Embedded newform 138.3.h.a.7.6

• Label: 138.3.h.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	138.7
Dual form			138.3.h.a.79.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.926113 - 1.06879i) q^{2} +(-1.45709 + 0.936417i) q^{3} +(-0.284630 - 1.97964i) q^{4} +(2.51088 - 1.14668i) q^{5} +(-0.348599 + 2.42456i) q^{6} +(-2.32390 - 7.91446i) q^{7} +(-2.37942 - 1.52916i) q^{8} +(1.24625 - 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	2.51088	−	1.14668i	0.502176	−	0.229336i								−0.148188	−	0.988959i	\(-0.547344\pi\)
							0.650363	+	0.759623i	\(0.274617\pi\)
\(7\)	−2.32390	−	7.91446i	−0.331985	−	1.13064i	−0.941261	−	0.337679i	\(-0.890358\pi\)
							0.609276	−	0.792958i	\(-0.291460\pi\)
\(11\)	12.0633	−	10.4529i	1.09666	−	0.950261i	0.0976703	−	0.995219i	\(-0.468861\pi\)
							0.998989	+	0.0449582i	\(0.0143155\pi\)
\(13\)	3.87002	+	1.13634i	0.297694	+	0.0874108i	0.427169	−	0.904172i	\(-0.359511\pi\)
							−0.129475	+	0.991583i	\(0.541329\pi\)
\(17\)	−18.1734	−	2.61294i	−1.06902	−	0.153702i	−0.414731	−	0.909944i	\(-0.636124\pi\)
							−0.654292	+	0.756242i	\(0.727033\pi\)
\(19\)	22.4781	−	3.23187i	1.18306	−	0.170098i	0.477424	−	0.878673i	\(-0.341570\pi\)
							0.705636	+	0.708575i	\(0.250661\pi\)
\(23\)	−16.1982	−	16.3285i	−0.704268	−	0.709934i
							\(29\)	−6.63215	+	46.1276i	−0.228695	+	1.59061i	0.474922	+	0.880028i	\(0.342476\pi\)
−0.703617	+	0.710580i	\(0.748433\pi\)
\(31\)	39.6015	+	25.4503i	1.27747	+	0.820978i	0.990573	−	0.136984i	\(-0.0437409\pi\)
							0.286894	+	0.957962i	\(0.407377\pi\)
\(37\)	3.60206	+	1.64501i	0.0973531	+	0.0444597i	0.463496	−	0.886099i	\(-0.346595\pi\)
							−0.366143	+	0.930558i	\(0.619322\pi\)
\(41\)	−9.84979	−	21.5680i	−0.240239	−	0.526050i	0.750655	−	0.660694i	\(-0.229738\pi\)
							−0.990894	+	0.134645i	\(0.957011\pi\)
\(43\)	41.9505	+	65.2762i	0.975593	+	1.51805i	0.850546	+	0.525901i	\(0.176272\pi\)
							0.125047	+	0.992151i	\(0.460092\pi\)
\(47\)	34.0179			0.723785			0.361893	−	0.932220i	\(-0.382131\pi\)
							0.361893	+	0.932220i	\(0.382131\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.7.6	✓	80
3.2	odd	2		414.3.l.b.145.2		80
23.10	odd	22	inner	138.3.h.a.79.6	yes	80
69.56	even	22		414.3.l.b.217.2		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.7.6	✓	80	1.1	even	1	trivial
138.3.h.a.79.6	yes	80	23.10	odd	22	inner
414.3.l.b.145.2		80	3.2	odd	2
414.3.l.b.217.2		80	69.56	even	22