Embedded newform 26.3.f.b.7.1

• Label: 26.3.f.b.7.1
• Level: $26$
• Weight: $3$
• Character: 26.7
• Analytic conductor: $0.708$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	26.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.708448687337\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	8.0.612074651904.1
Defining polynomial:	\( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			7.1
Root			\(4.71318 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.7
Dual form			26.3.f.b.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(-2.78960 + 4.83174i) q^{3} +(1.73205 - 1.00000i) q^{4} +(0.323893 + 0.323893i) q^{5} +(2.04213 - 7.62134i) q^{6} +(7.67890 + 2.05755i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-11.0638 - 19.1630i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{5} + 6 q^{6} - 2 q^{7} - 16 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36603	+	0.366025i	−0.683013	+	0.183013i
							\(3\)	−2.78960	+	4.83174i	−0.929868	+	1.61058i	−0.146330	+	0.989236i	\(0.546746\pi\)
−0.783539	+	0.621343i	\(0.786587\pi\)
\(5\)	0.323893	+	0.323893i	0.0647786	+	0.0647786i	0.738754	−	0.673975i	\(-0.235415\pi\)
							−0.673975	+	0.738754i	\(0.735415\pi\)
\(7\)	7.67890	+	2.05755i	1.09699	+	0.293936i	0.761536	−	0.648122i	\(-0.224445\pi\)
							0.335449	+	0.942058i	\(0.391112\pi\)
\(11\)	1.44868	+	5.40655i	0.131698	+	0.491504i	0.999990	−	0.00455003i	\(-0.00144833\pi\)
							−0.868291	+	0.496054i	\(0.834782\pi\)
\(13\)	12.8550	+	1.93621i	0.988846	+	0.148939i
							\(17\)	−1.74285	+	1.00623i	−0.102520	+	0.0591902i	−0.550384	−	0.834912i	\(-0.685519\pi\)
0.447863	+	0.894102i	\(0.352185\pi\)
\(19\)	1.19911	−	4.47512i	0.0631108	−	0.235533i	−0.927164	−	0.374655i	\(-0.877761\pi\)
							0.990275	+	0.139122i	\(0.0444279\pi\)
\(23\)	−15.0976	−	8.71663i	−0.656419	−	0.378984i	0.134492	−	0.990915i	\(-0.457060\pi\)
							−0.790911	+	0.611931i	\(0.790393\pi\)
\(29\)	8.25026	−	14.2899i	0.284492	−	0.492754i	−0.687994	−	0.725716i	\(-0.741508\pi\)
							0.972486	+	0.232962i	\(0.0748418\pi\)
\(31\)	−9.27353	−	9.27353i	−0.299146	−	0.299146i	0.541533	−	0.840679i	\(-0.317844\pi\)
							−0.840679	+	0.541533i	\(0.817844\pi\)
\(37\)	9.12370	+	34.0501i	0.246586	+	0.920273i	0.972579	+	0.232571i	\(0.0747138\pi\)
							−0.725993	+	0.687702i	\(0.758620\pi\)
\(41\)	58.6217	−	15.7076i	1.42980	−	0.383113i	0.540849	−	0.841120i	\(-0.318103\pi\)
							0.888949	+	0.458007i	\(0.151436\pi\)
\(43\)	−45.2101	+	26.1020i	−1.05140	+	0.607024i	−0.923040	−	0.384704i	\(-0.874304\pi\)
							−0.128357	+	0.991728i	\(0.540970\pi\)
\(47\)	8.73527	−	8.73527i	0.185857	−	0.185857i	−0.608045	−	0.793902i	\(-0.708046\pi\)
							0.793902	+	0.608045i	\(0.208046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.3.f.b.7.1	✓	8
3.2	odd	2		234.3.bb.f.163.2		8
4.3	odd	2		208.3.bd.f.33.2		8
13.2	odd	12	inner	26.3.f.b.15.1	yes	8
13.3	even	3		338.3.f.h.89.1		8
13.4	even	6		338.3.d.f.99.4		8
13.5	odd	4		338.3.f.h.19.1		8
13.6	odd	12		338.3.d.g.239.4		8
13.7	odd	12		338.3.d.f.239.4		8
13.8	odd	4		338.3.f.j.19.1		8
13.9	even	3		338.3.d.g.99.4		8
13.10	even	6		338.3.f.j.89.1		8
13.11	odd	12		338.3.f.i.249.1		8
13.12	even	2		338.3.f.i.319.1		8
39.2	even	12		234.3.bb.f.145.2		8
52.15	even	12		208.3.bd.f.145.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.b.7.1	✓	8	1.1	even	1	trivial
26.3.f.b.15.1	yes	8	13.2	odd	12	inner
208.3.bd.f.33.2		8	4.3	odd	2
208.3.bd.f.145.2		8	52.15	even	12
234.3.bb.f.145.2		8	39.2	even	12
234.3.bb.f.163.2		8	3.2	odd	2
338.3.d.f.99.4		8	13.4	even	6
338.3.d.f.239.4		8	13.7	odd	12
338.3.d.g.99.4		8	13.9	even	3
338.3.d.g.239.4		8	13.6	odd	12
338.3.f.h.19.1		8	13.5	odd	4
338.3.f.h.89.1		8	13.3	even	3
338.3.f.i.249.1		8	13.11	odd	12
338.3.f.i.319.1		8	13.12	even	2
338.3.f.j.19.1		8	13.8	odd	4
338.3.f.j.89.1		8	13.10	even	6