Embedded newform 15.12.e.a.2.4

• Label: 15.12.e.a.2.4
• Level: $15$
• Weight: $12$
• Character: 15.2
• Analytic conductor: $11.525$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5251477084\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.4
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-43.6110 + 43.6110i) q^{2} +(-418.328 - 46.3532i) q^{3} -1755.85i q^{4} +(6653.83 - 2134.16i) q^{5} +(20265.2 - 16222.2i) q^{6} +(35410.1 + 35410.1i) q^{7} +(-12741.2 - 12741.2i) q^{8} +(172850. + 38781.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−43.6110	+	43.6110i	−0.963677	+	0.963677i	−0.999363	−	0.0356861i	\(-0.988638\pi\)
							0.0356861	+	0.999363i	\(0.488638\pi\)
\(3\)	−418.328	−	46.3532i	−0.993917	−	0.110132i
							\(5\)	6653.83	−	2134.16i	0.952219	−	0.305416i
\(7\)	35410.1	+	35410.1i	0.796321	+	0.796321i								0.982513	−	0.186192i	\(-0.0596147\pi\)
							−0.186192	+	0.982513i	\(0.559615\pi\)
\(11\)		−	524649.i		−	0.982220i	−0.871098	−	0.491110i	\(-0.836591\pi\)
							0.871098	−	0.491110i	\(-0.163409\pi\)
\(13\)	−409234.	+	409234.i	−0.305691	+	0.305691i	−0.843236	−	0.537544i	\(-0.819352\pi\)
							0.537544	+	0.843236i	\(0.319352\pi\)
\(17\)	−4.40971e6	+	4.40971e6i	−0.753253	+	0.753253i	−0.975085	−	0.221832i	\(-0.928796\pi\)
							0.221832	+	0.975085i	\(0.428796\pi\)
\(19\)			9.09734e6i			0.842888i	0.906855	+	0.421444i	\(0.138477\pi\)
							−0.906855	+	0.421444i	\(0.861523\pi\)
\(23\)	3.10110e7	+	3.10110e7i	1.00464	+	1.00464i	0.999989	+	0.00465491i	\(0.00148171\pi\)
							0.00465491	+	0.999989i	\(0.498518\pi\)
\(29\)	−1.53459e7			−0.138932			−0.0694661	−	0.997584i	\(-0.522130\pi\)
							−0.0694661	+	0.997584i	\(0.522130\pi\)
\(31\)	−1.36758e8			−0.857950			−0.428975	−	0.903316i	\(-0.641125\pi\)
							−0.428975	+	0.903316i	\(0.641125\pi\)
\(37\)	5.28154e8	+	5.28154e8i	1.25213	+	1.25213i	0.954762	+	0.297372i	\(0.0961102\pi\)
							0.297372	+	0.954762i	\(0.403890\pi\)
\(41\)			1.17968e8i			0.159021i	0.996834	+	0.0795104i	\(0.0253357\pi\)
							−0.996834	+	0.0795104i	\(0.974664\pi\)
\(43\)	−6.42326e8	+	6.42326e8i	−0.666315	+	0.666315i	−0.956861	−	0.290546i	\(-0.906163\pi\)
							0.290546	+	0.956861i	\(0.406163\pi\)
\(47\)	−1.78057e9	+	1.78057e9i	−1.13246	+	1.13246i	−0.142688	+	0.989768i	\(0.545574\pi\)
							−0.989768	+	0.142688i	\(0.954426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.4	✓	40
3.2	odd	2	inner	15.12.e.a.2.17	yes	40
5.2	odd	4		75.12.e.d.68.4		40
5.3	odd	4	inner	15.12.e.a.8.17	yes	40
5.4	even	2		75.12.e.d.32.17		40
15.2	even	4		75.12.e.d.68.17		40
15.8	even	4	inner	15.12.e.a.8.4	yes	40
15.14	odd	2		75.12.e.d.32.4		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.4	✓	40	1.1	even	1	trivial
15.12.e.a.2.17	yes	40	3.2	odd	2	inner
15.12.e.a.8.4	yes	40	15.8	even	4	inner
15.12.e.a.8.17	yes	40	5.3	odd	4	inner
75.12.e.d.32.4		40	15.14	odd	2
75.12.e.d.32.17		40	5.4	even	2
75.12.e.d.68.4		40	5.2	odd	4
75.12.e.d.68.17		40	15.2	even	4