Embedded newform 2160.1.r.a.1459.2

• Label: 2160.1.r.a.1459.2
• Level: $2160$
• Weight: $1$
• Character: 2160.1459
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $8$
• Projective image: $D_{12}$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2160.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.07798042729\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{24})\)
Defining polynomial:	\( x^{8} - x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

Embedding invariants

Embedding label			1459.2
Root			\(0.258819 + 0.965926i\) of defining polynomial
Character	\(\chi\)	\(=\)	2160.1459
Dual form			2160.1.r.a.379.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.707107 + 0.707107i) q^{5} +(0.707107 + 0.707107i) q^{8} +(0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{16} +0.517638i q^{17} +(0.366025 - 0.366025i) q^{19} +(-0.965926 - 0.258819i) q^{20} +1.93185i q^{23} +1.00000i q^{25} -1.00000i q^{31} +(-0.965926 - 0.258819i) q^{32} +(0.500000 - 0.133975i) q^{34} +(-0.448288 - 0.258819i) q^{38} +1.00000i q^{40} +(1.86603 - 0.500000i) q^{46} -1.41421 q^{47} +1.00000 q^{49} +(0.965926 - 0.258819i) q^{50} +(1.22474 + 1.22474i) q^{53} +(1.36603 + 1.36603i) q^{61} +(-0.965926 + 0.258819i) q^{62} +1.00000i q^{64} +(-0.258819 - 0.448288i) q^{68} +(-0.133975 + 0.500000i) q^{76} -1.73205i q^{79} +(0.965926 - 0.258819i) q^{80} +(0.707107 + 0.707107i) q^{83} +(-0.366025 + 0.366025i) q^{85} +(-0.965926 - 1.67303i) q^{92} +(0.366025 + 1.36603i) q^{94} +0.517638 q^{95} +(-0.258819 - 0.965926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{10} + 4 q^{16} - 4 q^{19} + 4 q^{34} + 8 q^{46} + 8 q^{49} + 4 q^{61} - 8 q^{76} + 4 q^{85} - 4 q^{94}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(1297\)	\(1621\)	\(2081\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.258819	−	0.965926i	−0.258819	−	0.965926i
							\(3\)		0			0
\(5\)	0.707107	+	0.707107i	0.707107	+	0.707107i
							\(7\)		0			0		1.00000			\(0\)
−1.00000			\(\pi\)
\(11\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(13\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(17\)			0.517638i			0.517638i	0.965926	+	0.258819i	\(0.0833333\pi\)
							−0.965926	+	0.258819i	\(0.916667\pi\)
\(19\)	0.366025	−	0.366025i	0.366025	−	0.366025i	−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(23\)			1.93185i			1.93185i	0.258819	+	0.965926i	\(0.416667\pi\)
							−0.258819	+	0.965926i	\(0.583333\pi\)
\(29\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(31\)		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	−	0.500000i	\(-0.166667\pi\)
\(37\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(41\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(43\)		0			0		−0.707107	−	0.707107i	\(-0.750000\pi\)
							0.707107	+	0.707107i	\(0.250000\pi\)
\(47\)	−1.41421			−1.41421			−0.707107	−	0.707107i	\(-0.750000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(53\)	1.22474	+	1.22474i	1.22474	+	1.22474i	0.965926	+	0.258819i	\(0.0833333\pi\)
							0.258819	+	0.965926i	\(0.416667\pi\)
\(59\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(61\)	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	\(0.166667\pi\)
							0.500000	+	0.866025i	\(0.333333\pi\)
\(67\)		0			0		0.707107	−	0.707107i	\(-0.250000\pi\)
							−0.707107	+	0.707107i	\(0.750000\pi\)
\(71\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(73\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(79\)		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	\(-0.666667\pi\)
							0.500000	−	0.866025i	\(-0.333333\pi\)
\(83\)	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	\(-0.0833333\pi\)
							−0.258819	+	0.965926i	\(0.583333\pi\)
\(89\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)
\(97\)		0			0		1.00000			\(0\)
							−1.00000			\(\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.1.r.a.1459.2	yes	8
3.2	odd	2	inner	2160.1.r.a.1459.3	yes	8
5.4	even	2	inner	2160.1.r.a.1459.3	yes	8
15.14	odd	2	CM	2160.1.r.a.1459.2	yes	8
16.11	odd	4	inner	2160.1.r.a.379.3	yes	8
48.11	even	4	inner	2160.1.r.a.379.2	✓	8
80.59	odd	4	inner	2160.1.r.a.379.2	✓	8
240.59	even	4	inner	2160.1.r.a.379.3	yes	8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.1.r.a.379.2	✓	8	48.11	even	4	inner
2160.1.r.a.379.2	✓	8	80.59	odd	4	inner
2160.1.r.a.379.3	yes	8	16.11	odd	4	inner
2160.1.r.a.379.3	yes	8	240.59	even	4	inner
2160.1.r.a.1459.2	yes	8	1.1	even	1	trivial
2160.1.r.a.1459.2	yes	8	15.14	odd	2	CM
2160.1.r.a.1459.3	yes	8	3.2	odd	2	inner
2160.1.r.a.1459.3	yes	8	5.4	even	2	inner