Embedded newform 15.10.e.a.2.2

• Label: 15.10.e.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.1261 + 24.1261i) q^{2} +(21.4940 - 138.640i) q^{3} -652.142i q^{4} +(844.923 + 1113.21i) q^{5} +(2826.28 + 3863.41i) q^{6} +(1781.62 + 1781.62i) q^{7} +(3381.08 + 3381.08i) q^{8} +(-18759.0 - 5959.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.1261	+	24.1261i	−1.06624	+	1.06624i	−0.0685902	+	0.997645i	\(0.521850\pi\)
							−0.997645	+	0.0685902i	\(0.978150\pi\)
\(3\)	21.4940	−	138.640i	0.153204	−	0.988195i
							\(5\)	844.923	+	1113.21i	0.604578	+	0.796546i
\(7\)	1781.62	+	1781.62i	0.280462	+	0.280462i								0.833293	−	0.552832i	\(-0.186453\pi\)
							−0.552832	+	0.833293i	\(0.686453\pi\)
\(11\)			20586.3i			0.423946i	0.977275	+	0.211973i	\(0.0679890\pi\)
							−0.977275	+	0.211973i	\(0.932011\pi\)
\(13\)	−105880.	+	105880.i	−1.02818	+	1.02818i	−0.0285841	+	0.999591i	\(0.509100\pi\)
							−0.999591	+	0.0285841i	\(0.990900\pi\)
\(17\)	−269375.	+	269375.i	−0.782236	+	0.782236i	−0.980208	−	0.197972i	\(-0.936565\pi\)
							0.197972	+	0.980208i	\(0.436565\pi\)
\(19\)			779592.i			1.37238i	0.727420	+	0.686192i	\(0.240719\pi\)
							−0.727420	+	0.686192i	\(0.759281\pi\)
\(23\)	−758052.	−	758052.i	−0.564838	−	0.564838i	0.365840	−	0.930678i	\(-0.380782\pi\)
							−0.930678	+	0.365840i	\(0.880782\pi\)
\(29\)	3.30998e6			0.869029			0.434515	−	0.900665i	\(-0.356920\pi\)
							0.434515	+	0.900665i	\(0.356920\pi\)
\(31\)	5.40032e6			1.05025			0.525124	−	0.851026i	\(-0.324019\pi\)
							0.525124	+	0.851026i	\(0.324019\pi\)
\(37\)	−9.21797e6	−	9.21797e6i	−0.808589	−	0.808589i	0.175832	−	0.984420i	\(-0.443739\pi\)
							−0.984420	+	0.175832i	\(0.943739\pi\)
\(41\)		−	2.96554e6i		−	0.163899i	−0.996636	−	0.0819494i	\(-0.973885\pi\)
							0.996636	−	0.0819494i	\(-0.0261146\pi\)
\(43\)	−2.26603e7	+	2.26603e7i	−1.01078	+	1.01078i	−0.0108415	+	0.999941i	\(0.503451\pi\)
							−0.999941	+	0.0108415i	\(0.996549\pi\)
\(47\)	2.16395e7	−	2.16395e7i	0.646856	−	0.646856i	−0.305376	−	0.952232i	\(-0.598782\pi\)
							0.952232	+	0.305376i	\(0.0987821\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.2	✓	32
3.2	odd	2	inner	15.10.e.a.2.15	yes	32
5.2	odd	4		75.10.e.f.68.2		32
5.3	odd	4	inner	15.10.e.a.8.15	yes	32
5.4	even	2		75.10.e.f.32.15		32
15.2	even	4		75.10.e.f.68.15		32
15.8	even	4	inner	15.10.e.a.8.2	yes	32
15.14	odd	2		75.10.e.f.32.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.2	✓	32	1.1	even	1	trivial
15.10.e.a.2.15	yes	32	3.2	odd	2	inner
15.10.e.a.8.2	yes	32	15.8	even	4	inner
15.10.e.a.8.15	yes	32	5.3	odd	4	inner
75.10.e.f.32.2		32	15.14	odd	2
75.10.e.f.32.15		32	5.4	even	2
75.10.e.f.68.2		32	5.2	odd	4
75.10.e.f.68.15		32	15.2	even	4