Embedded newform 15.4.e.a.2.3

• Label: 15.4.e.a.2.3
• Level: $15$
• Weight: $4$
• Character: 15.2
• Analytic conductor: $0.885$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.885028650086\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	8.0.28356903014400.8
Defining polynomial:	\( x^{8} + 209x^{4} + 1600 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.3
Root			\(1.18766 + 1.18766i\) of defining polynomial
Character	\(\chi\)	\(=\)	15.2
Dual form			15.4.e.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18766 - 1.18766i) q^{2} +(-0.932827 - 5.11173i) q^{3} +5.17891i q^{4} +(2.48157 + 10.9015i) q^{5} +(-7.17891 - 4.96314i) q^{6} +(-13.3578 - 13.3578i) q^{7} +(15.6521 + 15.6521i) q^{8} +(-25.2597 + 9.53673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 12 q^{6} - 16 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18766	−	1.18766i	0.419903	−	0.419903i	−0.465267	−	0.885170i	\(-0.654042\pi\)
							0.885170	+	0.465267i	\(0.154042\pi\)
\(3\)	−0.932827	−	5.11173i	−0.179523	−	0.983754i
							\(5\)	2.48157	+	10.9015i	0.221958	+	0.975056i
\(7\)	−13.3578	−	13.3578i	−0.721254	−	0.721254i								0.247606	−	0.968861i	\(-0.420356\pi\)
							−0.968861	+	0.247606i	\(0.920356\pi\)
\(11\)		−	28.7164i		−	0.787121i	−0.919299	−	0.393560i	\(-0.871243\pi\)
							0.919299	−	0.393560i	\(-0.128757\pi\)
\(13\)	14.1789	−	14.1789i	0.302502	−	0.302502i	−0.539490	−	0.841992i	\(-0.681383\pi\)
							0.841992	+	0.539490i	\(0.181383\pi\)
\(17\)	18.5587	−	18.5587i	0.264773	−	0.264773i	−0.562217	−	0.826990i	\(-0.690051\pi\)
							0.826990	+	0.562217i	\(0.190051\pi\)
\(19\)			49.0735i			0.592538i	0.955105	+	0.296269i	\(0.0957425\pi\)
							−0.955105	+	0.296269i	\(0.904258\pi\)
\(23\)	−37.7738	−	37.7738i	−0.342451	−	0.342451i	0.514837	−	0.857288i	\(-0.327852\pi\)
							−0.857288	+	0.514837i	\(0.827852\pi\)
\(29\)	125.854			0.805882			0.402941	−	0.915226i	\(-0.367988\pi\)
							0.402941	+	0.915226i	\(0.367988\pi\)
\(31\)	247.367			1.43318			0.716588	−	0.697496i	\(-0.245703\pi\)
							0.716588	+	0.697496i	\(0.245703\pi\)
\(37\)	−127.463	−	127.463i	−0.566347	−	0.566347i	0.364756	−	0.931103i	\(-0.381152\pi\)
							−0.931103	+	0.364756i	\(0.881152\pi\)
\(41\)			390.328i			1.48680i	0.668845	+	0.743402i	\(0.266789\pi\)
							−0.668845	+	0.743402i	\(0.733211\pi\)
\(43\)	−39.3993	+	39.3993i	−0.139729	+	0.139729i	−0.773511	−	0.633783i	\(-0.781501\pi\)
							0.633783	+	0.773511i	\(0.281501\pi\)
\(47\)	124.560	−	124.560i	0.386575	−	0.386575i	−0.486889	−	0.873464i	\(-0.661868\pi\)
							0.873464	+	0.486889i	\(0.161868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.4.e.a.2.3	yes	8
3.2	odd	2	inner	15.4.e.a.2.2	✓	8
4.3	odd	2		240.4.v.c.17.2		8
5.2	odd	4		75.4.e.c.68.3		8
5.3	odd	4	inner	15.4.e.a.8.2	yes	8
5.4	even	2		75.4.e.c.32.2		8
12.11	even	2		240.4.v.c.17.1		8
15.2	even	4		75.4.e.c.68.2		8
15.8	even	4	inner	15.4.e.a.8.3	yes	8
15.14	odd	2		75.4.e.c.32.3		8
20.3	even	4		240.4.v.c.113.1		8
60.23	odd	4		240.4.v.c.113.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.e.a.2.2	✓	8	3.2	odd	2	inner
15.4.e.a.2.3	yes	8	1.1	even	1	trivial
15.4.e.a.8.2	yes	8	5.3	odd	4	inner
15.4.e.a.8.3	yes	8	15.8	even	4	inner
75.4.e.c.32.2		8	5.4	even	2
75.4.e.c.32.3		8	15.14	odd	2
75.4.e.c.68.2		8	15.2	even	4
75.4.e.c.68.3		8	5.2	odd	4
240.4.v.c.17.1		8	12.11	even	2
240.4.v.c.17.2		8	4.3	odd	2
240.4.v.c.113.1		8	20.3	even	4
240.4.v.c.113.2		8	60.23	odd	4