Embedded newform 15.8.e.a.2.8

• Label: 15.8.e.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	15.2
Dual form			15.8.e.a.8.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.42850 - 5.42850i) q^{2} +(44.8885 - 13.1156i) q^{3} +69.0629i q^{4} +(275.850 - 45.0737i) q^{5} +(172.479 - 314.875i) q^{6} +(-775.470 - 775.470i) q^{7} +(1069.75 + 1069.75i) q^{8} +(1842.96 - 1177.48i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.42850	−	5.42850i	0.479816	−	0.479816i	−0.425257	−	0.905073i	\(-0.639816\pi\)
							0.905073	+	0.425257i	\(0.139816\pi\)
\(3\)	44.8885	−	13.1156i	0.959867	−	0.280455i
							\(5\)	275.850	−	45.0737i	0.986912	−	0.161260i
\(7\)	−775.470	−	775.470i	−0.854520	−	0.854520i								0.136166	−	0.990686i	\(-0.456522\pi\)
							−0.990686	+	0.136166i	\(0.956522\pi\)
\(11\)		−	1977.76i		−	0.448023i	−0.974587	−	0.224011i	\(-0.928085\pi\)
							0.974587	−	0.224011i	\(-0.0719152\pi\)
\(13\)	−6601.06	+	6601.06i	−0.833320	+	0.833320i	−0.987969	−	0.154649i	\(-0.950575\pi\)
							0.154649	+	0.987969i	\(0.450575\pi\)
\(17\)	−23341.3	+	23341.3i	−1.15227	+	1.15227i	−0.166169	+	0.986097i	\(0.553140\pi\)
							−0.986097	+	0.166169i	\(0.946860\pi\)
\(19\)			9983.78i			0.333932i	0.985963	+	0.166966i	\(0.0533970\pi\)
							−0.985963	+	0.166966i	\(0.946603\pi\)
\(23\)	−24556.6	−	24556.6i	−0.420844	−	0.420844i	0.464650	−	0.885494i	\(-0.346180\pi\)
							−0.885494	+	0.464650i	\(0.846180\pi\)
\(29\)	135024.			1.02806			0.514028	−	0.857773i	\(-0.328153\pi\)
							0.514028	+	0.857773i	\(0.328153\pi\)
\(31\)	−128063.			−0.772071			−0.386035	−	0.922484i	\(-0.626156\pi\)
							−0.386035	+	0.922484i	\(0.626156\pi\)
\(37\)	−274562.	−	274562.i	−0.891116	−	0.891116i	0.103512	−	0.994628i	\(-0.466992\pi\)
							−0.994628	+	0.103512i	\(0.966992\pi\)
\(41\)		−	21856.1i		−	0.0495256i	−0.999693	−	0.0247628i	\(-0.992117\pi\)
							0.999693	−	0.0247628i	\(-0.00788305\pi\)
\(43\)	82226.5	−	82226.5i	0.157715	−	0.157715i	−0.623839	−	0.781553i	\(-0.714428\pi\)
							0.781553	+	0.623839i	\(0.214428\pi\)
\(47\)	400763.	−	400763.i	0.563048	−	0.563048i	−0.367124	−	0.930172i	\(-0.619658\pi\)
							0.930172	+	0.367124i	\(0.119658\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.e.a.2.8	yes	24
3.2	odd	2	inner	15.8.e.a.2.5	✓	24
5.2	odd	4		75.8.e.d.68.8		24
5.3	odd	4	inner	15.8.e.a.8.5	yes	24
5.4	even	2		75.8.e.d.32.5		24
15.2	even	4		75.8.e.d.68.5		24
15.8	even	4	inner	15.8.e.a.8.8	yes	24
15.14	odd	2		75.8.e.d.32.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.e.a.2.5	✓	24	3.2	odd	2	inner
15.8.e.a.2.8	yes	24	1.1	even	1	trivial
15.8.e.a.8.5	yes	24	5.3	odd	4	inner
15.8.e.a.8.8	yes	24	15.8	even	4	inner
75.8.e.d.32.5		24	5.4	even	2
75.8.e.d.32.8		24	15.14	odd	2
75.8.e.d.68.5		24	15.2	even	4
75.8.e.d.68.8		24	5.2	odd	4