Embedded newform 13.12.c.a.3.10

• Label: 13.12.c.a.3.10
• 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			3.10
Character	\(\chi\)	\(=\)	13.3
Dual form			13.12.c.a.9.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(31.0971 + 53.8618i) q^{2} +(289.297 + 501.076i) q^{3} +(-910.060 + 1576.27i) q^{4} -11626.3 q^{5} +(-17992.6 + 31164.1i) q^{6} +(6582.83 - 11401.8i) q^{7} +14172.8 q^{8} +(-78811.5 + 136506. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 31 q^{2} - 244 q^{3} - 11265 q^{4} - 10436 q^{5} - 11082 q^{6} - 2928 q^{7} - 422334 q^{8} - 681618 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	31.0971	+	53.8618i	0.687155	+	1.19019i	0.972754	+	0.231839i	\(0.0744742\pi\)
							−0.285599	+	0.958349i	\(0.592192\pi\)
\(3\)	289.297	+	501.076i	0.687348	+	1.19052i	0.972693	+	0.232096i	\(0.0745584\pi\)
							−0.285345	+	0.958425i	\(0.592108\pi\)
\(5\)	−11626.3			−1.66382			−0.831911	−	0.554908i	\(-0.812753\pi\)
							−0.831911	+	0.554908i	\(0.812753\pi\)
\(7\)	6582.83	−	11401.8i	0.148038	−	0.256410i	−0.782464	−	0.622696i	\(-0.786037\pi\)
							0.930502	+	0.366286i	\(0.119371\pi\)
\(11\)	463960.	+	803603.i	0.868603	+	1.50446i	0.863425	+	0.504477i	\(0.168315\pi\)
							0.00517785	+	0.999987i	\(0.498352\pi\)
\(13\)	−556395.	−	1.21761e6i	−0.415618	−	0.909539i
							\(17\)	−3.53793e6	+	6.12787e6i	−0.604338	+	1.04674i	0.387817	+	0.921736i	\(0.373229\pi\)
−0.992156	+	0.125008i	\(0.960104\pi\)
\(19\)	−923707.	+	1.59991e6i	−0.0855834	+	0.148235i	−0.905640	−	0.424048i	\(-0.860609\pi\)
							0.820056	+	0.572283i	\(0.193942\pi\)
\(23\)	−6.89575e6	−	1.19438e7i	−0.223397	−	0.386936i	0.732440	−	0.680832i	\(-0.238381\pi\)
							−0.955837	+	0.293896i	\(0.905048\pi\)
\(29\)	4.03656e7	+	6.99153e7i	0.365445	+	0.632970i	0.988848	−	0.148931i	\(-0.0475833\pi\)
							−0.623402	+	0.781901i	\(0.714250\pi\)
\(31\)	4.18311e7			0.262428			0.131214	−	0.991354i	\(-0.458113\pi\)
							0.131214	+	0.991354i	\(0.458113\pi\)
\(37\)	6.94504e7	+	1.20292e8i	0.164651	+	0.285184i	0.936531	−	0.350584i	\(-0.114017\pi\)
							−0.771880	+	0.635768i	\(0.780683\pi\)
\(41\)	−9.35552e7	−	1.62042e8i	−0.126112	−	0.218432i	0.796055	−	0.605224i	\(-0.206917\pi\)
							−0.922167	+	0.386792i	\(0.873583\pi\)
\(43\)	5.14745e8	−	8.91564e8i	0.533968	−	0.924860i	−0.465244	−	0.885182i	\(-0.654034\pi\)
							0.999213	−	0.0396777i	\(-0.0126331\pi\)
\(47\)	−2.51459e8			−0.159929			−0.0799647	−	0.996798i	\(-0.525481\pi\)
							−0.0799647	+	0.996798i	\(0.525481\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.12.c.a.3.10	✓	24
13.3	even	3		169.12.a.d.1.3		12
13.9	even	3	inner	13.12.c.a.9.10	yes	24
13.10	even	6		169.12.a.f.1.10		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.12.c.a.3.10	✓	24	1.1	even	1	trivial
13.12.c.a.9.10	yes	24	13.9	even	3	inner
169.12.a.d.1.3		12	13.3	even	3
169.12.a.f.1.10		12	13.10	even	6