			22.1744i			1.03053i	0.857031	+	0.515265i	$0.172306\pi$
$463$			22.1744i			1.03053i	−0.857031	+	0.515265i	$0.827694\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	1.94768i		−	0.0901280i	−0.998984	−	0.0450640i	$-0.985651\pi$
$467$		−	1.94768i		−	0.0901280i	0.998984	−	0.0450640i	$-0.0143492\pi$
$468$		0			0
$469$	16.8431	+	9.72439i	0.777744	+	0.449031i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	29.5519	+	51.1854i	1.35880	+	2.35351i
$474$		0			0
$475$	−8.21039	−	8.09874i	−0.376718	−	0.371596i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.2097	−	6.47191i	−0.512183	−	0.295709i	0.221547	−	0.975150i	$-0.428889\pi$
$479$	−11.2097	−	6.47191i	−0.512183	−	0.295709i	−0.733731	+	0.679440i	$0.762223\pi$
$480$		0			0
$481$	−3.96863	+	6.87386i	−0.180954	+	0.313421i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.3633	−	6.56058i	−0.515979	−	0.297901i
$486$		0			0
$487$	3.61840			0.163966			0.0819828	−	0.996634i	$-0.473875\pi$
$487$	3.61840			0.163966			0.0819828	+	0.996634i	$0.473875\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.9856	+	8.07461i	−0.631163	+	0.364402i	−0.781202	−	0.624278i	$-0.785393\pi$
$491$	−13.9856	+	8.07461i	−0.631163	+	0.364402i	0.150039	+	0.988680i	$0.452060\pi$
$492$		0			0
$493$		−	46.0431i		−	2.07368i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.4062	+	18.1324i	−1.40876	+	0.813347i
$498$		0			0
$499$	−27.2192	+	15.7150i	−1.21850	+	0.703501i	−0.964597	−	0.263728i	$-0.915048\pi$
$499$	−27.2192	+	15.7150i	−1.21850	+	0.703501i	−0.253903	+	0.967230i	$0.581714\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.8218	+	12.5988i	0.972987	+	0.561754i	0.900146	−	0.435589i	$-0.143460\pi$
$503$	21.8218	+	12.5988i	0.972987	+	0.561754i	0.0728416	+	0.997344i	$0.476793\pi$
$504$		0			0
$505$	−30.4575			−1.35534
$506$		0			0
$507$		0			0
$508$		0			0
$509$	32.6687	+	18.8613i	1.44802	+	0.836013i	0.998363	−	0.0571902i	$-0.0182142\pi$
$509$	32.6687	+	18.8613i	1.44802	+	0.836013i	0.449653	+	0.893203i	$0.351547\pi$
$510$		0			0
$511$	−27.2192	+	15.7150i	−1.20411	+	0.695192i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.298763	−	0.517473i	−0.0131651	−	0.0228026i
$516$		0			0
$517$	−5.35425	+	9.27383i	−0.235480	+	0.407863i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	22.9768i		−	1.00663i	−0.864102	−	0.503316i	$-0.832113\pi$
$521$		−	22.9768i		−	1.00663i	0.864102	−	0.503316i	$-0.167887\pi$
$522$		0			0
$523$	8.21039	−	14.2208i	0.359015	−	0.621832i	−0.628781	−	0.777582i	$-0.716446\pi$
$523$	8.21039	−	14.2208i	0.359015	−	0.621832i	0.987797	+	0.155750i	$0.0497793\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.3591	−	30.0669i	−0.756175	−	1.30973i
$528$		0			0
$529$	−5.19738	−	9.00213i	−0.225973	−	0.391397i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	31.0381			1.34441
$534$		0			0
$535$	14.6116	+	25.3080i	0.631713	+	1.09416i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			9.04845i			0.389744i
$540$		0			0
$541$	15.7915	−	27.3517i	0.678930	−	1.17594i	−0.296374	−	0.955072i	$-0.595777\pi$
$541$	15.7915	−	27.3517i	0.678930	−	1.17594i	0.975304	−	0.220869i	$-0.0708892\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.39082	−	4.84444i	−0.359423	−	0.207513i
$546$		0			0
$547$	11.1189	−	19.2585i	0.475410	−	0.823434i	−0.524193	−	0.851599i	$-0.675633\pi$
