Embedded newform 15.8.e.a.2.10

• Label: 15.8.e.a.2.10
• Level: $15$
• Weight: $8$
• Character: 15.2
• Analytic conductor: $4.686$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.68577538226\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.10
Character	\(\chi\)	\(=\)	15.2
Dual form			15.8.e.a.8.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.36735 - 9.36735i) q^{2} +(-40.3227 - 23.6872i) q^{3} -47.4943i q^{4} +(-125.932 - 249.532i) q^{5} +(-599.602 + 155.830i) q^{6} +(-696.811 - 696.811i) q^{7} +(754.124 + 754.124i) q^{8} +(1064.84 + 1910.26i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.36735	−	9.36735i	0.827964	−	0.827964i	−0.159271	−	0.987235i	\(-0.550914\pi\)
							0.987235	+	0.159271i	\(0.0509143\pi\)
\(3\)	−40.3227	−	23.6872i	−0.862234	−	0.506511i
							\(5\)	−125.932	−	249.532i	−0.450548	−	0.892752i
\(7\)	−696.811	−	696.811i	−0.767842	−	0.767842i								0.209884	−	0.977726i	\(-0.432691\pi\)
							−0.977726	+	0.209884i	\(0.932691\pi\)
\(11\)		−	3593.12i		−	0.813948i	−0.913440	−	0.406974i	\(-0.866584\pi\)
							0.913440	−	0.406974i	\(-0.133416\pi\)
\(13\)	7096.38	−	7096.38i	0.895849	−	0.895849i	−0.0992166	−	0.995066i	\(-0.531634\pi\)
							0.995066	+	0.0992166i	\(0.0316337\pi\)
\(17\)	12294.3	−	12294.3i	0.606922	−	0.606922i	−0.335219	−	0.942140i	\(-0.608810\pi\)
							0.942140	+	0.335219i	\(0.108810\pi\)
\(19\)			29903.4i			1.00019i	0.865970	+	0.500096i	\(0.166702\pi\)
							−0.865970	+	0.500096i	\(0.833298\pi\)
\(23\)	−21102.4	−	21102.4i	−0.361647	−	0.361647i	0.502772	−	0.864419i	\(-0.332314\pi\)
							−0.864419	+	0.502772i	\(0.832314\pi\)
\(29\)	−126124.			−0.960299			−0.480149	−	0.877187i	\(-0.659418\pi\)
							−0.480149	+	0.877187i	\(0.659418\pi\)
\(31\)	−4590.30			−0.0276742			−0.0138371	−	0.999904i	\(-0.504405\pi\)
							−0.0138371	+	0.999904i	\(0.504405\pi\)
\(37\)	298795.	+	298795.i	0.969765	+	0.969765i	0.999556	−	0.0297909i	\(-0.00948413\pi\)
							−0.0297909	+	0.999556i	\(0.509484\pi\)
\(41\)		−	453523.i		−	1.02767i	−0.857888	−	0.513837i	\(-0.828224\pi\)
							0.857888	−	0.513837i	\(-0.171776\pi\)
\(43\)	239641.	−	239641.i	0.459644	−	0.459644i	−0.438895	−	0.898538i	\(-0.644630\pi\)
							0.898538	+	0.438895i	\(0.144630\pi\)
\(47\)	−101235.	+	101235.i	−0.142229	+	0.142229i	−0.774636	−	0.632407i	\(-0.782067\pi\)
							0.632407	+	0.774636i	\(0.282067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.e.a.2.10	yes	24
3.2	odd	2	inner	15.8.e.a.2.3	✓	24
5.2	odd	4		75.8.e.d.68.10		24
5.3	odd	4	inner	15.8.e.a.8.3	yes	24
5.4	even	2		75.8.e.d.32.3		24
15.2	even	4		75.8.e.d.68.3		24
15.8	even	4	inner	15.8.e.a.8.10	yes	24
15.14	odd	2		75.8.e.d.32.10		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.e.a.2.3	✓	24	3.2	odd	2	inner
15.8.e.a.2.10	yes	24	1.1	even	1	trivial
15.8.e.a.8.3	yes	24	5.3	odd	4	inner
15.8.e.a.8.10	yes	24	15.8	even	4	inner
75.8.e.d.32.3		24	5.4	even	2
75.8.e.d.32.10		24	15.14	odd	2
75.8.e.d.68.3		24	15.2	even	4
75.8.e.d.68.10		24	5.2	odd	4