Embedded newform 138.3.h.a.37.3

• Label: 138.3.h.a.37.3
• Level: $138$
• Weight: $3$
• Character: 138.37
• 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			37.3
Character	\(\chi\)	\(=\)	138.37
Dual form			138.3.h.a.97.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35693 + 0.398430i) q^{2} +(1.13425 + 1.30900i) q^{3} +(1.68251 - 1.08128i) q^{4} +(-3.60161 + 0.517833i) q^{5} +(-2.06064 - 1.32429i) q^{6} +(-10.7332 - 4.90168i) q^{7} +(-1.85223 + 2.13758i) q^{8} +(-0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35693	+	0.398430i	−0.678464	+	0.199215i
							\(3\)	1.13425	+	1.30900i	0.378084	+	0.436332i
\(5\)	−3.60161	+	0.517833i	−0.720321	+	0.103567i								−0.492720	−	0.870188i	\(-0.663997\pi\)
							−0.227601	+	0.973754i	\(0.573088\pi\)
\(7\)	−10.7332	−	4.90168i	−1.53331	−	0.700239i	−0.543080	−	0.839681i	\(-0.682742\pi\)
							−0.990230	+	0.139442i	\(0.955469\pi\)
\(11\)	−1.94118	+	6.61105i	−0.176471	+	0.601004i	0.822986	+	0.568061i	\(0.192306\pi\)
							−0.999457	+	0.0329432i	\(0.989512\pi\)
\(13\)	−6.19818	−	13.5721i	−0.476783	−	1.04401i	−0.983336	−	0.181800i	\(-0.941808\pi\)
							0.506553	−	0.862209i	\(-0.330920\pi\)
\(17\)	−4.14877	+	6.45562i	−0.244046	+	0.379742i	−0.941569	−	0.336819i	\(-0.890649\pi\)
							0.697524	+	0.716562i	\(0.254285\pi\)
\(19\)	−4.76797	−	7.41911i	−0.250946	−	0.390479i	0.692811	−	0.721119i	\(-0.256372\pi\)
							−0.943757	+	0.330640i	\(0.892736\pi\)
\(23\)	−22.9605	+	1.34801i	−0.998281	+	0.0586092i
							\(29\)	−5.87374	−	3.77482i	−0.202543	−	0.130166i	0.435438	−	0.900219i	\(-0.356594\pi\)
−0.637980	+	0.770053i	\(0.720230\pi\)
\(31\)	12.5670	−	14.5031i	0.405386	−	0.467841i	−0.515944	−	0.856623i	\(-0.672559\pi\)
							0.921330	+	0.388782i	\(0.127104\pi\)
\(37\)	62.6120	+	9.00224i	1.69222	+	0.243304i	0.919961	−	0.392009i	\(-0.128220\pi\)
							0.772255	+	0.635313i	\(0.219129\pi\)
\(41\)	4.26387	+	29.6558i	0.103997	+	0.723313i	0.973384	+	0.229181i	\(0.0736049\pi\)
							−0.869387	+	0.494132i	\(0.835486\pi\)
\(43\)	−54.0605	+	46.8437i	−1.25722	+	1.08939i	−0.265085	+	0.964225i	\(0.585400\pi\)
							−0.992136	+	0.125163i	\(0.960055\pi\)
\(47\)	31.4133			0.668368			0.334184	−	0.942508i	\(-0.391539\pi\)
							0.334184	+	0.942508i	\(0.391539\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.37.3	✓	80
3.2	odd	2		414.3.l.b.37.7		80
23.5	odd	22	inner	138.3.h.a.97.3	yes	80
69.5	even	22		414.3.l.b.235.7		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.37.3	✓	80	1.1	even	1	trivial
138.3.h.a.97.3	yes	80	23.5	odd	22	inner
414.3.l.b.37.7		80	3.2	odd	2
414.3.l.b.235.7		80	69.5	even	22