Newform orbit 140.2.o.a

• Label: 140.2.o.a
• Level: $140$
• Weight: $2$
• Character orbit: 140.o
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.39328	+	0.242400i	−0.406021	+	0.703249i	1.88248	−	0.675465i	0.866025	−	0.500000i	0.395235	−	1.07825i	0.336411	−	2.62428i	−2.45910	+	1.39743i	1.17029	+	2.02701i	−1.08542	+	0.906567i
31.2	−1.27092	+	0.620297i	−0.331177	+	0.573616i	1.23046	−	1.57669i	−0.866025	+	0.500000i	0.0650866	−	0.934447i	1.68748	+	2.03775i	−0.585797	+	2.76710i	1.28064	+	2.21814i	0.790498	−	1.17265i
31.3	−1.26796	−	0.626319i	−1.49907	+	2.59647i	1.21545	+	1.58830i	−0.866025	+	0.500000i	3.52698	−	2.35332i	−2.06101	−	1.65899i	−0.546365	−	2.77516i	−2.99443	−	5.18651i	1.41125	−	0.0915727i
31.4	−0.950668	+	1.04701i	1.36859	−	2.37047i	−0.192463	−	1.99072i	−0.866025	+	0.500000i	1.18083	+	3.68646i	−1.02440	−	2.43939i	2.26727	+	1.69100i	−2.24609	−	3.89033i	0.299797	−	1.38207i
31.5	−0.894275	−	1.09557i	0.895374	−	1.55083i	−0.400544	+	1.95948i	0.866025	−	0.500000i	−2.49976	+	0.405928i	0.644798	−	2.56598i	2.50494	−	1.31349i	−0.103389	−	0.179074i	−1.32225	−	0.501653i
31.6	−0.501653	−	1.32225i	−0.895374	+	1.55083i	−1.49669	+	1.32662i	0.866025	−	0.500000i	2.49976	+	0.405928i	−0.644798	+	2.56598i	2.50494	+	1.31349i	−0.103389	−	0.179074i	−1.09557	−	0.894275i
31.7	−0.397222	+	1.35728i	0.556469	−	0.963833i	−1.68443	−	1.07828i	0.866025	−	0.500000i	1.08715	+	1.13814i	2.32410	+	1.26433i	2.13263	−	1.85793i	0.880685	+	1.52539i	0.334637	+	1.37405i
31.8	0.0915727	−	1.41125i	1.49907	−	2.59647i	−1.98323	−	0.258463i	−0.866025	+	0.500000i	−3.52698	−	2.35332i	2.06101	+	1.65899i	−0.546365	+	2.77516i	−2.99443	−	5.18651i	0.626319	+	1.26796i
31.9	0.288532	+	1.38447i	−0.450639	+	0.780530i	−1.83350	+	0.798926i	−0.866025	+	0.500000i	−1.21064	−	0.398687i	−2.29962	+	1.30833i	−1.63511	−	2.30790i	1.09385	+	1.89460i	−0.942109	−	1.05472i
31.10	0.569639	+	1.29442i	−1.51353	+	2.62152i	−1.35102	+	1.47470i	0.866025	−	0.500000i	−4.25550	−	0.465823i	2.57616	−	0.602834i	−2.67847	−	0.908739i	−3.08156	−	5.33743i	1.14053	+	0.836177i
31.11	0.836177	+	1.14053i	1.51353	−	2.62152i	−0.601615	+	1.90737i	0.866025	−	0.500000i	4.25550	−	0.465823i	−2.57616	+	0.602834i	−2.67847	+	0.908739i	−3.08156	−	5.33743i	1.29442	+	0.569639i
31.12	0.906567	−	1.08542i	0.406021	−	0.703249i	−0.356272	−	1.96801i	0.866025	−	0.500000i	−0.395235	−	1.07825i	−0.336411	+	2.62428i	−2.45910	−	1.39743i	1.17029	+	2.02701i	0.242400	−	1.39328i
31.13	1.05472	+	0.942109i	0.450639	−	0.780530i	0.224860	+	1.98732i	−0.866025	+	0.500000i	1.21064	−	0.398687i	2.29962	−	1.30833i	−1.63511	+	2.30790i	1.09385	+	1.89460i	−1.38447	−	0.288532i
