Newform orbit 784.2.bg.b

• Label: 784.2.bg.b
• Level: $784$
• Weight: $2$
• Character orbit: 784.bg
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.bg (of order \(21\), degree \(12\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(2\) over \(\Q(\zeta_{21})\)
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 7 q^{3} + 19 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
65.1		0		−0.125927	−	1.68037i		0		1.87725	−	1.27989i		0		2.59910	−	0.494638i		0		0.158698	−	0.0239199i		0
65.2		0		−0.0722934	−	0.964688i		0		−3.07015	+	2.09319i		0		−2.50800	+	0.842585i		0		2.04110	−	0.307646i		0
81.1		0		−1.95066	−	0.601698i		0		0.958118	−	0.889004i		0		2.43754	+	1.02877i		0		0.964304	+	0.657452i		0
81.2		0		2.39605	+	0.739084i		0		−2.18585	+	2.02817i		0		2.12971	−	1.56982i		0		2.71611	+	1.85181i		0
193.1		0		−0.125927	+	1.68037i		0		1.87725	+	1.27989i		0		2.59910	+	0.494638i		0		0.158698	+	0.0239199i		0
193.2		0		−0.0722934	+	0.964688i		0		−3.07015	−	2.09319i		0		−2.50800	−	0.842585i		0		2.04110	+	0.307646i		0
289.1		0		−1.88469	−	1.28496i		0		0.269665	−	3.59844i		0		−0.415648	−	2.61290i		0		0.804920	+	2.05090i		0
289.2		0		0.934966	+	0.637449i		0		0.0772188	−	1.03041i		0		−1.36748	+	2.26495i		0		−0.628203	−	1.60064i		0
305.1		0		−1.10442	+	0.166465i		0		0.578784	−	1.47472i		0		−2.58946	−	0.542854i		0		−1.67467	+	0.516569i		0
305.2		0		2.81578	−	0.424410i		0		0.904721	−	2.30519i		0		1.30835	+	2.29961i		0		4.88176	−	1.50582i		0
401.1		0		−1.10442	−	0.166465i		0		0.578784	+	1.47472i		0		−2.58946	+	0.542854i		0		−1.67467	−	0.516569i		0
401.2		0		2.81578	+	0.424410i		0		0.904721	+	2.30519i		0		1.30835	−	2.29961i		0		4.88176	+	1.50582i		0
417.1		0		−0.427316	−	1.08878i		0		−0.0821245	−	0.0123783i		0		2.45519	−	0.985929i		0		1.19630	−	1.11001i		0
417.2		0		0.784496	+	1.99886i		0		−2.07983	−	0.313484i		0		−2.53097	−	0.770831i		0		−1.18087	+	1.09568i		0
513.1		0		−1.95066	+	0.601698i		0		0.958118	+	0.889004i		0		2.43754	−	1.02877i		0		0.964304	−	0.657452i		0
513.2		0		2.39605	−	0.739084i		0		−2.18585	−	2.02817i		0		2.12971	+	1.56982i		0		2.71611	−	1.85181i		0
529.1		0		−1.88469	+	1.28496i		0		0.269665	+	3.59844i		0		−0.415648	+	2.61290i		0		0.804920	−	2.05090i		0
529.2		0		0.934966	−	0.637449i		0		0.0772188	+	1.03041i		0		−1.36748	−	2.26495i		0		−0.628203	+	1.60064i		0
625.1		0		−0.0584109	+	0.0541974i		0		−0.427005	−	0.131714i		0		−1.60505	+	2.10329i		0		−0.223716	+	2.98528i		0
625.2		0		2.19243	−	2.03428i		0		3.17919	+	0.980650i		0		0.0867132	−	2.64433i		0		0.444274	−	5.92843i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.bg.b		24
4.b	odd	2	1		98.2.g.b	✓	24
12.b	even	2	1		882.2.z.b		24
28.d	even	2	1		686.2.g.e		24
28.f	even	6	1		686.2.e.d		24
28.f	even	6	1		686.2.g.d		24
28.g	odd	6	1		686.2.e.c		24
28.g	odd	6	1		686.2.g.f		24
49.g	even	21	1	inner	784.2.bg.b		24
196.j	even	14	1		686.2.g.d		24
196.k	odd	14	1		686.2.g.f		24
196.o	odd	42	1		98.2.g.b	✓	24
196.o	odd	42	1		686.2.e.c		24
196.o	odd	42	1		4802.2.a.o		12
196.p	even	42	1		686.2.e.d		24
196.p	even	42	1		686.2.g.e		24
196.p	even	42	1		4802.2.a.l		12
588.bb	even	42	1		882.2.z.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.2.g.b	✓	24	4.b	odd	2	1
98.2.g.b	✓	24	196.o	odd	42	1
686.2.e.c		24	28.g	odd	6	1
686.2.e.c		24	196.o	odd	42	1
686.2.e.d		24	28.f	even	6	1
686.2.e.d		24	196.p	even	42	1
686.2.g.d		24	28.f	even	6	1
686.2.g.d		24	196.j	even	14	1
686.2.g.e		24	28.d	even	2	1
686.2.g.e		24	196.p	even	42	1
686.2.g.f		24	28.g	odd	6	1
686.2.g.f		24	196.k	odd	14	1
784.2.bg.b		24	1.a	even	1	1	trivial
784.2.bg.b		24	49.g	even	21	1	inner
882.2.z.b		24	12.b	even	2	1
882.2.z.b		24	588.bb	even	42	1
4802.2.a.l		12	196.p	even	42	1
4802.2.a.o		12	196.o	odd	42	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^24 - 7*T3^23 + 12*T3^22 + 28*T3^21 - 105*T3^20 - 63*T3^19 + 617*T3^18 - 756*T3^17 - 464*T3^16 + 3983*T3^15 - 3927*T3^14 - 4998*T3^13 + 7925*T3^12 - 53459*T3^11 + 59017*T3^10 + 103362*T3^9 + 101076*T3^8 + 133280*T3^7 - 100988*T3^6 + 77147*T3^5 + 188335*T3^4 + 167510*T3^3 + 282957*T3^2 + 31857*T3 + 1681