Embedded newform 13.12.e.a.10.9

• Label: 13.12.e.a.10.9
• 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			10.9
Character	\(\chi\)	\(=\)	13.10
Dual form			13.12.e.a.4.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(42.3913 - 24.4746i) q^{2} +(-27.6563 - 47.9021i) q^{3} +(174.016 - 301.405i) q^{4} +10452.4i q^{5} +(-2344.78 - 1353.76i) q^{6} +(39866.5 + 23016.9i) q^{7} +83212.2i q^{8} +(87043.8 - 150764. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 242 q^{3} + 8191 q^{4} + 45528 q^{6} + 128496 q^{7} - 322217 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	42.3913	−	24.4746i	0.936725	−	0.540818i	0.0477927	−	0.998857i	\(-0.484781\pi\)
							0.888932	+	0.458039i	\(0.151448\pi\)
\(3\)	−27.6563	−	47.9021i	−0.0657094	−	0.113812i	0.831299	−	0.555825i	\(-0.187598\pi\)
							−0.897008	+	0.442013i	\(0.854264\pi\)
\(5\)			10452.4i			1.49583i	0.663793	+	0.747916i	\(0.268946\pi\)
							−0.663793	+	0.747916i	\(0.731054\pi\)
\(7\)	39866.5	+	23016.9i	0.896539	+	0.517617i	0.876076	−	0.482174i	\(-0.160153\pi\)
							0.0204632	+	0.999791i	\(0.493486\pi\)
\(11\)	−291569.	+	168337.i	−0.545860	+	0.315152i	−0.747450	−	0.664318i	\(-0.768722\pi\)
							0.201591	+	0.979470i	\(0.435389\pi\)
\(13\)	481786.	−	1.24902e6i	0.359886	−	0.932996i
							\(17\)	−4.91347e6	+	8.51037e6i	−0.839303	+	1.45372i	0.0511748	+	0.998690i	\(0.483703\pi\)
−0.890478	+	0.455026i	\(0.849630\pi\)
\(19\)	1.51259e7	+	8.73294e6i	1.40145	+	0.809125i	0.994541	−	0.104346i	\(-0.0332749\pi\)
							0.406904	+	0.913471i	\(0.366608\pi\)
\(23\)	−1.95644e7	−	3.38866e7i	−0.633817	−	1.09780i	−0.986764	−	0.162161i	\(-0.948154\pi\)
							0.352947	−	0.935643i	\(-0.385180\pi\)
\(29\)	−4.33540e7	−	7.50913e7i	−0.392500	−	0.679830i	0.600278	−	0.799791i	\(-0.295056\pi\)
							−0.992779	+	0.119961i	\(0.961723\pi\)
\(31\)		−	6.58000e7i		−	0.412797i	−0.978468	−	0.206398i	\(-0.933826\pi\)
							0.978468	−	0.206398i	\(-0.0661743\pi\)
\(37\)	6.64675e8	−	3.83751e8i	1.57580	−	0.909787i	0.580360	−	0.814360i	\(-0.302912\pi\)
							0.995436	−	0.0954265i	\(-0.0304215\pi\)
\(41\)	1.97192e8	−	1.13849e8i	0.265814	−	0.153468i	−0.361170	−	0.932500i	\(-0.617623\pi\)
							0.626984	+	0.779032i	\(0.284289\pi\)
\(43\)	−1.09327e8	+	1.89361e8i	−0.113410	+	0.196432i	−0.917143	−	0.398558i	\(-0.869511\pi\)
							0.803733	+	0.594990i	\(0.202844\pi\)
\(47\)		−	1.30598e9i		−	0.830615i	−0.909681	−	0.415307i	\(-0.863674\pi\)
							0.909681	−	0.415307i	\(-0.136326\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.e.a.10.9	yes	22
13.2	odd	12		169.12.a.g.1.17		22
13.4	even	6	inner	13.12.e.a.4.9	✓	22
13.11	odd	12		169.12.a.g.1.6		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.e.a.4.9	✓	22	13.4	even	6	inner
13.12.e.a.10.9	yes	22	1.1	even	1	trivial
169.12.a.g.1.6		22	13.11	odd	12
169.12.a.g.1.17		22	13.2	odd	12