Newform orbit 980.2.s.g

• Label: 980.2.s.g
• Level: $980$
• Weight: $2$
• Character orbit: 980.s
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $96$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 16 q^{4} + 64 q^{9} - 16 q^{16} + 16 q^{25} - 96 q^{29} + 8 q^{30} + 352 q^{36} + 48 q^{44} + 32 q^{46} + 64 q^{50} - 24 q^{60} - 160 q^{64} + 16 q^{65} + 112 q^{74} + 48 q^{81} - 128 q^{85} + 112 q^{86}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.40659	−	0.146688i	−2.16636	+	1.25075i	1.95697	+	0.412657i	2.17545	+	0.517106i	3.23064	−	1.44151i		0		−2.69211	−	0.867500i	1.62875	−	2.82107i	−2.98411	−	1.04647i
19.2	−1.40659	−	0.146688i	2.16636	−	1.25075i	1.95697	+	0.412657i	−2.17545	−	0.517106i	−3.23064	+	1.44151i		0		−2.69211	−	0.867500i	1.62875	−	2.82107i	2.98411	+	1.04647i
19.3	−1.33248	−	0.473800i	−2.75938	+	1.59313i	1.55103	+	1.26266i	−0.236883	−	2.22349i	4.43165	−	0.815425i		0		−1.46847	−	2.41735i	3.57612	−	6.19402i	−0.737844	+	3.07499i
19.4	−1.33248	−	0.473800i	2.75938	−	1.59313i	1.55103	+	1.26266i	0.236883	+	2.22349i	−4.43165	+	0.815425i		0		−1.46847	−	2.41735i	3.57612	−	6.19402i	0.737844	−	3.07499i
19.5	−1.28974	+	0.580158i	−1.31926	+	0.761677i	1.32683	−	1.49650i	−1.98305	−	1.03321i	1.25961	−	1.74774i		0		−0.843059	+	2.69986i	−0.339697	+	0.588372i	3.15703	+	0.182089i
19.6	−1.28974	+	0.580158i	1.31926	−	0.761677i	1.32683	−	1.49650i	1.98305	+	1.03321i	−1.25961	+	1.74774i		0		−0.843059	+	2.69986i	−0.339697	+	0.588372i	−3.15703	−	0.182089i
19.7	−1.21565	+	0.722628i	−1.80140	+	1.04004i	0.955617	−	1.75693i	0.656871	+	2.13741i	1.43832	−	2.56607i		0		0.107908	+	2.82637i	0.663371	−	1.14899i	−2.34308	−	2.12367i
19.8	−1.21565	+	0.722628i	1.80140	−	1.04004i	0.955617	−	1.75693i	−0.656871	−	2.13741i	−1.43832	+	2.56607i		0		0.107908	+	2.82637i	0.663371	−	1.14899i	2.34308	+	2.12367i
19.9	−1.07656	−	0.917065i	−2.75938	+	1.59313i	0.317983	+	1.97456i	0.236883	+	2.22349i	4.43165	+	0.815425i		0		1.46847	−	2.41735i	3.57612	−	6.19402i	1.78406	−	2.61096i
19.10	−1.07656	−	0.917065i	2.75938	−	1.59313i	0.317983	+	1.97456i	−0.236883	−	2.22349i	−4.43165	−	0.815425i		0		1.46847	−	2.41735i	3.57612	−	6.19402i	−1.78406	+	2.61096i
19.11	−1.01395	+	0.985849i	−0.366665	+	0.211694i	0.0562045	−	1.99921i	−2.08328	+	0.812373i	0.163083	−	0.576124i		0		1.91393	+	2.08252i	−1.41037	+	2.44283i	1.31147	−	2.87751i
19.12	−1.01395	+	0.985849i	0.366665	−	0.211694i	0.0562045	−	1.99921i	2.08328	−	0.812373i	−0.163083	+	0.576124i		0		1.91393	+	2.08252i	−1.41037	+	2.44283i	−1.31147	+	2.87751i
19.13	−0.947463	+	1.04991i	−1.43970	+	0.831213i	−0.204628	−	1.98950i	−1.30355	+	1.81680i	0.491366	−	2.29911i		0		2.28268	+	1.67014i	−0.118168	+	0.204674i	−0.672404	−	3.08996i
19.14	−0.947463	+	1.04991i	1.43970	−	0.831213i	−0.204628	−	1.98950i	1.30355	−	1.81680i	−0.491366	+	2.29911i		0		2.28268	+	1.67014i	−0.118168	+	0.204674i	0.672404	+	3.08996i
19.15	−0.830328	−	1.14480i	−2.16636	+	1.25075i	−0.621111	+	1.90111i	−2.17545	−	0.517106i	3.23064	+	1.44151i		0		2.69211	−	0.867500i	1.62875	−	2.82107i	1.21436	+	2.91982i
19.16	−0.830328	−	1.14480i	2.16636	−	1.25075i	−0.621111	+	1.90111i	2.17545	+	0.517106i	−3.23064	−	1.44151i		0		2.69211	−	0.867500i	1.62875	−	2.82107i	−1.21436	−	2.91982i
19.17	−0.435518	+	1.34548i	−1.43970	+	0.831213i	−1.62065	−	1.17196i	0.921614	−	2.03731i	−0.491366	−	2.29911i		0		2.28268	−	1.67014i	−0.118168	+	0.204674i	2.33978	+	2.12730i
19.18	−0.435518	+	1.34548i	1.43970	−	0.831213i	−1.62065	−	1.17196i	−0.921614	+	2.03731i	0.491366	+	2.29911i		0		2.28268	−	1.67014i	−0.118168	+	0.204674i	−2.33978	−	2.12730i
19.19	−0.346793	+	1.37103i	−0.366665	+	0.211694i	−1.75947	−	0.950931i	−0.338104	−	2.21036i	−0.163083	−	0.576124i		0		1.91393	−	2.08252i	−1.41037	+	2.44283i	3.14773	+	0.302985i
19.20	−0.346793	+	1.37103i	0.366665	−	0.211694i	−1.75947	−	0.950931i	0.338104	+	2.21036i	0.163083	+	0.576124i		0		1.91393	−	2.08252i	−1.41037	+	2.44283i	−3.14773	−	0.302985i

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
7.c	even	3	1	inner
7.d	odd	6	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner
35.c	odd	2	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner
140.c	even	2	1	inner
140.p	odd	6	1	inner
140.s	even	6	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^24 - 26*T3^22 + 425*T3^20 - 4254*T3^18 + 31009*T3^16 - 159592*T3^14 + 618304*T3^12 - 1709896*T3^10 + 3489772*T3^8 - 4606656*T3^6 + 3924128*T3^4 - 685088*T3^2 + 99856