Embedded newform 13.15.d.a.8.7

• Label: 13.15.d.a.8.7
• Level: $13$
• Weight: $15$
• Character: 13.8
• Analytic conductor: $16.163$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	13.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.1627658597\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.7
Character	\(\chi\)	\(=\)	13.8
Dual form			13.15.d.a.5.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.7406 - 20.7406i) q^{2} +2104.15 q^{3} -15523.7i q^{4} +(-63748.0 - 63748.0i) q^{5} +(-43641.4 - 43641.4i) q^{6} +(-524901. + 524901. i) q^{7} +(-661784. + 661784. i) q^{8} -355511. q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 126 q^{2} - 4 q^{3} + 40308 q^{5} - 215296 q^{6} - 342172 q^{7} + 2132400 q^{8} + 51018332 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.7406	−	20.7406i	−0.162036	−	0.162036i	0.621432	−	0.783468i	\(-0.286551\pi\)
							−0.783468	+	0.621432i	\(0.786551\pi\)
\(3\)	2104.15			0.962118			0.481059	−	0.876688i	\(-0.340252\pi\)
							0.481059	+	0.876688i	\(0.340252\pi\)
\(5\)	−63748.0	−	63748.0i	−0.815975	−	0.815975i	0.169547	−	0.985522i	\(-0.445769\pi\)
							−0.985522	+	0.169547i	\(0.945769\pi\)
\(7\)	−524901.	+	524901.i	−0.637370	+	0.637370i	−0.949906	−	0.312536i	\(-0.898821\pi\)
							0.312536	+	0.949906i	\(0.398821\pi\)
\(11\)	1.16790e7	−	1.16790e7i	0.599316	−	0.599316i	−0.340815	−	0.940130i	\(-0.610703\pi\)
							0.940130	+	0.340815i	\(0.110703\pi\)
\(13\)	−3.21691e6	+	6.26660e7i	−0.0512667	+	0.998685i
							\(17\)			7.70187e8i			1.87695i	0.345342	+	0.938477i	\(0.387763\pi\)
−0.345342	+	0.938477i	\(0.612237\pi\)
\(19\)	−8.30024e8	−	8.30024e8i	−0.928571	−	0.928571i	0.0690424	−	0.997614i	\(-0.478006\pi\)
							−0.997614	+	0.0690424i	\(0.978006\pi\)
\(23\)		−	2.80800e9i		−	0.824713i	−0.911023	−	0.412356i	\(-0.864706\pi\)
							0.911023	−	0.412356i	\(-0.135294\pi\)
\(29\)	−1.62459e10			−0.941796			−0.470898	−	0.882188i	\(-0.656070\pi\)
							−0.470898	+	0.882188i	\(0.656070\pi\)
\(31\)	−1.57349e10	−	1.57349e10i	−0.571915	−	0.571915i	0.360748	−	0.932663i	\(-0.382521\pi\)
							−0.932663	+	0.360748i	\(0.882521\pi\)
\(37\)	9.20918e10	−	9.20918e10i	0.970083	−	0.970083i	−0.0294825	−	0.999565i	\(-0.509386\pi\)
							0.999565	+	0.0294825i	\(0.00938595\pi\)
\(41\)	−5.99153e10	−	5.99153e10i	−0.307646	−	0.307646i	0.536350	−	0.843996i	\(-0.319803\pi\)
							−0.843996	+	0.536350i	\(0.819803\pi\)
\(43\)		−	6.67554e10i		−	0.245588i	−0.992432	−	0.122794i	\(-0.960815\pi\)
							0.992432	−	0.122794i	\(-0.0391855\pi\)
\(47\)	2.70022e11	−	2.70022e11i	0.532985	−	0.532985i	−0.388475	−	0.921459i	\(-0.626998\pi\)
							0.921459	+	0.388475i	\(0.126998\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.15.d.a.8.7	yes	32
13.5	odd	4	inner	13.15.d.a.5.7	✓	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.15.d.a.5.7	✓	32	13.5	odd	4	inner
13.15.d.a.8.7	yes	32	1.1	even	1	trivial