Embedded newform 15.8.e.a.8.2

• Label: 15.8.e.a.8.2
• Level: $15$
• Weight: $8$
• Character: 15.8
• 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			8.2
Character	\(\chi\)	\(=\)	15.8
Dual form			15.8.e.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.2671 - 13.2671i) q^{2} +(42.4163 + 19.6942i) q^{3} +224.030i q^{4} +(122.889 + 251.044i) q^{5} +(-301.455 - 824.023i) q^{6} +(796.332 - 796.332i) q^{7} +(1274.03 - 1274.03i) q^{8} +(1411.28 + 1670.71i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.2671	−	13.2671i	−1.17265	−	1.17265i	−0.981575	−	0.191079i	\(-0.938801\pi\)
							−0.191079	−	0.981575i	\(-0.561199\pi\)
\(3\)	42.4163	+	19.6942i	0.907001	+	0.421128i
							\(5\)	122.889	+	251.044i	0.439662	+	0.898163i
\(7\)	796.332	−	796.332i	0.877508	−	0.877508i								−0.115769	−	0.993276i	\(-0.536933\pi\)
							0.993276	+	0.115769i	\(0.0369331\pi\)
\(11\)		−	2194.99i		−	0.497232i	−0.968602	−	0.248616i	\(-0.920024\pi\)
							0.968602	−	0.248616i	\(-0.0799757\pi\)
\(13\)	5051.41	+	5051.41i	0.637691	+	0.637691i	0.949985	−	0.312294i	\(-0.101097\pi\)
							−0.312294	+	0.949985i	\(0.601097\pi\)
\(17\)	17341.7	+	17341.7i	0.856093	+	0.856093i	0.990875	−	0.134782i	\(-0.0430334\pi\)
							−0.134782	+	0.990875i	\(0.543033\pi\)
\(19\)		−	36559.5i		−	1.22282i	−0.791314	−	0.611410i	\(-0.790603\pi\)
							0.791314	−	0.611410i	\(-0.209397\pi\)
\(23\)	−9551.69	+	9551.69i	−0.163694	+	0.163694i	−0.784201	−	0.620507i	\(-0.786927\pi\)
							0.620507	+	0.784201i	\(0.286927\pi\)
\(29\)	−35370.9			−0.269310			−0.134655	−	0.990893i	\(-0.542993\pi\)
							−0.134655	+	0.990893i	\(0.542993\pi\)
\(31\)	−237710.			−1.43312			−0.716559	−	0.697526i	\(-0.754284\pi\)
							−0.716559	+	0.697526i	\(0.754284\pi\)
\(37\)	64138.1	−	64138.1i	0.208166	−	0.208166i	−0.595321	−	0.803488i	\(-0.702975\pi\)
							0.803488	+	0.595321i	\(0.202975\pi\)
\(41\)			9927.62i			0.0224958i	0.999937	+	0.0112479i	\(0.00358040\pi\)
							−0.999937	+	0.0112479i	\(0.996420\pi\)
\(43\)	−190060.	−	190060.i	−0.364545	−	0.364545i	0.500938	−	0.865483i	\(-0.332988\pi\)
							−0.865483	+	0.500938i	\(0.832988\pi\)
\(47\)	−213766.	−	213766.i	−0.300328	−	0.300328i	0.540814	−	0.841142i	\(-0.318116\pi\)
							−0.841142	+	0.540814i	\(0.818116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.e.a.8.2	yes	24
3.2	odd	2	inner	15.8.e.a.8.11	yes	24
5.2	odd	4	inner	15.8.e.a.2.11	yes	24
5.3	odd	4		75.8.e.d.32.2		24
5.4	even	2		75.8.e.d.68.11		24
15.2	even	4	inner	15.8.e.a.2.2	✓	24
15.8	even	4		75.8.e.d.32.11		24
15.14	odd	2		75.8.e.d.68.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.e.a.2.2	✓	24	15.2	even	4	inner
15.8.e.a.2.11	yes	24	5.2	odd	4	inner
15.8.e.a.8.2	yes	24	1.1	even	1	trivial
15.8.e.a.8.11	yes	24	3.2	odd	2	inner
75.8.e.d.32.2		24	5.3	odd	4
75.8.e.d.32.11		24	15.8	even	4
75.8.e.d.68.2		24	15.14	odd	2
75.8.e.d.68.11		24	5.4	even	2