Embedded newform 13.12.c.a.3.7

• Label: 13.12.c.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.17841 + 3.77312i) q^{2} +(374.885 + 649.321i) q^{3} +(1014.51 - 1757.18i) q^{4} +7470.18 q^{5} +(-1633.31 + 2828.98i) q^{6} +(19231.4 - 33309.7i) q^{7} +17762.8 q^{8} +(-192505. + 333428. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.17841	+	3.77312i	0.0481366	+	0.0833750i	0.889090	−	0.457733i	\(-0.151338\pi\)
							−0.840953	+	0.541108i	\(0.818005\pi\)
\(3\)	374.885	+	649.321i	0.890701	+	1.54274i	0.839037	+	0.544074i	\(0.183119\pi\)
							0.0516633	+	0.998665i	\(0.483548\pi\)
\(5\)	7470.18			1.06905			0.534523	−	0.845154i	\(-0.320491\pi\)
							0.534523	+	0.845154i	\(0.320491\pi\)
\(7\)	19231.4	−	33309.7i	0.432485	−	0.749086i	−0.564602	−	0.825364i	\(-0.690970\pi\)
							0.997087	+	0.0762774i	\(0.0243035\pi\)
\(11\)	−112539.	−	194924.i	−0.210690	−	0.364927i	0.741240	−	0.671240i	\(-0.234238\pi\)
							−0.951931	+	0.306313i	\(0.900905\pi\)
\(13\)	1.17064e6	+	649430.i	0.874451	+	0.485114i
							\(17\)	−1.78352e6	+	3.08915e6i	−0.304656	+	0.527680i	−0.977185	−	0.212391i	\(-0.931875\pi\)
0.672529	+	0.740071i	\(0.265208\pi\)
\(19\)	−8.78534e6	+	1.52167e7i	−0.813980	+	1.40986i	0.0960773	+	0.995374i	\(0.469370\pi\)
							−0.910058	+	0.414482i	\(0.863963\pi\)
\(23\)	−2.75384e7	−	4.76978e7i	−0.892144	−	1.54524i	−0.837300	−	0.546744i	\(-0.815867\pi\)
							−0.0548443	−	0.998495i	\(-0.517466\pi\)
\(29\)	1.49068e7	+	2.58194e7i	0.134957	+	0.233753i	0.925581	−	0.378549i	\(-0.123577\pi\)
							−0.790624	+	0.612302i	\(0.790244\pi\)
\(31\)	2.79740e7			0.175495			0.0877476	−	0.996143i	\(-0.472033\pi\)
							0.0877476	+	0.996143i	\(0.472033\pi\)
\(37\)	5.11452e7	+	8.85861e7i	0.121254	+	0.210018i	0.920262	−	0.391302i	\(-0.127975\pi\)
							−0.799009	+	0.601320i	\(0.794642\pi\)
\(41\)	−5.12595e8	−	8.87840e8i	−0.690976	−	1.19680i	−0.971519	−	0.236963i	\(-0.923848\pi\)
							0.280543	−	0.959841i	\(-0.409486\pi\)
\(43\)	5.77045e8	−	9.99471e8i	0.598595	−	1.03680i	−0.394434	−	0.918924i	\(-0.629059\pi\)
							0.993029	−	0.117872i	\(-0.0376073\pi\)
\(47\)	1.48396e9			0.943809			0.471905	−	0.881650i	\(-0.343567\pi\)
							0.471905	+	0.881650i	\(0.343567\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.7	✓	24
13.3	even	3		169.12.a.d.1.6		12
13.9	even	3	inner	13.12.c.a.9.7	yes	24
13.10	even	6		169.12.a.f.1.7		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.7	✓	24	1.1	even	1	trivial
13.12.c.a.9.7	yes	24	13.9	even	3	inner
169.12.a.d.1.6		12	13.3	even	3
169.12.a.f.1.7		12	13.10	even	6