Newform orbit 504.2.bk.b

• Label: 504.2.bk.b
• Level: $504$
• Weight: $2$
• Character orbit: 504.bk
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−1.40711	−	0.141583i		0		1.95991	+	0.398446i	−1.91923	+	3.32420i		0		1.55724	+	2.13893i	−2.70139	−	0.838146i		0		3.17121	−	4.40578i
19.2	−1.35459	−	0.406311i		0		1.66982	+	1.10077i	1.09169	−	1.89086i		0		1.40064	−	2.24459i	−1.81467	−	2.16955i		0		−2.24706	+	2.11777i
19.3	−1.28822	+	0.583515i		0		1.31902	−	1.50339i	−0.245316	+	0.424900i		0		−2.09582	−	1.61479i	−0.821939	+	2.70637i		0		0.0680858	−	0.690511i
19.4	−1.02917	−	0.969953i		0		0.118381	+	1.99649i	1.09169	−	1.89086i		0		−1.40064	+	2.24459i	1.81467	−	2.16955i		0		−2.95757	+	0.887128i
19.5	−0.826169	−	1.14780i		0		−0.634890	+	1.89655i	−1.91923	+	3.32420i		0		−1.55724	−	2.13893i	2.70139	−	0.838146i		0		5.40112	−	0.543460i
19.6	−0.809233	+	1.15980i		0		−0.690284	−	1.87710i	−1.03180	+	1.78713i		0		2.39181	−	1.13104i	2.73567	+	0.718419i		0		−1.23775	−	2.64289i
19.7	−0.599802	+	1.28072i		0		−1.28048	−	1.53635i	1.03180	−	1.78713i		0		−2.39181	+	1.13104i	2.73567	−	0.718419i		0		1.66993	+	2.39337i
19.8	−0.138771	−	1.40739i		0		−1.96148	+	0.390611i	−0.245316	+	0.424900i		0		2.09582	+	1.61479i	0.821939	+	2.70637i		0		0.632043	+	0.286291i
19.9	0.138771	+	1.40739i		0		−1.96148	+	0.390611i	0.245316	−	0.424900i		0		2.09582	+	1.61479i	−0.821939	−	2.70637i		0		0.632043	+	0.286291i
19.10	0.599802	−	1.28072i		0		−1.28048	−	1.53635i	−1.03180	+	1.78713i		0		−2.39181	+	1.13104i	−2.73567	+	0.718419i		0		1.66993	+	2.39337i
19.11	0.809233	−	1.15980i		0		−0.690284	−	1.87710i	1.03180	−	1.78713i		0		2.39181	−	1.13104i	−2.73567	−	0.718419i		0		−1.23775	−	2.64289i
19.12	0.826169	+	1.14780i		0		−0.634890	+	1.89655i	1.91923	−	3.32420i		0		−1.55724	−	2.13893i	−2.70139	+	0.838146i		0		5.40112	−	0.543460i
19.13	1.02917	+	0.969953i		0		0.118381	+	1.99649i	−1.09169	+	1.89086i		0		−1.40064	+	2.24459i	−1.81467	+	2.16955i		0		−2.95757	+	0.887128i
19.14	1.28822	−	0.583515i		0		1.31902	−	1.50339i	0.245316	−	0.424900i		0		−2.09582	−	1.61479i	0.821939	−	2.70637i		0		0.0680858	−	0.690511i
19.15	1.35459	+	0.406311i		0		1.66982	+	1.10077i	−1.09169	+	1.89086i		0		1.40064	−	2.24459i	1.81467	+	2.16955i		0		−2.24706	+	2.11777i
19.16	1.40711	+	0.141583i		0		1.95991	+	0.398446i	1.91923	−	3.32420i		0		1.55724	+	2.13893i	2.70139	+	0.838146i		0		3.17121	−	4.40578i
451.1	−1.40711	+	0.141583i		0		1.95991	−	0.398446i	−1.91923	−	3.32420i		0		1.55724	−	2.13893i	−2.70139	+	0.838146i		0		3.17121	+	4.40578i
451.2	−1.35459	+	0.406311i		0		1.66982	−	1.10077i	1.09169	+	1.89086i		0		1.40064	+	2.24459i	−1.81467	+	2.16955i		0		−2.24706	−	2.11777i
451.3	−1.28822	−	0.583515i		0		1.31902	+	1.50339i	−0.245316	−	0.424900i		0		−2.09582	+	1.61479i	−0.821939	−	2.70637i		0		0.0680858	+	0.690511i
451.4	−1.02917	+	0.969953i		0		0.118381	−	1.99649i	1.09169	+	1.89086i		0		−1.40064	−	2.24459i	1.81467	+	2.16955i		0		−2.95757	−	0.887128i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
8.d	odd	2	1	inner
21.g	even	6	1	inner
24.f	even	2	1	inner
56.m	even	6	1	inner
168.be	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.bk.b	✓	32
3.b	odd	2	1	inner	504.2.bk.b	✓	32
4.b	odd	2	1		2016.2.bs.b		32
7.d	odd	6	1	inner	504.2.bk.b	✓	32
8.b	even	2	1		2016.2.bs.b		32
8.d	odd	2	1	inner	504.2.bk.b	✓	32
12.b	even	2	1		2016.2.bs.b		32
21.g	even	6	1	inner	504.2.bk.b	✓	32
24.f	even	2	1	inner	504.2.bk.b	✓	32
24.h	odd	2	1		2016.2.bs.b		32
28.f	even	6	1		2016.2.bs.b		32
56.j	odd	6	1		2016.2.bs.b		32
56.m	even	6	1	inner	504.2.bk.b	✓	32
84.j	odd	6	1		2016.2.bs.b		32
168.ba	even	6	1		2016.2.bs.b		32
168.be	odd	6	1	inner	504.2.bk.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.bk.b	✓	32	1.a	even	1	1	trivial
504.2.bk.b	✓	32	3.b	odd	2	1	inner
504.2.bk.b	✓	32	7.d	odd	6	1	inner
504.2.bk.b	✓	32	8.d	odd	2	1	inner
504.2.bk.b	✓	32	21.g	even	6	1	inner
504.2.bk.b	✓	32	24.f	even	2	1	inner
504.2.bk.b	✓	32	56.m	even	6	1	inner
504.2.bk.b	✓	32	168.be	odd	6	1	inner
2016.2.bs.b		32	4.b	odd	2	1
2016.2.bs.b		32	8.b	even	2	1
2016.2.bs.b		32	12.b	even	2	1
2016.2.bs.b		32	24.h	odd	2	1
2016.2.bs.b		32	28.f	even	6	1
2016.2.bs.b		32	56.j	odd	6	1
2016.2.bs.b		32	84.j	odd	6	1
2016.2.bs.b		32	168.ba	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^16 + 24*T5^14 + 417*T5^12 + 3144*T5^10 + 17145*T5^8 + 49968*T5^6 + 101448*T5^4 + 24192*T5^2 + 5184