$547$	11.1189	−	19.2585i	0.475410	−	0.823434i	0.999603	+	0.0281650i	$0.00896637\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.1953	−	25.5427i	1.07336	−	1.08815i
$552$		0			0
$553$	−28.5000	+	16.4545i	−1.21194	+	0.699716i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.7221	+	22.0354i	−0.539053	+	0.933668i	0.459902	+	0.887970i	$0.347885\pi$
$557$	−12.7221	+	22.0354i	−0.539053	+	0.933668i	−0.998955	+	0.0456982i	$0.985449\pi$
$558$		0			0
$559$	−49.2623			−2.08357
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.74697			0.284351			0.142176	−	0.989841i	$-0.454590\pi$
$563$	6.74697			0.284351			0.142176	+	0.989841i	$0.454590\pi$
$564$		0			0
$565$	−20.2804	+	11.7089i	−0.853202	+	0.492597i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			9.68889i			0.406179i	0.979160	+	0.203090i	$0.0650983\pi$
$569$			9.68889i			0.406179i	−0.979160	+	0.203090i	$0.934902\pi$
$570$		0			0
$571$			28.4416i			1.19024i	0.803635	+	0.595122i	$0.202896\pi$
$571$			28.4416i			1.19024i	−0.803635	+	0.595122i	$0.797104\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.13495	+	4.69672i	−0.339251	+	0.195867i
$576$		0			0
$577$	−22.3542			−0.930620			−0.465310	−	0.885148i	$-0.654057\pi$
$577$	−22.3542			−0.930620			−0.465310	+	0.885148i	$0.654057\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.50679			0.186973
$582$		0			0
$583$	−25.9252	+	44.9038i	−1.07371	+	1.85972i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.07472	+	1.77519i	−0.126907	+	0.0732700i	−0.562110	−	0.827063i	$-0.690010\pi$
$587$	−3.07472	+	1.77519i	−0.126907	+	0.0732700i	0.435202	+	0.900333i	$0.356677\pi$
$588$		0			0
$589$	6.82288	−	26.1789i	0.281132	−	1.07868i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.0145	+	27.7379i	−0.657635	+	1.13906i	0.323591	+	0.946197i	$0.395110\pi$
$593$	−16.0145	+	27.7379i	−0.657635	+	1.13906i	−0.981226	+	0.192861i	$0.938223\pi$
$594$		0			0
$595$	−17.1996	−	9.93022i	−0.705116	−	0.407099i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.44821	−	11.1686i	0.263467	−	0.456338i	−0.703694	−	0.710503i	$-0.748467\pi$
$599$	6.44821	−	11.1686i	0.263467	−	0.456338i	0.967161	+	0.254165i	$0.0818007\pi$
$600$		0			0
$601$		−	31.0682i		−	1.26730i	−0.773620	−	0.633650i	$-0.781556\pi$
$601$		−	31.0682i		−	1.26730i	0.773620	−	0.633650i	$-0.218444\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.7519	+	25.5510i	0.599749	+	1.03880i
$606$		0			0
$607$	−47.0637			−1.91026			−0.955128	−	0.296193i	$-0.904283\pi$
$607$	−47.0637			−1.91026			−0.955128	+	0.296193i	$0.904283\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.46270	−	7.72963i	−0.180542	−	0.312707i
$612$		0			0
$613$	−8.76013	−	15.1730i	−0.353818	−	0.612831i	0.633097	−	0.774073i	$-0.281784\pi$
$613$	−8.76013	−	15.1730i	−0.353818	−	0.612831i	−0.986915	+	0.161241i	$0.948450\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	16.8298	−	29.1500i	0.677541	−	1.17354i	−0.298178	−	0.954510i	$-0.596379\pi$
$617$	16.8298	−	29.1500i	0.677541	−	1.17354i	0.975719	−	0.219026i	$-0.0702879\pi$
$618$		0			0
$619$			19.1859i			0.771147i	0.922677	+	0.385574i	$0.125996\pi$
$619$			19.1859i			0.771147i	−0.922677	+	0.385574i	$0.874004\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.44821	+	11.1686i	−0.258342	+	0.447462i
$624$		0			0
