Newform orbit 980.2.bu.a

• Label: 980.2.bu.a
• Level: $980$
• Weight: $2$
• Character orbit: 980.bu
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.bu (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(28\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 672 q - 6 q^{5} - 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−1.55727	−	2.94650i		0		−2.18883	−	0.457195i		0		−2.19928	−	1.47078i		0		−4.56680	+	6.69827i		0
17.2		0		−1.49295	−	2.82480i		0		0.835809	+	2.07399i		0		2.05029	−	1.67221i		0		−4.06063	+	5.95586i		0
17.3		0		−1.38581	−	2.62207i		0		1.92873	−	1.13137i		0		0.720232	+	2.54583i		0		−3.26485	+	4.78866i		0
17.4		0		−1.24059	−	2.34730i		0		−1.85944	−	1.24197i		0		1.52770	+	2.16013i		0		−2.28082	+	3.34534i		0
17.5		0		−0.926658	−	1.75332i		0		2.10222	+	0.762021i		0		−1.71116	−	2.01790i		0		−0.525483	+	0.770741i		0
17.6		0		−0.915530	−	1.73227i		0		−1.47288	+	1.68244i		0		2.64567	−	0.0213163i		0		−0.472596	+	0.693171i		0
17.7		0		−0.900380	−	1.70360i		0		0.914213	−	2.04064i		0		−0.534311	−	2.59124i		0		−0.401617	+	0.589064i		0
17.8		0		−0.810418	−	1.53339i		0		−0.373728	−	2.20462i		0		−2.45153	+	0.994984i		0		−0.00453352	+	0.00664945i		0
17.9		0		−0.795590	−	1.50533i		0		1.06111	+	1.96826i		0		−0.714974	+	2.54731i		0		0.0569079	−	0.0834686i		0
17.10		0		−0.769287	−	1.45556i		0		−1.43210	+	1.71729i		0		−2.36057	+	1.19487i		0		0.163104	−	0.239229i		0
17.11		0		−0.312253	−	0.590811i		0		−1.76140	−	1.37749i		0		1.39089	−	2.25065i		0		1.43840	−	2.10975i		0
17.12		0		−0.214069	−	0.405037i		0		2.19322	−	0.435632i		0		2.62921	−	0.295383i		0		1.57173	−	2.30530i		0
17.13		0		−0.0661857	−	0.125230i		0		0.825782	−	2.07800i		0		1.99284	+	1.74028i		0		1.67866	−	2.46214i		0
17.14		0		0.130449	+	0.246822i		0		0.583307	+	2.15865i		0		1.29366	+	2.30791i		0		1.64606	−	2.41432i		0
17.15		0		0.214765	+	0.406354i		0		2.07702	−	0.828240i		0		−2.23299	+	1.41906i		0		1.57096	−	2.30418i		0
17.16		0		0.215491	+	0.407729i		0		−1.96406	+	1.06886i		0		−1.58234	−	2.12043i		0		1.57015	−	2.30299i		0
17.17		0		0.243030	+	0.459835i		0		0.0580237	+	2.23532i		0		1.33599	−	2.28366i		0		1.53758	−	2.25521i		0
17.18		0		0.414748	+	0.784742i		0		−2.09832	+	0.772685i		0		−1.72140	+	2.00918i		0		1.24616	−	1.82778i		0
17.19		0		0.437640	+	0.828056i		0		−2.14093	−	0.645319i		0		1.86491	+	1.87673i		0		1.19581	−	1.75394i		0
17.20		0		0.545111	+	1.03140i		0		2.17148	−	0.533555i		0		−2.64530	+	0.0486299i		0		0.923320	−	1.35426i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
49.h	odd	42	1	inner
245.x	even	84	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	980.2.bu.a	✓	672
5.c	odd	4	1	inner	980.2.bu.a	✓	672
49.h	odd	42	1	inner	980.2.bu.a	✓	672
245.x	even	84	1	inner	980.2.bu.a	✓	672

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
980.2.bu.a	✓	672	1.a	even	1	1	trivial
980.2.bu.a	✓	672	5.c	odd	4	1	inner
980.2.bu.a	✓	672	49.h	odd	42	1	inner
980.2.bu.a	✓	672	245.x	even	84	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(980, [\chi])\).