Embedded newform 13.7.f.a.6.3

• Label: 13.7.f.a.6.3
• Level: $13$
• Weight: $7$
• Character: 13.6
• Analytic conductor: $2.991$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	13.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.99070308706\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.3
Character	\(\chi\)	\(=\)	13.6
Dual form			13.7.f.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.397723 - 1.48432i) q^{2} +(-0.503519 + 0.872121i) q^{3} +(53.3806 - 30.8193i) q^{4} +(84.5480 - 84.5480i) q^{5} +(1.49477 + 0.400522i) q^{6} +(41.0022 - 153.022i) q^{7} +(-136.519 - 136.519i) q^{8} +(363.993 + 630.454i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.397723	−	1.48432i	−0.0497153	−	0.185540i	0.936603	−	0.350393i	\(-0.113952\pi\)
							−0.986318	+	0.164853i	\(0.947285\pi\)
\(3\)	−0.503519	+	0.872121i	−0.0186489	+	0.0323008i	−0.875199	−	0.483763i	\(-0.839270\pi\)
							0.856550	+	0.516063i	\(0.172603\pi\)
\(5\)	84.5480	−	84.5480i	0.676384	−	0.676384i	−0.282796	−	0.959180i	\(-0.591262\pi\)
							0.959180	+	0.282796i	\(0.0912619\pi\)
\(7\)	41.0022	−	153.022i	0.119540	−	0.446130i	−0.880046	−	0.474888i	\(-0.842489\pi\)
							0.999586	+	0.0287583i	\(0.00915530\pi\)
\(11\)	−1010.62	+	270.795i	−0.759295	+	0.203452i	−0.617637	−	0.786463i	\(-0.711910\pi\)
							−0.141658	+	0.989916i	\(0.545243\pi\)
\(13\)	−1383.55	+	1706.63i	−0.629745	+	0.776802i
							\(17\)	−1495.03	+	863.153i	−0.304300	+	0.175688i	−0.644373	−	0.764711i	\(-0.722882\pi\)
0.340073	+	0.940399i	\(0.389548\pi\)
\(19\)	6576.65	+	1762.21i	0.958835	+	0.256919i	0.704107	−	0.710094i	\(-0.251347\pi\)
							0.254728	+	0.967013i	\(0.418014\pi\)
\(23\)	−3373.09	−	1947.46i	−0.277233	−	0.160061i	0.354937	−	0.934890i	\(-0.384502\pi\)
							−0.632170	+	0.774830i	\(0.717836\pi\)
\(29\)	−17563.9	+	30421.6i	−0.720158	+	1.24735i	0.240778	+	0.970580i	\(0.422598\pi\)
							−0.960936	+	0.276770i	\(0.910736\pi\)
\(31\)	−12140.0	+	12140.0i	−0.407507	+	0.407507i	−0.880868	−	0.473361i	\(-0.843041\pi\)
							0.473361	+	0.880868i	\(0.343041\pi\)
\(37\)	55534.6	−	14880.5i	1.09637	−	0.293773i	0.335087	−	0.942187i	\(-0.391234\pi\)
							0.761287	+	0.648415i	\(0.224568\pi\)
\(41\)	−19534.1	−	72902.3i	−0.283428	−	1.05777i	−0.949981	−	0.312309i	\(-0.898898\pi\)
							0.666553	−	0.745458i	\(-0.267769\pi\)
\(43\)	56257.3	−	32480.2i	0.707577	−	0.408520i	−0.102586	−	0.994724i	\(-0.532712\pi\)
							0.810163	+	0.586204i	\(0.199378\pi\)
\(47\)	−64283.9	−	64283.9i	−0.619168	−	0.619168i	0.326150	−	0.945318i	\(-0.394248\pi\)
							−0.945318	+	0.326150i	\(0.894248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.7.f.a.6.3	✓	24
13.11	odd	12	inner	13.7.f.a.11.3	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.7.f.a.6.3	✓	24	1.1	even	1	trivial
13.7.f.a.11.3	yes	24	13.11	odd	12	inner