Embedded newform 15.12.e.a.2.13

• Label: 15.12.e.a.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	15.2
Dual form			15.12.e.a.8.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(21.4344 - 21.4344i) q^{2} +(205.234 - 367.459i) q^{3} +1129.13i q^{4} +(6189.39 + 3243.38i) q^{5} +(-3477.19 - 12275.3i) q^{6} +(45393.8 + 45393.8i) q^{7} +(68099.9 + 68099.9i) q^{8} +(-92905.0 - 150830. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 504 q^{3} + 2940 q^{6} + 31504 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	21.4344	−	21.4344i	0.473638	−	0.473638i	−0.429452	−	0.903090i	\(-0.641293\pi\)
							0.903090	+	0.429452i	\(0.141293\pi\)
\(3\)	205.234	−	367.459i	0.487621	−	0.873055i
							\(5\)	6189.39	+	3243.38i	0.885754	+	0.464155i
\(7\)	45393.8	+	45393.8i	1.02084	+	1.02084i								0.999778	+	0.0210609i	\(0.00670439\pi\)
							0.0210609	+	0.999778i	\(0.493296\pi\)
\(11\)		−	645995.i		−	1.20940i	−0.796454	−	0.604699i	\(-0.793293\pi\)
							0.796454	−	0.604699i	\(-0.206707\pi\)
\(13\)	−946612.	+	946612.i	−0.707104	+	0.707104i	−0.965925	−	0.258821i	\(-0.916666\pi\)
							0.258821	+	0.965925i	\(0.416666\pi\)
\(17\)	7.24571e6	−	7.24571e6i	1.23769	−	1.23769i	0.276748	−	0.960942i	\(-0.410743\pi\)
							0.960942	−	0.276748i	\(-0.0892569\pi\)
\(19\)			4.63621e6i			0.429554i	0.976663	+	0.214777i	\(0.0689025\pi\)
							−0.976663	+	0.214777i	\(0.931097\pi\)
\(23\)	−7.22464e6	−	7.22464e6i	−0.234052	−	0.234052i	0.580329	−	0.814382i	\(-0.302924\pi\)
							−0.814382	+	0.580329i	\(0.802924\pi\)
\(29\)	−7.09879e7			−0.642681			−0.321341	−	0.946964i	\(-0.604133\pi\)
							−0.321341	+	0.946964i	\(0.604133\pi\)
\(31\)	−1.10632e8			−0.694054			−0.347027	−	0.937855i	\(-0.612809\pi\)
							−0.347027	+	0.937855i	\(0.612809\pi\)
\(37\)	1.47033e7	+	1.47033e7i	0.0348582	+	0.0348582i	0.724321	−	0.689463i	\(-0.242153\pi\)
							−0.689463	+	0.724321i	\(0.742153\pi\)
\(41\)		−	5.09469e8i		−	0.686763i	−0.939196	−	0.343381i	\(-0.888428\pi\)
							0.939196	−	0.343381i	\(-0.111572\pi\)
\(43\)	9.36283e8	−	9.36283e8i	0.971249	−	0.971249i	−0.0283489	−	0.999598i	\(-0.509025\pi\)
							0.999598	+	0.0283489i	\(0.00902494\pi\)
\(47\)	−1.38733e9	+	1.38733e9i	−0.882354	+	0.882354i	−0.993774	−	0.111419i	\(-0.964460\pi\)
							0.111419	+	0.993774i	\(0.464460\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.e.a.2.13	yes	40
3.2	odd	2	inner	15.12.e.a.2.8	✓	40
5.2	odd	4		75.12.e.d.68.13		40
5.3	odd	4	inner	15.12.e.a.8.8	yes	40
5.4	even	2		75.12.e.d.32.8		40
15.2	even	4		75.12.e.d.68.8		40
15.8	even	4	inner	15.12.e.a.8.13	yes	40
15.14	odd	2		75.12.e.d.32.13		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.e.a.2.8	✓	40	3.2	odd	2	inner
15.12.e.a.2.13	yes	40	1.1	even	1	trivial
15.12.e.a.8.8	yes	40	5.3	odd	4	inner
15.12.e.a.8.13	yes	40	15.8	even	4	inner
75.12.e.d.32.8		40	5.4	even	2
75.12.e.d.32.13		40	15.14	odd	2
75.12.e.d.68.8		40	15.2	even	4
75.12.e.d.68.13		40	5.2	odd	4