Embedded newform 13.12.c.a.3.8

• Label: 13.12.c.a.3.8
• 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			3.8
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.6130 + 21.8464i) q^{2} +(-378.076 - 654.847i) q^{3} +(705.823 - 1222.52i) q^{4} -3120.93 q^{5} +(9537.36 - 16519.2i) q^{6} +(-34271.3 + 59359.7i) q^{7} +87273.2 q^{8} +(-197309. + 341750. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.6130	+	21.8464i	0.278711	+	0.482742i	0.971065	−	0.238817i	\(-0.0767595\pi\)
							−0.692354	+	0.721558i	\(0.743426\pi\)
\(3\)	−378.076	−	654.847i	−0.898281	−	1.55587i	−0.829691	−	0.558223i	\(-0.811483\pi\)
							−0.0685896	−	0.997645i	\(-0.521850\pi\)
\(5\)	−3120.93			−0.446631			−0.223315	−	0.974746i	\(-0.571688\pi\)
							−0.223315	+	0.974746i	\(0.571688\pi\)
\(7\)	−34271.3	+	59359.7i	−0.770712	+	1.33491i	0.166462	+	0.986048i	\(0.446766\pi\)
							−0.937173	+	0.348864i	\(0.886568\pi\)
\(11\)	−56374.9	−	97644.1i	−0.105542	−	0.182804i	0.808417	−	0.588610i	\(-0.200324\pi\)
							−0.913960	+	0.405805i	\(0.866991\pi\)
\(13\)	393875.	−	1.27946e6i	0.294218	−	0.955738i
							\(17\)	−5.20271e6	+	9.01136e6i	−0.888711	+	1.53929i	−0.0473106	+	0.998880i	\(0.515065\pi\)
−0.841400	+	0.540412i	\(0.818268\pi\)
\(19\)	−2.21520e6	+	3.83684e6i	−0.205243	+	0.355491i	−0.950210	−	0.311610i	\(-0.899132\pi\)
							0.744967	+	0.667101i	\(0.232465\pi\)
\(23\)	−1.49805e7	−	2.59470e7i	−0.485315	−	0.840590i	0.514543	−	0.857465i	\(-0.327962\pi\)
							−0.999858	+	0.0168747i	\(0.994628\pi\)
\(29\)	−1.58006e7	−	2.73674e7i	−0.143049	−	0.247768i	0.785595	−	0.618742i	\(-0.212357\pi\)
							−0.928643	+	0.370974i	\(0.879024\pi\)
\(31\)	−9.05990e7			−0.568374			−0.284187	−	0.958769i	\(-0.591724\pi\)
							−0.284187	+	0.958769i	\(0.591724\pi\)
\(37\)	−1.60733e8	−	2.78398e8i	−0.381062	−	0.660019i	0.610152	−	0.792284i	\(-0.291108\pi\)
							−0.991214	+	0.132265i	\(0.957775\pi\)
\(41\)	1.22838e8	+	2.12762e8i	0.165586	+	0.286803i	0.936863	−	0.349696i	\(-0.113715\pi\)
							−0.771278	+	0.636499i	\(0.780382\pi\)
\(43\)	−7.30503e7	+	1.26527e8i	−0.0757784	+	0.131252i	−0.901424	−	0.432936i	\(-0.857477\pi\)
							0.825646	+	0.564188i	\(0.190811\pi\)
\(47\)	−1.13024e9			−0.718843			−0.359421	−	0.933175i	\(-0.617026\pi\)
							−0.359421	+	0.933175i	\(0.617026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.8	✓	24
13.3	even	3		169.12.a.d.1.5		12
13.9	even	3	inner	13.12.c.a.9.8	yes	24
13.10	even	6		169.12.a.f.1.8		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.8	✓	24	1.1	even	1	trivial
13.12.c.a.9.8	yes	24	13.9	even	3	inner
169.12.a.d.1.5		12	13.3	even	3
169.12.a.f.1.8		12	13.10	even	6