Embedded newform 138.3.h.a.7.4

• Label: 138.3.h.a.7.4
• Level: $138$
• Weight: $3$
• Character: 138.7
• Analytic conductor: $3.760$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76022764817\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	138.7
Dual form			138.3.h.a.79.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.926113 + 1.06879i) q^{2} +(1.45709 - 0.936417i) q^{3} +(-0.284630 - 1.97964i) q^{4} +(0.0820350 - 0.0374641i) q^{5} +(-0.348599 + 2.42456i) q^{6} +(2.03512 + 6.93100i) q^{7} +(2.37942 + 1.52916i) q^{8} +(1.24625 - 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.926113	+	1.06879i	−0.463056	+	0.534396i
							\(3\)	1.45709	−	0.936417i	0.485698	−	0.312139i
\(5\)	0.0820350	−	0.0374641i	0.0164070	−	0.00749282i								−0.407195	−	0.913341i	\(-0.633493\pi\)
							0.423602	+	0.905849i	\(0.360766\pi\)
\(7\)	2.03512	+	6.93100i	0.290732	+	0.990142i	0.967273	+	0.253737i	\(0.0816599\pi\)
							−0.676541	+	0.736405i	\(0.736522\pi\)
\(11\)	9.00264	−	7.80083i	0.818422	−	0.709167i	−0.141346	−	0.989960i	\(-0.545143\pi\)
							0.959768	+	0.280794i	\(0.0905977\pi\)
\(13\)	23.4551	+	6.88703i	1.80424	+	0.529772i	0.998080	−	0.0619306i	\(-0.0197257\pi\)
							0.806157	+	0.591702i	\(0.201544\pi\)
\(17\)	−13.7847	−	1.98195i	−0.810867	−	0.116585i	−0.275605	−	0.961271i	\(-0.588878\pi\)
							−0.535262	+	0.844686i	\(0.679787\pi\)
\(19\)	22.9589	−	3.30099i	1.20836	−	0.173736i	0.491455	−	0.870903i	\(-0.336465\pi\)
							0.716909	+	0.697167i	\(0.245556\pi\)
\(23\)	−6.36933	+	22.1005i	−0.276928	+	0.960891i
							\(29\)	2.59769	−	18.0673i	0.0895754	−	0.623011i	−0.894739	−	0.446589i	\(-0.852639\pi\)
0.984315	−	0.176422i	\(-0.0564522\pi\)
\(31\)	−20.8636	−	13.4082i	−0.673019	−	0.432523i	0.158994	−	0.987280i	\(-0.449175\pi\)
							−0.832013	+	0.554756i	\(0.812811\pi\)
\(37\)	19.5296	+	8.91886i	0.527826	+	0.241050i	0.661460	−	0.749981i	\(-0.269937\pi\)
							−0.133633	+	0.991031i	\(0.542664\pi\)
\(41\)	−17.3320	−	37.9517i	−0.422731	−	0.925652i	−0.994451	−	0.105203i	\(-0.966451\pi\)
							0.571720	−	0.820449i	\(-0.306276\pi\)
\(43\)	−23.4649	−	36.5122i	−0.545696	−	0.849120i	0.453414	−	0.891300i	\(-0.350206\pi\)
							−0.999110	+	0.0421803i	\(0.986570\pi\)
\(47\)	−75.9543			−1.61605			−0.808025	−	0.589148i	\(-0.799463\pi\)
							−0.808025	+	0.589148i	\(0.799463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.7.4	✓	80
3.2	odd	2		414.3.l.b.145.7		80
23.10	odd	22	inner	138.3.h.a.79.4	yes	80
69.56	even	22		414.3.l.b.217.7		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.7.4	✓	80	1.1	even	1	trivial
138.3.h.a.79.4	yes	80	23.10	odd	22	inner
414.3.l.b.145.7		80	3.2	odd	2
414.3.l.b.217.7		80	69.56	even	22