Embedded newform 37.10.c.a.10.5

• Label: 37.10.c.a.10.5
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.5
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.2809 + 28.1993i) q^{2} +(-30.6895 - 53.1557i) q^{3} +(-274.135 - 474.815i) q^{4} +(1243.37 + 2153.58i) q^{5} +1998.61 q^{6} +(-3568.14 - 6180.20i) q^{7} +1181.00 q^{8} +(7957.81 - 13783.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.2809	+	28.1993i	−0.719520	+	1.24625i	0.241669	+	0.970359i	\(0.422305\pi\)
							−0.961190	+	0.275887i	\(0.911028\pi\)
\(3\)	−30.6895	−	53.1557i	−0.218748	−	0.378882i	0.735678	−	0.677332i	\(-0.236864\pi\)
							−0.954425	+	0.298450i	\(0.903531\pi\)
\(5\)	1243.37	+	2153.58i	0.889682	+	1.54097i	0.840251	+	0.542197i	\(0.182407\pi\)
							0.0494305	+	0.998778i	\(0.484259\pi\)
\(7\)	−3568.14	−	6180.20i	−0.561695	−	0.972885i	−0.997349	−	0.0727702i	\(-0.976816\pi\)
							0.435654	−	0.900114i	\(-0.356517\pi\)
\(11\)	53246.8			1.09654			0.548272	−	0.836300i	\(-0.315286\pi\)
							0.548272	+	0.836300i	\(0.315286\pi\)
\(13\)	−62938.4	−	109012.i	−0.611182	−	1.05860i	−0.991042	−	0.133554i	\(-0.957361\pi\)
							0.379860	−	0.925044i	\(-0.375972\pi\)
\(17\)	184372.	−	319341.i	0.535395	−	0.927331i	−0.463749	−	0.885967i	\(-0.653496\pi\)
							0.999144	−	0.0413647i	\(-0.0131706\pi\)
\(19\)	224252.	+	388416.i	0.394771	+	0.683763i	0.993072	−	0.117508i	\(-0.0374907\pi\)
							−0.598301	+	0.801271i	\(0.704157\pi\)
\(23\)	−777294.			−0.579175			−0.289587	−	0.957152i	\(-0.593518\pi\)
							−0.289587	+	0.957152i	\(0.593518\pi\)
\(29\)	5.50671e6			1.44578			0.722889	−	0.690964i	\(-0.242814\pi\)
							0.722889	+	0.690964i	\(0.242814\pi\)
\(31\)	5.16497e6			1.00448			0.502239	−	0.864729i	\(-0.332510\pi\)
							0.502239	+	0.864729i	\(0.332510\pi\)
\(37\)	−1.04057e7	−	4.65641e6i	−0.912779	−	0.408454i
							\(41\)	8.87492e6	+	1.53718e7i	0.490498	+	0.849567i	0.999940	−	0.0109376i	\(-0.00348161\pi\)
−0.509442	+	0.860505i	\(0.670148\pi\)
\(43\)	2.69907e7			1.20394			0.601972	−	0.798518i	\(-0.294382\pi\)
							0.601972	+	0.798518i	\(0.294382\pi\)
\(47\)	4.14065e6			0.123774			0.0618869	−	0.998083i	\(-0.480288\pi\)
							0.0618869	+	0.998083i	\(0.480288\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.5	✓	54
37.26	even	3	inner	37.10.c.a.26.5	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.5	✓	54	1.1	even	1	trivial
37.10.c.a.26.5	yes	54	37.26	even	3	inner