Embedded newform 138.4.e.b.73.2

• Label: 138.4.e.b.73.2
• Level: $138$
• Weight: $4$
• Character: 138.73
• 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			73.2
Character	\(\chi\)	\(=\)	138.73
Dual form			138.4.e.b.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.830830 + 1.81926i) q^{2} +(2.87848 - 0.845198i) q^{3} +(-2.61944 + 3.02300i) q^{4} +(-8.49581 - 5.45993i) q^{5} +(3.92916 + 4.53450i) q^{6} +(4.18194 + 29.0860i) q^{7} +(-7.67594 - 2.25386i) q^{8} +(7.57128 - 4.86577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.830830	+	1.81926i	0.293743	+	0.643207i
							\(3\)	2.87848	−	0.845198i	0.553964	−	0.162658i
\(5\)	−8.49581	−	5.45993i	−0.759889	−	0.488351i								0.102415	−	0.994742i	\(-0.467343\pi\)
							−0.862304	+	0.506391i	\(0.830979\pi\)
\(7\)	4.18194	+	29.0860i	0.225803	+	1.57050i	0.715505	+	0.698607i	\(0.246197\pi\)
							−0.489702	+	0.871890i	\(0.662894\pi\)
\(11\)	−8.00375	+	17.5258i	−0.219384	+	0.480383i	−0.987039	−	0.160479i	\(-0.948696\pi\)
							0.767655	+	0.640863i	\(0.221423\pi\)
\(13\)	−12.3589	+	85.9581i	−0.263673	+	1.83388i	0.240942	+	0.970540i	\(0.422544\pi\)
							−0.504615	+	0.863345i	\(0.668365\pi\)
\(17\)	31.7802	+	36.6763i	0.453402	+	0.523253i	0.935720	−	0.352742i	\(-0.114751\pi\)
							−0.482319	+	0.875996i	\(0.660205\pi\)
\(19\)	5.59778	−	6.46019i	0.0675905	−	0.0780036i	−0.720947	−	0.692990i	\(-0.756293\pi\)
							0.788538	+	0.614986i	\(0.210838\pi\)
\(23\)	81.0921	−	74.7734i	0.735169	−	0.677884i
							\(29\)	39.7282	+	45.8488i	0.254391	+	0.293583i	0.868552	−	0.495598i	\(-0.165051\pi\)
−0.614161	+	0.789181i	\(0.710505\pi\)
\(31\)	−296.853	−	87.1640i	−1.71988	−	0.505004i	−0.734978	−	0.678091i	\(-0.762808\pi\)
							−0.984907	+	0.173087i	\(0.944626\pi\)
\(37\)	−16.3782	+	10.5256i	−0.0727720	+	0.0467677i	−0.576521	−	0.817082i	\(-0.695590\pi\)
							0.503749	+	0.863850i	\(0.331954\pi\)
\(41\)	297.757	+	191.357i	1.13419	+	0.728901i	0.966431	−	0.256927i	\(-0.0827099\pi\)
							0.167761	+	0.985828i	\(0.446346\pi\)
\(43\)	302.550	−	88.8366i	1.07299	−	0.315057i	0.302916	−	0.953017i	\(-0.402040\pi\)
							0.770069	+	0.637960i	\(0.220222\pi\)
\(47\)	−104.009			−0.322794			−0.161397	−	0.986890i	\(-0.551600\pi\)
							−0.161397	+	0.986890i	\(0.551600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.73.2	✓	30
23.6	even	11	inner	138.4.e.b.121.2	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.73.2	✓	30	1.1	even	1	trivial
138.4.e.b.121.2	yes	30	23.6	even	11	inner