Newform orbit 720.2.bg.a

• Label: 720.2.bg.a
• Level: $720$
• Weight: $2$
• Character orbit: 720.bg
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.bg (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1	−1.41227	+	0.0740368i		0		1.98904	−	0.209121i	1.57418	−	1.58807i		0		0.639411	+	0.639411i	−2.79358	+	0.442598i		0		−2.10560	+	2.35933i
53.2	−1.40993	+	0.109955i		0		1.97582	−	0.310058i	−1.03889	+	1.98008i		0		2.57554	+	2.57554i	−2.75168	+	0.654411i		0		1.24705	−	2.90601i
53.3	−1.40193	+	0.185953i		0		1.93084	−	0.521389i	0.103242	+	2.23368i		0		−1.05564	−	1.05564i	−2.60996	+	1.09000i		0		−0.560100	−	3.11228i
53.4	−1.36491	−	0.370150i		0		1.72598	+	1.01045i	1.84310	+	1.26609i		0		−0.461154	−	0.461154i	−1.98179	−	2.01804i		0		−2.04703	−	2.41033i
53.5	−1.34802	−	0.427613i		0		1.63429	+	1.15286i	−1.95099	−	1.09255i		0		1.70979	+	1.70979i	−1.71008	−	2.25292i		0		2.16277	+	2.30704i
53.6	−1.32770	+	0.487044i		0		1.52558	−	1.29330i	−1.95786	−	1.08017i		0		−0.362166	−	0.362166i	−1.39562	+	2.46013i		0		3.12555	+	0.480582i
53.7	−1.29317	+	0.572467i		0		1.34456	−	1.48059i	2.23555	+	0.0482170i		0		−2.70126	−	2.70126i	−0.891154	+	2.68437i		0		−2.91854	+	1.21743i
53.8	−1.24910	−	0.663127i		0		1.12053	+	1.65663i	−1.07871	−	1.95867i		0		−1.03362	−	1.03362i	−0.301100	−	2.81235i		0		0.0485690	+	3.16190i
53.9	−1.09241	−	0.898132i		0		0.386719	+	1.96226i	−1.82168	+	1.29671i		0		−2.16011	−	2.16011i	1.33991	−	2.49091i		0		3.15465	+	0.219567i
53.10	−1.08211	+	0.910521i		0		0.341905	−	1.97056i	−0.897076	−	2.04823i		0		−3.15125	−	3.15125i	1.42426	+	2.44366i		0		2.83569	+	1.39960i
53.11	−1.01816	−	0.981505i		0		0.0732970	+	1.99866i	1.34765	+	1.78433i		0		3.52140	+	3.52140i	1.88706	−	2.10689i		0		0.379207	−	3.13946i
53.12	−0.977925	+	1.02160i		0		−0.0873238	−	1.99809i	1.41125	−	1.73447i		0		2.32146	+	2.32146i	2.12664	+	1.86478i		0		0.391831	+	3.13791i
53.13	−0.959172	+	1.03923i		0		−0.159980	−	1.99359i	2.23299	−	0.117317i		0		0.303980	+	0.303980i	2.22524	+	1.74594i		0		−2.01990	+	2.43311i
53.14	−0.850328	−	1.13002i		0		−0.553884	+	1.92177i	−2.05344	+	0.885098i		0		2.95475	+	2.95475i	2.64262	−	1.00824i		0		2.74627	+	1.56780i
53.15	−0.797498	+	1.16790i		0		−0.727993	−	1.86280i	−0.948701	+	2.02484i		0		−1.06871	−	1.06871i	2.75614	+	0.635356i		0		−1.60823	−	2.72279i
53.16	−0.787263	−	1.17483i		0		−0.760434	+	1.84979i	−0.0422885	−	2.23567i		0		−1.80532	−	1.80532i	2.77185	−	0.562896i		0		−2.59323	+	1.80974i
53.17	−0.745171	−	1.20197i		0		−0.889441	+	1.79134i	2.18604	−	0.470363i		0		0.177389	+	0.177389i	2.81591	−	0.265776i		0		−2.19433	−	2.27704i
53.18	−0.480016	+	1.33026i		0		−1.53917	−	1.27709i	−0.859295	−	2.06437i		0		1.97431	+	1.97431i	2.43768	−	1.43447i		0		3.15862	−	0.152155i
53.19	−0.434580	+	1.34579i		0		−1.62228	−	1.16970i	−1.40237	+	1.74165i		0		−2.18248	−	2.18248i	2.27918	−	1.67491i		0		−1.73445	−	2.64418i
53.20	−0.237394	−	1.39415i		0		−1.88729	+	0.661925i	−2.22117	+	0.257710i		0		−3.41086	−	3.41086i	1.37085	+	2.47402i		0		0.886579	+	3.03545i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
80.t	odd	4	1	inner
240.bf	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.bg.a	yes	96
3.b	odd	2	1	inner	720.2.bg.a	yes	96
4.b	odd	2	1		2880.2.bg.a		96
5.c	odd	4	1		720.2.bc.a	✓	96
12.b	even	2	1		2880.2.bg.a		96
15.e	even	4	1		720.2.bc.a	✓	96
16.e	even	4	1		720.2.bc.a	✓	96
16.f	odd	4	1		2880.2.bc.a		96
20.e	even	4	1		2880.2.bc.a		96
48.i	odd	4	1		720.2.bc.a	✓	96
48.k	even	4	1		2880.2.bc.a		96
60.l	odd	4	1		2880.2.bc.a		96
80.j	even	4	1		2880.2.bg.a		96
80.t	odd	4	1	inner	720.2.bg.a	yes	96
240.bd	odd	4	1		2880.2.bg.a		96
240.bf	even	4	1	inner	720.2.bg.a	yes	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.2.bc.a	✓	96	5.c	odd	4	1
720.2.bc.a	✓	96	15.e	even	4	1
720.2.bc.a	✓	96	16.e	even	4	1
720.2.bc.a	✓	96	48.i	odd	4	1
720.2.bg.a	yes	96	1.a	even	1	1	trivial
720.2.bg.a	yes	96	3.b	odd	2	1	inner
720.2.bg.a	yes	96	80.t	odd	4	1	inner
720.2.bg.a	yes	96	240.bf	even	4	1	inner
2880.2.bc.a		96	16.f	odd	4	1
2880.2.bc.a		96	20.e	even	4	1
2880.2.bc.a		96	48.k	even	4	1
2880.2.bc.a		96	60.l	odd	4	1
2880.2.bg.a		96	4.b	odd	2	1
2880.2.bg.a		96	12.b	even	2	1
2880.2.bg.a		96	80.j	even	4	1
2880.2.bg.a		96	240.bd	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).