Embedded newform 38.7.f.a.3.4

• Label: 38.7.f.a.3.4
• Level: $38$
• Weight: $7$
• Character: 38.3
• Analytic conductor: $8.742$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.74205517755\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	38.3
Dual form			38.7.f.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.63616 + 4.33340i) q^{2} +(10.0117 + 27.5069i) q^{3} +(-5.55674 - 31.5138i) q^{4} +(26.4694 - 150.115i) q^{5} +(-155.603 - 56.6347i) q^{6} +(-232.836 - 403.285i) q^{7} +(156.767 + 90.5097i) q^{8} +(-97.9501 + 82.1899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 30 q^{3} - 240 q^{6} + 432 q^{7} + 690 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.63616	+	4.33340i	−0.454519	+	0.541675i
							\(3\)	10.0117	+	27.5069i	0.370804	+	1.01877i	0.975052	+	0.221978i	\(0.0712515\pi\)
−0.604248	+	0.796796i	\(0.706526\pi\)
\(5\)	26.4694	−	150.115i	0.211755	−	1.20092i	−0.674695	−	0.738097i	\(-0.735725\pi\)
							0.886450	−	0.462825i	\(-0.153164\pi\)
\(7\)	−232.836	−	403.285i	−0.678823	−	1.17576i	−0.975335	−	0.220728i	\(-0.929157\pi\)
							0.296512	−	0.955029i	\(-0.404177\pi\)
\(11\)	−50.2851	+	87.0963i	−0.0377799	+	0.0654367i	−0.884297	−	0.466925i	\(-0.845362\pi\)
							0.846517	+	0.532361i	\(0.178695\pi\)
\(13\)	484.248	−	1330.46i	0.220413	−	0.605580i	−0.779367	−	0.626568i	\(-0.784459\pi\)
							0.999780	+	0.0209882i	\(0.00668123\pi\)
\(17\)	−702.241	−	589.250i	−0.142935	−	0.119937i	0.568516	−	0.822672i	\(-0.307518\pi\)
							−0.711452	+	0.702735i	\(0.751962\pi\)
\(19\)	4046.66	−	5538.09i	0.589978	−	0.807419i
							\(23\)	397.585	+	2254.81i	0.0326773	+	0.185322i	0.996777	−	0.0802168i	\(-0.0255612\pi\)
−0.964100	+	0.265539i	\(0.914450\pi\)
\(29\)	−10701.3	−	12753.4i	−0.438777	−	0.522914i	0.500656	−	0.865646i	\(-0.333092\pi\)
							−0.939433	+	0.342732i	\(0.888648\pi\)
\(31\)	−33711.4	+	19463.3i	−1.13160	+	0.653327i	−0.944336	−	0.328984i	\(-0.893294\pi\)
							−0.187260	+	0.982310i	\(0.559961\pi\)
\(37\)		−	67366.8i		−	1.32997i	−0.746858	−	0.664983i	\(-0.768439\pi\)
							0.746858	−	0.664983i	\(-0.231561\pi\)
\(41\)	−31132.5	−	85535.9i	−0.451713	−	1.24107i	−0.931518	−	0.363696i	\(-0.881515\pi\)
							0.479804	−	0.877375i	\(-0.340708\pi\)
\(43\)	−16148.2	+	91580.7i	−0.203104	+	1.15186i	0.697293	+	0.716787i	\(0.254388\pi\)
							−0.900396	+	0.435071i	\(0.856723\pi\)
\(47\)	95929.1	−	80494.1i	0.923968	−	0.775301i	−0.0507567	−	0.998711i	\(-0.516163\pi\)
							0.974725	+	0.223410i	\(0.0717189\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.7.f.a.3.4	✓	60
19.13	odd	18	inner	38.7.f.a.13.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.f.a.3.4	✓	60	1.1	even	1	trivial
38.7.f.a.13.4	yes	60	19.13	odd	18	inner