Embedded newform 13.12.c.a.3.2

• Label: 13.12.c.a.3.2
• Level: $13$
• Weight: $12$
• Character: 13.3
• 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			3.2
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-37.0263 - 64.1314i) q^{2} +(226.124 + 391.658i) q^{3} +(-1717.89 + 2975.47i) q^{4} +250.396 q^{5} +(16745.0 - 29003.3i) q^{6} +(22976.7 - 39796.8i) q^{7} +102768. q^{8} +(-13690.5 + 23712.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−37.0263	−	64.1314i	−0.818172	−	1.41712i	−0.907027	−	0.421072i	\(-0.861654\pi\)
							0.0888551	−	0.996045i	\(-0.471679\pi\)
\(3\)	226.124	+	391.658i	0.537254	+	0.930551i	0.999051	+	0.0435650i	\(0.0138716\pi\)
							−0.461797	+	0.886986i	\(0.652795\pi\)
\(5\)	250.396			0.0358337			0.0179169	−	0.999839i	\(-0.494297\pi\)
							0.0179169	+	0.999839i	\(0.494297\pi\)
\(7\)	22976.7	−	39796.8i	0.516712	−	0.894972i	−0.483100	−	0.875565i	\(-0.660489\pi\)
							0.999812	−	0.0194062i	\(-0.00617758\pi\)
\(11\)	−106579.	−	184600.i	−0.199532	−	0.345599i	0.748845	−	0.662745i	\(-0.230609\pi\)
							−0.948377	+	0.317146i	\(0.897275\pi\)
\(13\)	−1.04961e6	−	830955.i	−0.784040	−	0.620711i
							\(17\)	958873.	−	1.66082e6i	0.163792	−	0.283696i	−0.772434	−	0.635095i	\(-0.780961\pi\)
0.936226	+	0.351400i	\(0.114294\pi\)
\(19\)	6.72303e6	−	1.16446e7i	0.622902	−	1.07890i	−0.366040	−	0.930599i	\(-0.619287\pi\)
							0.988942	−	0.148300i	\(-0.0473800\pi\)
\(23\)	−1.88522e7	−	3.26530e7i	−0.610743	−	1.05784i	−0.991115	−	0.133005i	\(-0.957537\pi\)
							0.380372	−	0.924834i	\(-0.375796\pi\)
\(29\)	−9.10548e7	−	1.57712e8i	−0.824354	−	1.42782i	−0.902412	−	0.430875i	\(-0.858205\pi\)
							0.0780575	−	0.996949i	\(-0.475128\pi\)
\(31\)	1.30823e8			0.820719			0.410360	−	0.911924i	\(-0.365403\pi\)
							0.410360	+	0.911924i	\(0.365403\pi\)
\(37\)	3.75634e8	+	6.50617e8i	0.890544	+	1.54247i	0.839225	+	0.543785i	\(0.183009\pi\)
							0.0513190	+	0.998682i	\(0.483657\pi\)
\(41\)	−1.64511e8	−	2.84941e8i	−0.221760	−	0.384099i	0.733583	−	0.679600i	\(-0.237847\pi\)
							−0.955342	+	0.295501i	\(0.904513\pi\)
\(43\)	−1.63452e8	+	2.83107e8i	−0.169556	+	0.293680i	−0.938264	−	0.345920i	\(-0.887567\pi\)
							0.768708	+	0.639600i	\(0.220900\pi\)
\(47\)	−3.33809e8			−0.212305			−0.106152	−	0.994350i	\(-0.533853\pi\)
							−0.106152	+	0.994350i	\(0.533853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.2	✓	24
13.3	even	3		169.12.a.d.1.11		12
13.9	even	3	inner	13.12.c.a.9.2	yes	24
13.10	even	6		169.12.a.f.1.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.2	✓	24	1.1	even	1	trivial
13.12.c.a.9.2	yes	24	13.9	even	3	inner
169.12.a.d.1.11		12	13.3	even	3
169.12.a.f.1.2		12	13.10	even	6