Embedded newform 138.4.e.c.55.3

• Label: 138.4.e.c.55.3
• Level: $138$
• Weight: $4$
• Character: 138.55
• 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			55.3
Character	\(\chi\)	\(=\)	138.55
Dual form			138.4.e.c.133.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 - 0.563465i) q^{2} +(-1.96458 - 2.26725i) q^{3} +(3.36501 - 2.16256i) q^{4} +(1.75662 + 12.2176i) q^{5} +(-5.04752 - 3.24384i) q^{6} +(12.8585 - 28.1562i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.91899	−	0.563465i	0.678464	−	0.199215i
							\(3\)	−1.96458	−	2.26725i	−0.378084	−	0.436332i
\(5\)	1.75662	+	12.2176i	0.157117	+	1.09277i								0.903912	+	0.427717i	\(0.140682\pi\)
							−0.746796	+	0.665053i	\(0.768409\pi\)
\(7\)	12.8585	−	28.1562i	0.694293	−	1.52029i	−0.152463	−	0.988309i	\(-0.548721\pi\)
							0.846756	−	0.531981i	\(-0.178552\pi\)
\(11\)	38.1759	+	11.2095i	1.04641	+	0.307253i	0.759365	−	0.650665i	\(-0.225510\pi\)
							0.287042	+	0.957918i	\(0.407328\pi\)
\(13\)	−28.5267	−	62.4647i	−0.608605	−	1.33266i	−0.923524	−	0.383541i	\(-0.874704\pi\)
							0.314918	−	0.949119i	\(-0.398023\pi\)
\(17\)	73.7494	+	47.3958i	1.05217	+	0.676187i	0.947967	−	0.318369i	\(-0.103135\pi\)
							0.104201	+	0.994556i	\(0.466772\pi\)
\(19\)	53.8170	−	34.5861i	0.649815	−	0.417610i	−0.173784	−	0.984784i	\(-0.555599\pi\)
							0.823598	+	0.567173i	\(0.191963\pi\)
\(23\)	−25.3912	−	107.342i	−0.230193	−	0.973145i
							\(29\)	−192.734	−	123.863i	−1.23413	−	0.793128i	−0.249601	−	0.968349i	\(-0.580300\pi\)
−0.984529	+	0.175221i	\(0.943936\pi\)
\(31\)	−115.579	+	133.385i	−0.669633	+	0.772797i	−0.984319	−	0.176398i	\(-0.943555\pi\)
							0.314686	+	0.949196i	\(0.398101\pi\)
\(37\)	−51.7947	+	360.240i	−0.230135	+	1.60062i	0.467382	+	0.884056i	\(0.345197\pi\)
							−0.697517	+	0.716569i	\(0.745712\pi\)
\(41\)	30.4498	+	211.783i	0.115987	+	0.806707i	0.961903	+	0.273391i	\(0.0881452\pi\)
							−0.845916	+	0.533316i	\(0.820946\pi\)
\(43\)	301.091	+	347.478i	1.06781	+	1.23232i	0.971517	+	0.236970i	\(0.0761544\pi\)
							0.0962971	+	0.995353i	\(0.469300\pi\)
\(47\)	78.1421			0.242515			0.121257	−	0.992621i	\(-0.461307\pi\)
							0.121257	+	0.992621i	\(0.461307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.55.3	✓	30
23.18	even	11	inner	138.4.e.c.133.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.55.3	✓	30	1.1	even	1	trivial
138.4.e.c.133.3	yes	30	23.18	even	11	inner