Embedded newform 135.4.q.a.113.10

• Label: 135.4.q.a.113.10
• Level: $135$
• Weight: $4$
• Character: 135.113
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(52\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.10
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.93122 - 0.343937i) q^{2} +(-1.16700 - 5.06341i) q^{3} +(7.45776 + 1.31500i) q^{4} +(11.0891 - 1.42519i) q^{5} +(2.84625 + 20.3068i) q^{6} +(4.64813 + 3.25465i) q^{7} +(1.62835 + 0.436314i) q^{8} +(-24.2762 + 11.8180i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.93122	−	0.343937i	−1.38990	−	0.121600i	−0.632503	−	0.774558i	\(-0.717972\pi\)
							−0.757394	+	0.652958i	\(0.773528\pi\)
\(3\)	−1.16700	−	5.06341i	−0.224590	−	0.974453i
							\(5\)	11.0891	−	1.42519i	0.991842	−	0.127473i
\(7\)	4.64813	+	3.25465i	0.250975	+	0.175735i								0.692294	−	0.721616i	\(-0.256600\pi\)
							−0.441319	+	0.897351i	\(0.645489\pi\)
\(11\)	−8.46726	+	23.2636i	−0.232089	+	0.637658i	−0.999996	−	0.00290278i	\(-0.999076\pi\)
							0.767907	+	0.640561i	\(0.221298\pi\)
\(13\)	7.46611	+	85.3381i	0.159287	+	1.82066i	0.484251	+	0.874929i	\(0.339092\pi\)
							−0.324964	+	0.945726i	\(0.605352\pi\)
\(17\)	118.619	−	31.7839i	1.69232	−	0.453454i	0.721329	−	0.692592i	\(-0.243531\pi\)
							0.970986	+	0.239138i	\(0.0768647\pi\)
\(19\)	−25.4887	+	14.7159i	−0.307764	+	0.177687i	−0.645925	−	0.763401i	\(-0.723528\pi\)
							0.338162	+	0.941088i	\(0.390195\pi\)
\(23\)	30.2793	+	43.2433i	0.274507	+	0.392037i	0.932607	−	0.360894i	\(-0.117528\pi\)
							−0.658100	+	0.752931i	\(0.728640\pi\)
\(29\)	189.421	−	158.943i	1.21292	−	1.01776i	0.213753	−	0.976888i	\(-0.431431\pi\)
							0.999164	−	0.0408713i	\(-0.0130134\pi\)
\(31\)	−47.1930	+	267.645i	−0.273423	+	1.55066i	0.470503	+	0.882398i	\(0.344072\pi\)
							−0.743927	+	0.668261i	\(0.767039\pi\)
\(37\)	−92.0179	−	343.416i	−0.408856	−	1.52587i	−0.796832	−	0.604201i	\(-0.793493\pi\)
							0.387977	−	0.921669i	\(-0.373174\pi\)
\(41\)	−18.5447	+	22.1007i	−0.0706388	+	0.0841840i	−0.800207	−	0.599725i	\(-0.795277\pi\)
							0.729568	+	0.683909i	\(0.239721\pi\)
\(43\)	42.6402	+	91.4423i	0.151223	+	0.324298i	0.967296	−	0.253650i	\(-0.0816313\pi\)
							−0.816073	+	0.577948i	\(0.803853\pi\)
\(47\)	25.9749	−	37.0959i	0.0806132	−	0.115128i	−0.776820	−	0.629722i	\(-0.783169\pi\)
							0.857434	+	0.514595i	\(0.172058\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.10	yes	624
5.2	odd	4	inner	135.4.q.a.32.43	✓	624
27.11	odd	18	inner	135.4.q.a.38.43	yes	624
135.92	even	36	inner	135.4.q.a.92.10	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.43	✓	624	5.2	odd	4	inner
135.4.q.a.38.43	yes	624	27.11	odd	18	inner
135.4.q.a.92.10	yes	624	135.92	even	36	inner
135.4.q.a.113.10	yes	624	1.1	even	1	trivial