Embedded newform 15.12.e.a.2.1

• Label: 15.12.e.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-61.8751 + 61.8751i) q^{2} +(3.69942 - 420.872i) q^{3} -5609.07i q^{4} +(-4347.55 - 5470.55i) q^{5} +(25812.6 + 26270.4i) q^{6} +(-11712.7 - 11712.7i) q^{7} +(220342. + 220342. i) q^{8} +(-177120. - 3113.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−61.8751	+	61.8751i	−1.36726	+	1.36726i	−0.502938	+	0.864322i	\(0.667748\pi\)
							−0.864322	+	0.502938i	\(0.832252\pi\)
\(3\)	3.69942	−	420.872i	0.00878954	−	0.999961i
							\(5\)	−4347.55	−	5470.55i	−0.622171	−	0.782882i
\(7\)	−11712.7	−	11712.7i	−0.263401	−	0.263401i								0.563033	−	0.826434i	\(-0.309634\pi\)
							−0.826434	+	0.563033i	\(0.809634\pi\)
\(11\)		−	435843.i		−	0.815964i	−0.912990	−	0.407982i	\(-0.866233\pi\)
							0.912990	−	0.407982i	\(-0.133767\pi\)
\(13\)	−468596.	+	468596.i	−0.350034	+	0.350034i	−0.860122	−	0.510088i	\(-0.829613\pi\)
							0.510088	+	0.860122i	\(0.329613\pi\)
\(17\)	2.28332e6	−	2.28332e6i	0.390030	−	0.390030i	−0.484668	−	0.874698i	\(-0.661060\pi\)
							0.874698	+	0.484668i	\(0.161060\pi\)
\(19\)			1.22767e6i			0.113746i	0.998381	+	0.0568731i	\(0.0181130\pi\)
							−0.998381	+	0.0568731i	\(0.981887\pi\)
\(23\)	1.59097e7	+	1.59097e7i	0.515417	+	0.515417i	0.916181	−	0.400764i	\(-0.131255\pi\)
							−0.400764	+	0.916181i	\(0.631255\pi\)
\(29\)	−1.86070e8			−1.68457			−0.842283	−	0.539036i	\(-0.818789\pi\)
							−0.842283	+	0.539036i	\(0.818789\pi\)
\(31\)	−3.77714e7			−0.236959			−0.118480	−	0.992956i	\(-0.537802\pi\)
							−0.118480	+	0.992956i	\(0.537802\pi\)
\(37\)	3.83394e8	+	3.83394e8i	0.908942	+	0.908942i	0.996187	−	0.0872453i	\(-0.0278064\pi\)
							−0.0872453	+	0.996187i	\(0.527806\pi\)
\(41\)			5.80541e8i			0.782567i	0.920270	+	0.391283i	\(0.127969\pi\)
							−0.920270	+	0.391283i	\(0.872031\pi\)
\(43\)	1.11553e8	−	1.11553e8i	0.115719	−	0.115719i	−0.646876	−	0.762595i	\(-0.723925\pi\)
							0.762595	+	0.646876i	\(0.223925\pi\)
\(47\)	3.45632e8	−	3.45632e8i	0.219824	−	0.219824i	−0.588600	−	0.808424i	\(-0.700321\pi\)
							0.808424	+	0.588600i	\(0.200321\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.1	✓	40
3.2	odd	2	inner	15.12.e.a.2.20	yes	40
5.2	odd	4		75.12.e.d.68.1		40
5.3	odd	4	inner	15.12.e.a.8.20	yes	40
5.4	even	2		75.12.e.d.32.20		40
15.2	even	4		75.12.e.d.68.20		40
15.8	even	4	inner	15.12.e.a.8.1	yes	40
15.14	odd	2		75.12.e.d.32.1		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.1	✓	40	1.1	even	1	trivial
15.12.e.a.2.20	yes	40	3.2	odd	2	inner
15.12.e.a.8.1	yes	40	15.8	even	4	inner
15.12.e.a.8.20	yes	40	5.3	odd	4	inner
75.12.e.d.32.1		40	15.14	odd	2
75.12.e.d.32.20		40	5.4	even	2
75.12.e.d.68.1		40	5.2	odd	4
75.12.e.d.68.20		40	15.2	even	4