Embedded newform 138.3.h.a.37.1

• Label: 138.3.h.a.37.1
• 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			37.1
Character	\(\chi\)	\(=\)	138.37
Dual form			138.3.h.a.97.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35693 + 0.398430i) q^{2} +(-1.13425 - 1.30900i) q^{3} +(1.68251 - 1.08128i) q^{4} +(-3.69869 + 0.531791i) q^{5} +(2.06064 + 1.32429i) q^{6} +(1.55981 + 0.712339i) q^{7} +(-1.85223 + 2.13758i) q^{8} +(-0.426945 + 2.96946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 16 q^{4} - 24 q^{9}+O(q^{10}) \)

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
\(5\)	−3.69869	+	0.531791i	−0.739738	+	0.106358i								−0.501874	−	0.864940i	\(-0.667356\pi\)
							−0.237863	+	0.971299i	\(0.576447\pi\)
\(7\)	1.55981	+	0.712339i	0.222829	+	0.101763i	0.523702	−	0.851902i	\(-0.324551\pi\)
							−0.300873	+	0.953664i	\(0.597278\pi\)
\(11\)	−2.18728	+	7.44919i	−0.198844	+	0.677199i	0.798340	+	0.602206i	\(0.205712\pi\)
							−0.997184	+	0.0749929i	\(0.976107\pi\)
\(13\)	5.92513	+	12.9742i	0.455779	+	0.998017i	0.988429	+	0.151682i	\(0.0484689\pi\)
							−0.532650	+	0.846335i	\(0.678804\pi\)
\(17\)	−10.0045	+	15.5673i	−0.588499	+	0.915722i	0.411492	+	0.911413i	\(0.365008\pi\)
							−0.999991	+	0.00430852i	\(0.998629\pi\)
\(19\)	6.25448	+	9.73216i	0.329183	+	0.512219i	0.965913	−	0.258868i	\(-0.0833496\pi\)
							−0.636730	+	0.771087i	\(0.719713\pi\)
\(23\)	−5.02484	+	22.4444i	−0.218471	+	0.975843i
							\(29\)	−6.64977	−	4.27355i	−0.229303	−	0.147364i	0.420945	−	0.907086i	\(-0.361698\pi\)
−0.650248	+	0.759722i	\(0.725335\pi\)
\(31\)	−2.18826	+	2.52538i	−0.0705889	+	0.0814639i	−0.789946	−	0.613177i	\(-0.789891\pi\)
							0.719357	+	0.694641i	\(0.244437\pi\)
\(37\)	18.1732	+	2.61291i	0.491168	+	0.0706192i	0.383449	−	0.923562i	\(-0.374736\pi\)
							0.107719	+	0.994181i	\(0.465645\pi\)
\(41\)	−5.38361	−	37.4438i	−0.131308	−	0.913264i	−0.943853	−	0.330366i	\(-0.892828\pi\)
							0.812545	−	0.582898i	\(-0.198081\pi\)
\(43\)	27.9934	−	24.2564i	0.651010	−	0.564103i	−0.265502	−	0.964110i	\(-0.585538\pi\)
							0.916512	+	0.400007i	\(0.130992\pi\)
\(47\)	26.3563			0.560772			0.280386	−	0.959887i	\(-0.409537\pi\)
							0.280386	+	0.959887i	\(0.409537\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.h.a.37.1	✓	80
3.2	odd	2		414.3.l.b.37.8		80
23.5	odd	22	inner	138.3.h.a.97.1	yes	80
69.5	even	22		414.3.l.b.235.8		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.h.a.37.1	✓	80	1.1	even	1	trivial
138.3.h.a.97.1	yes	80	23.5	odd	22	inner
414.3.l.b.37.8		80	3.2	odd	2
414.3.l.b.235.8		80	69.5	even	22