Embedded newform 37.4.e.a.11.4

• Label: 37.4.e.a.11.4
• 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			11.4
Root			\(0.575927i\) of defining polynomial
Character	\(\chi\)	\(=\)	37.11
Dual form			37.4.e.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.498768 + 0.287964i) q^{2} +(-1.51417 + 2.62262i) q^{3} +(-3.83415 + 6.64095i) q^{4} +(-6.67348 - 3.85293i) q^{5} -1.74410i q^{6} +(-10.3028 + 17.8450i) q^{7} -9.02381i q^{8} +(8.91459 + 15.4405i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.498768	+	0.287964i	−0.176341	+	0.101811i	−0.585572	−	0.810620i	\(-0.699130\pi\)
							0.409231	+	0.912431i	\(0.365797\pi\)
\(3\)	−1.51417	+	2.62262i	−0.291402	+	0.504723i	−0.974141	−	0.225939i	\(-0.927455\pi\)
							0.682740	+	0.730662i	\(0.260788\pi\)
\(5\)	−6.67348	−	3.85293i	−0.596894	−	0.344617i	0.170925	−	0.985284i	\(-0.445324\pi\)
							−0.767819	+	0.640667i	\(0.778658\pi\)
\(7\)	−10.3028	+	17.8450i	−0.556298	+	0.963537i	0.441503	+	0.897260i	\(0.354445\pi\)
							−0.997801	+	0.0662770i	\(0.978888\pi\)
\(11\)	20.9419			0.574021			0.287010	−	0.957927i	\(-0.407339\pi\)
							0.287010	+	0.957927i	\(0.407339\pi\)
\(13\)	−4.26314	−	2.46133i	−0.0909525	−	0.0525114i	0.453834	−	0.891086i	\(-0.350056\pi\)
							−0.544786	+	0.838575i	\(0.683389\pi\)
\(17\)	42.2040	−	24.3665i	0.602116	−	0.347632i	−0.167758	−	0.985828i	\(-0.553653\pi\)
							0.769873	+	0.638197i	\(0.220319\pi\)
\(19\)	62.6916	+	36.1950i	0.756971	+	0.437037i	0.828207	−	0.560422i	\(-0.189361\pi\)
							−0.0712365	+	0.997459i	\(0.522694\pi\)
\(23\)			149.439i			1.35479i	0.735617	+	0.677397i	\(0.236892\pi\)
							−0.735617	+	0.677397i	\(0.763108\pi\)
\(29\)			132.515i			0.848532i	0.905538	+	0.424266i	\(0.139468\pi\)
							−0.905538	+	0.424266i	\(0.860532\pi\)
\(31\)			133.601i			0.774049i	0.922069	+	0.387024i	\(0.126497\pi\)
							−0.922069	+	0.387024i	\(0.873503\pi\)
\(37\)	−138.484	+	177.412i	−0.615315	+	0.788282i
							\(41\)	54.2927	−	94.0377i	0.206807	−	0.358201i	−0.743900	−	0.668291i	\(-0.767026\pi\)
0.950707	+	0.310091i	\(0.100359\pi\)
\(43\)		−	331.680i		−	1.17630i	−0.808754	−	0.588148i	\(-0.799857\pi\)
							0.808754	−	0.588148i	\(-0.200143\pi\)
\(47\)	491.766			1.52620			0.763101	−	0.646280i	\(-0.223676\pi\)
							0.763101	+	0.646280i	\(0.223676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.e.a.11.4	✓	16
3.2	odd	2		333.4.s.c.307.5		16
37.8	odd	12		1369.4.a.g.1.7		16
37.27	even	6	inner	37.4.e.a.27.4	yes	16
37.29	odd	12		1369.4.a.g.1.10		16
111.101	odd	6		333.4.s.c.64.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.e.a.11.4	✓	16	1.1	even	1	trivial
37.4.e.a.27.4	yes	16	37.27	even	6	inner
333.4.s.c.64.5		16	111.101	odd	6
333.4.s.c.307.5		16	3.2	odd	2
1369.4.a.g.1.7		16	37.8	odd	12
1369.4.a.g.1.10		16	37.29	odd	12