Newform orbit 980.2.c.e

• Label: 980.2.c.e
• Level: $980$
• Weight: $2$
• Character orbit: 980.c
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 16 q^{4} - 64 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
979.1	−1.38298	−	0.295571i		−	1.66243i	1.82528	+	0.817539i	2.22517	−	0.220514i	−0.491366	+	2.29911i		0		−2.28268	−	1.67014i	0.236337			−3.14254	−	0.352730i
979.2	−1.38298	−	0.295571i			1.66243i	1.82528	+	0.817539i	−2.22517	+	0.220514i	0.491366	−	2.29911i		0		−2.28268	−	1.67014i	0.236337			3.14254	+	0.352730i
979.3	−1.38298	+	0.295571i		−	1.66243i	1.82528	−	0.817539i	−2.22517	−	0.220514i	0.491366	+	2.29911i		0		−2.28268	+	1.67014i	0.236337			3.14254	−	0.352730i
979.4	−1.38298	+	0.295571i			1.66243i	1.82528	−	0.817539i	2.22517	+	0.220514i	−0.491366	−	2.29911i		0		−2.28268	+	1.67014i	0.236337			−3.14254	+	0.352730i
979.5	−1.36075	−	0.385185i		−	0.423388i	1.70326	+	1.04828i	1.74517	−	1.39799i	−0.163083	+	0.576124i		0		−1.91393	−	2.08252i	2.82074			−2.91323	+	1.23009i
979.6	−1.36075	−	0.385185i			0.423388i	1.70326	+	1.04828i	−1.74517	+	1.39799i	0.163083	−	0.576124i		0		−1.91393	−	2.08252i	2.82074			2.91323	−	1.23009i
979.7	−1.36075	+	0.385185i		−	0.423388i	1.70326	−	1.04828i	−1.74517	−	1.39799i	0.163083	+	0.576124i		0		−1.91393	+	2.08252i	2.82074			2.91323	+	1.23009i
979.8	−1.36075	+	0.385185i			0.423388i	1.70326	−	1.04828i	1.74517	+	1.39799i	−0.163083	−	0.576124i		0		−1.91393	+	2.08252i	2.82074			−2.91323	−	1.23009i
979.9	−1.23364	−	0.691471i		−	2.08008i	1.04374	+	1.70605i	1.52262	+	1.63757i	−1.43832	+	2.56607i		0		−0.107908	−	2.82637i	−1.32674			−0.746026	−	3.07302i
979.10	−1.23364	−	0.691471i			2.08008i	1.04374	+	1.70605i	−1.52262	−	1.63757i	1.43832	−	2.56607i		0		−0.107908	−	2.82637i	−1.32674			0.746026	+	3.07302i
979.11	−1.23364	+	0.691471i		−	2.08008i	1.04374	−	1.70605i	−1.52262	+	1.63757i	1.43832	+	2.56607i		0		−0.107908	+	2.82637i	−1.32674			0.746026	−	3.07302i
979.12	−1.23364	+	0.691471i			2.08008i	1.04374	−	1.70605i	1.52262	−	1.63757i	−1.43832	−	2.56607i		0		−0.107908	+	2.82637i	−1.32674			−0.746026	+	3.07302i
979.13	−1.14730	−	0.826865i		−	1.52335i	0.632590	+	1.89732i	0.0967363	−	2.23397i	−1.25961	+	1.74774i		0		0.843059	−	2.69986i	0.679393			−1.95818	+	2.48305i
979.14	−1.14730	−	0.826865i			1.52335i	0.632590	+	1.89732i	−0.0967363	+	2.23397i	1.25961	−	1.74774i		0		0.843059	−	2.69986i	0.679393			1.95818	−	2.48305i
979.15	−1.14730	+	0.826865i		−	1.52335i	0.632590	−	1.89732i	−0.0967363	−	2.23397i	1.25961	+	1.74774i		0		0.843059	+	2.69986i	0.679393			1.95818	+	2.48305i
979.16	−1.14730	+	0.826865i			1.52335i	0.632590	−	1.89732i	0.0967363	+	2.23397i	−1.25961	−	1.74774i		0		0.843059	+	2.69986i	0.679393			−1.95818	−	2.48305i
979.17	−0.576258	−	1.29148i		−	2.50150i	−1.33585	+	1.48845i	−0.639901	+	2.14255i	−3.23064	+	1.44151i		0		2.69211	+	0.867500i	−3.25749			3.13582	−	0.408241i
979.18	−0.576258	−	1.29148i			2.50150i	−1.33585	+	1.48845i	0.639901	−	2.14255i	3.23064	−	1.44151i		0		2.69211	+	0.867500i	−3.25749			−3.13582	+	0.408241i
979.19	−0.576258	+	1.29148i		−	2.50150i	−1.33585	−	1.48845i	0.639901	+	2.14255i	3.23064	+	1.44151i		0		2.69211	−	0.867500i	−3.25749			−3.13582	−	0.408241i
979.20	−0.576258	+	1.29148i			2.50150i	−1.33585	−	1.48845i	−0.639901	−	2.14255i	−3.23064	−	1.44151i		0		2.69211	−	0.867500i	−3.25749			3.13582	+	0.408241i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
7.b	odd	2	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
35.c	odd	2	1	inner
140.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.c.e	✓	48
4.b	odd	2	1	inner	980.2.c.e	✓	48
5.b	even	2	1	inner	980.2.c.e	✓	48
7.b	odd	2	1	inner	980.2.c.e	✓	48
7.c	even	3	2		980.2.s.g		96
7.d	odd	6	2		980.2.s.g		96
20.d	odd	2	1	inner	980.2.c.e	✓	48
28.d	even	2	1	inner	980.2.c.e	✓	48
28.f	even	6	2		980.2.s.g		96
28.g	odd	6	2		980.2.s.g		96
35.c	odd	2	1	inner	980.2.c.e	✓	48
35.i	odd	6	2		980.2.s.g		96
35.j	even	6	2		980.2.s.g		96
140.c	even	2	1	inner	980.2.c.e	✓	48
140.p	odd	6	2		980.2.s.g		96
140.s	even	6	2		980.2.s.g		96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.2.c.e	✓	48	1.a	even	1	1	trivial
980.2.c.e	✓	48	4.b	odd	2	1	inner
980.2.c.e	✓	48	5.b	even	2	1	inner
980.2.c.e	✓	48	7.b	odd	2	1	inner
980.2.c.e	✓	48	20.d	odd	2	1	inner
980.2.c.e	✓	48	28.d	even	2	1	inner
980.2.c.e	✓	48	35.c	odd	2	1	inner
980.2.c.e	✓	48	140.c	even	2	1	inner
980.2.s.g		96	7.c	even	3	2
980.2.s.g		96	7.d	odd	6	2
980.2.s.g		96	28.f	even	6	2
980.2.s.g		96	28.g	odd	6	2
980.2.s.g		96	35.i	odd	6	2
980.2.s.g		96	35.j	even	6	2
980.2.s.g		96	140.p	odd	6	2
980.2.s.g		96	140.s	even	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 26T_{3}^{10} + 251T_{3}^{8} + 1136T_{3}^{6} + 2456T_{3}^{4} + 2168T_{3}^{2} + 316 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).