Newform orbit 980.2.g.a

• Label: 980.2.g.a
• Level: $980$
• Weight: $2$
• Character orbit: 980.g
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 140)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
391.1	−1.40582	−	0.153886i	−3.02707			1.95264	+	0.432671i		−	1.00000i	4.25550	+	0.465823i		0		−2.67847	−	0.908739i	6.16313			−0.153886	+	1.40582i
391.2	−1.40582	−	0.153886i	3.02707			1.95264	+	0.432671i			1.00000i	−4.25550	−	0.465823i		0		−2.67847	−	0.908739i	6.16313			0.153886	−	1.40582i
391.3	−1.40582	+	0.153886i	−3.02707			1.95264	−	0.432671i			1.00000i	4.25550	−	0.465823i		0		−2.67847	+	0.908739i	6.16313			−0.153886	−	1.40582i
391.4	−1.40582	+	0.153886i	3.02707			1.95264	−	0.432671i		−	1.00000i	−4.25550	+	0.465823i		0		−2.67847	+	0.908739i	6.16313			0.153886	+	1.40582i
391.5	−1.34325	−	0.442358i	−0.901278			1.60864	+	1.18839i			1.00000i	1.21064	+	0.398687i		0		−1.63511	−	2.30790i	−2.18770			0.442358	−	1.34325i
391.6	−1.34325	−	0.442358i	0.901278			1.60864	+	1.18839i		−	1.00000i	−1.21064	−	0.398687i		0		−1.63511	−	2.30790i	−2.18770			−0.442358	+	1.34325i
391.7	−1.34325	+	0.442358i	−0.901278			1.60864	−	1.18839i		−	1.00000i	1.21064	−	0.398687i		0		−1.63511	+	2.30790i	−2.18770			0.442358	+	1.34325i
391.8	−1.34325	+	0.442358i	0.901278			1.60864	−	1.18839i			1.00000i	−1.21064	+	0.398687i		0		−1.63511	+	2.30790i	−2.18770			−0.442358	−	1.34325i
391.9	−0.976830	−	1.02265i	−1.11294			−0.0916073	+	1.99790i			1.00000i	1.08715	+	1.13814i		0		2.13263	−	1.85793i	−1.76137			1.02265	−	0.976830i
391.10	−0.976830	−	1.02265i	1.11294			−0.0916073	+	1.99790i		−	1.00000i	−1.08715	−	1.13814i		0		2.13263	−	1.85793i	−1.76137			−1.02265	+	0.976830i
391.11	−0.976830	+	1.02265i	−1.11294			−0.0916073	−	1.99790i		−	1.00000i	1.08715	−	1.13814i		0		2.13263	+	1.85793i	−1.76137			1.02265	+	0.976830i
391.12	−0.976830	+	1.02265i	1.11294			−0.0916073	−	1.99790i			1.00000i	−1.08715	+	1.13814i		0		2.13263	+	1.85793i	−1.76137			−1.02265	−	0.976830i
391.13	−0.431404	−	1.34681i	−2.73718			−1.62778	+	1.16204i		−	1.00000i	1.18083	+	3.68646i		0		2.26727	+	1.69100i	4.49217			−1.34681	+	0.431404i
391.14	−0.431404	−	1.34681i	2.73718			−1.62778	+	1.16204i			1.00000i	−1.18083	−	3.68646i		0		2.26727	+	1.69100i	4.49217			1.34681	−	0.431404i
391.15	−0.431404	+	1.34681i	−2.73718			−1.62778	−	1.16204i			1.00000i	1.18083	−	3.68646i		0		2.26727	−	1.69100i	4.49217			−1.34681	−	0.431404i
391.16	−0.431404	+	1.34681i	2.73718			−1.62778	−	1.16204i		−	1.00000i	−1.18083	+	3.68646i		0		2.26727	−	1.69100i	4.49217			1.34681	+	0.431404i
391.17	0.0982654	−	1.41080i	−0.662355			−1.98069	−	0.277265i			1.00000i	−0.0650866	+	0.934447i		0		−0.585797	+	2.76710i	−2.56129			1.41080	+	0.0982654i
391.18	0.0982654	−	1.41080i	0.662355			−1.98069	−	0.277265i		−	1.00000i	0.0650866	−	0.934447i		0		−0.585797	+	2.76710i	−2.56129			−1.41080	−	0.0982654i
391.19	0.0982654	+	1.41080i	−0.662355			−1.98069	+	0.277265i		−	1.00000i	−0.0650866	−	0.934447i		0		−0.585797	−	2.76710i	−2.56129			1.41080	−	0.0982654i
391.20	0.0982654	+	1.41080i	0.662355			−1.98069	+	0.277265i			1.00000i	0.0650866	+	0.934447i		0		−0.585797	−	2.76710i	−2.56129			−1.41080	+	0.0982654i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
28.d	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.g.a		32
4.b	odd	2	1	inner	980.2.g.a		32
7.b	odd	2	1	inner	980.2.g.a		32
7.c	even	3	1		140.2.o.a	✓	32
7.c	even	3	1		980.2.o.f		32
7.d	odd	6	1		140.2.o.a	✓	32
7.d	odd	6	1		980.2.o.f		32
28.d	even	2	1	inner	980.2.g.a		32
28.f	even	6	1		140.2.o.a	✓	32
28.f	even	6	1		980.2.o.f		32
28.g	odd	6	1		140.2.o.a	✓	32
28.g	odd	6	1		980.2.o.f		32
35.i	odd	6	1		700.2.p.c		32
35.j	even	6	1		700.2.p.c		32
35.k	even	12	1		700.2.t.c		32
35.k	even	12	1		700.2.t.d		32
35.l	odd	12	1		700.2.t.c		32
35.l	odd	12	1		700.2.t.d		32
140.p	odd	6	1		700.2.p.c		32
140.s	even	6	1		700.2.p.c		32
140.w	even	12	1		700.2.t.c		32
140.w	even	12	1		700.2.t.d		32
140.x	odd	12	1		700.2.t.c		32
140.x	odd	12	1		700.2.t.d		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.o.a	✓	32	7.c	even	3	1
140.2.o.a	✓	32	7.d	odd	6	1
140.2.o.a	✓	32	28.f	even	6	1
140.2.o.a	✓	32	28.g	odd	6	1
700.2.p.c		32	35.i	odd	6	1
700.2.p.c		32	35.j	even	6	1
700.2.p.c		32	140.p	odd	6	1
700.2.p.c		32	140.s	even	6	1
700.2.t.c		32	35.k	even	12	1
700.2.t.c		32	35.l	odd	12	1
700.2.t.c		32	140.w	even	12	1
700.2.t.c		32	140.x	odd	12	1
700.2.t.d		32	35.k	even	12	1
700.2.t.d		32	35.l	odd	12	1
700.2.t.d		32	140.w	even	12	1
700.2.t.d		32	140.x	odd	12	1
980.2.g.a		32	1.a	even	1	1	trivial
980.2.g.a		32	4.b	odd	2	1	inner
980.2.g.a		32	7.b	odd	2	1	inner
980.2.g.a		32	28.d	even	2	1	inner
980.2.o.f		32	7.c	even	3	1
980.2.o.f		32	7.d	odd	6	1
980.2.o.f		32	28.f	even	6	1
980.2.o.f		32	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 32T_{3}^{14} + 395T_{3}^{12} - 2368T_{3}^{10} + 7243T_{3}^{8} - 11424T_{3}^{6} + 9345T_{3}^{4} - 3744T_{3}^{2} + 576 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).