Embedded newform 37.10.f.a.7.9

• Label: 37.10.f.a.7.9
• Level: $37$
• Weight: $10$
• Character: 37.7
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $162$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(162\)
Relative dimension:	\(27\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.9
Character	\(\chi\)	\(=\)	37.7
Dual form			37.10.f.a.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-19.0907 + 6.94843i) q^{2} +(-243.959 - 88.7940i) q^{3} +(-76.0425 + 63.8072i) q^{4} +(-175.069 - 992.868i) q^{5} +5274.32 q^{6} +(314.940 + 1786.11i) q^{7} +(6209.20 - 10754.6i) q^{8} +(36553.8 + 30672.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−19.0907	+	6.94843i	−0.843696	+	0.307080i	−0.727467	−	0.686142i	\(-0.759303\pi\)
							−0.116228	+	0.993223i	\(0.537080\pi\)
\(3\)	−243.959	−	88.7940i	−1.73889	−	0.632904i	−0.739691	−	0.672946i	\(-0.765028\pi\)
							−0.999198	+	0.0400423i	\(0.987251\pi\)
\(5\)	−175.069	−	992.868i	−0.125269	−	0.710438i	−0.981147	−	0.193261i	\(-0.938094\pi\)
							0.855878	−	0.517178i	\(-0.173017\pi\)
\(7\)	314.940	+	1786.11i	0.0495778	+	0.281169i	0.999511	−	0.0312850i	\(-0.00995994\pi\)
							−0.949933	+	0.312454i	\(0.898849\pi\)
\(11\)	−6421.96	+	11123.2i	−0.132251	+	0.229066i	−0.924544	−	0.381075i	\(-0.875554\pi\)
							0.792293	+	0.610141i	\(0.208887\pi\)
\(13\)	109213.	−	91640.8i	1.06055	−	0.889906i	0.0663851	−	0.997794i	\(-0.478853\pi\)
							0.994163	+	0.107889i	\(0.0344090\pi\)
\(17\)	−208527.	−	174975.i	−0.605539	−	0.508108i	0.287682	−	0.957726i	\(-0.407115\pi\)
							−0.893221	+	0.449619i	\(0.851560\pi\)
\(19\)	240555.	+	87554.7i	0.423470	+	0.154130i	0.544960	−	0.838462i	\(-0.316545\pi\)
							−0.121490	+	0.992593i	\(0.538767\pi\)
\(23\)	−298856.	−	517635.i	−0.222683	−	0.385699i	0.732939	−	0.680295i	\(-0.238148\pi\)
							−0.955622	+	0.294596i	\(0.904815\pi\)
\(29\)	857750.	−	1.48567e6i	0.225201	−	0.390059i	−0.731179	−	0.682186i	\(-0.761030\pi\)
							0.956380	+	0.292127i	\(0.0943628\pi\)
\(31\)	−2.38453e6			−0.463740			−0.231870	−	0.972747i	\(-0.574484\pi\)
							−0.231870	+	0.972747i	\(0.574484\pi\)
\(37\)	8.09134e6	−	8.03069e6i	0.709762	−	0.704441i
							\(41\)	−2.49870e7	+	2.09666e7i	−1.38098	+	1.15878i	−0.412129	+	0.911125i	\(0.635215\pi\)
−0.968849	+	0.247653i	\(0.920341\pi\)
\(43\)	1.10063e7			0.490948			0.245474	−	0.969403i	\(-0.421056\pi\)
							0.245474	+	0.969403i	\(0.421056\pi\)
\(47\)	2.42364e7	+	4.19786e7i	0.724482	+	1.25484i	0.959187	+	0.282772i	\(0.0912541\pi\)
							−0.234706	+	0.972067i	\(0.575413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.f.a.7.9	✓	162
37.16	even	9	inner	37.10.f.a.16.9	yes	162

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.f.a.7.9	✓	162	1.1	even	1	trivial
37.10.f.a.16.9	yes	162	37.16	even	9	inner