Embedded newform 13.12.c.a.3.12

• Label: 13.12.c.a.3.12
• Level: $13$
• Weight: $12$
• Character: 13.3
• Analytic conductor: $9.988$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• 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			3.12
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(43.8927 + 76.0244i) q^{2} +(-272.285 - 471.612i) q^{3} +(-2829.14 + 4900.21i) q^{4} -4938.39 q^{5} +(23902.7 - 41400.6i) q^{6} +(24711.6 - 42801.8i) q^{7} -316930. q^{8} +(-59704.8 + 103412. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	43.8927	+	76.0244i	0.969901	+	1.67992i	0.695827	+	0.718209i	\(0.255038\pi\)
							0.274074	+	0.961709i	\(0.411629\pi\)
\(3\)	−272.285	−	471.612i	−0.646930	−	1.12051i	−0.983852	−	0.178983i	\(-0.942719\pi\)
							0.336923	−	0.941532i	\(-0.390614\pi\)
\(5\)	−4938.39			−0.706725			−0.353362	−	0.935487i	\(-0.614962\pi\)
							−0.353362	+	0.935487i	\(0.614962\pi\)
\(7\)	24711.6	−	42801.8i	0.555728	−	0.962549i	−0.442118	−	0.896957i	\(-0.645773\pi\)
							0.997846	−	0.0655925i	\(-0.0208937\pi\)
\(11\)	−242243.	−	419577.i	−0.453514	−	0.785510i	0.545087	−	0.838379i	\(-0.316497\pi\)
							−0.998601	+	0.0528696i	\(0.983163\pi\)
\(13\)	−1.30684e6	+	290377.i	−0.976192	+	0.216907i
							\(17\)	−1.75097e6	+	3.03278e6i	−0.299096	+	0.518050i	−0.975929	−	0.218087i	\(-0.930018\pi\)
0.676833	+	0.736136i	\(0.263352\pi\)
\(19\)	−5.89646e6	+	1.02130e7i	−0.546319	+	0.946253i	0.452203	+	0.891915i	\(0.350638\pi\)
							−0.998523	+	0.0543379i	\(0.982695\pi\)
\(23\)	1.38468e7	+	2.39833e7i	0.448586	+	0.776974i	0.998294	−	0.0583828i	\(-0.0185944\pi\)
							−0.549708	+	0.835357i	\(0.685261\pi\)
\(29\)	−7.11521e7	−	1.23239e8i	−0.644167	−	1.11573i	−0.984493	−	0.175423i	\(-0.943871\pi\)
							0.340326	−	0.940308i	\(-0.389463\pi\)
\(31\)	9.57989e7			0.600995			0.300498	−	0.953783i	\(-0.402847\pi\)
							0.300498	+	0.953783i	\(0.402847\pi\)
\(37\)	1.86584e7	+	3.23173e7i	0.0442349	+	0.0766171i	0.887295	−	0.461202i	\(-0.152582\pi\)
							−0.843060	+	0.537819i	\(0.819248\pi\)
\(41\)	2.35345e8	+	4.07630e8i	0.317244	+	0.549483i	0.979912	−	0.199430i	\(-0.0639091\pi\)
							−0.662668	+	0.748914i	\(0.730576\pi\)
\(43\)	7.60071e8	−	1.31648e9i	0.788456	−	1.36565i	−0.138456	−	0.990369i	\(-0.544214\pi\)
							0.926912	−	0.375278i	\(-0.122453\pi\)
\(47\)	−3.60203e6			−0.00229092			−0.00114546	−	0.999999i	\(-0.500365\pi\)
							−0.00114546	+	0.999999i	\(0.500365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.12	✓	24
13.3	even	3		169.12.a.d.1.1		12
13.9	even	3	inner	13.12.c.a.9.12	yes	24
13.10	even	6		169.12.a.f.1.12		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.12	✓	24	1.1	even	1	trivial
13.12.c.a.9.12	yes	24	13.9	even	3	inner
169.12.a.d.1.1		12	13.3	even	3
169.12.a.f.1.12		12	13.10	even	6