$625$	2.38562	+	4.13202i	0.0954249	+	0.165281i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.39082	+	4.84444i	−0.334564	+	0.193161i
$630$		0			0
$631$	10.0196	+	5.78481i	0.398874	+	0.230290i	0.685998	−	0.727604i	$-0.259366\pi$
$631$	10.0196	+	5.78481i	0.398874	+	0.230290i	−0.287124	+	0.957893i	$0.592699\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.0459			−0.755812
$636$		0			0
$637$	−6.53137	−	3.77089i	−0.258782	−	0.149408i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.31292	+	3.64477i	−0.249345	+	0.143960i	−0.619464	−	0.785025i	$-0.712650\pi$
$641$	−6.31292	+	3.64477i	−0.249345	+	0.143960i	0.370119	+	0.928984i	$0.379317\pi$
$642$		0			0
$643$	−19.2036	+	11.0872i	−0.757314	+	0.437236i	−0.828331	−	0.560240i	$-0.810709\pi$
$643$	−19.2036	+	11.0872i	−0.757314	+	0.437236i	0.0710164	+	0.997475i	$0.477376\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			25.1977i			0.990623i	0.868715	+	0.495311i	$0.164946\pi$
$647$			25.1977i			0.990623i	−0.868715	+	0.495311i	$0.835054\pi$
$648$		0			0
$649$	45.3431	−	26.1789i	1.77987	−	1.02761i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.61226			−0.141358			−0.0706792	−	0.997499i	$-0.522517\pi$
$653$	−3.61226			−0.141358			−0.0706792	+	0.997499i	$0.522517\pi$
$654$		0			0
$655$	−26.6351	−	15.3778i	−1.04072	−	0.600859i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.4338	−	35.3925i	0.795990	−	1.37869i	−0.126219	−	0.992002i	$-0.540284\pi$
$659$	20.4338	−	35.3925i	0.795990	−	1.37869i	0.922209	−	0.386692i	$-0.126382\pi$
$660$		0			0
$661$	27.0000	+	15.5885i	1.05018	+	0.606321i	0.922699	−	0.385521i	$-0.125978\pi$
$661$	27.0000	+	15.5885i	1.05018	+	0.606321i	0.127479	+	0.991841i	$0.459311\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.10766	−	14.9207i	−0.159288	−	0.578599i
$666$		0			0
$667$	−14.6116	−	25.3080i	−0.565762	−	0.979929i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22.1206	−	12.7713i	−0.853956	−	0.493032i
$672$		0			0
$673$			40.2334i			1.55088i	0.631419	+	0.775442i	$0.282473\pi$
$673$			40.2334i			1.55088i	−0.631419	+	0.775442i	$0.717527\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	14.0626i		−	0.540470i	−0.962794	−	0.270235i	$-0.912899\pi$
$677$		−	14.0626i		−	0.540470i	0.962794	−	0.270235i	$-0.0871014\pi$
$678$		0			0
$679$	9.89385	+	17.1367i	0.379691	+	0.657644i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	25.1953			0.964072			0.482036	−	0.876151i	$-0.339897\pi$
$683$	25.1953			0.964072			0.482036	+	0.876151i	$0.339897\pi$
$684$		0			0
$685$	9.41699			0.359805
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.6083	−	37.4268i	−0.823212	−	1.42585i
$690$		0			0
$691$			13.8835i			0.528155i	0.964501	+	0.264078i	$0.0850675\pi$
$691$			13.8835i			0.528155i	−0.964501	+	0.264078i	$0.914933\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	30.6957i		−	1.16436i
$696$		0			0
$697$	32.8118	+	18.9439i	1.24283	+	0.717551i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.5672	−	26.9632i	−0.587965	−	1.01839i	−0.994499	−	0.104749i	$-0.966596\pi$
$701$	−15.5672	−	26.9632i	−0.587965	−	1.01839i	0.406534	−	0.913636i	$-0.366737\pi$
$702$		0			0
$703$	−7.30579	−	1.90407i	−0.275543	−	0.0718135i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	39.7785	+	22.9661i	1.49602	+	0.863730i
$708$		0			0
$709$	−11.5516	+	20.0080i	−0.433831	+	0.751417i	−0.997199	−	0.0747888i	$-0.976172\pi$
