Embedded newform 37.4.e.a.11.1

• Label: 37.4.e.a.11.1
• Level: $37$
• Weight: $4$
• Character: 37.11
• Analytic conductor: $2.183$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.18307067021\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 82*x^14 + 2679*x^12 + 44392*x^10 + 392767*x^8 + 1779258*x^6 + 3438825*x^4 + 1208748*x^2 + 82944
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.1
Root			\(4.65090i\) of defining polynomial
Character	\(\chi\)	\(=\)	37.11
Dual form			37.4.e.a.27.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.02780 + 2.32545i) q^{2} +(1.78194 - 3.08642i) q^{3} +(6.81543 - 11.8047i) q^{4} +(11.2971 + 6.52240i) q^{5} +16.5753i q^{6} +(-10.9405 + 18.9495i) q^{7} +26.1886i q^{8} +(7.14934 + 12.3830i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.02780	+	2.32545i	−1.42404	+	0.822171i	−0.996641	−	0.0818919i	\(-0.973904\pi\)
							−0.427400	+	0.904063i	\(0.640570\pi\)
\(3\)	1.78194	−	3.08642i	0.342935	−	0.593982i	−0.642041	−	0.766670i	\(-0.721912\pi\)
							0.984976	+	0.172689i	\(0.0552454\pi\)
\(5\)	11.2971	+	6.52240i	1.01045	+	0.583381i	0.911322	−	0.411694i	\(-0.135063\pi\)
							0.0991237	+	0.995075i	\(0.468396\pi\)
\(7\)	−10.9405	+	18.9495i	−0.590733	+	1.02318i	0.403401	+	0.915023i	\(0.367828\pi\)
							−0.994134	+	0.108156i	\(0.965505\pi\)
\(11\)	64.2890			1.76217			0.881085	−	0.472957i	\(-0.156814\pi\)
							0.881085	+	0.472957i	\(0.156814\pi\)
\(13\)	4.00733	+	2.31363i	0.0854948	+	0.0493605i	0.542138	−	0.840290i	\(-0.317615\pi\)
							−0.456643	+	0.889650i	\(0.650948\pi\)
\(17\)	−17.3220	+	10.0009i	−0.247130	+	0.142681i	−0.618449	−	0.785825i	\(-0.712239\pi\)
							0.371319	+	0.928505i	\(0.378906\pi\)
\(19\)	−78.5690	−	45.3618i	−0.948682	−	0.547722i	−0.0560106	−	0.998430i	\(-0.517838\pi\)
							−0.892671	+	0.450708i	\(0.851171\pi\)
\(23\)		−	98.6344i		−	0.894204i	−0.894483	−	0.447102i	\(-0.852456\pi\)
							0.894483	−	0.447102i	\(-0.147544\pi\)
\(29\)			176.614i			1.13091i	0.824778	+	0.565456i	\(0.191300\pi\)
							−0.824778	+	0.565456i	\(0.808700\pi\)
\(31\)		−	131.680i		−	0.762919i	−0.924386	−	0.381460i	\(-0.875422\pi\)
							0.924386	−	0.381460i	\(-0.124578\pi\)
\(37\)	−220.324	−	45.9387i	−0.978947	−	0.204115i
							\(41\)	171.976	−	297.870i	0.655075	−	1.13462i	−0.326800	−	0.945093i	\(-0.605970\pi\)
0.981875	−	0.189529i	\(-0.0606962\pi\)
\(43\)		−	18.1916i		−	0.0645161i	−0.999480	−	0.0322581i	\(-0.989730\pi\)
							0.999480	−	0.0322581i	\(-0.0102698\pi\)
\(47\)	−598.471			−1.85736			−0.928681	−	0.370881i	\(-0.879056\pi\)
							−0.928681	+	0.370881i	\(0.879056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.e.a.11.1	✓	16
3.2	odd	2		333.4.s.c.307.8		16
37.8	odd	12		1369.4.a.g.1.1		16
37.27	even	6	inner	37.4.e.a.27.1	yes	16
37.29	odd	12		1369.4.a.g.1.16		16
111.101	odd	6		333.4.s.c.64.8		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.e.a.11.1	✓	16	1.1	even	1	trivial
37.4.e.a.27.1	yes	16	37.27	even	6	inner
333.4.s.c.64.8		16	111.101	odd	6
333.4.s.c.307.8		16	3.2	odd	2
1369.4.a.g.1.1		16	37.8	odd	12
1369.4.a.g.1.16		16	37.29	odd	12