Newform orbit 525.2.bf.f

• Label: 525.2.bf.f
• Level: $525$
• Weight: $2$
• Character orbit: 525.bf
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} - 24 q^{6} + 12 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−2.35640	−	0.631395i	1.54878	−	0.775426i	3.42191	+	1.97564i		0		−4.13914	+	0.849321i	1.91891	−	1.82148i	−3.36596	−	3.36596i	1.79743	−	2.40193i		0
32.2	−2.17249	−	0.582118i	0.245221	+	1.71460i	2.64881	+	1.52929i		0		0.465359	−	3.86771i	2.38227	+	1.15099i	−1.68355	−	1.68355i	−2.87973	+	0.840915i		0
32.3	−1.46015	−	0.391246i	0.852980	−	1.50746i	0.246919	+	0.142558i		0		−1.83527	+	1.86739i	−2.36949	+	1.17707i	1.83305	+	1.83305i	−1.54485	−	2.57166i		0
32.4	−1.26950	−	0.340162i	−1.71749	+	0.224146i	−0.236127	−	0.136328i		0		2.25660	+	0.299670i	−1.25943	−	2.32676i	2.11207	+	2.11207i	2.89952	−	0.769934i		0
32.5	−0.907300	−	0.243110i	−1.12459	−	1.31730i	−0.967960	−	0.558852i		0		0.700094	+	1.46859i	2.64571	−	0.0144144i	2.07075	+	2.07075i	−0.470578	+	2.96286i		0
32.6	−0.298314	−	0.0799329i	1.64547	+	0.540759i	−1.64945	−	0.952310i		0		−0.447643	−	0.292843i	−0.951942	+	2.46856i	0.852694	+	0.852694i	2.41516	+	1.77961i		0
32.7	0.298314	+	0.0799329i	−1.15464	+	1.29105i	−1.64945	−	0.952310i		0		−0.447643	+	0.292843i	−0.951942	+	2.46856i	−0.852694	−	0.852694i	−0.333606	−	2.98139i		0
32.8	0.907300	+	0.243110i	0.315275	−	1.70312i	−0.967960	−	0.558852i		0		0.700094	−	1.46859i	2.64571	−	0.0144144i	−2.07075	−	2.07075i	−2.80120	−	1.07390i		0
32.9	1.26950	+	0.340162i	1.59946	−	0.664627i	−0.236127	−	0.136328i		0		2.25660	−	0.299670i	−1.25943	−	2.32676i	−2.11207	−	2.11207i	2.11654	−	2.12609i		0
32.10	1.46015	+	0.391246i	−1.49243	−	0.879005i	0.246919	+	0.142558i		0		−1.83527	−	1.86739i	−2.36949	+	1.17707i	−1.83305	−	1.83305i	1.45470	+	2.62371i		0
32.11	2.17249	+	0.582118i	0.644934	+	1.60750i	2.64881	+	1.52929i		0		0.465359	+	3.86771i	2.38227	+	1.15099i	1.68355	+	1.68355i	−2.16812	+	2.07346i		0
32.12	2.35640	+	0.631395i	−1.72899	+	0.102851i	3.42191	+	1.97564i		0		−4.13914	−	0.849321i	1.91891	−	1.82148i	3.36596	+	3.36596i	2.97884	−	0.355658i		0
107.1	−0.631395	−	2.35640i	0.775426	−	1.54878i	−3.42191	+	1.97564i		0		−4.13914	−	0.849321i	−1.82148	+	1.91891i	3.36596	+	3.36596i	−1.79743	−	2.40193i		0
107.2	−0.582118	−	2.17249i	−1.71460	−	0.245221i	−2.64881	+	1.52929i		0		0.465359	+	3.86771i	1.15099	+	2.38227i	1.68355	+	1.68355i	2.87973	+	0.840915i		0
107.3	−0.391246	−	1.46015i	1.50746	−	0.852980i	−0.246919	+	0.142558i		0		−1.83527	−	1.86739i	1.17707	−	2.36949i	−1.83305	−	1.83305i	1.54485	−	2.57166i		0
107.4	−0.340162	−	1.26950i	−0.224146	+	1.71749i	0.236127	−	0.136328i		0		2.25660	−	0.299670i	−2.32676	−	1.25943i	−2.11207	−	2.11207i	−2.89952	−	0.769934i		0
107.5	−0.243110	−	0.907300i	1.31730	+	1.12459i	0.967960	−	0.558852i		0		0.700094	−	1.46859i	−0.0144144	+	2.64571i	−2.07075	−	2.07075i	0.470578	+	2.96286i		0
107.6	−0.0799329	−	0.298314i	−0.540759	−	1.64547i	1.64945	−	0.952310i		0		−0.447643	+	0.292843i	2.46856	−	0.951942i	−0.852694	−	0.852694i	−2.41516	+	1.77961i		0
