Embedded newform 138.5.h.a.7.6

• Label: 138.5.h.a.7.6
• Level: $138$
• Weight: $5$
• Character: 138.7
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.6
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85223 + 2.13758i) q^{2} +(4.37128 - 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(2.65171 - 1.21100i) q^{5} +(-2.09159 + 14.5473i) q^{6} +(-17.2943 - 58.8990i) q^{7} +(19.0354 + 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85223	+	2.13758i	−0.463056	+	0.534396i
							\(3\)	4.37128	−	2.80925i	0.485698	−	0.312139i
\(5\)	2.65171	−	1.21100i	0.106068	−	0.0484398i								−0.361674	−	0.932305i	\(-0.617795\pi\)
							0.467743	+	0.883865i	\(0.345067\pi\)
\(7\)	−17.2943	−	58.8990i	−0.352945	−	1.20202i	−0.924412	−	0.381395i	\(-0.875444\pi\)
							0.571467	−	0.820625i	\(-0.306374\pi\)
\(11\)	−136.402	+	118.193i	−1.12729	+	0.976805i	−0.999887	−	0.0150462i	\(-0.995210\pi\)
							−0.127406	+	0.991851i	\(0.540665\pi\)
\(13\)	−37.4431	−	10.9943i	−0.221557	−	0.0650549i	0.169070	−	0.985604i	\(-0.445924\pi\)
							−0.390627	+	0.920549i	\(0.627742\pi\)
\(17\)	−187.624	−	26.9763i	−0.649218	−	0.0933435i	−0.190165	−	0.981752i	\(-0.560902\pi\)
							−0.459053	+	0.888409i	\(0.651811\pi\)
\(19\)	−47.4827	+	6.82698i	−0.131531	+	0.0189113i	−0.207766	−	0.978179i	\(-0.566619\pi\)
							0.0762349	+	0.997090i	\(0.475710\pi\)
\(23\)	−186.932	+	494.871i	−0.353369	+	0.935484i
							\(29\)	81.1901	−	564.690i	0.0965400	−	0.671451i	−0.882877	−	0.469604i	\(-0.844397\pi\)
0.979417	−	0.201847i	\(-0.0646942\pi\)
\(31\)	−194.800	−	125.191i	−0.202706	−	0.130271i	0.435350	−	0.900261i	\(-0.356625\pi\)
							−0.638056	+	0.769990i	\(0.720261\pi\)
\(37\)	−2042.61	−	932.830i	−1.49205	−	0.681395i	−0.508337	−	0.861158i	\(-0.669740\pi\)
							−0.983710	+	0.179763i	\(0.942467\pi\)
\(41\)	−1312.35	−	2873.65i	−0.780698	−	1.70949i	−0.701541	−	0.712629i	\(-0.747504\pi\)
							−0.0791571	−	0.996862i	\(-0.525223\pi\)
\(43\)	−1882.44	−	2929.13i	−1.01809	−	1.58417i	−0.792429	−	0.609964i	\(-0.791184\pi\)
							−0.225656	−	0.974207i	\(-0.572453\pi\)
\(47\)	2998.58			1.35744			0.678719	−	0.734398i	\(-0.262535\pi\)
							0.678719	+	0.734398i	\(0.262535\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.6	✓	160
23.10	odd	22	inner	138.5.h.a.79.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.6	✓	160	1.1	even	1	trivial
138.5.h.a.79.6	yes	160	23.10	odd	22	inner