Embedded newform 138.4.e.b.85.1

• Label: 138.4.e.b.85.1
• Level: $138$
• Weight: $4$
• Character: 138.85
• 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			85.1
Character	\(\chi\)	\(=\)	138.85
Dual form			138.4.e.b.13.1

$q$-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 \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{11}\right)\)

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								0.660083	−	0.751192i	\(-0.270521\pi\)
							−0.999976	+	0.00693150i	\(0.997794\pi\)
\(7\)	2.38181	−	0.699363i	0.128606	−	0.0377620i	−0.216796	−	0.976217i	\(-0.569561\pi\)
							0.345402	+	0.938455i	\(0.387743\pi\)
\(11\)	36.6788	+	42.3296i	1.00537	+	1.16026i	0.987048	+	0.160427i	\(0.0512871\pi\)
							0.0183225	+	0.999832i	\(0.494167\pi\)
\(13\)	−6.64053	−	1.94984i	−0.141673	−	0.0415990i	0.210127	−	0.977674i	\(-0.432612\pi\)
							−0.351800	+	0.936075i	\(0.614430\pi\)
\(17\)	−5.76838	+	40.1200i	−0.0822964	+	0.572384i	0.906396	+	0.422428i	\(0.138822\pi\)
							−0.988693	+	0.149956i	\(0.952087\pi\)
\(19\)	3.74037	+	26.0149i	0.0451632	+	0.314117i	0.999863	+	0.0165465i	\(0.00526717\pi\)
							−0.954700	+	0.297570i	\(0.903824\pi\)
\(23\)	−16.1352	+	109.118i	−0.146279	+	0.989243i
							\(29\)	−38.6083	+	268.527i	−0.247220	+	1.71945i	0.366916	+	0.930254i	\(0.380414\pi\)
−0.614136	+	0.789200i	\(0.710496\pi\)
\(31\)	136.569	+	87.7677i	0.791244	+	0.508502i	0.872748	−	0.488171i	\(-0.162336\pi\)
							−0.0815039	+	0.996673i	\(0.525972\pi\)
\(37\)	62.9324	−	137.803i	0.279622	−	0.612287i	−0.716755	−	0.697325i	\(-0.754374\pi\)
							0.996378	+	0.0850372i	\(0.0271009\pi\)
\(41\)	149.334	+	326.995i	0.568830	+	1.24556i	0.947418	+	0.319998i	\(0.103682\pi\)
							−0.378588	+	0.925565i	\(0.623590\pi\)
\(43\)	263.221	−	169.162i	0.933507	−	0.599928i	0.0169600	−	0.999856i	\(-0.494601\pi\)
							0.916547	+	0.399928i	\(0.130965\pi\)
\(47\)	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.85.1	yes	30
23.13	even	11	inner	138.4.e.b.13.1	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.13.1	✓	30	23.13	even	11	inner
138.4.e.b.85.1	yes	30	1.1	even	1	trivial