Newform orbit 720.2.bm.g

• Label: 720.2.bm.g
• Level: $720$
• Weight: $2$
• Character orbit: 720.bm
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) =$ $32 q + 16 q^{4} - 24 q^{10} - 64 q^{16} + 32 q^{19} + 32 q^{31} - 72 q^{40} - 32 q^{46} + 128 q^{49} - 32 q^{64} - 104 q^{70} - 32 q^{76} + 224 q^{79} - 48 q^{85} - 32 q^{91} - 48 q^{94}+O(q^{100})$

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	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
109.1	−1.29731	−	0.563016i		0		1.36603	+	1.46081i	−0.496304	+	2.18029i		0		−0.862231			−0.949697	−	2.66422i		0		1.87140	−	2.54909i
109.2	−1.29731	−	0.563016i		0		1.36603	+	1.46081i	2.18029	−	0.496304i		0		0.862231			−0.949697	−	2.66422i		0		−3.10794	−	0.583681i
109.3	−1.29731	+	0.563016i		0		1.36603	−	1.46081i	0.679792	+	2.13023i		0		−3.29610			−0.949697	+	2.66422i		0		−2.08126	−	2.38084i
109.4	−1.29731	+	0.563016i		0		1.36603	−	1.46081i	2.13023	+	0.679792i		0		3.29610			−0.949697	+	2.66422i		0		−3.14630	+	0.317454i
109.5	−0.903873	−	1.08766i		0		−0.366025	+	1.96622i	−2.18690	−	0.466314i		0		−4.79660			2.46943	−	1.37910i		0		1.46949	+	2.80011i
109.6	−0.903873	−	1.08766i		0		−0.366025	+	1.96622i	−0.466314	−	2.18690i		0		4.79660			2.46943	−	1.37910i		0		−1.95713	+	2.48388i
109.7	−0.903873	+	1.08766i		0		−0.366025	−	1.96622i	−1.80193	+	1.32403i		0		3.06349			2.46943	+	1.37910i		0		0.188608	−	3.15665i
109.8	−0.903873	+	1.08766i		0		−0.366025	−	1.96622i	1.32403	−	1.80193i		0		−3.06349			2.46943	+	1.37910i		0		0.763129	+	3.06882i
109.9	0.903873	−	1.08766i		0		−0.366025	−	1.96622i	−1.32403	+	1.80193i		0		−3.06349			−2.46943	−	1.37910i		0		0.763129	+	3.06882i
109.10	0.903873	−	1.08766i		0		−0.366025	−	1.96622i	1.80193	−	1.32403i		0		3.06349			−2.46943	−	1.37910i		0		0.188608	−	3.15665i
109.11	0.903873	+	1.08766i		0		−0.366025	+	1.96622i	0.466314	+	2.18690i		0		4.79660			−2.46943	+	1.37910i		0		−1.95713	+	2.48388i
109.12	0.903873	+	1.08766i		0		−0.366025	+	1.96622i	2.18690	+	0.466314i		0		−4.79660			−2.46943	+	1.37910i		0		1.46949	+	2.80011i
109.13	1.29731	−	0.563016i		0		1.36603	−	1.46081i	−2.13023	−	0.679792i		0		3.29610			0.949697	−	2.66422i		0		−3.14630	+	0.317454i
109.14	1.29731	−	0.563016i		0		1.36603	−	1.46081i	−0.679792	−	2.13023i		0		−3.29610			0.949697	−	2.66422i		0		−2.08126	−	2.38084i
109.15	1.29731	+	0.563016i		0		1.36603	+	1.46081i	−2.18029	+	0.496304i		0		0.862231			0.949697	+	2.66422i		0		−3.10794	−	0.583681i
109.16	1.29731	+	0.563016i		0		1.36603	+	1.46081i	0.496304	−	2.18029i		0		−0.862231			0.949697	+	2.66422i		0		1.87140	−	2.54909i
469.1	−1.29731	−	0.563016i		0		1.36603	+	1.46081i	0.679792	−	2.13023i		0		−3.29610			−0.949697	−	2.66422i		0		−2.08126	+	2.38084i
469.2	−1.29731	−	0.563016i		0		1.36603	+	1.46081i	2.13023	−	0.679792i		0		3.29610			−0.949697	−	2.66422i		0		−3.14630	−	0.317454i
469.3	−1.29731	+	0.563016i		0		1.36603	−	1.46081i	−0.496304	−	2.18029i		0		−0.862231			−0.949697	+	2.66422i		0		1.87140	+	2.54909i
469.4	−1.29731	+	0.563016i		0		1.36603	−	1.46081i	2.18029	+	0.496304i		0		0.862231			−0.949697	+	2.66422i		0		−3.10794	+	0.583681i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
16.e	even	4	1	inner
48.i	odd	4	1	inner
80.q	even	4	1	inner
240.bm	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.bm.g	✓	32
3.b	odd	2	1	inner	720.2.bm.g	✓	32
5.b	even	2	1	inner	720.2.bm.g	✓	32
15.d	odd	2	1	inner	720.2.bm.g	✓	32
16.e	even	4	1	inner	720.2.bm.g	✓	32
48.i	odd	4	1	inner	720.2.bm.g	✓	32
80.q	even	4	1	inner	720.2.bm.g	✓	32
240.bm	odd	4	1	inner	720.2.bm.g	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
720.2.bm.g	✓	32	1.a	even	1	1	trivial
720.2.bm.g	✓	32	3.b	odd	2	1	inner
720.2.bm.g	✓	32	5.b	even	2	1	inner
720.2.bm.g	✓	32	15.d	odd	2	1	inner
720.2.bm.g	✓	32	16.e	even	4	1	inner
720.2.bm.g	✓	32	48.i	odd	4	1	inner
720.2.bm.g	✓	32	80.q	even	4	1	inner
720.2.bm.g	✓	32	240.bm	odd	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(720, [\chi])$:

$T_{7}^{8} - 44T_{7}^{6} + 600T_{7}^{4} - 2768T_{7}^{2} + 1744$

$T_{11}^{16} + 1792T_{11}^{12} + 839616T_{11}^{8} + 55447552T_{11}^{4} + 48664576$

$T_{13}^{16} + 1696T_{13}^{12} + 86112T_{13}^{8} + 957952T_{13}^{4} + 3041536$