Newform orbit 630.2.bk.c

• Label: 630.2.bk.c
• Level: $630$
• Weight: $2$
• Character orbit: 630.bk
• Analytic conductor: $5.031$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.03057532734\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{3} - 32 q^{4} + 16 q^{5} - 2 q^{6} - 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
101.1		−	1.00000i	−1.69553	−	0.353784i	−1.00000			0.500000	−	0.866025i	−0.353784	+	1.69553i	−1.70202	−	2.02561i			1.00000i	2.74967	+	1.19970i	−0.866025	−	0.500000i
101.2		−	1.00000i	−1.61337	−	0.630115i	−1.00000			0.500000	−	0.866025i	−0.630115	+	1.61337i	2.57812	+	0.594391i			1.00000i	2.20591	+	2.03321i	−0.866025	−	0.500000i
101.3		−	1.00000i	−0.546603	−	1.64354i	−1.00000			0.500000	−	0.866025i	−1.64354	+	0.546603i	−0.852286	+	2.50472i			1.00000i	−2.40245	+	1.79673i	−0.866025	−	0.500000i
101.4		−	1.00000i	−0.275993	+	1.70992i	−1.00000			0.500000	−	0.866025i	1.70992	+	0.275993i	−2.63940	+	0.183152i			1.00000i	−2.84766	−	0.943852i	−0.866025	−	0.500000i
101.5		−	1.00000i	0.271725	+	1.71060i	−1.00000			0.500000	−	0.866025i	1.71060	−	0.271725i	1.13531	+	2.38979i			1.00000i	−2.85233	+	0.929628i	−0.866025	−	0.500000i
101.6		−	1.00000i	1.24270	−	1.20653i	−1.00000			0.500000	−	0.866025i	−1.20653	−	1.24270i	−2.37260	−	1.17080i			1.00000i	0.0885938	−	2.99869i	−0.866025	−	0.500000i
101.7		−	1.00000i	1.29195	+	1.15363i	−1.00000			0.500000	−	0.866025i	1.15363	−	1.29195i	1.14063	−	2.38725i			1.00000i	0.338261	+	2.98087i	−0.866025	−	0.500000i
101.8		−	1.00000i	1.69115	−	0.374168i	−1.00000			0.500000	−	0.866025i	−0.374168	−	1.69115i	1.34623	+	2.27765i			1.00000i	2.72000	−	1.26555i	−0.866025	−	0.500000i
101.9			1.00000i	−1.63983	+	0.557633i	−1.00000			0.500000	−	0.866025i	−0.557633	−	1.63983i	2.23703	−	1.41269i		−	1.00000i	2.37809	−	1.82885i	0.866025	+	0.500000i
101.10			1.00000i	−1.61346	+	0.629881i	−1.00000			0.500000	−	0.866025i	−0.629881	−	1.61346i	−0.166511	+	2.64051i		−	1.00000i	2.20650	−	2.03258i	0.866025	+	0.500000i
101.11			1.00000i	−1.14223	−	1.30204i	−1.00000			0.500000	−	0.866025i	1.30204	−	1.14223i	2.06989	+	1.64790i		−	1.00000i	−0.390607	+	2.97446i	0.866025	+	0.500000i
101.12			1.00000i	−0.560056	+	1.63900i	−1.00000			0.500000	−	0.866025i	−1.63900	−	0.560056i	0.0477786	−	2.64532i		−	1.00000i	−2.37267	−	1.83587i	0.866025	+	0.500000i
101.13			1.00000i	−0.117327	−	1.72807i	−1.00000			0.500000	−	0.866025i	1.72807	−	0.117327i	−2.63765	+	0.206895i		−	1.00000i	−2.97247	+	0.405499i	0.866025	+	0.500000i
101.14			1.00000i	0.661104	+	1.60092i	−1.00000			0.500000	−	0.866025i	−1.60092	+	0.661104i	−2.57921	+	0.589657i		−	1.00000i	−2.12588	+	2.11675i	0.866025	+	0.500000i
101.15			1.00000i	1.51441	−	0.840572i	−1.00000			0.500000	−	0.866025i	0.840572	+	1.51441i	−1.15079	−	2.38237i		−	1.00000i	1.58688	−	2.54594i	0.866025	+	0.500000i
101.16			1.00000i	1.53137	+	0.809270i	−1.00000			0.500000	−	0.866025i	−0.809270	+	1.53137i	2.54549	+	0.721453i		−	1.00000i	1.69016	+	2.47858i	0.866025	+	0.500000i
131.1		−	1.00000i	−1.63983	−	0.557633i	−1.00000			0.500000	+	0.866025i	−0.557633	+	1.63983i	2.23703	+	1.41269i			1.00000i	2.37809	+	1.82885i	0.866025	−	0.500000i
131.2		−	1.00000i	−1.61346	−	0.629881i	−1.00000			0.500000	+	0.866025i	−0.629881	+	1.61346i	−0.166511	−	2.64051i			1.00000i	2.20650	+	2.03258i	0.866025	−	0.500000i
131.3		−	1.00000i	−1.14223	+	1.30204i	−1.00000			0.500000	+	0.866025i	1.30204	+	1.14223i	2.06989	−	1.64790i			1.00000i	−0.390607	−	2.97446i	0.866025	−	0.500000i
131.4		−	1.00000i	−0.560056	−	1.63900i	−1.00000			0.500000	+	0.866025i	−1.63900	+	0.560056i	0.0477786	+	2.64532i			1.00000i	−2.37267	+	1.83587i	0.866025	−	0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	630.2.bk.c	yes	32
3.b	odd	2	1		1890.2.bk.c		32
7.d	odd	6	1		630.2.t.c	✓	32
9.c	even	3	1		1890.2.t.c		32
9.d	odd	6	1		630.2.t.c	✓	32
21.g	even	6	1		1890.2.t.c		32
63.i	even	6	1	inner	630.2.bk.c	yes	32
63.t	odd	6	1		1890.2.bk.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.t.c	✓	32	7.d	odd	6	1
630.2.t.c	✓	32	9.d	odd	6	1
630.2.bk.c	yes	32	1.a	even	1	1	trivial
630.2.bk.c	yes	32	63.i	even	6	1	inner
1890.2.t.c		32	9.c	even	3	1
1890.2.t.c		32	21.g	even	6	1
1890.2.bk.c		32	3.b	odd	2	1
1890.2.bk.c		32	63.t	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T11^32 + 6*T11^31 - 75*T11^30 - 522*T11^29 + 3837*T11^28 + 27120*T11^27 - 116464*T11^26 - 871158*T11^25 + 2714322*T11^24 + 19669806*T11^23 - 43647258*T11^22 - 311083236*T11^21 + 549963012*T11^20 + 3510623412*T11^19 - 4436357925*T11^18 - 28031285250*T11^17 + 25728571968*T11^16 + 157809350682*T11^15 - 71866305412*T11^14 - 559094576838*T11^13 + 190053472554*T11^12 + 1335489670152*T11^11 - 284619463494*T11^10 - 1930301876892*T11^9 + 748497750037*T11^8 + 1537893376626*T11^7 - 824343591318*T11^6 - 761752025028*T11^5 + 711174412296*T11^4 - 61416558072*T11^3 - 79306667352*T11^2 + 6022210032*T11 + 7925984784