Embedded newform 15.10.e.a.2.12

• Label: 15.10.e.a.2.12
• 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			2.12
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.8737 - 13.8737i) q^{2} +(-128.756 + 55.7208i) q^{3} +127.039i q^{4} +(919.994 - 1052.02i) q^{5} +(-1013.28 + 2559.39i) q^{6} +(6797.28 + 6797.28i) q^{7} +(8865.86 + 8865.86i) q^{8} +(13473.4 - 14348.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.8737	−	13.8737i	0.613138	−	0.613138i	−0.330624	−	0.943762i	\(-0.607259\pi\)
							0.943762	+	0.330624i	\(0.107259\pi\)
\(3\)	−128.756	+	55.7208i	−0.917747	+	0.397166i
							\(5\)	919.994	−	1052.02i	0.658294	−	0.752761i
\(7\)	6797.28	+	6797.28i	1.07002	+	1.07002i								0.997356	+	0.0726688i	\(0.0231516\pi\)
							0.0726688	+	0.997356i	\(0.476848\pi\)
\(11\)			28401.8i			0.584895i	0.956282	+	0.292448i	\(0.0944698\pi\)
							−0.956282	+	0.292448i	\(0.905530\pi\)
\(13\)	28203.0	−	28203.0i	0.273874	−	0.273874i	−0.556784	−	0.830658i	\(-0.687965\pi\)
							0.830658	+	0.556784i	\(0.187965\pi\)
\(17\)	−21103.5	+	21103.5i	−0.0612822	+	0.0612822i	−0.737084	−	0.675802i	\(-0.763798\pi\)
							0.675802	+	0.737084i	\(0.263798\pi\)
\(19\)			759794.i			1.33753i	0.743472	+	0.668767i	\(0.233178\pi\)
							−0.743472	+	0.668767i	\(0.766822\pi\)
\(23\)	−1.18196e6	−	1.18196e6i	−0.880698	−	0.880698i	0.112907	−	0.993606i	\(-0.463984\pi\)
							−0.993606	+	0.112907i	\(0.963984\pi\)
\(29\)	6.93799e6			1.82156			0.910779	−	0.412895i	\(-0.135482\pi\)
							0.910779	+	0.412895i	\(0.135482\pi\)
\(31\)	−1.45656e6			−0.283270			−0.141635	−	0.989919i	\(-0.545236\pi\)
							−0.141635	+	0.989919i	\(0.545236\pi\)
\(37\)	−8.18291e6	−	8.18291e6i	−0.717795	−	0.717795i	0.250359	−	0.968153i	\(-0.419452\pi\)
							−0.968153	+	0.250359i	\(0.919452\pi\)
\(41\)		−	7.44760e6i		−	0.411613i	−0.978593	−	0.205806i	\(-0.934018\pi\)
							0.978593	−	0.205806i	\(-0.0659817\pi\)
\(43\)	6.21407e6	−	6.21407e6i	0.277184	−	0.277184i	−0.554800	−	0.831984i	\(-0.687205\pi\)
							0.831984	+	0.554800i	\(0.187205\pi\)
\(47\)	1.95906e7	−	1.95906e7i	0.585610	−	0.585610i	−0.350830	−	0.936439i	\(-0.614100\pi\)
							0.936439	+	0.350830i	\(0.114100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.12	yes	32
3.2	odd	2	inner	15.10.e.a.2.5	✓	32
5.2	odd	4		75.10.e.f.68.12		32
5.3	odd	4	inner	15.10.e.a.8.5	yes	32
5.4	even	2		75.10.e.f.32.5		32
15.2	even	4		75.10.e.f.68.5		32
15.8	even	4	inner	15.10.e.a.8.12	yes	32
15.14	odd	2		75.10.e.f.32.12		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.5	✓	32	3.2	odd	2	inner
15.10.e.a.2.12	yes	32	1.1	even	1	trivial
15.10.e.a.8.5	yes	32	5.3	odd	4	inner
15.10.e.a.8.12	yes	32	15.8	even	4	inner
75.10.e.f.32.5		32	5.4	even	2
75.10.e.f.32.12		32	15.14	odd	2
75.10.e.f.68.5		32	15.2	even	4
75.10.e.f.68.12		32	5.2	odd	4