$709$	−11.5516	+	20.0080i	−0.433831	+	0.751417i	0.563369	+	0.826206i	$0.309505\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−19.0832	−	11.0177i	−0.714671	−	0.412615i
$714$		0			0
$715$	−38.6586			−1.44575
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.76147	+	2.74903i	−0.177573	+	0.102522i	−0.586152	−	0.810201i	$-0.699358\pi$
$719$	−4.76147	+	2.74903i	−0.177573	+	0.102522i	0.408579	+	0.912723i	$0.366024\pi$
$720$		0			0
$721$			0.901115i			0.0335593i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.8595	−	10.8886i	0.700426	−	0.404391i
$726$		0			0
$727$	2.71380	−	1.56681i	0.100649	−	0.0581099i	−0.448830	−	0.893617i	$-0.648159\pi$
$727$	2.71380	−	1.56681i	0.100649	−	0.0581099i	0.549480	+	0.835507i	$0.314826\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−52.0774	−	30.0669i	−1.92615	−	1.11206i
$732$		0			0
$733$	−35.6458			−1.31661			−0.658303	−	0.752753i	$-0.728725\pi$
$733$	−35.6458			−1.31661			−0.658303	+	0.752753i	$0.728725\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	40.0206	+	23.1059i	1.47418	+	0.851117i
$738$		0			0
$739$	10.0196	−	5.78481i	0.368577	−	0.212798i	−0.304260	−	0.952589i	$-0.598409\pi$
$739$	10.0196	−	5.78481i	0.368577	−	0.212798i	0.672836	+	0.739791i	$0.265076\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.4941	+	44.1570i	0.935287	+	1.61996i	0.774121	+	0.633037i	$0.218192\pi$
$743$	25.4941	+	44.1570i	0.935287	+	1.61996i	0.161166	+	0.986927i	$0.448475\pi$
$744$		0			0
$745$	0.760130	−	1.31658i	0.0278490	−	0.0482359i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	44.0707i		−	1.61031i
$750$		0			0
$751$	−14.8063	+	25.6452i	−0.540289	+	0.935808i	0.458598	+	0.888644i	$0.348352\pi$
$751$	−14.8063	+	25.6452i	−0.540289	+	0.935808i	−0.998887	+	0.0471644i	$0.984982\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.37349	−	5.84305i	−0.122774	−	0.212650i
$756$		0			0
$757$	3.38562	+	5.86407i	0.123053	+	0.213133i	0.920970	−	0.389633i	$-0.127398\pi$
$757$	3.38562	+	5.86407i	0.123053	+	0.213133i	−0.797917	+	0.602767i	$0.794065\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.21532			−0.297805			−0.148903	−	0.988852i	$-0.547574\pi$
$761$	−8.21532			−0.297805			−0.148903	+	0.988852i	$0.547574\pi$
$762$		0			0
$763$	7.30579	+	12.6540i	0.264487	+	0.458105i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			43.6396i			1.57573i
$768$		0			0
$769$	16.0203	−	27.7479i	0.577705	−	1.00061i	−0.418037	−	0.908430i	$-0.637282\pi$
$769$	16.0203	−	27.7479i	0.577705	−	1.00061i	0.995742	−	0.0921848i	$-0.0293850\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.5466	−	7.24379i	−0.451271	−	0.260541i	0.257096	−	0.966386i	$-0.417234\pi$
$773$	−12.5466	−	7.24379i	−0.451271	−	0.260541i	−0.708367	+	0.705845i	$0.750568\pi$
$774$		0			0
$775$	8.21039	−	14.2208i	0.294926	−	0.510826i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.83619	+	28.4642i	0.280761	+	1.01984i
$780$		0			0
$781$	−74.6235	+	43.0839i	−2.67024	+	1.54166i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.35187	+	12.7338i	−0.262400	+	0.454489i
$786$		0			0
$787$	15.6419			0.557574			0.278787	−	0.960353i	$-0.410068\pi$
$787$	15.6419			0.557574			0.278787	+	0.960353i	$0.410068\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	35.3158			1.25568
$792$		0			0