31.14	1.17265	−	0.790498i	0.331177	−	0.573616i	0.750225	−	1.85396i	−0.866025	+	0.500000i	−0.0650866	−	0.934447i	−1.68748	−	2.03775i	−0.585797	−	2.76710i	1.28064	+	2.21814i	−0.620297	+	1.27092i
31.15	1.37405	+	0.334637i	−0.556469	+	0.963833i	1.77604	+	0.919616i	0.866025	−	0.500000i	−1.08715	+	1.13814i	−2.32410	−	1.26433i	2.13263	+	1.85793i	0.880685	+	1.52539i	1.35728	−	0.397222i
31.16	1.38207	−	0.299797i	−1.36859	+	2.37047i	1.82024	−	0.828682i	−0.866025	+	0.500000i	−1.18083	+	3.68646i	1.02440	+	2.43939i	2.26727	−	1.69100i	−2.24609	−	3.89033i	−1.04701	+	0.950668i
131.1	−1.39328	−	0.242400i	−0.406021	−	0.703249i	1.88248	+	0.675465i	0.866025	+	0.500000i	0.395235	+	1.07825i	0.336411	+	2.62428i	−2.45910	−	1.39743i	1.17029	−	2.02701i	−1.08542	−	0.906567i
131.2	−1.27092	−	0.620297i	−0.331177	−	0.573616i	1.23046	+	1.57669i	−0.866025	−	0.500000i	0.0650866	+	0.934447i	1.68748	−	2.03775i	−0.585797	−	2.76710i	1.28064	−	2.21814i	0.790498	+	1.17265i
131.3	−1.26796	+	0.626319i	−1.49907	−	2.59647i	1.21545	−	1.58830i	−0.866025	−	0.500000i	3.52698	+	2.35332i	−2.06101	+	1.65899i	−0.546365	+	2.77516i	−2.99443	+	5.18651i	1.41125	+	0.0915727i
131.4	−0.950668	−	1.04701i	1.36859	+	2.37047i	−0.192463	+	1.99072i	−0.866025	−	0.500000i	1.18083	−	3.68646i	−1.02440	+	2.43939i	2.26727	−	1.69100i	−2.24609	+	3.89033i	0.299797	+	1.38207i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	140.2.o.a	✓	32
4.b	odd	2	1	inner	140.2.o.a	✓	32
5.b	even	2	1		700.2.p.c		32
5.c	odd	4	1		700.2.t.c		32
5.c	odd	4	1		700.2.t.d		32
7.b	odd	2	1		980.2.o.f		32
7.c	even	3	1		980.2.g.a		32
7.c	even	3	1		980.2.o.f		32
7.d	odd	6	1	inner	140.2.o.a	✓	32
7.d	odd	6	1		980.2.g.a		32
20.d	odd	2	1		700.2.p.c		32
20.e	even	4	1		700.2.t.c		32
20.e	even	4	1		700.2.t.d		32
28.d	even	2	1		980.2.o.f		32
28.f	even	6	1	inner	140.2.o.a	✓	32
28.f	even	6	1		980.2.g.a		32
28.g	odd	6	1		980.2.g.a		32
28.g	odd	6	1		980.2.o.f		32
35.i	odd	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
140.s	even	6	1		700.2.p.c		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	1.a	even	1	1	trivial
140.2.o.a	✓	32	4.b	odd	2	1	inner
140.2.o.a	✓	32	7.d	odd	6	1	inner
140.2.o.a	✓	32	28.f	even	6	1	inner
700.2.p.c		32	5.b	even	2	1
700.2.p.c		32	20.d	odd	2	1
700.2.p.c		32	35.i	odd	6	1
700.2.p.c		32	140.s	even	6	1
700.2.t.c		32	5.c	odd	4	1
700.2.t.c		32	20.e	even	4	1
700.2.t.c		32	35.k	even	12	1
700.2.t.c		32	140.x	odd	12	1
700.2.t.d		32	5.c	odd	4	1
700.2.t.d		32	20.e	even	4	1
700.2.t.d		32	35.k	even	12	1
700.2.t.d		32	140.x	odd	12	1
980.2.g.a		32	7.c	even	3	1
980.2.g.a		32	7.d	odd	6	1
980.2.g.a		32	28.f	even	6	1
980.2.g.a		32	28.g	odd	6	1
980.2.o.f		32	7.b	odd	2	1
980.2.o.f		32	7.c	even	3	1
980.2.o.f		32	28.d	even	2	1
980.2.o.f		32	28.g	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).