Embedded newform 138.4.e.b.13.1

• Label: 138.4.e.b.13.1
• Level: $138$
• Weight: $4$
• Character: 138.13
• 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			13.1
Character	\(\chi\)	\(=\)	138.13
Dual form			138.4.e.b.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30972 - 1.51150i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-0.569259 + 3.95929i) q^{4} +(-3.80011 + 8.32109i) q^{5} +(0.853889 + 5.93893i) q^{6} +(2.38181 + 0.699363i) q^{7} +(6.73003 - 4.32513i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30972	−	1.51150i	−0.463056	−	0.534396i
							\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
\(5\)	−3.80011	+	8.32109i	−0.339892	+	0.744261i								−0.999976	−	0.00693150i	\(-0.997794\pi\)
							0.660083	+	0.751192i	\(0.270521\pi\)
\(7\)	2.38181	+	0.699363i	0.128606	+	0.0377620i	0.345402	−	0.938455i	\(-0.387743\pi\)
							−0.216796	+	0.976217i	\(0.569561\pi\)
\(11\)	36.6788	−	42.3296i	1.00537	−	1.16026i	0.0183225	−	0.999832i	\(-0.494167\pi\)
							0.987048	−	0.160427i	\(-0.0512871\pi\)
\(13\)	−6.64053	+	1.94984i	−0.141673	+	0.0415990i	−0.351800	−	0.936075i	\(-0.614430\pi\)
							0.210127	+	0.977674i	\(0.432612\pi\)
\(17\)	−5.76838	−	40.1200i	−0.0822964	−	0.572384i	−0.988693	−	0.149956i	\(-0.952087\pi\)
							0.906396	−	0.422428i	\(-0.138822\pi\)
\(19\)	3.74037	−	26.0149i	0.0451632	−	0.314117i	−0.954700	−	0.297570i	\(-0.903824\pi\)
							0.999863	−	0.0165465i	\(-0.00526717\pi\)
\(23\)	−16.1352	−	109.118i	−0.146279	−	0.989243i
							\(29\)	−38.6083	−	268.527i	−0.247220	−	1.71945i	−0.614136	−	0.789200i	\(-0.710496\pi\)
0.366916	−	0.930254i	\(-0.380414\pi\)
\(31\)	136.569	−	87.7677i	0.791244	−	0.508502i	−0.0815039	−	0.996673i	\(-0.525972\pi\)
							0.872748	+	0.488171i	\(0.162336\pi\)
\(37\)	62.9324	+	137.803i	0.279622	+	0.612287i	0.996378	−	0.0850372i	\(-0.0271009\pi\)
							−0.716755	+	0.697325i	\(0.754374\pi\)
\(41\)	149.334	−	326.995i	0.568830	−	1.24556i	−0.378588	−	0.925565i	\(-0.623590\pi\)
							0.947418	−	0.319998i	\(-0.103682\pi\)
\(43\)	263.221	+	169.162i	0.933507	+	0.599928i	0.916547	−	0.399928i	\(-0.130965\pi\)
							0.0169600	+	0.999856i	\(0.494601\pi\)
\(47\)	132.472			0.411129			0.205565	−	0.978644i	\(-0.434097\pi\)
							0.205565	+	0.978644i	\(0.434097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.13.1	✓	30
23.16	even	11	inner	138.4.e.b.85.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.13.1	✓	30	1.1	even	1	trivial
138.4.e.b.85.1	yes	30	23.16	even	11	inner