Embedded newform 37.11.g.a.8.20

• Label: 37.11.g.a.8.20
• Level: $37$
• Weight: $11$
• Character: 37.8
• Analytic conductor: $23.508$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5082183489\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.20
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25.9310 + 6.94820i) q^{2} +(385.296 + 222.451i) q^{3} +(-262.669 - 151.652i) q^{4} +(3458.18 - 926.615i) q^{5} +(8445.50 + 8445.50i) q^{6} +(-8871.91 + 15366.6i) q^{7} +(-25196.0 - 25196.0i) q^{8} +(69444.4 + 120281. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25.9310	+	6.94820i	0.810345	+	0.217131i	0.640121	−	0.768274i	\(-0.278884\pi\)
							0.170224	+	0.985405i	\(0.445551\pi\)
\(3\)	385.296	+	222.451i	1.58558	+	0.915436i	0.994023	+	0.109174i	\(0.0348207\pi\)
							0.591559	+	0.806262i	\(0.298513\pi\)
\(5\)	3458.18	−	926.615i	1.10662	−	0.296517i	0.341160	−	0.940005i	\(-0.389180\pi\)
							0.765456	+	0.643488i	\(0.222513\pi\)
\(7\)	−8871.91	+	15366.6i	−0.527870	+	0.914297i	0.471602	+	0.881811i	\(0.343676\pi\)
							−0.999472	+	0.0324859i	\(0.989658\pi\)
\(11\)			138779.i			0.861710i	0.902421	+	0.430855i	\(0.141788\pi\)
							−0.902421	+	0.430855i	\(0.858212\pi\)
\(13\)	318911.	−	85451.8i	0.858919	−	0.230147i	0.197629	−	0.980277i	\(-0.436676\pi\)
							0.661290	+	0.750130i	\(0.270009\pi\)
\(17\)	−374306.	+	1.39693e6i	−0.263622	+	0.983851i	0.699466	+	0.714666i	\(0.253421\pi\)
							−0.963088	+	0.269185i	\(0.913246\pi\)
\(19\)	4.23614e6	−	1.13507e6i	1.71081	−	0.458411i	0.735190	−	0.677861i	\(-0.237093\pi\)
							0.975624	+	0.219450i	\(0.0704264\pi\)
\(23\)	−8.22255e6	−	8.22255e6i	−1.27752	−	1.27752i	−0.942052	−	0.335466i	\(-0.891106\pi\)
							−0.335466	−	0.942052i	\(-0.608894\pi\)
\(29\)	311283.	−	311283.i	0.0151763	−	0.0151763i	−0.699478	−	0.714654i	\(-0.746584\pi\)
							0.714654	+	0.699478i	\(0.246584\pi\)
\(31\)	2.81273e7	−	2.81273e7i	0.982470	−	0.982470i	−0.0173793	−	0.999849i	\(-0.505532\pi\)
							0.999849	+	0.0173793i	\(0.00553228\pi\)
\(37\)	−6.93208e7	+	1.79212e6i	−0.999666	+	0.0258439i
							\(41\)	2.97181e7	+	1.71577e7i	0.256508	+	0.148095i	0.622741	−	0.782428i	\(-0.286019\pi\)
−0.366232	+	0.930523i	\(0.619353\pi\)
\(43\)	−1.41935e6	−	1.41935e6i	−0.00965488	−	0.00965488i	0.702263	−	0.711918i	\(-0.252173\pi\)
							−0.711918	+	0.702263i	\(0.752173\pi\)
\(47\)	−6.10140e6			−0.0266036			−0.0133018	−	0.999912i	\(-0.504234\pi\)
							−0.0133018	+	0.999912i	\(0.504234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.20	✓	120
37.14	odd	12	inner	37.11.g.a.14.20	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.20	✓	120	1.1	even	1	trivial
37.11.g.a.14.20	yes	120	37.14	odd	12	inner