Embedded newform 15.10.e.a.2.1

• Label: 15.10.e.a.2.1
• Level: $15$
• Weight: $10$
• Character: 15.2
• Analytic conductor: $7.726$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72553754246\)
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			2.1
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-30.5207 + 30.5207i) q^{2} +(-54.5047 + 129.276i) q^{3} -1351.03i q^{4} +(1261.11 - 602.260i) q^{5} +(-2282.07 - 5609.11i) q^{6} +(-2749.16 - 2749.16i) q^{7} +(25607.7 + 25607.7i) q^{8} +(-13741.5 - 14092.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−30.5207	+	30.5207i	−1.34884	+	1.34884i	−0.461911	+	0.886926i	\(0.652836\pi\)
							−0.886926	+	0.461911i	\(0.847164\pi\)
\(3\)	−54.5047	+	129.276i	−0.388498	+	0.921450i
							\(5\)	1261.11	−	602.260i	0.902379	−	0.430942i
\(7\)	−2749.16	−	2749.16i	−0.432772	−	0.432772i								0.456798	−	0.889570i	\(-0.348996\pi\)
							−0.889570	+	0.456798i	\(0.848996\pi\)
\(11\)		−	4815.81i		−	0.0991750i	−0.998770	−	0.0495875i	\(-0.984209\pi\)
							0.998770	−	0.0495875i	\(-0.0157907\pi\)
\(13\)	66741.4	−	66741.4i	0.648113	−	0.648113i	−0.304424	−	0.952537i	\(-0.598464\pi\)
							0.952537	+	0.304424i	\(0.0984639\pi\)
\(17\)	−34639.8	+	34639.8i	−0.100590	+	0.100590i	−0.755611	−	0.655021i	\(-0.772660\pi\)
							0.655021	+	0.755611i	\(0.272660\pi\)
\(19\)		−	427206.i		−	0.752050i	−0.926610	−	0.376025i	\(-0.877291\pi\)
							0.926610	−	0.376025i	\(-0.122709\pi\)
\(23\)	670596.	+	670596.i	0.499673	+	0.499673i	0.911336	−	0.411663i	\(-0.135052\pi\)
							−0.411663	+	0.911336i	\(0.635052\pi\)
\(29\)	883484.			0.231957			0.115979	−	0.993252i	\(-0.463000\pi\)
							0.115979	+	0.993252i	\(0.463000\pi\)
\(31\)	7.22393e6			1.40490			0.702451	−	0.711733i	\(-0.252089\pi\)
							0.702451	+	0.711733i	\(0.252089\pi\)
\(37\)	−5.43918e6	−	5.43918e6i	−0.477118	−	0.477118i	0.427091	−	0.904209i	\(-0.359538\pi\)
							−0.904209	+	0.427091i	\(0.859538\pi\)
\(41\)		−	2.76695e7i		−	1.52923i	−0.644485	−	0.764617i	\(-0.722928\pi\)
							0.644485	−	0.764617i	\(-0.277072\pi\)
\(43\)	−1.38521e7	+	1.38521e7i	−0.617885	+	0.617885i	−0.944988	−	0.327104i	\(-0.893927\pi\)
							0.327104	+	0.944988i	\(0.393927\pi\)
\(47\)	1.97125e6	−	1.97125e6i	0.0589253	−	0.0589253i	−0.677030	−	0.735955i	\(-0.736733\pi\)
							0.735955	+	0.677030i	\(0.236733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.1	✓	32
3.2	odd	2	inner	15.10.e.a.2.16	yes	32
5.2	odd	4		75.10.e.f.68.1		32
5.3	odd	4	inner	15.10.e.a.8.16	yes	32
5.4	even	2		75.10.e.f.32.16		32
15.2	even	4		75.10.e.f.68.16		32
15.8	even	4	inner	15.10.e.a.8.1	yes	32
15.14	odd	2		75.10.e.f.32.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.1	✓	32	1.1	even	1	trivial
15.10.e.a.2.16	yes	32	3.2	odd	2	inner
15.10.e.a.8.1	yes	32	15.8	even	4	inner
15.10.e.a.8.16	yes	32	5.3	odd	4	inner
75.10.e.f.32.1		32	15.14	odd	2
75.10.e.f.32.16		32	5.4	even	2
75.10.e.f.68.1		32	5.2	odd	4
75.10.e.f.68.16		32	15.2	even	4