Embedded newform 13.12.c.a.9.5

• Label: 13.12.c.a.9.5
• Level: $13$
• Weight: $12$
• Character: 13.9
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.98846134727\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.5
Character	\(\chi\)	\(=\)	13.9
Dual form			13.12.c.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.8467 + 22.2512i) q^{2} +(-179.645 + 311.153i) q^{3} +(693.923 + 1201.91i) q^{4} +11234.1 q^{5} +(-4615.69 - 7994.60i) q^{6} +(14221.5 + 24632.4i) q^{7} -88278.7 q^{8} +(24029.2 + 41619.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.8467	+	22.2512i	−0.283875	+	0.491686i	−0.972336	−	0.233588i	\(-0.924953\pi\)
							0.688461	+	0.725274i	\(0.258287\pi\)
\(3\)	−179.645	+	311.153i	−0.426822	+	0.739278i	−0.996589	−	0.0825288i	\(-0.973700\pi\)
							0.569766	+	0.821807i	\(0.307034\pi\)
\(5\)	11234.1			1.60770			0.803848	−	0.594835i	\(-0.202783\pi\)
							0.803848	+	0.594835i	\(0.202783\pi\)
\(7\)	14221.5	+	24632.4i	0.319821	+	0.553946i	0.980451	−	0.196766i	\(-0.0630439\pi\)
							−0.660629	+	0.750712i	\(0.729711\pi\)
\(11\)	120580.	−	208851.i	0.225744	−	0.391001i	−0.730798	−	0.682594i	\(-0.760852\pi\)
							0.956543	+	0.291593i	\(0.0941852\pi\)
\(13\)	−1.06865e6	+	806322.i	−0.798262	+	0.602310i
							\(17\)	−1.72928e6	−	2.99521e6i	−0.295391	−	0.511632i	0.679685	−	0.733504i	\(-0.262117\pi\)
−0.975076	+	0.221872i	\(0.928783\pi\)
\(19\)	−390714.	−	676736.i	−0.0362004	−	0.0627010i	0.847358	−	0.531023i	\(-0.178192\pi\)
							−0.883558	+	0.468322i	\(0.844859\pi\)
\(23\)	1.37324e7	−	2.37853e7i	0.444882	−	0.770558i	−0.553162	−	0.833074i	\(-0.686579\pi\)
							0.998044	+	0.0625157i	\(0.0199124\pi\)
\(29\)	−1.05289e7	+	1.82367e7i	−0.0953225	+	0.165103i	−0.909743	−	0.415172i	\(-0.863722\pi\)
							0.814421	+	0.580275i	\(0.197055\pi\)
\(31\)	2.43179e8			1.52558			0.762792	−	0.646644i	\(-0.223828\pi\)
							0.762792	+	0.646644i	\(0.223828\pi\)
\(37\)	−3.41191e8	+	5.90960e8i	−0.808887	+	1.40103i	0.104748	+	0.994499i	\(0.466596\pi\)
							−0.913635	+	0.406535i	\(0.866737\pi\)
\(41\)	−4.15813e8	+	7.20209e8i	−0.560515	+	0.970840i	0.436937	+	0.899492i	\(0.356063\pi\)
							−0.997452	+	0.0713475i	\(0.977270\pi\)
\(43\)	−6.87730e8	−	1.19118e9i	−0.713414	−	1.23567i	−0.963568	−	0.267463i	\(-0.913815\pi\)
							0.250154	−	0.968206i	\(-0.419519\pi\)
\(47\)	1.55148e9			0.986749			0.493374	−	0.869817i	\(-0.335763\pi\)
							0.493374	+	0.869817i	\(0.335763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.9.5	yes	24
13.3	even	3	inner	13.12.c.a.3.5	✓	24
13.4	even	6		169.12.a.f.1.5		12
13.9	even	3		169.12.a.d.1.8		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.5	✓	24	13.3	even	3	inner
13.12.c.a.9.5	yes	24	1.1	even	1	trivial
169.12.a.d.1.8		12	13.9	even	3
169.12.a.f.1.5		12	13.4	even	6