Embedded newform 15.12.e.a.2.16

• Label: 15.12.e.a.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(37.4902 - 37.4902i) q^{2} +(201.005 - 369.789i) q^{3} -763.031i q^{4} +(-6987.38 + 67.8135i) q^{5} +(-6327.78 - 21399.2i) q^{6} +(-46363.7 - 46363.7i) q^{7} +(48173.8 + 48173.8i) q^{8} +(-96341.3 - 148659. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	37.4902	−	37.4902i	0.828424	−	0.828424i	−0.158874	−	0.987299i	\(-0.550787\pi\)
							0.987299	+	0.158874i	\(0.0507865\pi\)
\(3\)	201.005	−	369.789i	0.477572	−	0.878593i
							\(5\)	−6987.38	+	67.8135i	−0.999953	+	0.00970467i
\(7\)	−46363.7	−	46363.7i	−1.04265	−	1.04265i								−0.999049	−	0.0436027i	\(-0.986116\pi\)
							−0.0436027	−	0.999049i	\(-0.513884\pi\)
\(11\)		−	55617.1i		−	0.104123i	−0.998644	−	0.0520617i	\(-0.983421\pi\)
							0.998644	−	0.0520617i	\(-0.0165793\pi\)
\(13\)	631358.	−	631358.i	0.471615	−	0.471615i	−0.430822	−	0.902437i	\(-0.641776\pi\)
							0.902437	+	0.430822i	\(0.141776\pi\)
\(17\)	−2.79681e6	+	2.79681e6i	−0.477743	+	0.477743i	−0.904409	−	0.426666i	\(-0.859688\pi\)
							0.426666	+	0.904409i	\(0.359688\pi\)
\(19\)		−	1.01926e7i		−	0.944368i	−0.881500	−	0.472184i	\(-0.843466\pi\)
							0.881500	−	0.472184i	\(-0.156534\pi\)
\(23\)	−1.26689e7	−	1.26689e7i	−0.410426	−	0.410426i	0.471461	−	0.881887i	\(-0.343727\pi\)
							−0.881887	+	0.471461i	\(0.843727\pi\)
\(29\)	1.48130e8			1.34108			0.670539	−	0.741874i	\(-0.266063\pi\)
							0.670539	+	0.741874i	\(0.266063\pi\)
\(31\)	2.61818e8			1.64252			0.821258	−	0.570557i	\(-0.193273\pi\)
							0.821258	+	0.570557i	\(0.193273\pi\)
\(37\)	−3.41740e8	−	3.41740e8i	−0.810189	−	0.810189i	0.174473	−	0.984662i	\(-0.444178\pi\)
							−0.984662	+	0.174473i	\(0.944178\pi\)
\(41\)		−	1.32229e9i		−	1.78244i	−0.453567	−	0.891222i	\(-0.649849\pi\)
							0.453567	−	0.891222i	\(-0.350151\pi\)
\(43\)	−6.29639e7	+	6.29639e7i	−0.0653153	+	0.0653153i	−0.739010	−	0.673695i	\(-0.764706\pi\)
							0.673695	+	0.739010i	\(0.264706\pi\)
\(47\)	−1.38825e9	+	1.38825e9i	−0.882939	+	0.882939i	−0.993832	−	0.110894i	\(-0.964629\pi\)
							0.110894	+	0.993832i	\(0.464629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.16	yes	40
3.2	odd	2	inner	15.12.e.a.2.5	✓	40
5.2	odd	4		75.12.e.d.68.16		40
5.3	odd	4	inner	15.12.e.a.8.5	yes	40
5.4	even	2		75.12.e.d.32.5		40
15.2	even	4		75.12.e.d.68.5		40
15.8	even	4	inner	15.12.e.a.8.16	yes	40
15.14	odd	2		75.12.e.d.32.16		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.5	✓	40	3.2	odd	2	inner
15.12.e.a.2.16	yes	40	1.1	even	1	trivial
15.12.e.a.8.5	yes	40	5.3	odd	4	inner
15.12.e.a.8.16	yes	40	15.8	even	4	inner
75.12.e.d.32.5		40	5.4	even	2
75.12.e.d.32.16		40	15.14	odd	2
75.12.e.d.68.5		40	15.2	even	4
75.12.e.d.68.16		40	5.2	odd	4