Embedded newform 138.4.e.c.73.3

• Label: 138.4.e.c.73.3
• Level: $138$
• Weight: $4$
• Character: 138.73
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			73.3
Character	\(\chi\)	\(=\)	138.73
Dual form			138.4.e.c.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.830830 - 1.81926i) q^{2} +(-2.87848 + 0.845198i) q^{3} +(-2.61944 + 3.02300i) q^{4} +(3.54028 + 2.27520i) q^{5} +(3.92916 + 4.53450i) q^{6} +(-0.484390 - 3.36901i) q^{7} +(7.67594 + 2.25386i) q^{8} +(7.57128 - 4.86577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.830830	−	1.81926i	−0.293743	−	0.643207i
							\(3\)	−2.87848	+	0.845198i	−0.553964	+	0.162658i
\(5\)	3.54028	+	2.27520i	0.316652	+	0.203500i								0.689306	−	0.724470i	\(-0.257916\pi\)
							−0.372653	+	0.927971i	\(0.621552\pi\)
\(7\)	−0.484390	−	3.36901i	−0.0261546	−	0.181909i	0.972556	−	0.232667i	\(-0.0747453\pi\)
							−0.998711	+	0.0507581i	\(0.983836\pi\)
\(11\)	1.72851	−	3.78491i	0.0473787	−	0.103745i	−0.884462	−	0.466611i	\(-0.845475\pi\)
							0.931841	+	0.362867i	\(0.118202\pi\)
\(13\)	5.88375	−	40.9224i	0.125528	−	0.873064i	−0.825598	−	0.564259i	\(-0.809162\pi\)
							0.951125	−	0.308805i	\(-0.0999291\pi\)
\(17\)	−58.2358	−	67.2077i	−0.830839	−	0.958839i	0.168801	−	0.985650i	\(-0.446010\pi\)
							−0.999641	+	0.0268107i	\(0.991465\pi\)
\(19\)	33.8758	−	39.0947i	0.409033	−	0.472050i	−0.513432	−	0.858131i	\(-0.671626\pi\)
							0.922465	+	0.386081i	\(0.126171\pi\)
\(23\)	−98.8248	−	48.9966i	−0.895930	−	0.444196i
							\(29\)	−82.2694	−	94.9439i	−0.526795	−	0.607953i	0.428524	−	0.903530i	\(-0.359034\pi\)
−0.955319	+	0.295577i	\(0.904488\pi\)
\(31\)	118.111	+	34.6805i	0.684302	+	0.200929i	0.605362	−	0.795950i	\(-0.293028\pi\)
							0.0789401	+	0.996879i	\(0.474846\pi\)
\(37\)	58.4593	−	37.5695i	0.259747	−	0.166929i	−0.404286	−	0.914633i	\(-0.632480\pi\)
							0.664033	+	0.747703i	\(0.268843\pi\)
\(41\)	−115.765	−	74.3977i	−0.440963	−	0.283389i	0.301261	−	0.953542i	\(-0.402592\pi\)
							−0.742224	+	0.670152i	\(0.766229\pi\)
\(43\)	225.540	−	66.2247i	0.799874	−	0.234864i	0.143846	−	0.989600i	\(-0.454053\pi\)
							0.656029	+	0.754736i	\(0.272235\pi\)
\(47\)	40.0435			0.124275			0.0621377	−	0.998068i	\(-0.480208\pi\)
							0.0621377	+	0.998068i	\(0.480208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.73.3	✓	30
23.6	even	11	inner	138.4.e.c.121.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.73.3	✓	30	1.1	even	1	trivial
138.4.e.c.121.3	yes	30	23.6	even	11	inner