Embedded newform 15.10.e.a.2.9

• Label: 15.10.e.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.02674 - 4.02674i) q^{2} +(130.186 - 52.2930i) q^{3} +479.571i q^{4} +(-216.015 + 1380.75i) q^{5} +(313.655 - 734.796i) q^{6} +(17.2800 + 17.2800i) q^{7} +(3992.80 + 3992.80i) q^{8} +(14213.9 - 13615.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.02674	−	4.02674i	0.177958	−	0.177958i	−0.612507	−	0.790465i	\(-0.709839\pi\)
							0.790465	+	0.612507i	\(0.209839\pi\)
\(3\)	130.186	−	52.2930i	0.927939	−	0.372733i
							\(5\)	−216.015	+	1380.75i	−0.154568	+	0.987982i
\(7\)	17.2800	+	17.2800i	0.00272021	+	0.00272021i								0.708466	−	0.705745i	\(-0.249388\pi\)
							−0.705745	+	0.708466i	\(0.749388\pi\)
\(11\)			88470.9i			1.82194i	0.412476	+	0.910969i	\(0.364664\pi\)
							−0.412476	+	0.910969i	\(0.635336\pi\)
\(13\)	106827.	−	106827.i	1.03737	−	1.03737i	0.0380967	−	0.999274i	\(-0.487871\pi\)
							0.999274	−	0.0380967i	\(-0.0121295\pi\)
\(17\)	221082.	−	221082.i	0.641999	−	0.641999i	−0.309048	−	0.951046i	\(-0.600010\pi\)
							0.951046	+	0.309048i	\(0.100010\pi\)
\(19\)			138075.i			0.243065i	0.992587	+	0.121533i	\(0.0387809\pi\)
							−0.992587	+	0.121533i	\(0.961219\pi\)
\(23\)	−1.10129e6	−	1.10129e6i	−0.820592	−	0.820592i	0.165601	−	0.986193i	\(-0.447044\pi\)
							−0.986193	+	0.165601i	\(0.947044\pi\)
\(29\)	−2.89861e6			−0.761026			−0.380513	−	0.924776i	\(-0.624253\pi\)
							−0.380513	+	0.924776i	\(0.624253\pi\)
\(31\)	3.99131e6			0.776224			0.388112	−	0.921612i	\(-0.373127\pi\)
							0.388112	+	0.921612i	\(0.373127\pi\)
\(37\)	4.00364e6	+	4.00364e6i	0.351194	+	0.351194i	0.860554	−	0.509360i	\(-0.170118\pi\)
							−0.509360	+	0.860554i	\(0.670118\pi\)
\(41\)		−	9.96490e6i		−	0.550739i	−0.961339	−	0.275369i	\(-0.911200\pi\)
							0.961339	−	0.275369i	\(-0.0888001\pi\)
\(43\)	1.56894e6	−	1.56894e6i	0.0699839	−	0.0699839i	−0.671248	−	0.741232i	\(-0.734242\pi\)
							0.741232	+	0.671248i	\(0.234242\pi\)
\(47\)	1.81354e7	−	1.81354e7i	0.542111	−	0.542111i	−0.382037	−	0.924147i	\(-0.624777\pi\)
							0.924147	+	0.382037i	\(0.124777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.9	yes	32
3.2	odd	2	inner	15.10.e.a.2.8	✓	32
5.2	odd	4		75.10.e.f.68.9		32
5.3	odd	4	inner	15.10.e.a.8.8	yes	32
5.4	even	2		75.10.e.f.32.8		32
15.2	even	4		75.10.e.f.68.8		32
15.8	even	4	inner	15.10.e.a.8.9	yes	32
15.14	odd	2		75.10.e.f.32.9		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.8	✓	32	3.2	odd	2	inner
15.10.e.a.2.9	yes	32	1.1	even	1	trivial
15.10.e.a.8.8	yes	32	5.3	odd	4	inner
15.10.e.a.8.9	yes	32	15.8	even	4	inner
75.10.e.f.32.8		32	5.4	even	2
75.10.e.f.32.9		32	15.14	odd	2
75.10.e.f.68.8		32	15.2	even	4
75.10.e.f.68.9		32	5.2	odd	4