Embedded newform 138.4.e.c.55.2

• Label: 138.4.e.c.55.2
• 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			55.2
Character	\(\chi\)	\(=\)	138.55
Dual form			138.4.e.c.133.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.91899 - 0.563465i) q^{2} +(-1.96458 - 2.26725i) q^{3} +(3.36501 - 2.16256i) q^{4} +(1.45978 + 10.1530i) q^{5} +(-5.04752 - 3.24384i) q^{6} +(-13.4424 + 29.4347i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-1.28083 + 8.90839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	1.45978	+	10.1530i	0.130567	+	0.908110i								0.944817	+	0.327597i	\(0.106239\pi\)
							−0.814251	+	0.580513i	\(0.802852\pi\)
\(7\)	−13.4424	+	29.4347i	−0.725820	+	1.58932i	0.0797449	+	0.996815i	\(0.474589\pi\)
							−0.805565	+	0.592508i	\(0.798138\pi\)
\(11\)	42.6703	+	12.5291i	1.16960	+	0.343425i	0.808156	−	0.588969i	\(-0.200466\pi\)
							0.361444	+	0.932394i	\(0.382284\pi\)
\(13\)	13.6990	+	29.9967i	0.292264	+	0.639968i	0.997625	−	0.0688796i	\(-0.0219424\pi\)
							−0.705361	+	0.708848i	\(0.749215\pi\)
\(17\)	−51.9310	−	33.3740i	−0.740890	−	0.476141i	0.114957	−	0.993370i	\(-0.463327\pi\)
							−0.855847	+	0.517230i	\(0.826963\pi\)
\(19\)	66.5011	−	42.7377i	0.802969	−	0.516037i	−0.0736145	−	0.997287i	\(-0.523453\pi\)
							0.876583	+	0.481250i	\(0.159817\pi\)
\(23\)	−95.7715	+	54.7249i	−0.868250	+	0.496128i
							\(29\)	174.731	+	112.293i	1.11885	+	0.719043i	0.963205	−	0.268769i	\(-0.0866169\pi\)
0.155649	+	0.987812i	\(0.450253\pi\)
\(31\)	−142.621	+	164.594i	−0.826308	+	0.953610i	−0.999511	−	0.0312760i	\(-0.990043\pi\)
							0.173203	+	0.984886i	\(0.444588\pi\)
\(37\)	17.8556	−	124.188i	0.0793362	−	0.551795i	−0.910925	−	0.412572i	\(-0.864630\pi\)
							0.990261	−	0.139223i	\(-0.0444605\pi\)
\(41\)	57.2594	+	398.248i	0.218108	+	1.51697i	0.745014	+	0.667049i	\(0.232443\pi\)
							−0.526906	+	0.849923i	\(0.676648\pi\)
\(43\)	−303.464	−	350.217i	−1.07623	−	1.24204i	−0.968805	−	0.247823i	\(-0.920285\pi\)
							−0.107425	−	0.994213i	\(-0.534260\pi\)
\(47\)	−150.426			−0.466850			−0.233425	−	0.972375i	\(-0.574993\pi\)
							−0.233425	+	0.972375i	\(0.574993\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.55.2	✓	30
23.18	even	11	inner	138.4.e.c.133.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.55.2	✓	30	1.1	even	1	trivial
138.4.e.c.133.2	yes	30	23.18	even	11	inner