Embedded newform 26.4.c.b.9.2

• Label: 26.4.c.b.9.2
• Level: $26$
• Weight: $4$
• Character: 26.9
• Analytic conductor: $1.534$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	26.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.53404966015\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{217})\)
Defining polynomial:	\( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.2
Root			\(3.93273 - 6.81169i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.9
Dual form			26.4.c.b.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(4.43273 - 7.67771i) q^{3} +(-2.00000 - 3.46410i) q^{4} +3.86546 q^{5} +(8.86546 + 15.3554i) q^{6} +(7.56727 + 13.1069i) q^{7} +8.00000 q^{8} +(-25.7982 - 44.6838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 3 q^{3} - 8 q^{4} - 14 q^{5} + 6 q^{6} + 45 q^{7} + 32 q^{8} - 59 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	+	1.73205i	−0.353553	+	0.612372i
							\(3\)	4.43273	−	7.67771i	0.853079	−	1.47758i	−0.0253362	−	0.999679i	\(-0.508066\pi\)
0.878416	−	0.477898i	\(-0.158601\pi\)
\(5\)	3.86546			0.345737			0.172869	−	0.984945i	\(-0.444696\pi\)
							0.172869	+	0.984945i	\(0.444696\pi\)
\(7\)	7.56727	+	13.1069i	0.408594	+	0.707706i	0.994732	−	0.102505i	\(-0.0326858\pi\)
							−0.586138	+	0.810211i	\(0.699352\pi\)
\(11\)	−27.0291	+	46.8158i	−0.740871	+	1.28323i	0.211228	+	0.977437i	\(0.432254\pi\)
							−0.952099	+	0.305790i	\(0.901080\pi\)
\(13\)	−19.7982	−	42.4857i	−0.422387	−	0.906416i
							\(17\)	61.9618	+	107.321i	0.883997	+	1.53113i	0.846859	+	0.531818i	\(0.178491\pi\)
0.0371386	+	0.999310i	\(0.488176\pi\)
\(19\)	−4.43273	−	7.67771i	−0.0535231	−	0.0927046i	0.838023	−	0.545636i	\(-0.183712\pi\)
							−0.891546	+	0.452931i	\(0.850378\pi\)
\(23\)	−19.2982	+	33.4254i	−0.174954	+	0.303030i	−0.940145	−	0.340773i	\(-0.889311\pi\)
							0.765191	+	0.643803i	\(0.222644\pi\)
\(29\)	93.6928	−	162.281i	0.599942	−	1.03913i	−0.392887	−	0.919587i	\(-0.628524\pi\)
							0.992829	−	0.119543i	\(-0.0381429\pi\)
\(31\)	−36.7710			−0.213041			−0.106521	−	0.994311i	\(-0.533971\pi\)
							−0.106521	+	0.994311i	\(0.533971\pi\)
\(37\)	160.769	−	278.460i	0.714332	−	1.23726i	−0.248885	−	0.968533i	\(-0.580064\pi\)
							0.963217	−	0.268726i	\(-0.0866025\pi\)
\(41\)	13.1546	−	22.7844i	0.0501074	−	0.0867886i	−0.839884	−	0.542766i	\(-0.817377\pi\)
							0.889991	+	0.455978i	\(0.150710\pi\)
\(43\)	−68.8182	−	119.197i	−0.244062	−	0.422729i	0.717805	−	0.696244i	\(-0.245147\pi\)
							−0.961868	+	0.273515i	\(0.911814\pi\)
\(47\)	−300.466			−0.932499			−0.466249	−	0.884653i	\(-0.654395\pi\)
							−0.466249	+	0.884653i	\(0.654395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.4.c.b.9.2	yes	4
3.2	odd	2		234.4.h.h.217.1		4
4.3	odd	2		208.4.i.d.113.1		4
13.2	odd	12		338.4.e.f.23.2		8
13.3	even	3	inner	26.4.c.b.3.2	✓	4
13.4	even	6		338.4.a.g.1.1		2
13.5	odd	4		338.4.e.f.147.4		8
13.6	odd	12		338.4.b.e.337.1		4
13.7	odd	12		338.4.b.e.337.3		4
13.8	odd	4		338.4.e.f.147.2		8
13.9	even	3		338.4.a.h.1.1		2
13.10	even	6		338.4.c.j.315.2		4
13.11	odd	12		338.4.e.f.23.4		8
13.12	even	2		338.4.c.j.191.2		4
39.29	odd	6		234.4.h.h.55.1		4
52.3	odd	6		208.4.i.d.81.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.c.b.3.2	✓	4	13.3	even	3	inner
26.4.c.b.9.2	yes	4	1.1	even	1	trivial
208.4.i.d.81.1		4	52.3	odd	6
208.4.i.d.113.1		4	4.3	odd	2
234.4.h.h.55.1		4	39.29	odd	6
234.4.h.h.217.1		4	3.2	odd	2
338.4.a.g.1.1		2	13.4	even	6
338.4.a.h.1.1		2	13.9	even	3
338.4.b.e.337.1		4	13.6	odd	12
338.4.b.e.337.3		4	13.7	odd	12
338.4.c.j.191.2		4	13.12	even	2
338.4.c.j.315.2		4	13.10	even	6
338.4.e.f.23.2		8	13.2	odd	12
338.4.e.f.23.4		8	13.11	odd	12
338.4.e.f.147.2		8	13.8	odd	4
338.4.e.f.147.4		8	13.5	odd	4