Embedded newform 37.4.e.a.11.6

• Label: 37.4.e.a.11.6
• 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			11.6
Root			\(-2.31004i\) of defining polynomial
Character	\(\chi\)	\(=\)	37.11
Dual form			37.4.e.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00056 - 1.15502i) q^{2} +(3.80485 - 6.59020i) q^{3} +(-1.33185 + 2.30684i) q^{4} +(-1.68208 - 0.971152i) q^{5} -17.5787i q^{6} +(-1.95249 + 3.38181i) q^{7} +24.6336i q^{8} +(-15.4538 - 26.7667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00056	−	1.15502i	0.707303	−	0.408362i	−0.102759	−	0.994706i	\(-0.532767\pi\)
							0.810062	+	0.586345i	\(0.199434\pi\)
\(3\)	3.80485	−	6.59020i	0.732244	−	1.26828i	−0.223678	−	0.974663i	\(-0.571806\pi\)
							0.955922	−	0.293621i	\(-0.0948602\pi\)
\(5\)	−1.68208	−	0.971152i	−0.150450	−	0.0868624i	0.422885	−	0.906183i	\(-0.361017\pi\)
							−0.573335	+	0.819321i	\(0.694351\pi\)
\(7\)	−1.95249	+	3.38181i	−0.105424	+	0.182600i	−0.913912	−	0.405913i	\(-0.866953\pi\)
							0.808487	+	0.588514i	\(0.200287\pi\)
\(11\)	30.7619			0.843189			0.421594	−	0.906785i	\(-0.361471\pi\)
							0.421594	+	0.906785i	\(0.361471\pi\)
\(13\)	20.1419	+	11.6289i	0.429719	+	0.248098i	0.699227	−	0.714900i	\(-0.253528\pi\)
							−0.269508	+	0.962998i	\(0.586861\pi\)
\(17\)	−73.7899	+	42.6026i	−1.05275	+	0.607803i	−0.923417	−	0.383799i	\(-0.874616\pi\)
							−0.129329	+	0.991602i	\(0.541282\pi\)
\(19\)	−69.7392	−	40.2640i	−0.842067	−	0.486168i	0.0158993	−	0.999874i	\(-0.494939\pi\)
							−0.857966	+	0.513706i	\(0.828272\pi\)
\(23\)			133.760i			1.21265i	0.795217	+	0.606325i	\(0.207357\pi\)
							−0.795217	+	0.606325i	\(0.792643\pi\)
\(29\)		−	287.139i		−	1.83864i	−0.393515	−	0.919318i	\(-0.628741\pi\)
							0.393515	−	0.919318i	\(-0.371259\pi\)
\(31\)			102.082i			0.591432i	0.955276	+	0.295716i	\(0.0955582\pi\)
							−0.955276	+	0.295716i	\(0.904442\pi\)
\(37\)	167.205	−	150.650i	0.742929	−	0.669370i
							\(41\)	−91.5077	+	158.496i	−0.348563	+	0.603729i	−0.985994	−	0.166778i	\(-0.946664\pi\)
0.637431	+	0.770507i	\(0.279997\pi\)
\(43\)		−	349.272i		−	1.23868i	−0.785121	−	0.619342i	\(-0.787399\pi\)
							0.785121	−	0.619342i	\(-0.212601\pi\)
\(47\)	−201.328			−0.624825			−0.312412	−	0.949947i	\(-0.601137\pi\)
							−0.312412	+	0.949947i	\(0.601137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.e.a.11.6	✓	16
3.2	odd	2		333.4.s.c.307.3		16
37.8	odd	12		1369.4.a.g.1.11		16
37.27	even	6	inner	37.4.e.a.27.6	yes	16
37.29	odd	12		1369.4.a.g.1.6		16
111.101	odd	6		333.4.s.c.64.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.e.a.11.6	✓	16	1.1	even	1	trivial
37.4.e.a.27.6	yes	16	37.27	even	6	inner
333.4.s.c.64.3		16	111.101	odd	6
333.4.s.c.307.3		16	3.2	odd	2
1369.4.a.g.1.6		16	37.29	odd	12
1369.4.a.g.1.11		16	37.8	odd	12