Embedded newform 138.4.e.c.31.1

• Label: 138.4.e.c.31.1
• Level: $138$
• Weight: $4$
• Character: 138.31
• 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			31.1
Character	\(\chi\)	\(=\)	138.31
Dual form			138.4.e.c.49.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.284630 - 1.97964i) q^{2} +(1.24625 + 2.72890i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-7.03109 - 8.11432i) q^{5} +(5.75696 - 1.69040i) q^{6} +(-5.50559 - 3.53823i) q^{7} +(-3.32332 + 7.27706i) q^{8} +(-5.89375 + 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.284630	−	1.97964i	0.100632	−	0.699909i
							\(3\)	1.24625	+	2.72890i	0.239840	+	0.525176i
\(5\)	−7.03109	−	8.11432i	−0.628880	−	0.725766i								0.348488	−	0.937313i	\(-0.386695\pi\)
							−0.977368	+	0.211547i	\(0.932150\pi\)
\(7\)	−5.50559	−	3.53823i	−0.297274	−	0.191046i	0.383498	−	0.923542i	\(-0.374719\pi\)
							−0.680772	+	0.732495i	\(0.738356\pi\)
\(11\)	−0.141989	−	0.987554i	−0.00389193	−	0.0270690i	0.987783	−	0.155836i	\(-0.0498071\pi\)
							−0.991675	+	0.128767i	\(0.958898\pi\)
\(13\)	−47.0714	+	30.2509i	−1.00425	+	0.645392i	−0.935899	−	0.352269i	\(-0.885410\pi\)
							−0.0683514	+	0.997661i	\(0.521774\pi\)
\(17\)	−63.0414	+	18.5106i	−0.899399	+	0.264087i	−0.698572	−	0.715539i	\(-0.746181\pi\)
							−0.200827	+	0.979627i	\(0.564363\pi\)
\(19\)	−132.331	−	38.8558i	−1.59783	−	0.469165i	−0.642887	−	0.765961i	\(-0.722264\pi\)
							−0.954941	+	0.296796i	\(0.904082\pi\)
\(23\)	−34.5210	−	104.763i	−0.312962	−	0.949766i
							\(29\)	164.965	−	48.4380i	1.05632	−	0.310162i	0.292949	−	0.956128i	\(-0.405363\pi\)
0.763366	+	0.645966i	\(0.223545\pi\)
\(31\)	−101.009	+	221.179i	−0.585218	+	1.28145i	0.353071	+	0.935596i	\(0.385137\pi\)
							−0.938289	+	0.345852i	\(0.887590\pi\)
\(37\)	84.3861	−	97.3868i	0.374946	−	0.432710i	−0.536646	−	0.843808i	\(-0.680309\pi\)
							0.911592	+	0.411097i	\(0.134854\pi\)
\(41\)	162.344	+	187.355i	0.618386	+	0.713656i	0.975400	−	0.220443i	\(-0.0707504\pi\)
							−0.357013	+	0.934099i	\(0.616205\pi\)
\(43\)	−16.3365	−	35.7720i	−0.0579371	−	0.126865i	0.878449	−	0.477836i	\(-0.158579\pi\)
							−0.936386	+	0.350971i	\(0.885851\pi\)
\(47\)	269.657			0.836883			0.418441	−	0.908244i	\(-0.362577\pi\)
							0.418441	+	0.908244i	\(0.362577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.31.1	✓	30
23.3	even	11	inner	138.4.e.c.49.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.31.1	✓	30	1.1	even	1	trivial
138.4.e.c.49.1	yes	30	23.3	even	11	inner