Embedded newform 13.12.c.a.3.5

• Label: 13.12.c.a.3.5
• Level: $13$
• Weight: $12$
• Character: 13.3
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• 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			3.5
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.5

$q$-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 \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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	0.688461	−	0.725274i	\(-0.258287\pi\)
							−0.972336	+	0.233588i	\(0.924953\pi\)
\(3\)	−179.645	−	311.153i	−0.426822	−	0.739278i	0.569766	−	0.821807i	\(-0.307034\pi\)
							−0.996589	+	0.0825288i	\(0.973700\pi\)
\(5\)	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	−0.660629	−	0.750712i	\(-0.729711\pi\)
							0.980451	+	0.196766i	\(0.0630439\pi\)
\(11\)	120580.	+	208851.i	0.225744	+	0.391001i	0.956543	−	0.291593i	\(-0.0941852\pi\)
							−0.730798	+	0.682594i	\(0.760852\pi\)
\(13\)	−1.06865e6	−	806322.i	−0.798262	−	0.602310i
							\(17\)	−1.72928e6	+	2.99521e6i	−0.295391	+	0.511632i	−0.975076	−	0.221872i	\(-0.928783\pi\)
0.679685	+	0.733504i	\(0.262117\pi\)
\(19\)	−390714.	+	676736.i	−0.0362004	+	0.0627010i	−0.883558	−	0.468322i	\(-0.844859\pi\)
							0.847358	+	0.531023i	\(0.178192\pi\)
\(23\)	1.37324e7	+	2.37853e7i	0.444882	+	0.770558i	0.998044	−	0.0625157i	\(-0.0199124\pi\)
							−0.553162	+	0.833074i	\(0.686579\pi\)
\(29\)	−1.05289e7	−	1.82367e7i	−0.0953225	−	0.165103i	0.814421	−	0.580275i	\(-0.197055\pi\)
							−0.909743	+	0.415172i	\(0.863722\pi\)
\(31\)	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	−0.913635	−	0.406535i	\(-0.866737\pi\)
							0.104748	−	0.994499i	\(-0.466596\pi\)
\(41\)	−4.15813e8	−	7.20209e8i	−0.560515	−	0.970840i	−0.997452	−	0.0713475i	\(-0.977270\pi\)
							0.436937	−	0.899492i	\(-0.356063\pi\)
\(43\)	−6.87730e8	+	1.19118e9i	−0.713414	+	1.23567i	0.250154	+	0.968206i	\(0.419519\pi\)
							−0.963568	+	0.267463i	\(0.913815\pi\)
\(47\)	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.3.5	✓	24
13.3	even	3		169.12.a.d.1.8		12
13.9	even	3	inner	13.12.c.a.9.5	yes	24
13.10	even	6		169.12.a.f.1.5		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.5	✓	24	1.1	even	1	trivial
13.12.c.a.9.5	yes	24	13.9	even	3	inner
169.12.a.d.1.8		12	13.3	even	3
169.12.a.f.1.5		12	13.10	even	6