Embedded newform 37.11.g.a.8.12

• Label: 37.11.g.a.8.12
• 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			8.12
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.8168 - 5.30990i) q^{2} +(278.697 + 160.906i) q^{3} +(-522.299 - 301.550i) q^{4} +(4955.27 - 1327.76i) q^{5} +(-4668.48 - 4668.48i) q^{6} +(8535.82 - 14784.5i) q^{7} +(23604.2 + 23604.2i) q^{8} +(22256.7 + 38549.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.8168	−	5.30990i	−0.619275	−	0.165934i	−0.0644770	−	0.997919i	\(-0.520538\pi\)
							−0.554798	+	0.831985i	\(0.687205\pi\)
\(3\)	278.697	+	160.906i	1.14690	+	0.662163i	0.948130	−	0.317884i	\(-0.102972\pi\)
							0.198769	+	0.980046i	\(0.436305\pi\)
\(5\)	4955.27	−	1327.76i	1.58569	−	0.424883i	0.645006	−	0.764178i	\(-0.276855\pi\)
							0.940680	+	0.339295i	\(0.110188\pi\)
\(7\)	8535.82	−	14784.5i	0.507873	−	0.879662i	−0.492085	−	0.870547i	\(-0.663765\pi\)
							0.999958	−	0.00911514i	\(-0.00290148\pi\)
\(11\)		−	299076.i		−	1.85703i	−0.371296	−	0.928515i	\(-0.621086\pi\)
							0.371296	−	0.928515i	\(-0.378914\pi\)
\(13\)	−397064.	+	106393.i	−1.06941	+	0.286547i	−0.750250	−	0.661154i	\(-0.770067\pi\)
							−0.319158	+	0.947701i	\(0.603400\pi\)
\(17\)	−264052.	+	985454.i	−0.185971	+	0.694051i	0.808450	+	0.588565i	\(0.200307\pi\)
							−0.994421	+	0.105487i	\(0.966360\pi\)
\(19\)	238290.	−	63849.5i	0.0962359	−	0.0257863i	−0.210380	−	0.977620i	\(-0.567470\pi\)
							0.306616	+	0.951833i	\(0.400803\pi\)
\(23\)	−4.19071e6	−	4.19071e6i	−0.651101	−	0.651101i	0.302157	−	0.953258i	\(-0.402293\pi\)
							−0.953258	+	0.302157i	\(0.902293\pi\)
\(29\)	−2.14158e7	+	2.14158e7i	−1.04410	+	1.04410i	−0.0451230	+	0.998981i	\(0.514368\pi\)
							−0.998981	+	0.0451230i	\(0.985632\pi\)
\(31\)	3.51816e7	−	3.51816e7i	1.22887	−	1.22887i	0.264482	−	0.964391i	\(-0.414799\pi\)
							0.964391	−	0.264482i	\(-0.0852009\pi\)
\(37\)	5.13058e7	−	4.66508e7i	0.739875	−	0.672745i
							\(41\)	9.17034e7	+	5.29450e7i	0.791528	+	0.456989i	0.840500	−	0.541811i	\(-0.182261\pi\)
−0.0489722	+	0.998800i	\(0.515595\pi\)
\(43\)	−1.04853e8	−	1.04853e8i	−0.713245	−	0.713245i	0.253968	−	0.967213i	\(-0.418264\pi\)
							−0.967213	+	0.253968i	\(0.918264\pi\)
\(47\)	9.35830e7			0.408045			0.204022	−	0.978966i	\(-0.434599\pi\)
							0.204022	+	0.978966i	\(0.434599\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.12	✓	120
37.14	odd	12	inner	37.11.g.a.14.12	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.12	✓	120	1.1	even	1	trivial
37.11.g.a.14.12	yes	120	37.14	odd	12	inner