Embedded newform 13.12.e.a.10.2

• Label: 13.12.e.a.10.2
• Level: $13$
• Weight: $12$
• Character: 13.10
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.98846134727\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	13.10
Dual form			13.12.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-57.1912 + 33.0193i) q^{2} +(212.159 + 367.469i) q^{3} +(1156.55 - 2003.21i) q^{4} -3712.50i q^{5} +(-24267.2 - 14010.7i) q^{6} +(-35175.6 - 20308.7i) q^{7} +17507.1i q^{8} +(-1448.99 + 2509.72i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−57.1912	+	33.0193i	−1.26376	+	0.729631i	−0.973799	−	0.227408i	\(-0.926975\pi\)
							−0.289958	+	0.957039i	\(0.593641\pi\)
\(3\)	212.159	+	367.469i	0.504073	+	0.873080i	0.999989	+	0.00470972i	\(0.00149916\pi\)
							−0.495916	+	0.868371i	\(0.665168\pi\)
\(5\)		−	3712.50i		−	0.531290i	−0.964071	−	0.265645i	\(-0.914415\pi\)
							0.964071	−	0.265645i	\(-0.0855849\pi\)
\(7\)	−35175.6	−	20308.7i	−0.791048	−	0.456712i	0.0492836	−	0.998785i	\(-0.484306\pi\)
							−0.840331	+	0.542073i	\(0.817640\pi\)
\(11\)	190359.	−	109904.i	0.356380	−	0.205756i	−0.311112	−	0.950373i	\(-0.600701\pi\)
							0.667491	+	0.744617i	\(0.267368\pi\)
\(13\)	1.19151e6	−	610298.i	0.890040	−	0.455883i
							\(17\)	−1.25859e6	+	2.17994e6i	−0.214988	+	0.372371i	−0.953269	−	0.302123i	\(-0.902305\pi\)
0.738281	+	0.674494i	\(0.235638\pi\)
\(19\)	−3.98413e6	−	2.30024e6i	−0.369138	−	0.213122i	0.303944	−	0.952690i	\(-0.401696\pi\)
							−0.673082	+	0.739568i	\(0.735030\pi\)
\(23\)	−1.16368e7	−	2.01556e7i	−0.376991	−	0.652968i	0.613631	−	0.789593i	\(-0.289708\pi\)
							−0.990623	+	0.136624i	\(0.956375\pi\)
\(29\)	5.31313e6	+	9.20261e6i	0.0481018	+	0.0833148i	0.889074	−	0.457764i	\(-0.151349\pi\)
							−0.840972	+	0.541079i	\(0.818016\pi\)
\(31\)		−	1.89667e8i		−	1.18988i	−0.803771	−	0.594939i	\(-0.797176\pi\)
							0.803771	−	0.594939i	\(-0.202824\pi\)
\(37\)	4.73582e8	−	2.73423e8i	1.12276	−	0.648224i	0.180653	−	0.983547i	\(-0.442179\pi\)
							0.942103	+	0.335323i	\(0.108845\pi\)
\(41\)	−5.75287e8	+	3.32142e8i	−0.775485	+	0.447727i	−0.834828	−	0.550511i	\(-0.814433\pi\)
							0.0593427	+	0.998238i	\(0.481100\pi\)
\(43\)	3.27347e8	−	5.66982e8i	0.339572	−	0.588156i	−0.644780	−	0.764368i	\(-0.723051\pi\)
							0.984352	+	0.176212i	\(0.0563844\pi\)
\(47\)		−	2.26926e9i		−	1.44326i	−0.692277	−	0.721632i	\(-0.743392\pi\)
							0.692277	−	0.721632i	\(-0.256608\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.e.a.10.2	yes	22
13.2	odd	12		169.12.a.g.1.3		22
13.4	even	6	inner	13.12.e.a.4.2	✓	22
13.11	odd	12		169.12.a.g.1.20		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.2	✓	22	13.4	even	6	inner
13.12.e.a.10.2	yes	22	1.1	even	1	trivial
169.12.a.g.1.3		22	13.2	odd	12
169.12.a.g.1.20		22	13.11	odd	12