Embedded newform 15.10.e.a.2.7

• Label: 15.10.e.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.35582 + 6.35582i) q^{2} +(-140.296 - 0.342183i) q^{3} +431.207i q^{4} +(1052.08 + 919.921i) q^{5} +(893.869 - 889.519i) q^{6} +(-7866.00 - 7866.00i) q^{7} +(-5994.85 - 5994.85i) q^{8} +(19682.8 + 96.0135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.35582	+	6.35582i	−0.280890	+	0.280890i	−0.833464	−	0.552574i	\(-0.813646\pi\)
							0.552574	+	0.833464i	\(0.313646\pi\)
\(3\)	−140.296	−	0.342183i	−0.999997	−	0.00243900i
							\(5\)	1052.08	+	919.921i	0.752807	+	0.658242i
\(7\)	−7866.00	−	7866.00i	−1.23826	−	1.23826i								−0.960711	−	0.277552i	\(-0.910477\pi\)
							−0.277552	−	0.960711i	\(-0.589523\pi\)
\(11\)		−	50710.2i		−	1.04431i	−0.852852	−	0.522153i	\(-0.825129\pi\)
							0.852852	−	0.522153i	\(-0.174871\pi\)
\(13\)	−4944.61	+	4944.61i	−0.0480161	+	0.0480161i	−0.730707	−	0.682691i	\(-0.760810\pi\)
							0.682691	+	0.730707i	\(0.260810\pi\)
\(17\)	174648.	−	174648.i	0.507159	−	0.507159i	−0.406494	−	0.913653i	\(-0.633249\pi\)
							0.913653	+	0.406494i	\(0.133249\pi\)
\(19\)			36012.9i			0.0633968i	0.999497	+	0.0316984i	\(0.0100916\pi\)
							−0.999497	+	0.0316984i	\(0.989908\pi\)
\(23\)	−995471.	−	995471.i	−0.741743	−	0.741743i	0.231170	−	0.972913i	\(-0.425745\pi\)
							−0.972913	+	0.231170i	\(0.925745\pi\)
\(29\)	−1.37690e6			−0.361502			−0.180751	−	0.983529i	\(-0.557853\pi\)
							−0.180751	+	0.983529i	\(0.557853\pi\)
\(31\)	−5.36776e6			−1.04392			−0.521958	−	0.852971i	\(-0.674798\pi\)
							−0.521958	+	0.852971i	\(0.674798\pi\)
\(37\)	−4.18640e6	−	4.18640e6i	−0.367225	−	0.367225i	0.499239	−	0.866464i	\(-0.333613\pi\)
							−0.866464	+	0.499239i	\(0.833613\pi\)
\(41\)		−	1.69845e7i		−	0.938695i	−0.883014	−	0.469347i	\(-0.844489\pi\)
							0.883014	−	0.469347i	\(-0.155511\pi\)
\(43\)	4.04862e6	−	4.04862e6i	0.180592	−	0.180592i	−0.611022	−	0.791614i	\(-0.709241\pi\)
							0.791614	+	0.611022i	\(0.209241\pi\)
\(47\)	−3.93367e7	+	3.93367e7i	−1.17587	+	1.17587i	−0.195078	+	0.980788i	\(0.562496\pi\)
							−0.980788	+	0.195078i	\(0.937504\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.7	✓	32
3.2	odd	2	inner	15.10.e.a.2.10	yes	32
5.2	odd	4		75.10.e.f.68.7		32
5.3	odd	4	inner	15.10.e.a.8.10	yes	32
5.4	even	2		75.10.e.f.32.10		32
15.2	even	4		75.10.e.f.68.10		32
15.8	even	4	inner	15.10.e.a.8.7	yes	32
15.14	odd	2		75.10.e.f.32.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.7	✓	32	1.1	even	1	trivial
15.10.e.a.2.10	yes	32	3.2	odd	2	inner
15.10.e.a.8.7	yes	32	15.8	even	4	inner
15.10.e.a.8.10	yes	32	5.3	odd	4	inner
75.10.e.f.32.7		32	15.14	odd	2
75.10.e.f.32.10		32	5.4	even	2
75.10.e.f.68.7		32	5.2	odd	4
75.10.e.f.68.10		32	15.2	even	4