Embedded newform 138.5.h.a.7.14

• Label: 138.5.h.a.7.14
• 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			7.14
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.85223 - 2.13758i) q^{2} +(4.37128 - 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(-22.2415 + 10.1573i) q^{5} +(2.09159 - 14.5473i) q^{6} +(-7.02266 - 23.9170i) q^{7} +(-19.0354 - 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

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
\(5\)	−22.2415	+	10.1573i	−0.889659	+	0.406294i								−0.807184	−	0.590300i	\(-0.799010\pi\)
							−0.0824746	+	0.996593i	\(0.526282\pi\)
\(7\)	−7.02266	−	23.9170i	−0.143320	−	0.488102i	0.856277	−	0.516518i	\(-0.172772\pi\)
							−0.999596	+	0.0284157i	\(0.990954\pi\)
\(11\)	16.4323	−	14.2386i	0.135804	−	0.117675i	−0.584310	−	0.811530i	\(-0.698635\pi\)
							0.720114	+	0.693856i	\(0.244089\pi\)
\(13\)	−245.106	−	71.9697i	−1.45033	−	0.425856i	−0.540681	−	0.841227i	\(-0.681834\pi\)
							−0.909652	+	0.415371i	\(0.863652\pi\)
\(17\)	−220.000	−	31.6313i	−0.761247	−	0.109451i	−0.249251	−	0.968439i	\(-0.580184\pi\)
							−0.511996	+	0.858988i	\(0.671094\pi\)
\(19\)	−403.822	+	58.0608i	−1.11862	+	0.160833i	−0.676727	−	0.736234i	\(-0.736602\pi\)
							−0.441893	+	0.897068i	\(0.645693\pi\)
\(23\)	448.185	−	281.018i	0.847231	−	0.531224i
							\(29\)	4.81256	−	33.4721i	0.00572243	−	0.0398004i	−0.986760	−	0.162186i	\(-0.948145\pi\)
0.992483	+	0.122386i	\(0.0390545\pi\)
\(31\)	449.224	+	288.699i	0.467455	+	0.300415i	0.753083	−	0.657926i	\(-0.228566\pi\)
							−0.285628	+	0.958340i	\(0.592202\pi\)
\(37\)	417.527	+	190.678i	0.304987	+	0.139283i	0.562029	−	0.827117i	\(-0.310021\pi\)
							−0.257043	+	0.966400i	\(0.582748\pi\)
\(41\)	−1231.64	−	2696.91i	−0.732682	−	1.60435i	−0.795230	−	0.606308i	\(-0.792650\pi\)
							0.0625482	−	0.998042i	\(-0.480077\pi\)
\(43\)	407.162	+	633.556i	0.220206	+	0.342648i	0.933727	−	0.357987i	\(-0.116537\pi\)
							−0.713520	+	0.700635i	\(0.752900\pi\)
\(47\)	−389.249			−0.176210			−0.0881052	−	0.996111i	\(-0.528081\pi\)
							−0.0881052	+	0.996111i	\(0.528081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.14	✓	160
23.10	odd	22	inner	138.5.h.a.79.14	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.14	✓	160	1.1	even	1	trivial
138.5.h.a.79.14	yes	160	23.10	odd	22	inner