Embedded newform 15.10.e.a.2.11

• Label: 15.10.e.a.2.11
• 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			2.11
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.0225 - 10.0225i) q^{2} +(-84.4162 - 112.058i) q^{3} +311.097i q^{4} +(-1254.86 + 615.193i) q^{5} +(-1969.17 - 277.038i) q^{6} +(2707.78 + 2707.78i) q^{7} +(8249.53 + 8249.53i) q^{8} +(-5430.82 + 18919.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.0225	−	10.0225i	0.442938	−	0.442938i	−0.450060	−	0.892998i	\(-0.648597\pi\)
							0.892998	+	0.450060i	\(0.148597\pi\)
\(3\)	−84.4162	−	112.058i	−0.601700	−	0.798722i
							\(5\)	−1254.86	+	615.193i	−0.897902	+	0.440196i
\(7\)	2707.78	+	2707.78i	0.426257	+	0.426257i								0.887351	−	0.461094i	\(-0.152543\pi\)
							−0.461094	+	0.887351i	\(0.652543\pi\)
\(11\)		−	25188.1i		−	0.518715i	−0.965781	−	0.259358i	\(-0.916489\pi\)
							0.965781	−	0.259358i	\(-0.0835109\pi\)
\(13\)	−131855.	+	131855.i	−1.28042	+	1.28042i	−0.339984	+	0.940431i	\(0.610422\pi\)
							−0.940431	+	0.339984i	\(0.889578\pi\)
\(17\)	−274335.	+	274335.i	−0.796639	+	0.796639i	−0.982564	−	0.185925i	\(-0.940472\pi\)
							0.185925	+	0.982564i	\(0.440472\pi\)
\(19\)		−	504083.i		−	0.887383i	−0.896180	−	0.443691i	\(-0.853669\pi\)
							0.896180	−	0.443691i	\(-0.146331\pi\)
\(23\)	−357367.	−	357367.i	−0.266280	−	0.266280i	0.561319	−	0.827599i	\(-0.310294\pi\)
							−0.827599	+	0.561319i	\(0.810294\pi\)
\(29\)	−1.44978e6			−0.380636			−0.190318	−	0.981722i	\(-0.560952\pi\)
							−0.190318	+	0.981722i	\(0.560952\pi\)
\(31\)	−1.57256e6			−0.305831			−0.152915	−	0.988239i	\(-0.548866\pi\)
							−0.152915	+	0.988239i	\(0.548866\pi\)
\(37\)	1.41364e7	+	1.41364e7i	1.24002	+	1.24002i	0.959990	+	0.280034i	\(0.0903458\pi\)
							0.280034	+	0.959990i	\(0.409654\pi\)
\(41\)			1.52870e7i			0.844878i	0.906391	+	0.422439i	\(0.138826\pi\)
							−0.906391	+	0.422439i	\(0.861174\pi\)
\(43\)	1.45503e7	−	1.45503e7i	0.649027	−	0.649027i	−0.303731	−	0.952758i	\(-0.598232\pi\)
							0.952758	+	0.303731i	\(0.0982323\pi\)
\(47\)	−2.49426e7	+	2.49426e7i	−0.745592	+	0.745592i	−0.973648	−	0.228056i	\(-0.926763\pi\)
							0.228056	+	0.973648i	\(0.426763\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.11	yes	32
3.2	odd	2	inner	15.10.e.a.2.6	✓	32
5.2	odd	4		75.10.e.f.68.11		32
5.3	odd	4	inner	15.10.e.a.8.6	yes	32
5.4	even	2		75.10.e.f.32.6		32
15.2	even	4		75.10.e.f.68.6		32
15.8	even	4	inner	15.10.e.a.8.11	yes	32
15.14	odd	2		75.10.e.f.32.11		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.6	✓	32	3.2	odd	2	inner
15.10.e.a.2.11	yes	32	1.1	even	1	trivial
15.10.e.a.8.6	yes	32	5.3	odd	4	inner
15.10.e.a.8.11	yes	32	15.8	even	4	inner
75.10.e.f.32.6		32	5.4	even	2
75.10.e.f.32.11		32	15.14	odd	2
75.10.e.f.68.6		32	15.2	even	4
75.10.e.f.68.11		32	5.2	odd	4