Embedded newform 13.12.e.a.4.2

• Label: 13.12.e.a.4.2
• Level: $13$
• Weight: $12$
• Character: 13.4
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $22$
• 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			4.2
Character	\(\chi\)	\(=\)	13.4
Dual form			13.12.e.a.10.2

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.2	✓	22	1.1	even	1	trivial
13.12.e.a.10.2	yes	22	13.10	even	6	inner
169.12.a.g.1.3		22	13.7	odd	12
169.12.a.g.1.20		22	13.6	odd	12