Embedded newform 15.8.e.a.8.4

• Label: 15.8.e.a.8.4
• 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			8.4
Character	\(\chi\)	\(=\)	15.8
Dual form			15.8.e.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.70096 - 6.70096i) q^{2} +(-34.4749 - 31.5988i) q^{3} -38.1944i q^{4} +(8.44382 + 279.381i) q^{5} +(19.2725 + 442.757i) q^{6} +(-68.7247 + 68.7247i) q^{7} +(-1113.66 + 1113.66i) q^{8} +(190.033 + 2178.73i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 924 q^{6} + 1344 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.70096	−	6.70096i	−0.592286	−	0.592286i	0.345962	−	0.938249i	\(-0.387553\pi\)
							−0.938249	+	0.345962i	\(0.887553\pi\)
\(3\)	−34.4749	−	31.5988i	−0.737188	−	0.675688i
							\(5\)	8.44382	+	279.381i	0.0302095	+	0.999544i
\(7\)	−68.7247	+	68.7247i	−0.0757303	+	0.0757303i								−0.743957	−	0.668227i	\(-0.767053\pi\)
							0.668227	+	0.743957i	\(0.267053\pi\)
\(11\)			1961.40i			0.444315i	0.975011	+	0.222157i	\(0.0713099\pi\)
							−0.975011	+	0.222157i	\(0.928690\pi\)
\(13\)	−5694.05	−	5694.05i	−0.718819	−	0.718819i	0.249545	−	0.968363i	\(-0.419719\pi\)
							−0.968363	+	0.249545i	\(0.919719\pi\)
\(17\)	−19280.1	−	19280.1i	−0.951783	−	0.951783i	0.0471069	−	0.998890i	\(-0.485000\pi\)
							−0.998890	+	0.0471069i	\(0.985000\pi\)
\(19\)			26085.4i			0.872490i	0.899828	+	0.436245i	\(0.143692\pi\)
							−0.899828	+	0.436245i	\(0.856308\pi\)
\(23\)	−65997.5	+	65997.5i	−1.13105	+	1.13105i	−0.141042	+	0.990004i	\(0.545045\pi\)
							−0.990004	+	0.141042i	\(0.954955\pi\)
\(29\)	−139767.			−1.06417			−0.532084	−	0.846691i	\(-0.678591\pi\)
							−0.532084	+	0.846691i	\(0.678591\pi\)
\(31\)	82701.9			0.498597			0.249299	−	0.968427i	\(-0.419800\pi\)
							0.249299	+	0.968427i	\(0.419800\pi\)
\(37\)	215003.	−	215003.i	0.697812	−	0.697812i	−0.266126	−	0.963938i	\(-0.585744\pi\)
							0.963938	+	0.266126i	\(0.0857438\pi\)
\(41\)		−	707814.i		−	1.60389i	−0.597396	−	0.801947i	\(-0.703798\pi\)
							0.597396	−	0.801947i	\(-0.296202\pi\)
\(43\)	−434648.	−	434648.i	−0.833677	−	0.833677i	0.154340	−	0.988018i	\(-0.450675\pi\)
							−0.988018	+	0.154340i	\(0.950675\pi\)
\(47\)	380701.	+	380701.i	0.534861	+	0.534861i	0.922015	−	0.387154i	\(-0.126542\pi\)
							−0.387154	+	0.922015i	\(0.626542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.e.a.8.4	yes	24
3.2	odd	2	inner	15.8.e.a.8.9	yes	24
5.2	odd	4	inner	15.8.e.a.2.9	yes	24
5.3	odd	4		75.8.e.d.32.4		24
5.4	even	2		75.8.e.d.68.9		24
15.2	even	4	inner	15.8.e.a.2.4	✓	24
15.8	even	4		75.8.e.d.32.9		24
15.14	odd	2		75.8.e.d.68.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.e.a.2.4	✓	24	15.2	even	4	inner
15.8.e.a.2.9	yes	24	5.2	odd	4	inner
15.8.e.a.8.4	yes	24	1.1	even	1	trivial
15.8.e.a.8.9	yes	24	3.2	odd	2	inner
75.8.e.d.32.4		24	5.3	odd	4
75.8.e.d.32.9		24	15.8	even	4
75.8.e.d.68.4		24	15.14	odd	2
75.8.e.d.68.9		24	5.4	even	2