$793$	18.4373	−	10.6448i	0.654726	−	0.378006i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	6.77307i		−	0.239915i	−0.992779	−	0.119957i	$-0.961724\pi$
$797$		−	6.77307i		−	0.239915i	0.992779	−	0.119957i	$-0.0382758\pi$
$798$		0			0
$799$		−	10.8951i		−	0.385441i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−64.6750	+	37.3401i	−2.28233	+	1.31771i
$804$		0			0
$805$	−12.6052			−0.444276
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.7725			0.624846			0.312423	−	0.949943i	$-0.398859\pi$
$809$	17.7725			0.624846			0.312423	+	0.949943i	$0.398859\pi$
$810$		0			0
$811$	16.4208	−	28.4416i	0.576611	−	0.998720i	−0.419253	−	0.907869i	$-0.637708\pi$
$811$	16.4208	−	28.4416i	0.576611	−	0.998720i	0.995865	−	0.0908508i	$-0.0289586\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−25.4941	+	14.7190i	−0.893019	+	0.515585i
$816$		0			0
$817$	−12.4373	−	45.1771i	−0.435124	−	1.58055i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.9970	+	43.2960i	−0.872400	+	1.51104i	−0.0128935	+	0.999917i	$0.504104\pi$
$821$	−24.9970	+	43.2960i	−0.872400	+	1.51104i	−0.859507	+	0.511125i	$0.829229\pi$
$822$		0			0
$823$	17.1996	+	9.93022i	0.599542	+	0.346146i	0.768861	−	0.639416i	$-0.220824\pi$
$823$	17.1996	+	9.93022i	0.599542	+	0.346146i	−0.169320	+	0.985561i	$0.554157\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−22.4194	+	38.8315i	−0.779597	+	1.35030i	0.152577	+	0.988292i	$0.451243\pi$
$827$	−22.4194	+	38.8315i	−0.779597	+	1.35030i	−0.932174	+	0.362011i	$0.882090\pi$
$828$		0			0
$829$			34.3150i			1.19181i	0.803056	+	0.595904i	$0.203206\pi$
$829$			34.3150i			1.19181i	−0.803056	+	0.595904i	$0.796794\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.60307	−	7.97275i	−0.159487	−	0.276239i
$834$		0			0
$835$	9.43544			0.326527
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.07472	+	5.32558i	0.106151	+	0.183859i	0.914208	−	0.405245i	$-0.132814\pi$
$839$	3.07472	+	5.32558i	0.106151	+	0.183859i	−0.808057	+	0.589105i	$0.799481\pi$
$840$		0			0
$841$	19.3745	+	33.5576i	0.668086	+	1.15716i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.13742	−	10.6303i	0.211134	−	0.365695i
$846$		0			0
$847$		−	44.4939i		−	1.52883i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.07472	+	5.32558i	−0.105400	+	0.182558i
$852$		0			0
$853$	19.1458	+	33.1614i	0.655538	+	1.13543i	0.981759	+	0.190132i	$0.0608914\pi$
$853$	19.1458	+	33.1614i	0.655538	+	1.13543i	−0.326221	+	0.945294i	$0.605775\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	44.8473	−	25.8926i	1.53196	−	0.884475i	0.532684	−	0.846314i	$-0.321184\pi$
$857$	44.8473	−	25.8926i	1.53196	−	0.884475i	0.999272	−	0.0381604i	$-0.0121498\pi$
$858$		0			0
$859$	−7.43152	−	4.29059i	−0.253560	−	0.146393i	0.367833	−	0.929892i	$-0.380100\pi$
$859$	−7.43152	−	4.29059i	−0.253560	−	0.146393i	−0.621393	+	0.783499i	$0.713433\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	41.4652			1.41149			0.705746	−	0.708465i	$-0.250612\pi$
$863$	41.4652			1.41149			0.705746	+	0.708465i	$0.250612\pi$
$864$		0			0
$865$	7.74902	+	4.47390i	0.263474	+	0.152117i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−67.7183	+	39.0972i	−2.29718	+	1.32628i
$870$		0			0
$871$	−33.3567	+	19.2585i	−1.13025	+	0.652550i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	27.1454i		−	0.917681i
$876$		0			0
$877$	32.3745	−	18.6914i	1.09321	−	0.631165i	0.158780	−	0.987314i	$-0.449244\pi$
