Embedded newform 13.13.d.a.5.7

• Label: 13.13.d.a.5.7
• Level: $13$
• Weight: $13$
• Character: 13.5
• Analytic conductor: $11.882$
• Analytic rank: $0$
• Dimension: $26$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	13.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8819196246\)
Analytic rank:	\(0\)
Dimension:	\(26\)
Relative dimension:	\(13\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	13.5
Dual form			13.13.d.a.8.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.9679 + 11.9679i) q^{2} -712.886 q^{3} +3809.54i q^{4} +(-8528.30 + 8528.30i) q^{5} +(8531.76 - 8531.76i) q^{6} +(71048.2 + 71048.2i) q^{7} +(-94612.8 - 94612.8i) q^{8} -23234.8 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 2 q^{2} - 4 q^{3} + 5146 q^{5} + 64736 q^{6} - 114402 q^{7} - 262080 q^{8} + 3897230 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.9679	+	11.9679i	−0.186999	+	0.186999i	−0.794397	−	0.607399i	\(-0.792213\pi\)
							0.607399	+	0.794397i	\(0.292213\pi\)
\(3\)	−712.886			−0.977896			−0.488948	−	0.872313i	\(-0.662619\pi\)
							−0.488948	+	0.872313i	\(0.662619\pi\)
\(5\)	−8528.30	+	8528.30i	−0.545811	+	0.545811i	−0.925227	−	0.379415i	\(-0.876125\pi\)
							0.379415	+	0.925227i	\(0.376125\pi\)
\(7\)	71048.2	+	71048.2i	0.603900	+	0.603900i	0.941345	−	0.337445i	\(-0.109563\pi\)
							−0.337445	+	0.941345i	\(0.609563\pi\)
\(11\)	−1.60930e6	−	1.60930e6i	−0.908408	−	0.908408i	0.0877359	−	0.996144i	\(-0.472037\pi\)
							−0.996144	+	0.0877359i	\(0.972037\pi\)
\(13\)	4.47909e6	+	1.79884e6i	0.927961	+	0.372677i
							\(17\)		−	4.16079e7i		−	1.72378i	−0.507092	−	0.861892i	\(-0.669280\pi\)
0.507092	−	0.861892i	\(-0.330720\pi\)
\(19\)	2.18570e7	−	2.18570e7i	0.464589	−	0.464589i	−0.435567	−	0.900156i	\(-0.643452\pi\)
							0.900156	+	0.435567i	\(0.143452\pi\)
\(23\)			1.10918e8i			0.749267i	0.927173	+	0.374634i	\(0.122231\pi\)
							−0.927173	+	0.374634i	\(0.877769\pi\)
\(29\)	−3.79879e8			−0.638642			−0.319321	−	0.947647i	\(-0.603455\pi\)
							−0.319321	+	0.947647i	\(0.603455\pi\)
\(31\)	−2.50813e8	+	2.50813e8i	−0.282605	+	0.282605i	−0.834147	−	0.551542i	\(-0.814040\pi\)
							0.551542	+	0.834147i	\(0.314040\pi\)
\(37\)	−1.93651e8	−	1.93651e8i	−0.0754763	−	0.0754763i	0.668361	−	0.743837i	\(-0.266996\pi\)
							−0.743837	+	0.668361i	\(0.766996\pi\)
\(41\)	−2.11583e9	+	2.11583e9i	−0.445429	+	0.445429i	−0.893832	−	0.448403i	\(-0.851993\pi\)
							0.448403	+	0.893832i	\(0.351993\pi\)
\(43\)		−	1.14015e10i		−	1.80365i	−0.432101	−	0.901825i	\(-0.642228\pi\)
							0.432101	−	0.901825i	\(-0.357772\pi\)
\(47\)	−7.80839e9	−	7.80839e9i	−0.724393	−	0.724393i	0.245104	−	0.969497i	\(-0.421178\pi\)
							−0.969497	+	0.245104i	\(0.921178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.13.d.a.5.7	✓	26
13.8	odd	4	inner	13.13.d.a.8.7	yes	26

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.13.d.a.5.7	✓	26	1.1	even	1	trivial
13.13.d.a.8.7	yes	26	13.8	odd	4	inner