Embedded newform 13.15.d.a.5.13

• Label: 13.15.d.a.5.13
• Level: $13$
• Weight: $15$
• Character: 13.5
• 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			5.13
Character	\(\chi\)	\(=\)	13.5
Dual form			13.15.d.a.8.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(100.811 - 100.811i) q^{2} -1095.41 q^{3} -3941.67i q^{4} +(-6132.14 + 6132.14i) q^{5} +(-110429. + 110429. i) q^{6} +(-84903.1 - 84903.1i) q^{7} +(1.25432e6 + 1.25432e6i) q^{8} -3.58306e6 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 126 q^{2} - 4 q^{3} + 40308 q^{5} - 215296 q^{6} - 342172 q^{7} + 2132400 q^{8} + 51018332 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	100.811	−	100.811i	0.787585	−	0.787585i	−0.193513	−	0.981098i	\(-0.561988\pi\)
							0.981098	+	0.193513i	\(0.0619881\pi\)
\(3\)	−1095.41			−0.500871			−0.250436	−	0.968133i	\(-0.580574\pi\)
							−0.250436	+	0.968133i	\(0.580574\pi\)
\(5\)	−6132.14	+	6132.14i	−0.0784914	+	0.0784914i	−0.745263	−	0.666771i	\(-0.767676\pi\)
							0.666771	+	0.745263i	\(0.267676\pi\)
\(7\)	−84903.1	−	84903.1i	−0.103095	−	0.103095i	0.653678	−	0.756773i	\(-0.273225\pi\)
							−0.756773	+	0.653678i	\(0.773225\pi\)
\(11\)	2.64072e7	+	2.64072e7i	1.35510	+	1.35510i	0.879847	+	0.475258i	\(0.157645\pi\)
							0.475258	+	0.879847i	\(0.342355\pi\)
\(13\)	1.43089e7	+	6.10953e7i	0.228036	+	0.973653i
							\(17\)		−	2.93214e8i		−	0.714567i	−0.933996	−	0.357283i	\(-0.883703\pi\)
0.933996	−	0.357283i	\(-0.116297\pi\)
\(19\)	5.26914e8	−	5.26914e8i	0.589474	−	0.589474i	−0.348015	−	0.937489i	\(-0.613144\pi\)
							0.937489	+	0.348015i	\(0.113144\pi\)
\(23\)			4.39854e9i			1.29185i	0.763399	+	0.645927i	\(0.223529\pi\)
							−0.763399	+	0.645927i	\(0.776471\pi\)
\(29\)	−1.36510e10			−0.791368			−0.395684	−	0.918387i	\(-0.629492\pi\)
							−0.395684	+	0.918387i	\(0.629492\pi\)
\(31\)	−6.49603e9	+	6.49603e9i	−0.236111	+	0.236111i	−0.815238	−	0.579127i	\(-0.803394\pi\)
							0.579127	+	0.815238i	\(0.303394\pi\)
\(37\)	−5.07123e10	−	5.07123e10i	−0.534197	−	0.534197i	0.387622	−	0.921819i	\(-0.373297\pi\)
							−0.921819	+	0.387622i	\(0.873297\pi\)
\(41\)	−2.46469e11	+	2.46469e11i	−1.26554	+	1.26554i	−0.317171	+	0.948368i	\(0.602733\pi\)
							−0.948368	+	0.317171i	\(0.897267\pi\)
\(43\)			1.06941e11i			0.393430i	0.980461	+	0.196715i	\(0.0630273\pi\)
							−0.980461	+	0.196715i	\(0.936973\pi\)
\(47\)	−4.88003e10	−	4.88003e10i	−0.0963247	−	0.0963247i	0.657302	−	0.753627i	\(-0.271697\pi\)
							−0.753627	+	0.657302i	\(0.771697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.15.d.a.5.13	✓	32
13.8	odd	4	inner	13.15.d.a.8.13	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.15.d.a.5.13	✓	32	1.1	even	1	trivial
13.15.d.a.8.13	yes	32	13.8	odd	4	inner