Embedded newform 15.10.e.a.2.14

• Label: 15.10.e.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	15.2
Dual form			15.10.e.a.8.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(24.0812 - 24.0812i) q^{2} +(-17.1863 - 139.239i) q^{3} -647.806i q^{4} +(1397.54 - 2.62404i) q^{5} +(-3766.92 - 2939.18i) q^{6} +(-4153.03 - 4153.03i) q^{7} +(-3270.36 - 3270.36i) q^{8} +(-19092.3 + 4786.01i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	24.0812	−	24.0812i	1.06425	−	1.06425i	0.0664584	−	0.997789i	\(-0.478830\pi\)
							0.997789	−	0.0664584i	\(-0.0211700\pi\)
\(3\)	−17.1863	−	139.239i	−0.122500	−	0.992469i
							\(5\)	1397.54	−	2.62404i	0.999998	−	0.00187761i
\(7\)	−4153.03	−	4153.03i	−0.653768	−	0.653768i								0.300130	−	0.953898i	\(-0.402970\pi\)
							−0.953898	+	0.300130i	\(0.902970\pi\)
\(11\)			53366.2i			1.09900i	0.835492	+	0.549502i	\(0.185183\pi\)
							−0.835492	+	0.549502i	\(0.814817\pi\)
\(13\)	23289.8	−	23289.8i	0.226162	−	0.226162i	−0.584925	−	0.811087i	\(-0.698876\pi\)
							0.811087	+	0.584925i	\(0.198876\pi\)
\(17\)	252242.	−	252242.i	0.732482	−	0.732482i	−0.238629	−	0.971111i	\(-0.576698\pi\)
							0.971111	+	0.238629i	\(0.0766979\pi\)
\(19\)		−	588333.i		−	1.03570i	−0.855473	−	0.517848i	\(-0.826733\pi\)
							0.855473	−	0.517848i	\(-0.173267\pi\)
\(23\)	1.34627e6	+	1.34627e6i	1.00313	+	1.00313i	0.999995	+	0.00313759i	\(0.000998729\pi\)
							0.00313759	+	0.999995i	\(0.499001\pi\)
\(29\)	1.67471e6			0.439691			0.219846	−	0.975535i	\(-0.429445\pi\)
							0.219846	+	0.975535i	\(0.429445\pi\)
\(31\)	−9.03269e6			−1.75667			−0.878334	−	0.478048i	\(-0.841344\pi\)
							−0.878334	+	0.478048i	\(0.841344\pi\)
\(37\)	1.99373e6	+	1.99373e6i	0.174888	+	0.174888i	0.789123	−	0.614235i	\(-0.210535\pi\)
							−0.614235	+	0.789123i	\(0.710535\pi\)
\(41\)			1.85816e7i			1.02696i	0.858100	+	0.513482i	\(0.171645\pi\)
							−0.858100	+	0.513482i	\(0.828355\pi\)
\(43\)	−2.46212e7	+	2.46212e7i	−1.09825	+	1.09825i	−0.103635	+	0.994615i	\(0.533048\pi\)
							−0.994615	+	0.103635i	\(0.966952\pi\)
\(47\)	2.25080e6	−	2.25080e6i	0.0672816	−	0.0672816i	−0.672665	−	0.739947i	\(-0.734851\pi\)
							0.739947	+	0.672665i	\(0.234851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.10.e.a.2.14	yes	32
3.2	odd	2	inner	15.10.e.a.2.3	✓	32
5.2	odd	4		75.10.e.f.68.14		32
5.3	odd	4	inner	15.10.e.a.8.3	yes	32
5.4	even	2		75.10.e.f.32.3		32
15.2	even	4		75.10.e.f.68.3		32
15.8	even	4	inner	15.10.e.a.8.14	yes	32
15.14	odd	2		75.10.e.f.32.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.e.a.2.3	✓	32	3.2	odd	2	inner
15.10.e.a.2.14	yes	32	1.1	even	1	trivial
15.10.e.a.8.3	yes	32	5.3	odd	4	inner
15.10.e.a.8.14	yes	32	15.8	even	4	inner
75.10.e.f.32.3		32	5.4	even	2
75.10.e.f.32.14		32	15.14	odd	2
75.10.e.f.68.3		32	15.2	even	4
75.10.e.f.68.14		32	5.2	odd	4