Embedded newform 26.3.f.b.11.2

• Label: 26.3.f.b.11.2
• Level: $26$
• Weight: $3$
• Character: 26.11
• 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			11.2
Root			\(-3.90972 + 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.11
Dual form			26.3.f.b.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(2.38787 + 4.13592i) q^{3} +(-1.73205 - 1.00000i) q^{4} +(-5.88981 - 5.88981i) q^{5} +(6.52379 - 1.74804i) q^{6} +(0.0922225 + 0.344179i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-6.90386 + 11.9578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 6 q^{5} + 6 q^{6} - 2 q^{7} - 16 q^{8} - 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.366025	−	1.36603i	0.183013	−	0.683013i
							\(3\)	2.38787	+	4.13592i	0.795957	+	1.37864i	0.922230	+	0.386643i	\(0.126365\pi\)
−0.126272	+	0.991996i	\(0.540301\pi\)
\(5\)	−5.88981	−	5.88981i	−1.17796	−	1.17796i	−0.980263	−	0.197700i	\(-0.936653\pi\)
							−0.197700	−	0.980263i	\(-0.563347\pi\)
\(7\)	0.0922225	+	0.344179i	0.0131746	+	0.0491684i	0.972200	−	0.234151i	\(-0.0752310\pi\)
							−0.959026	+	0.283319i	\(0.908564\pi\)
\(11\)	0.715507	+	0.191720i	0.0650461	+	0.0174290i	0.291195	−	0.956664i	\(-0.405947\pi\)
							−0.226149	+	0.974093i	\(0.572614\pi\)
\(13\)	11.4820	+	6.09614i	0.883233	+	0.468933i
							\(17\)	−2.49470	−	1.44031i	−0.146747	−	0.0847244i	0.424829	−	0.905274i	\(-0.360334\pi\)
−0.571576	+	0.820549i	\(0.693668\pi\)
\(19\)	−10.4951	+	2.81216i	−0.552375	+	0.148009i	−0.524201	−	0.851594i	\(-0.675636\pi\)
							−0.0281743	+	0.999603i	\(0.508969\pi\)
\(23\)	19.8278	−	11.4476i	0.862076	−	0.497720i	−0.00263083	−	0.999997i	\(-0.500837\pi\)
							0.864707	+	0.502277i	\(0.167504\pi\)
\(29\)	−15.2092	−	26.3432i	−0.524457	−	0.908386i	−0.999595	−	0.0284745i	\(-0.990935\pi\)
							0.475138	−	0.879911i	\(-0.342398\pi\)
\(31\)	14.8631	+	14.8631i	0.479456	+	0.479456i	0.904958	−	0.425502i	\(-0.139902\pi\)
							−0.425502	+	0.904958i	\(0.639902\pi\)
\(37\)	−40.2792	−	10.7928i	−1.08863	−	0.291697i	−0.330502	−	0.943805i	\(-0.607218\pi\)
							−0.758126	+	0.652108i	\(0.773885\pi\)
\(41\)	−6.66907	+	24.8893i	−0.162660	+	0.607056i	0.835667	+	0.549237i	\(0.185081\pi\)
							−0.998327	+	0.0578194i	\(0.981585\pi\)
\(43\)	−25.3907	−	14.6593i	−0.590480	−	0.340914i	0.174807	−	0.984603i	\(-0.444070\pi\)
							−0.765287	+	0.643689i	\(0.777403\pi\)
\(47\)	21.2000	−	21.2000i	0.451065	−	0.451065i	−0.444643	−	0.895708i	\(-0.646670\pi\)
							0.895708	+	0.444643i	\(0.146670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.3.f.b.11.2	✓	8
3.2	odd	2		234.3.bb.f.37.2		8
4.3	odd	2		208.3.bd.f.193.1		8
13.2	odd	12		338.3.d.g.239.1		8
13.3	even	3		338.3.d.g.99.1		8
13.4	even	6		338.3.f.j.319.2		8
13.5	odd	4		338.3.f.h.249.2		8
13.6	odd	12	inner	26.3.f.b.19.2	yes	8
13.7	odd	12		338.3.f.i.19.2		8
13.8	odd	4		338.3.f.j.249.2		8
13.9	even	3		338.3.f.h.319.2		8
13.10	even	6		338.3.d.f.99.1		8
13.11	odd	12		338.3.d.f.239.1		8
13.12	even	2		338.3.f.i.89.2		8
39.32	even	12		234.3.bb.f.19.2		8
52.19	even	12		208.3.bd.f.97.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.b.11.2	✓	8	1.1	even	1	trivial
26.3.f.b.19.2	yes	8	13.6	odd	12	inner
208.3.bd.f.97.1		8	52.19	even	12
208.3.bd.f.193.1		8	4.3	odd	2
234.3.bb.f.19.2		8	39.32	even	12
234.3.bb.f.37.2		8	3.2	odd	2
338.3.d.f.99.1		8	13.10	even	6
338.3.d.f.239.1		8	13.11	odd	12
338.3.d.g.99.1		8	13.3	even	3
338.3.d.g.239.1		8	13.2	odd	12
338.3.f.h.249.2		8	13.5	odd	4
338.3.f.h.319.2		8	13.9	even	3
338.3.f.i.19.2		8	13.7	odd	12
338.3.f.i.89.2		8	13.12	even	2
338.3.f.j.249.2		8	13.8	odd	4
338.3.f.j.319.2		8	13.4	even	6