Embedded newform 26.3.f.b.19.2

• Label: 26.3.f.b.19.2
• Level: $26$
• Weight: $3$
• Character: 26.19
• 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			19.2
Root			\(-3.90972 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.19
Dual form			26.3.f.b.11.2

$q$-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 \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.b.11.2	✓	8	13.11	odd	12	inner
26.3.f.b.19.2	yes	8	1.1	even	1	trivial
208.3.bd.f.97.1		8	4.3	odd	2
208.3.bd.f.193.1		8	52.11	even	12
234.3.bb.f.19.2		8	3.2	odd	2
234.3.bb.f.37.2		8	39.11	even	12
338.3.d.f.99.1		8	13.6	odd	12
338.3.d.f.239.1		8	13.4	even	6
338.3.d.g.99.1		8	13.7	odd	12
338.3.d.g.239.1		8	13.9	even	3
338.3.f.h.249.2		8	13.3	even	3
338.3.f.h.319.2		8	13.8	odd	4
338.3.f.i.19.2		8	13.12	even	2
338.3.f.i.89.2		8	13.2	odd	12
338.3.f.j.249.2		8	13.10	even	6
338.3.f.j.319.2		8	13.5	odd	4