Embedded newform 13.12.c.a.3.1

• Label: 13.12.c.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-39.1473 - 67.8051i) q^{2} +(-245.228 - 424.747i) q^{3} +(-2041.02 + 3535.16i) q^{4} +3256.36 q^{5} +(-19200.0 + 33255.4i) q^{6} +(-27428.1 + 47506.9i) q^{7} +159255. q^{8} +(-31700.0 + 54905.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−39.1473	−	67.8051i	−0.865041	−	1.49830i	−0.867006	−	0.498298i	\(-0.833959\pi\)
							0.00196471	−	0.999998i	\(-0.499375\pi\)
\(3\)	−245.228	−	424.747i	−0.582644	−	1.00917i	−0.995165	−	0.0982203i	\(-0.968685\pi\)
							0.412521	−	0.910948i	\(-0.364648\pi\)
\(5\)	3256.36			0.466012			0.233006	−	0.972475i	\(-0.425144\pi\)
							0.233006	+	0.972475i	\(0.425144\pi\)
\(7\)	−27428.1	+	47506.9i	−0.616818	+	1.06836i	0.373245	+	0.927733i	\(0.378245\pi\)
							−0.990063	+	0.140627i	\(0.955088\pi\)
\(11\)	−363435.	−	629487.i	−0.680404	−	1.17849i	−0.974858	−	0.222828i	\(-0.928471\pi\)
							0.294454	−	0.955666i	\(-0.404862\pi\)
\(13\)	240815.	+	1.31688e6i	0.179885	+	0.983688i
							\(17\)	2.50228e6	−	4.33408e6i	0.427432	−	0.740334i	−0.569212	−	0.822191i	\(-0.692752\pi\)
0.996644	+	0.0818566i	\(0.0260850\pi\)
\(19\)	−6.01929e6	+	1.04257e7i	−0.557699	+	0.965964i	0.439989	+	0.898003i	\(0.354982\pi\)
							−0.997688	+	0.0679603i	\(0.978351\pi\)
\(23\)	354734.	+	614418.i	0.0114921	+	0.0199049i	0.871714	−	0.490015i	\(-0.163009\pi\)
							−0.860222	+	0.509919i	\(0.829675\pi\)
\(29\)	8.43609e7	+	1.46117e8i	0.763752	+	1.32286i	0.940904	+	0.338673i	\(0.109978\pi\)
							−0.177152	+	0.984183i	\(0.556688\pi\)
\(31\)	2.87854e8			1.80585			0.902927	−	0.429794i	\(-0.141414\pi\)
							0.902927	+	0.429794i	\(0.141414\pi\)
\(37\)	1.52234e8	+	2.63678e8i	0.360914	+	0.625121i	0.988111	−	0.153739i	\(-0.0491316\pi\)
							−0.627198	+	0.778860i	\(0.715798\pi\)
\(41\)	1.08542e7	+	1.87999e7i	0.0146314	+	0.0253423i	0.873248	−	0.487275i	\(-0.162009\pi\)
							−0.858617	+	0.512618i	\(0.828676\pi\)
\(43\)	−1.87902e8	+	3.25456e8i	−0.194919	+	0.337610i	−0.946874	−	0.321605i	\(-0.895778\pi\)
							0.751955	+	0.659215i	\(0.229111\pi\)
\(47\)	−2.54860e9			−1.62093			−0.810465	−	0.585788i	\(-0.800785\pi\)
							−0.810465	+	0.585788i	\(0.800785\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.1	✓	24
13.3	even	3		169.12.a.d.1.12		12
13.9	even	3	inner	13.12.c.a.9.1	yes	24
13.10	even	6		169.12.a.f.1.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.1	✓	24	1.1	even	1	trivial
13.12.c.a.9.1	yes	24	13.9	even	3	inner
169.12.a.d.1.12		12	13.3	even	3
169.12.a.f.1.1		12	13.10	even	6