$877$	32.3745	−	18.6914i	1.09321	−	0.631165i	0.934429	+	0.356149i	$0.115911\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0473			1.01232			0.506160	−	0.862440i	$-0.331065\pi$
$881$	30.0473			1.01232			0.506160	+	0.862440i	$0.331065\pi$
$882$		0			0
$883$	−19.9134	−	11.4970i	−0.670141	−	0.386906i	0.125989	−	0.992032i	$-0.459789\pi$
$883$	−19.9134	−	11.4970i	−0.670141	−	0.386906i	−0.796130	+	0.605126i	$0.793123\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.13495	+	14.0902i	−0.273145	+	0.473101i	−0.969665	−	0.244436i	$-0.921397\pi$
$887$	−8.13495	+	14.0902i	−0.273145	+	0.473101i	0.696520	+	0.717537i	$0.254731\pi$
$888$		0			0
$889$	24.8745	+	14.3613i	0.834264	+	0.481663i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.96193	−	6.04412i	0.199508	−	0.202259i
$894$		0			0
$895$	7.30579	+	12.6540i	0.244205	+	0.422976i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	44.2412	+	25.5427i	1.47553	+	0.851895i
$900$		0			0
$901$		−	52.7540i		−	1.75749i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			29.0667i			0.966208i
$906$		0			0
$907$	22.2378	+	38.5170i	0.738394	+	1.27894i	0.953218	+	0.302284i	$0.0977490\pi$
$907$	22.2378	+	38.5170i	0.738394	+	1.27894i	−0.214824	+	0.976653i	$0.568918\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.6724			−0.519249			−0.259625	−	0.965710i	$-0.583599\pi$
$911$	−15.6724			−0.519249			−0.259625	+	0.965710i	$0.583599\pi$
$912$		0			0
$913$	10.7085			0.354400
$914$		0			0
$915$		0			0
$916$		0			0
$917$	23.1908	+	40.1677i	0.765829	+	1.32645i
$918$		0			0
$919$		−	16.1975i		−	0.534305i	−0.963654	−	0.267153i	$-0.913917\pi$
$919$		−	16.1975i		−	0.534305i	0.963654	−	0.267153i	$-0.0860828\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	71.8199i		−	2.36398i
$924$		0			0
$925$	−3.96863	−	2.29129i	−0.130488	−	0.0753371i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.81752	−	10.0762i	−0.190866	−	0.330590i	0.754671	−	0.656103i	$-0.227796\pi$
$929$	−5.81752	−	10.0762i	−0.190866	−	0.330590i	−0.945538	+	0.325513i	$0.894463\pi$
$930$		0			0
$931$	1.80920	−	6.94177i	0.0592942	−	0.227507i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40.8677	−	23.5950i	−1.33652	−	0.771638i
$936$		0			0
$937$	−18.9059	+	32.7459i	−0.617628	+	1.06976i	0.372289	+	0.928117i	$0.378573\pi$
$937$	−18.9059	+	32.7459i	−0.617628	+	1.06976i	−0.989917	+	0.141647i	$0.954760\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$	24.0470			0.783079
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.1351	−	11.6250i	0.654302	−	0.377762i	−0.135800	−	0.990736i	$-0.543361\pi$
$947$	20.1351	−	11.6250i	0.654302	−	0.377762i	0.790103	+	0.612975i	$0.210027\pi$
$948$		0			0
$949$		−	62.2451i		−	2.02056i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−16.5580	+	9.55977i	−0.536366	+	0.309671i	−0.743605	−	0.668619i	$-0.766886\pi$
$953$	−16.5580	+	9.55977i	−0.536366	+	0.309671i	0.207239	+	0.978290i	$0.433552\pi$
$954$		0			0
$955$	24.5054	−	14.1482i	0.792977	−	0.457825i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.2989	−	7.10077i	−0.397152	−	0.229296i
$960$		0			0
$961$	7.52026			0.242589
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.71992	−	4.45710i	−0.248513	−	0.143479i
$966$		0			0
$967$	−6.72164	+	3.88074i	−0.216153	+	0.124796i	−0.604168	−	0.796857i	$-0.706494\pi$
