Embedded newform 138.4.e.d.13.1

• Label: 138.4.e.d.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.d.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30972 + 1.51150i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-0.569259 + 3.95929i) q^{4} +(-1.00284 + 2.19592i) q^{5} +(-0.853889 - 5.93893i) q^{6} +(19.8790 + 5.83701i) q^{7} +(-6.73003 + 4.32513i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	−1.00284	+	2.19592i	−0.0896969	+	0.196409i								−0.949164	−	0.314783i	\(-0.898068\pi\)
							0.859467	+	0.511191i	\(0.170796\pi\)
\(7\)	19.8790	+	5.83701i	1.07337	+	0.315169i	0.770222	−	0.637775i	\(-0.220145\pi\)
							0.303145	+	0.952944i	\(0.401963\pi\)
\(11\)	−20.6310	+	23.8094i	−0.565497	+	0.652619i	−0.964423	−	0.264365i	\(-0.914838\pi\)
							0.398925	+	0.916983i	\(0.369383\pi\)
\(13\)	−12.3363	+	3.62226i	−0.263190	+	0.0772794i	−0.410665	−	0.911786i	\(-0.634703\pi\)
							0.147475	+	0.989066i	\(0.452885\pi\)
\(17\)	15.3725	+	106.918i	0.219317	+	1.52538i	0.740570	+	0.671979i	\(0.234555\pi\)
							−0.521254	+	0.853402i	\(0.674535\pi\)
\(19\)	−21.9642	+	152.764i	−0.265207	+	1.84456i	0.226771	+	0.973948i	\(0.427183\pi\)
							−0.491978	+	0.870607i	\(0.663726\pi\)
\(23\)	−45.2576	−	100.592i	−0.410298	−	0.911951i
							\(29\)	−0.911865	−	6.34216i	−0.00583893	−	0.0406107i	0.986694	−	0.162590i	\(-0.0519849\pi\)
−0.992533	+	0.121980i	\(0.961076\pi\)
\(31\)	225.060	−	144.637i	1.30393	−	0.837987i	0.310298	−	0.950639i	\(-0.399571\pi\)
							0.993634	+	0.112652i	\(0.0359346\pi\)
\(37\)	−140.532	−	307.723i	−0.624415	−	1.36728i	−0.912264	−	0.409602i	\(-0.865668\pi\)
							0.287850	−	0.957676i	\(-0.407060\pi\)
\(41\)	24.1422	−	52.8641i	0.0919605	−	0.201365i	−0.858063	−	0.513545i	\(-0.828332\pi\)
							0.950023	+	0.312180i	\(0.101059\pi\)
\(43\)	198.966	+	127.868i	0.705630	+	0.453481i	0.843612	−	0.536954i	\(-0.180425\pi\)
							−0.137981	+	0.990435i	\(0.544061\pi\)
\(47\)	31.7939			0.0986728			0.0493364	−	0.998782i	\(-0.484289\pi\)
							0.0493364	+	0.998782i	\(0.484289\pi\)

(See \(a_n\) instead)

Twists

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

    

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