107.7	0.0799329	+	0.298314i	−1.29105	+	1.15464i	1.64945	−	0.952310i		0		−0.447643	−	0.292843i	2.46856	−	0.951942i	0.852694	+	0.852694i	0.333606	−	2.98139i		0
107.8	0.243110	+	0.907300i	1.70312	−	0.315275i	0.967960	−	0.558852i		0		0.700094	+	1.46859i	−0.0144144	+	2.64571i	2.07075	+	2.07075i	2.80120	−	1.07390i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
7.c	even	3	1	inner
15.e	even	4	1	inner
21.h	odd	6	1	inner
35.l	odd	12	1	inner
105.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.bf.f		48
3.b	odd	2	1	inner	525.2.bf.f		48
5.b	even	2	1		105.2.x.a	✓	48
5.c	odd	4	1		105.2.x.a	✓	48
5.c	odd	4	1	inner	525.2.bf.f		48
7.c	even	3	1	inner	525.2.bf.f		48
15.d	odd	2	1		105.2.x.a	✓	48
15.e	even	4	1		105.2.x.a	✓	48
15.e	even	4	1	inner	525.2.bf.f		48
21.h	odd	6	1	inner	525.2.bf.f		48
35.c	odd	2	1		735.2.y.i		48
35.f	even	4	1		735.2.y.i		48
35.i	odd	6	1		735.2.j.e		24
35.i	odd	6	1		735.2.y.i		48
35.j	even	6	1		105.2.x.a	✓	48
35.j	even	6	1		735.2.j.g		24
35.k	even	12	1		735.2.j.e		24
35.k	even	12	1		735.2.y.i		48
35.l	odd	12	1		105.2.x.a	✓	48
35.l	odd	12	1	inner	525.2.bf.f		48
35.l	odd	12	1		735.2.j.g		24
105.g	even	2	1		735.2.y.i		48
105.k	odd	4	1		735.2.y.i		48
105.o	odd	6	1		105.2.x.a	✓	48
105.o	odd	6	1		735.2.j.g		24
105.p	even	6	1		735.2.j.e		24
105.p	even	6	1		735.2.y.i		48
105.w	odd	12	1		735.2.j.e		24
105.w	odd	12	1		735.2.y.i		48
105.x	even	12	1		105.2.x.a	✓	48
105.x	even	12	1	inner	525.2.bf.f		48
105.x	even	12	1		735.2.j.g		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.x.a	✓	48	5.b	even	2	1
105.2.x.a	✓	48	5.c	odd	4	1
105.2.x.a	✓	48	15.d	odd	2	1
105.2.x.a	✓	48	15.e	even	4	1
105.2.x.a	✓	48	35.j	even	6	1
105.2.x.a	✓	48	35.l	odd	12	1
105.2.x.a	✓	48	105.o	odd	6	1
105.2.x.a	✓	48	105.x	even	12	1
525.2.bf.f		48	1.a	even	1	1	trivial
525.2.bf.f		48	3.b	odd	2	1	inner
525.2.bf.f		48	5.c	odd	4	1	inner
525.2.bf.f		48	7.c	even	3	1	inner
525.2.bf.f		48	15.e	even	4	1	inner
525.2.bf.f		48	21.h	odd	6	1	inner
525.2.bf.f		48	35.l	odd	12	1	inner
525.2.bf.f		48	105.x	even	12	1	inner
735.2.j.e		24	35.i	odd	6	1
735.2.j.e		24	35.k	even	12	1
735.2.j.e		24	105.p	even	6	1
735.2.j.e		24	105.w	odd	12	1
735.2.j.g		24	35.j	even	6	1
735.2.j.g		24	35.l	odd	12	1
735.2.j.g		24	105.o	odd	6	1
735.2.j.g		24	105.x	even	12	1
735.2.y.i		48	35.c	odd	2	1
735.2.y.i		48	35.f	even	4	1
735.2.y.i		48	35.i	odd	6	1
735.2.y.i		48	35.k	even	12	1
735.2.y.i		48	105.g	even	2	1
735.2.y.i		48	105.k	odd	4	1
735.2.y.i		48	105.p	even	6	1
735.2.y.i		48	105.w	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

T2^48 - 70*T2^44 + 3423*T2^40 - 84374*T2^36 + 1495233*T2^32 - 11151456*T2^28 + 58992592*T2^24 - 164139168*T2^20 + 323534556*T2^16 - 229926880*T2^12 + 122918800*T2^8 - 1118000*T2^4 + 10000

T13^12 - 4*T13^11 + 8*T13^10 - 26*T13^9 + 334*T13^8 - 1504*T13^7 + 3682*T13^6 - 4826*T13^5 + 6361*T13^4 - 15848*T13^3 + 39200*T13^2 - 49280*T13 + 30976