$967$	−6.72164	+	3.88074i	−0.216153	+	0.124796i	0.388015	+	0.921653i	$0.373161\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.06023	+	8.76458i	0.162391	+	0.281269i	0.935726	−	0.352729i	$-0.114746\pi$
$971$	5.06023	+	8.76458i	0.162391	+	0.281269i	−0.773335	+	0.633998i	$0.781413\pi$
$972$		0			0
$973$	−23.1458	+	40.0896i	−0.742019	+	1.28521i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			47.9280i			1.53335i	0.642034	+	0.766676i	$0.278091\pi$
$977$			47.9280i			1.53335i	−0.642034	+	0.766676i	$0.721909\pi$
$978$		0			0
$979$	−15.3215	+	26.5375i	−0.489676	+	0.848144i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−14.2844	−	24.7413i	−0.455602	−	0.789125i	0.543121	−	0.839654i	$-0.317243\pi$
$983$	−14.2844	−	24.7413i	−0.455602	−	0.789125i	−0.998723	+	0.0505293i	$0.983909\pi$
$984$		0			0
$985$	8.23987	+	14.2719i	0.262544	+	0.454740i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−38.1664			−1.21362
$990$		0			0
$991$	−17.7148	−	30.6829i	−0.562729	−	0.974676i	−0.997257	−	0.0740173i	$-0.976418\pi$
$991$	−17.7148	−	30.6829i	−0.562729	−	0.974676i	0.434528	−	0.900659i	$-0.356915\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	16.4942i		−	0.522901i
$996$		0			0
$997$	7.20850	−	12.4855i	0.228295	−	0.395419i	−0.729008	−	0.684506i	$-0.760018\pi$
$997$	7.20850	−	12.4855i	0.228295	−	0.395419i	0.957303	+	0.289086i	$0.0933515\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.t.1855.4	yes	16
3.2	odd	2	inner	2736.2.bm.t.1855.6	yes	16
4.3	odd	2	inner	2736.2.bm.t.1855.3	yes	16
12.11	even	2	inner	2736.2.bm.t.1855.5	yes	16
19.8	odd	6	inner	2736.2.bm.t.559.4	yes	16
57.8	even	6	inner	2736.2.bm.t.559.6	yes	16
76.27	even	6	inner	2736.2.bm.t.559.3	✓	16
228.179	odd	6	inner	2736.2.bm.t.559.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.bm.t.559.3	✓	16	76.27	even	6	inner
2736.2.bm.t.559.4	yes	16	19.8	odd	6	inner
2736.2.bm.t.559.5	yes	16	228.179	odd	6	inner
2736.2.bm.t.559.6	yes	16	57.8	even	6	inner
2736.2.bm.t.1855.3	yes	16	4.3	odd	2	inner
2736.2.bm.t.1855.4	yes	16	1.1	even	1	trivial
2736.2.bm.t.1855.5	yes	16	12.11	even	2	inner
2736.2.bm.t.1855.6	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.767178 - 1.32879i) q^{5} +2.31392i q^{7} +O(q^{10})\)	\(q+(-0.767178 - 1.32879i) q^{5} +2.31392i q^{7} +5.49807i q^{11} +(-3.96863 - 2.29129i) q^{13} +(-2.79694 - 4.84444i) q^{17} +(1.09932 - 4.21800i) q^{19} +(-3.07472 - 1.77519i) q^{23} +(1.32288 - 2.29129i) q^{25} +(7.12824 + 4.11549i) q^{29} +6.20647 q^{31} +(3.07472 - 1.77519i) q^{35} -1.73205i q^{37} +(-5.86565 + 3.38654i) q^{41} +(9.30970 - 5.37496i) q^{43} +(1.68674 + 0.973842i) q^{47} +1.64575 q^{49} +(8.16719 + 4.71533i) q^{53} +(7.30579 - 4.21800i) q^{55} +(-4.76147 - 8.24710i) q^{59} +(-2.32288 + 4.02334i) q^{61} +7.03130i q^{65} +(4.20255 - 7.27903i) q^{67} +(7.83619 + 13.5727i) q^{71} +(6.79150 + 11.7632i) q^{73} -12.7221 q^{77} +(7.11107 + 12.3167i) q^{79} -1.94768i q^{83} +(-4.29150 + 7.43310i) q^{85} +(4.82670 + 2.78670i) q^{89} +(5.30187 - 9.18310i) q^{91} +(-6.44821 + 1.77519i) q^{95} +(7.40588 - 4.27579i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{49} - 16 q^{61} + 24 q^{73} + 16 q^{85} - 72 q^{97}+O(q^{100}) \) 16 * q - 16 * q^49 - 16 * q^61 + 24 * q^73 + 16 * q^85 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more