Embedded newform 15.12.e.a.2.3

• Label: 15.12.e.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-48.4604 + 48.4604i) q^{2} +(-342.528 + 244.585i) q^{3} -2648.82i q^{4} +(-5084.66 + 4793.15i) q^{5} +(4746.35 - 28451.7i) q^{6} +(-47877.8 - 47877.8i) q^{7} +(29116.1 + 29116.1i) q^{8} +(57503.4 - 167554. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−48.4604	+	48.4604i	−1.07083	+	1.07083i	−0.0735418	+	0.997292i	\(0.523430\pi\)
							−0.997292	+	0.0735418i	\(0.976570\pi\)
\(3\)	−342.528	+	244.585i	−0.813821	+	0.581116i
							\(5\)	−5084.66	+	4793.15i	−0.727658	+	0.685940i
\(7\)	−47877.8	−	47877.8i	−1.07670	−	1.07670i								−0.996803	−	0.0798986i	\(-0.974540\pi\)
							−0.0798986	−	0.996803i	\(-0.525460\pi\)
\(11\)			840806.i			1.57411i	0.616880	+	0.787057i	\(0.288396\pi\)
							−0.616880	+	0.787057i	\(0.711604\pi\)
\(13\)	−1.15438e6	+	1.15438e6i	−0.862300	+	0.862300i	−0.991605	−	0.129305i	\(-0.958726\pi\)
							0.129305	+	0.991605i	\(0.458726\pi\)
\(17\)	−1.93597e6	+	1.93597e6i	−0.330696	+	0.330696i	−0.852851	−	0.522155i	\(-0.825128\pi\)
							0.522155	+	0.852851i	\(0.325128\pi\)
\(19\)			1.46302e6i			0.135552i	0.997701	+	0.0677759i	\(0.0215903\pi\)
							−0.997701	+	0.0677759i	\(0.978410\pi\)
\(23\)	−8.35631e6	−	8.35631e6i	−0.270714	−	0.270714i	0.558673	−	0.829388i	\(-0.311311\pi\)
							−0.829388	+	0.558673i	\(0.811311\pi\)
\(29\)	−2.19424e7			−0.198653			−0.0993267	−	0.995055i	\(-0.531669\pi\)
							−0.0993267	+	0.995055i	\(0.531669\pi\)
\(31\)	3.79450e7			0.238048			0.119024	−	0.992891i	\(-0.462023\pi\)
							0.119024	+	0.992891i	\(0.462023\pi\)
\(37\)	2.67377e8	+	2.67377e8i	0.633892	+	0.633892i	0.949042	−	0.315150i	\(-0.102055\pi\)
							−0.315150	+	0.949042i	\(0.602055\pi\)
\(41\)		−	4.76890e8i		−	0.642846i	−0.946936	−	0.321423i	\(-0.895839\pi\)
							0.946936	−	0.321423i	\(-0.104161\pi\)
\(43\)	6.84811e8	−	6.84811e8i	0.710386	−	0.710386i	−0.256230	−	0.966616i	\(-0.582480\pi\)
							0.966616	+	0.256230i	\(0.0824804\pi\)
\(47\)	7.25295e8	−	7.25295e8i	0.461293	−	0.461293i	−0.437786	−	0.899079i	\(-0.644237\pi\)
							0.899079	+	0.437786i	\(0.144237\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.3	✓	40
3.2	odd	2	inner	15.12.e.a.2.18	yes	40
5.2	odd	4		75.12.e.d.68.3		40
5.3	odd	4	inner	15.12.e.a.8.18	yes	40
5.4	even	2		75.12.e.d.32.18		40
15.2	even	4		75.12.e.d.68.18		40
15.8	even	4	inner	15.12.e.a.8.3	yes	40
15.14	odd	2		75.12.e.d.32.3		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.3	✓	40	1.1	even	1	trivial
15.12.e.a.2.18	yes	40	3.2	odd	2	inner
15.12.e.a.8.3	yes	40	15.8	even	4	inner
15.12.e.a.8.18	yes	40	5.3	odd	4	inner
75.12.e.d.32.3		40	15.14	odd	2
75.12.e.d.32.18		40	5.4	even	2
75.12.e.d.68.3		40	5.2	odd	4
75.12.e.d.68.18		40	15.2	even	4