Embedded newform 138.4.e.d.73.2

• Label: 138.4.e.d.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.d.121.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.830830 - 1.81926i) q^{2} +(2.87848 - 0.845198i) q^{3} +(-2.61944 + 3.02300i) q^{4} +(0.809240 + 0.520067i) q^{5} +(-3.92916 - 4.53450i) q^{6} +(-0.360364 - 2.50638i) q^{7} +(7.67594 + 2.25386i) q^{8} +(7.57128 - 4.86577i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

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

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
\(5\)	0.809240	+	0.520067i	0.0723806	+	0.0465162i								0.576331	−	0.817217i	\(-0.304484\pi\)
							−0.503950	+	0.863733i	\(0.668120\pi\)
\(7\)	−0.360364	−	2.50638i	−0.0194578	−	0.135332i	0.977777	−	0.209648i	\(-0.0672318\pi\)
							−0.997235	+	0.0743159i	\(0.976323\pi\)
\(11\)	23.1613	−	50.7162i	0.634854	−	1.39014i	−0.269352	−	0.963042i	\(-0.586810\pi\)
							0.904207	−	0.427095i	\(-0.140463\pi\)
\(13\)	4.95799	−	34.4836i	0.105777	−	0.735694i	−0.866043	−	0.499970i	\(-0.833345\pi\)
							0.971820	−	0.235725i	\(-0.0757464\pi\)
\(17\)	−3.05789	−	3.52899i	−0.0436263	−	0.0503474i	0.733517	−	0.679671i	\(-0.237877\pi\)
							−0.777143	+	0.629324i	\(0.783332\pi\)
\(19\)	14.5431	−	16.7837i	0.175601	−	0.202654i	−0.661125	−	0.750275i	\(-0.729921\pi\)
							0.836727	+	0.547621i	\(0.184466\pi\)
\(23\)	110.297	+	1.22001i	0.999939	+	0.0110604i
							\(29\)	6.48895	+	7.48865i	0.0415506	+	0.0479520i	0.776144	−	0.630555i	\(-0.217173\pi\)
−0.734594	+	0.678507i	\(0.762627\pi\)
\(31\)	−241.730	−	70.9783i	−1.40051	−	0.411228i	−0.507654	−	0.861561i	\(-0.669487\pi\)
							−0.892861	+	0.450333i	\(0.851305\pi\)
\(37\)	−91.9473	+	59.0909i	−0.408542	+	0.262554i	−0.728731	−	0.684800i	\(-0.759890\pi\)
							0.320190	+	0.947353i	\(0.396253\pi\)
\(41\)	89.4180	+	57.4654i	0.340603	+	0.218892i	0.699749	−	0.714389i	\(-0.253295\pi\)
							−0.359145	+	0.933282i	\(0.616932\pi\)
\(43\)	−11.5742	+	3.39848i	−0.0410476	+	0.0120526i	−0.302192	−	0.953247i	\(-0.597718\pi\)
							0.261144	+	0.965300i	\(0.415900\pi\)
\(47\)	459.716			1.42673			0.713366	−	0.700791i	\(-0.247170\pi\)
							0.713366	+	0.700791i	\(0.247170\pi\)

(See \(a_n\) instead)

Twists

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

    

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