Embedded newform 37.8.e.a.11.5

• Label: 37.8.e.a.11.5
• Level: $37$
• Weight: $8$
• Character: 37.11
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	37.11
Dual form			37.8.e.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.0545 + 5.80498i) q^{2} +(31.9798 - 55.3907i) q^{3} +(3.39554 - 5.88124i) q^{4} +(-274.161 - 158.287i) q^{5} +742.569i q^{6} +(-610.528 + 1057.47i) q^{7} -1407.23i q^{8} +(-951.919 - 1648.77i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 18 q^{2} - 42 q^{3} + 1114 q^{4} - 672 q^{5} + 156 q^{7} - 9128 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.0545	+	5.80498i	−0.888702	+	0.513092i	−0.873518	−	0.486792i	\(-0.838167\pi\)
							−0.0151844	+	0.999885i	\(0.504834\pi\)
\(3\)	31.9798	−	55.3907i	0.683836	−	1.18444i	−0.289966	−	0.957037i	\(-0.593644\pi\)
							0.973801	−	0.227401i	\(-0.0730228\pi\)
\(5\)	−274.161	−	158.287i	−0.980866	−	0.566304i	−0.0783348	−	0.996927i	\(-0.524960\pi\)
							−0.902532	+	0.430624i	\(0.858294\pi\)
\(7\)	−610.528	+	1057.47i	−0.672764	+	1.16526i	0.304354	+	0.952559i	\(0.401560\pi\)
							−0.977117	+	0.212702i	\(0.931774\pi\)
\(11\)	5065.81			1.14756			0.573779	−	0.819010i	\(-0.305477\pi\)
							0.573779	+	0.819010i	\(0.305477\pi\)
\(13\)	3988.05	+	2302.50i	0.503453	+	0.290669i	0.730138	−	0.683299i	\(-0.239456\pi\)
							−0.226686	+	0.973968i	\(0.572789\pi\)
\(17\)	−18535.9	+	10701.7i	−0.915047	+	0.528302i	−0.882051	−	0.471153i	\(-0.843838\pi\)
							−0.0329952	+	0.999456i	\(0.510505\pi\)
\(19\)	37226.0	+	21492.5i	1.24511	+	0.718867i	0.970131	−	0.242582i	\(-0.0779945\pi\)
							0.274983	+	0.961449i	\(0.411328\pi\)
\(23\)			79205.8i			1.35740i	0.734413	+	0.678702i	\(0.237457\pi\)
							−0.734413	+	0.678702i	\(0.762543\pi\)
\(29\)		−	92340.5i		−	0.703071i	−0.936175	−	0.351536i	\(-0.885660\pi\)
							0.936175	−	0.351536i	\(-0.114340\pi\)
\(31\)			234435.i			1.41338i	0.707526	+	0.706688i	\(0.249811\pi\)
							−0.707526	+	0.706688i	\(0.750189\pi\)
\(37\)	−307647.	−	16887.3i	−0.998497	−	0.0548091i
							\(41\)	190248.	−	329519.i	0.431098	−	0.746683i	−0.565870	−	0.824494i	\(-0.691460\pi\)
0.996968	+	0.0778111i	\(0.0247931\pi\)
\(43\)			597859.i			1.14673i	0.819302	+	0.573363i	\(0.194361\pi\)
							−0.819302	+	0.573363i	\(0.805639\pi\)
\(47\)	−227053.			−0.318995			−0.159498	−	0.987198i	\(-0.550987\pi\)
							−0.159498	+	0.987198i	\(0.550987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.e.a.11.5	✓	40
37.27	even	6	inner	37.8.e.a.27.5	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.e.a.11.5	✓	40	1.1	even	1	trivial
37.8.e.a.27.5	yes	40	37.27	even	6	inner