Embedded newform 138.4.e.b.31.2

• Label: 138.4.e.b.31.2
• 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			31.2
Character	\(\chi\)	\(=\)	138.31
Dual form			138.4.e.b.49.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.284630 + 1.97964i) q^{2} +(-1.24625 - 2.72890i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(1.48009 + 1.70811i) q^{5} +(5.75696 - 1.69040i) q^{6} +(3.78550 + 2.43279i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-5.89375 + 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	1.48009	+	1.70811i	0.132383	+	0.152778i								0.818071	−	0.575118i	\(-0.195044\pi\)
							−0.685688	+	0.727896i	\(0.740498\pi\)
\(7\)	3.78550	+	2.43279i	0.204398	+	0.131358i	0.638835	−	0.769344i	\(-0.279417\pi\)
							−0.434438	+	0.900702i	\(0.643053\pi\)
\(11\)	−2.10712	−	14.6553i	−0.0577564	−	0.401705i	−0.998107	−	0.0614984i	\(-0.980412\pi\)
							0.940351	−	0.340207i	\(-0.110497\pi\)
\(13\)	57.7261	−	37.0983i	1.23156	−	0.791478i	0.247430	−	0.968906i	\(-0.420414\pi\)
							0.984134	+	0.177427i	\(0.0567775\pi\)
\(17\)	65.6958	−	19.2900i	0.937269	−	0.275207i	0.222792	−	0.974866i	\(-0.428483\pi\)
							0.714477	+	0.699659i	\(0.246665\pi\)
\(19\)	109.546	+	32.1656i	1.32271	+	0.388384i	0.865471	−	0.500959i	\(-0.167019\pi\)
							0.457242	+	0.889342i	\(0.348837\pi\)
\(23\)	79.2673	+	76.7052i	0.718625	+	0.695398i
							\(29\)	−154.994	+	45.5103i	−0.992471	+	0.291416i	−0.737363	−	0.675497i	\(-0.763929\pi\)
−0.255108	+	0.966913i	\(0.582111\pi\)
\(31\)	19.3826	−	42.4419i	0.112297	−	0.245897i	−0.845136	−	0.534551i	\(-0.820481\pi\)
							0.957433	+	0.288655i	\(0.0932079\pi\)
\(37\)	46.5322	−	53.7010i	0.206753	−	0.238605i	−0.642897	−	0.765952i	\(-0.722268\pi\)
							0.849650	+	0.527347i	\(0.176813\pi\)
\(41\)	179.778	+	207.475i	0.684796	+	0.790297i	0.986615	−	0.163068i	\(-0.0521391\pi\)
							−0.301818	+	0.953365i	\(0.597594\pi\)
\(43\)	−52.4498	−	114.849i	−0.186012	−	0.407309i	0.793535	−	0.608524i	\(-0.208238\pi\)
							−0.979547	+	0.201215i	\(0.935511\pi\)
\(47\)	290.966			0.903015			0.451508	−	0.892267i	\(-0.350886\pi\)
							0.451508	+	0.892267i	\(0.350886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.31.2	✓	30
23.3	even	11	inner	138.4.e.b.49.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.31.2	✓	30	1.1	even	1	trivial
138.4.e.b.49.2	yes	30	23.3	even	11	inner