Embedded newform 15.12.e.a.8.14

• Label: 15.12.e.a.8.14
• Level: $15$
• Weight: $12$
• Character: 15.8
• Analytic conductor: $11.525$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• 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			8.14
Character	\(\chi\)	\(=\)	15.8
Dual form			15.12.e.a.2.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(26.1390 + 26.1390i) q^{2} +(389.593 - 159.262i) q^{3} -681.505i q^{4} +(2800.46 + 6402.00i) q^{5} +(14346.5 + 6020.60i) q^{6} +(5513.09 - 5513.09i) q^{7} +(71346.5 - 71346.5i) q^{8} +(126418. - 124095. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	26.1390	+	26.1390i	0.577596	+	0.577596i	0.934240	−	0.356644i	\(-0.116079\pi\)
							−0.356644	+	0.934240i	\(0.616079\pi\)
\(3\)	389.593	−	159.262i	0.925644	−	0.378396i
							\(5\)	2800.46	+	6402.00i	0.400769	+	0.916179i
\(7\)	5513.09	−	5513.09i	0.123981	−	0.123981i								−0.642394	−	0.766375i	\(-0.722059\pi\)
							0.766375	+	0.642394i	\(0.222059\pi\)
\(11\)		−	213111.i		−	0.398975i	−0.979900	−	0.199488i	\(-0.936072\pi\)
							0.979900	−	0.199488i	\(-0.0639278\pi\)
\(13\)	1.40784e6	+	1.40784e6i	1.05163	+	1.05163i	0.998592	+	0.0530403i	\(0.0168912\pi\)
							0.0530403	+	0.998592i	\(0.483109\pi\)
\(17\)	−2.24338e6	−	2.24338e6i	−0.383207	−	0.383207i	0.489049	−	0.872256i	\(-0.337344\pi\)
							−0.872256	+	0.489049i	\(0.837344\pi\)
\(19\)			1.02378e7i			0.948557i	0.880375	+	0.474279i	\(0.157291\pi\)
							−0.880375	+	0.474279i	\(0.842709\pi\)
\(23\)	1.46579e7	−	1.46579e7i	0.474863	−	0.474863i	−0.428622	−	0.903484i	\(-0.641001\pi\)
							0.903484	+	0.428622i	\(0.141001\pi\)
\(29\)	−1.52840e8			−1.38372			−0.691859	−	0.722033i	\(-0.743208\pi\)
							−0.691859	+	0.722033i	\(0.743208\pi\)
\(31\)	−2.64198e8			−1.65745			−0.828725	−	0.559656i	\(-0.810933\pi\)
							−0.828725	+	0.559656i	\(0.810933\pi\)
\(37\)	2.07043e8	−	2.07043e8i	0.490853	−	0.490853i	−0.417722	−	0.908575i	\(-0.637171\pi\)
							0.908575	+	0.417722i	\(0.137171\pi\)
\(41\)		−	1.29665e8i		−	0.174788i	−0.996174	−	0.0873939i	\(-0.972146\pi\)
							0.996174	−	0.0873939i	\(-0.0278539\pi\)
\(43\)	−2.00424e8	−	2.00424e8i	−0.207909	−	0.207909i	0.595469	−	0.803378i	\(-0.296966\pi\)
							−0.803378	+	0.595469i	\(0.796966\pi\)
\(47\)	−9.40322e8	−	9.40322e8i	−0.598051	−	0.598051i	0.341742	−	0.939794i	\(-0.388983\pi\)
							−0.939794	+	0.341742i	\(0.888983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.8.14	yes	40
3.2	odd	2	inner	15.12.e.a.8.7	yes	40
5.2	odd	4	inner	15.12.e.a.2.7	✓	40
5.3	odd	4		75.12.e.d.32.14		40
5.4	even	2		75.12.e.d.68.7		40
15.2	even	4	inner	15.12.e.a.2.14	yes	40
15.8	even	4		75.12.e.d.32.7		40
15.14	odd	2		75.12.e.d.68.14		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.7	✓	40	5.2	odd	4	inner
15.12.e.a.2.14	yes	40	15.2	even	4	inner
15.12.e.a.8.7	yes	40	3.2	odd	2	inner
15.12.e.a.8.14	yes	40	1.1	even	1	trivial
75.12.e.d.32.7		40	15.8	even	4
75.12.e.d.32.14		40	5.3	odd	4
75.12.e.d.68.7		40	5.4	even	2
75.12.e.d.68.14		40	15.14	odd	2