Embedded newform 138.3.h.a.7.8

• Label: 138.3.h.a.7.8
• 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			7.8
Character	\(\chi\)	\(=\)	138.7
Dual form			138.3.h.a.79.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.926113 - 1.06879i) q^{2} +(1.45709 - 0.936417i) q^{3} +(-0.284630 - 1.97964i) q^{4} +(4.69832 - 2.14565i) q^{5} +(0.348599 - 2.42456i) q^{6} +(1.48393 + 5.05379i) q^{7} +(-2.37942 - 1.52916i) q^{8} +(1.24625 - 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	4.69832	−	2.14565i	0.939663	−	0.429130i								0.114119	−	0.993467i	\(-0.463595\pi\)
							0.825544	+	0.564337i	\(0.190868\pi\)
\(7\)	1.48393	+	5.05379i	0.211990	+	0.721971i	0.994992	+	0.0999522i	\(0.0318690\pi\)
							−0.783003	+	0.622019i	\(0.786313\pi\)
\(11\)	1.33809	−	1.15946i	0.121644	−	0.105406i	−0.591914	−	0.806001i	\(-0.701628\pi\)
							0.713559	+	0.700595i	\(0.247082\pi\)
\(13\)	−11.1767	−	3.28179i	−0.859749	−	0.252445i	−0.177999	−	0.984031i	\(-0.556962\pi\)
							−0.681750	+	0.731586i	\(0.738781\pi\)
\(17\)	−9.71450	−	1.39673i	−0.571441	−	0.0821608i	−0.149464	−	0.988767i	\(-0.547755\pi\)
							−0.421977	+	0.906606i	\(0.638664\pi\)
\(19\)	9.43233	−	1.35616i	0.496438	−	0.0713771i	0.110452	−	0.993881i	\(-0.464770\pi\)
							0.385987	+	0.922504i	\(0.373861\pi\)
\(23\)	20.9101	+	9.57950i	0.909136	+	0.416500i
							\(29\)	−6.28278	+	43.6977i	−0.216648	+	1.50682i	0.533644	+	0.845709i	\(0.320822\pi\)
−0.750291	+	0.661107i	\(0.770087\pi\)
\(31\)	−22.8567	−	14.6891i	−0.737312	−	0.473842i	0.117308	−	0.993096i	\(-0.462574\pi\)
							−0.854620	+	0.519254i	\(0.826210\pi\)
\(37\)	−28.2773	−	12.9138i	−0.764251	−	0.349022i	−0.00514309	−	0.999987i	\(-0.501637\pi\)
							−0.759108	+	0.650965i	\(0.774364\pi\)
\(41\)	28.6907	+	62.8238i	0.699772	+	1.53229i	0.840247	+	0.542203i	\(0.182410\pi\)
							−0.140475	+	0.990084i	\(0.544863\pi\)
\(43\)	7.93792	+	12.3517i	0.184603	+	0.287248i	0.921204	−	0.389080i	\(-0.127207\pi\)
							−0.736601	+	0.676327i	\(0.763571\pi\)
\(47\)	12.7189			0.270616			0.135308	−	0.990804i	\(-0.456798\pi\)
							0.135308	+	0.990804i	\(0.456798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.7.8	✓	80
3.2	odd	2		414.3.l.b.145.1		80
23.10	odd	22	inner	138.3.h.a.79.8	yes	80
69.56	even	22		414.3.l.b.217.1		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.7.8	✓	80	1.1	even	1	trivial
138.3.h.a.79.8	yes	80	23.10	odd	22	inner
414.3.l.b.145.1		80	3.2	odd	2
414.3.l.b.217.1		80	69.56	even	22