Newform orbit 720.2.by.f

• Label: 720.2.by.f
• Level: $720$
• Weight: $2$
• Character orbit: 720.by
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.by (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 360)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1		0		−1.72704	+	0.131617i		0		−1.06189	−	1.96784i		0		−1.19102	+	0.687633i		0		2.96535	−	0.454616i		0
49.2		0		−1.56081	+	0.750911i		0		1.52089	+	1.63917i		0		2.51643	−	1.45286i		0		1.87227	−	2.34406i		0
49.3		0		−1.54437	−	0.784174i		0		1.59892	−	1.56315i		0		−0.608912	+	0.351555i		0		1.77014	+	2.42211i		0
49.4		0		−1.27766	+	1.16944i		0		−2.07203	+	0.840657i		0		−3.28422	+	1.89614i		0		0.264813	−	2.98829i		0
49.5		0		−1.19594	−	1.25289i		0		1.61222	+	1.54943i		0		−1.98012	+	1.14322i		0		−0.139458	+	2.99676i		0
49.6		0		−1.11432	−	1.32601i		0		−1.73442	+	1.41130i		0		2.19623	−	1.26799i		0		−0.516592	+	2.95519i		0
49.7		0		−0.557359	+	1.63992i		0		−2.18221	+	0.487830i		0		3.90215	−	2.25291i		0		−2.37870	−	1.82805i		0
49.8		0		−0.284761	−	1.70848i		0		1.07259	−	1.96203i		0		3.55262	−	2.05111i		0		−2.83782	+	0.973019i		0
49.9		0		0.284761	+	1.70848i		0		1.16287	−	1.90990i		0		−3.55262	+	2.05111i		0		−2.83782	+	0.973019i		0
49.10		0		0.557359	−	1.63992i		0		0.668629	+	2.13376i		0		−3.90215	+	2.25291i		0		−2.37870	−	1.82805i		0
49.11		0		1.11432	+	1.32601i		0		−0.355011	+	2.20771i		0		−2.19623	+	1.26799i		0		−0.516592	+	2.95519i		0
49.12		0		1.19594	+	1.25289i		0		−2.14796	−	0.621508i		0		1.98012	−	1.14322i		0		−0.139458	+	2.99676i		0
49.13		0		1.27766	−	1.16944i		0		0.307984	+	2.21476i		0		3.28422	−	1.89614i		0		0.264813	−	2.98829i		0
49.14		0		1.54437	+	0.784174i		0		0.554267	−	2.16628i		0		0.608912	−	0.351555i		0		1.77014	+	2.42211i		0
49.15		0		1.56081	−	0.750911i		0		−2.18001	−	0.497547i		0		−2.51643	+	1.45286i		0		1.87227	−	2.34406i		0
49.16		0		1.72704	−	0.131617i		0		2.23514	−	0.0643013i		0		1.19102	−	0.687633i		0		2.96535	−	0.454616i		0
529.1		0		−1.72704	−	0.131617i		0		−1.06189	+	1.96784i		0		−1.19102	−	0.687633i		0		2.96535	+	0.454616i		0
529.2		0		−1.56081	−	0.750911i		0		1.52089	−	1.63917i		0		2.51643	+	1.45286i		0		1.87227	+	2.34406i		0
529.3		0		−1.54437	+	0.784174i		0		1.59892	+	1.56315i		0		−0.608912	−	0.351555i		0		1.77014	−	2.42211i		0
529.4		0		−1.27766	−	1.16944i		0		−2.07203	−	0.840657i		0		−3.28422	−	1.89614i		0		0.264813	+	2.98829i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.by.f		32
3.b	odd	2	1		2160.2.by.f		32
4.b	odd	2	1		360.2.bi.b	✓	32
5.b	even	2	1	inner	720.2.by.f		32
9.c	even	3	1	inner	720.2.by.f		32
9.d	odd	6	1		2160.2.by.f		32
12.b	even	2	1		1080.2.bi.b		32
15.d	odd	2	1		2160.2.by.f		32
20.d	odd	2	1		360.2.bi.b	✓	32
36.f	odd	6	1		360.2.bi.b	✓	32
36.f	odd	6	1		3240.2.f.k		16
36.h	even	6	1		1080.2.bi.b		32
36.h	even	6	1		3240.2.f.i		16
45.h	odd	6	1		2160.2.by.f		32
45.j	even	6	1	inner	720.2.by.f		32
60.h	even	2	1		1080.2.bi.b		32
180.n	even	6	1		1080.2.bi.b		32
180.n	even	6	1		3240.2.f.i		16
180.p	odd	6	1		360.2.bi.b	✓	32
180.p	odd	6	1		3240.2.f.k		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.bi.b	✓	32	4.b	odd	2	1
360.2.bi.b	✓	32	20.d	odd	2	1
360.2.bi.b	✓	32	36.f	odd	6	1
360.2.bi.b	✓	32	180.p	odd	6	1
720.2.by.f		32	1.a	even	1	1	trivial
720.2.by.f		32	5.b	even	2	1	inner
720.2.by.f		32	9.c	even	3	1	inner
720.2.by.f		32	45.j	even	6	1	inner
1080.2.bi.b		32	12.b	even	2	1
1080.2.bi.b		32	36.h	even	6	1
1080.2.bi.b		32	60.h	even	2	1
1080.2.bi.b		32	180.n	even	6	1
2160.2.by.f		32	3.b	odd	2	1
2160.2.by.f		32	9.d	odd	6	1
2160.2.by.f		32	15.d	odd	2	1
2160.2.by.f		32	45.h	odd	6	1
3240.2.f.i		16	36.h	even	6	1
3240.2.f.i		16	180.n	even	6	1
3240.2.f.k		16	36.f	odd	6	1
3240.2.f.k		16	180.p	odd	6	1

Hecke kernels

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

T7^32 - 74*T7^30 + 3261*T7^28 - 94822*T7^26 + 2048402*T7^24 - 33354582*T7^22 + 423923245*T7^20 - 4180084802*T7^18 + 32418940194*T7^16 - 194012221418*T7^14 + 896385771385*T7^12 - 3050239933122*T7^10 + 7537800201365*T7^8 - 11787045399052*T7^6 + 12471023605692*T7^4 - 5434101872048*T7^2 + 1700843738896

T11^16 + 8*T11^15 + 84*T11^14 + 272*T11^13 + 2101*T11^12 + 5148*T11^11 + 40012*T11^10 + 56480*T11^9 + 307101*T11^8 + 652964*T11^7 + 1619512*T11^6 + 1807632*T11^5 + 1683532*T11^4 + 485024*T11^3 + 113376*T11^2 - 1312*T11 + 16