Embedded newform 37.11.g.a.8.6

• Label: 37.11.g.a.8.6
• 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			8.6
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-41.0206 - 10.9914i) q^{2} +(-196.274 - 113.319i) q^{3} +(675.070 + 389.752i) q^{4} +(3858.05 - 1033.76i) q^{5} +(6805.74 + 6805.74i) q^{6} +(-8257.62 + 14302.6i) q^{7} +(7342.02 + 7342.02i) q^{8} +(-3842.19 - 6654.87i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−41.0206	−	10.9914i	−1.28189	−	0.343483i	−0.447318	−	0.894375i	\(-0.647621\pi\)
							−0.834576	+	0.550892i	\(0.814288\pi\)
\(3\)	−196.274	−	113.319i	−0.807712	−	0.466333i	0.0384489	−	0.999261i	\(-0.487758\pi\)
							−0.846161	+	0.532928i	\(0.821092\pi\)
\(5\)	3858.05	−	1033.76i	1.23458	−	0.330804i	0.418217	−	0.908347i	\(-0.362655\pi\)
							0.816360	+	0.577543i	\(0.195989\pi\)
\(7\)	−8257.62	+	14302.6i	−0.491320	+	0.850992i	−0.999950	−	0.00999348i	\(-0.996819\pi\)
							0.508630	+	0.860985i	\(0.330152\pi\)
\(11\)		−	233864.i		−	1.45211i	−0.687637	−	0.726054i	\(-0.741352\pi\)
							0.687637	−	0.726054i	\(-0.258648\pi\)
\(13\)	−143415.	+	38428.0i	−0.386259	+	0.103498i	−0.446722	−	0.894673i	\(-0.647409\pi\)
							0.0604634	+	0.998170i	\(0.480742\pi\)
\(17\)	22666.3	−	84591.9i	0.0159638	−	0.0595778i	−0.957484	−	0.288485i	\(-0.906849\pi\)
							0.973448	+	0.228907i	\(0.0735152\pi\)
\(19\)	−3.04778e6	+	816649.i	−1.23088	+	0.329813i	−0.814921	−	0.579572i	\(-0.803220\pi\)
							−0.415957	+	0.909384i	\(0.636553\pi\)
\(23\)	−587587.	−	587587.i	−0.0912921	−	0.0912921i	0.659986	−	0.751278i	\(-0.270562\pi\)
							−0.751278	+	0.659986i	\(0.770562\pi\)
\(29\)	2.03098e7	−	2.03098e7i	0.990185	−	0.990185i	−0.00976736	−	0.999952i	\(-0.503109\pi\)
							0.999952	+	0.00976736i	\(0.00310910\pi\)
\(31\)	−1.06096e7	+	1.06096e7i	−0.370587	+	0.370587i	−0.867691	−	0.497104i	\(-0.834397\pi\)
							0.497104	+	0.867691i	\(0.334397\pi\)
\(37\)	−6.93136e7	−	2.05070e6i	−0.999563	−	0.0295729i
							\(41\)	4.31410e7	+	2.49074e7i	0.372366	+	0.214986i	0.674492	−	0.738282i	\(-0.264363\pi\)
−0.302125	+	0.953268i	\(0.597696\pi\)
\(43\)	9.77718e7	+	9.77718e7i	0.665076	+	0.665076i	0.956572	−	0.291496i	\(-0.0941530\pi\)
							−0.291496	+	0.956572i	\(0.594153\pi\)
\(47\)	1.39310e8			0.607424			0.303712	−	0.952764i	\(-0.401774\pi\)
							0.303712	+	0.952764i	\(0.401774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.6	✓	120
37.14	odd	12	inner	37.11.g.a.14.6	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.6	✓	120	1.1	even	1	trivial
37.11.g.a.14.6	yes	120	37.14	odd	12	inner