Embedded newform 37.8.f.a.7.5

• Label: 37.8.f.a.7.5
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.3313 + 4.85219i) q^{2} +(11.3243 + 4.12171i) q^{3} +(56.1255 - 47.0949i) q^{4} +(4.98573 + 28.2755i) q^{5} -170.967 q^{6} +(7.15684 + 40.5884i) q^{7} +(388.247 - 672.464i) q^{8} +(-1564.09 - 1312.43i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.3313	+	4.85219i	−1.17833	+	0.428877i	−0.855612	−	0.517618i	\(-0.826819\pi\)
							−0.322718	+	0.946495i	\(0.604596\pi\)
\(3\)	11.3243	+	4.12171i	0.242151	+	0.0881359i	0.460245	−	0.887792i	\(-0.347762\pi\)
							−0.218094	+	0.975928i	\(0.569984\pi\)
\(5\)	4.98573	+	28.2755i	0.0178375	+	0.101161i	0.992427	−	0.122838i	\(-0.0391996\pi\)
							−0.974589	+	0.224000i	\(0.928089\pi\)
\(7\)	7.15684	+	40.5884i	0.00788639	+	0.0447259i	0.988497	−	0.151240i	\(-0.0483268\pi\)
							−0.980611	+	0.195966i	\(0.937216\pi\)
\(11\)	889.520	−	1540.69i	0.201503	−	0.349013i	−0.747510	−	0.664251i	\(-0.768751\pi\)
							0.949013	+	0.315237i	\(0.102084\pi\)
\(13\)	1422.78	−	1193.85i	0.179612	−	0.150713i	−0.548549	−	0.836118i	\(-0.684820\pi\)
							0.728161	+	0.685406i	\(0.240375\pi\)
\(17\)	21184.2	+	17775.6i	1.04578	+	0.877514i	0.992644	−	0.121073i	\(-0.0386335\pi\)
							0.0531371	+	0.998587i	\(0.483078\pi\)
\(19\)	24418.7	+	8887.67i	0.816741	+	0.297269i	0.716405	−	0.697684i	\(-0.245786\pi\)
							0.100336	+	0.994954i	\(0.468008\pi\)
\(23\)	33013.9	+	57181.8i	0.565783	+	0.979964i	0.996976	+	0.0777049i	\(0.0247592\pi\)
							−0.431194	+	0.902259i	\(0.641907\pi\)
\(29\)	−27825.8	+	48195.7i	−0.211863	+	0.366957i	−0.952298	−	0.305171i	\(-0.901286\pi\)
							0.740435	+	0.672128i	\(0.234620\pi\)
\(31\)	2217.57			0.0133694			0.00668469	−	0.999978i	\(-0.497872\pi\)
							0.00668469	+	0.999978i	\(0.497872\pi\)
\(37\)	154786.	−	266408.i	0.502372	−	0.864651i
							\(41\)	−5043.33	+	4231.86i	−0.0114281	+	0.00958931i	−0.648484	−	0.761228i	\(-0.724597\pi\)
0.637056	+	0.770818i	\(0.280152\pi\)
\(43\)	255512.			0.490086			0.245043	−	0.969512i	\(-0.421198\pi\)
							0.245043	+	0.969512i	\(0.421198\pi\)
\(47\)	395251.	+	684595.i	0.555303	+	0.961814i	0.997880	+	0.0650828i	\(0.0207312\pi\)
							−0.442577	+	0.896731i	\(0.645936\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.5	✓	132
37.16	even	9	inner	37.8.f.a.16.5	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.5	✓	132	1.1	even	1	trivial
37.8.f.a.16.5	yes	132	37.16	even	9	inner