Embedded newform 138.4.e.b.31.1

• Label: 138.4.e.b.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.b.49.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.284630 + 1.97964i) q^{2} +(-1.24625 - 2.72890i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-11.4557 - 13.2206i) q^{5} +(5.75696 - 1.69040i) q^{6} +(16.3172 + 10.4864i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-5.89375 + 6.80175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	−11.4557	−	13.2206i	−1.02463	−	1.18249i								−0.983047	−	0.183354i	\(-0.941305\pi\)
							−0.0415852	−	0.999135i	\(-0.513241\pi\)
\(7\)	16.3172	+	10.4864i	0.881047	+	0.566215i	0.901113	−	0.433584i	\(-0.142751\pi\)
							−0.0200661	+	0.999799i	\(0.506388\pi\)
\(11\)	9.06537	+	63.0510i	0.248483	+	1.72824i	0.606990	+	0.794710i	\(0.292377\pi\)
							−0.358507	+	0.933527i	\(0.616714\pi\)
\(13\)	−51.1562	+	32.8761i	−1.09140	+	0.701398i	−0.957163	−	0.289551i	\(-0.906494\pi\)
							−0.134235	+	0.990950i	\(0.542858\pi\)
\(17\)	−79.9667	+	23.4803i	−1.14087	+	0.334989i	−0.796971	−	0.604017i	\(-0.793566\pi\)
							−0.343898	+	0.939007i	\(0.611748\pi\)
\(19\)	114.775	+	33.7009i	1.38585	+	0.406922i	0.887801	−	0.460227i	\(-0.152232\pi\)
							0.498048	+	0.867149i	\(0.334050\pi\)
\(23\)	24.3792	−	107.576i	0.221018	−	0.975270i
							\(29\)	−19.4968	+	5.72479i	−0.124844	+	0.0366574i	−0.343558	−	0.939132i	\(-0.611632\pi\)
0.218714	+	0.975789i	\(0.429814\pi\)
\(31\)	−106.543	+	233.296i	−0.617279	+	1.35165i	0.300203	+	0.953875i	\(0.402946\pi\)
							−0.917482	+	0.397777i	\(0.869782\pi\)
\(37\)	−55.5011	+	64.0517i	−0.246603	+	0.284596i	−0.865534	−	0.500850i	\(-0.833021\pi\)
							0.618931	+	0.785446i	\(0.287566\pi\)
\(41\)	36.2459	+	41.8299i	0.138065	+	0.159335i	0.820571	−	0.571544i	\(-0.193656\pi\)
							−0.682506	+	0.730880i	\(0.739110\pi\)
\(43\)	157.976	+	345.919i	0.560258	+	1.22679i	0.951824	+	0.306644i	\(0.0992062\pi\)
							−0.391566	+	0.920150i	\(0.628067\pi\)
\(47\)	−619.674			−1.92316			−0.961582	−	0.274517i	\(-0.911482\pi\)
							−0.961582	+	0.274517i	\(0.911482\pi\)

(See \(a_n\) instead)

Twists

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

    

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