Properties

Label 525.2.bc.d
Level 525
Weight 2
Character orbit 525.bc
Analytic conductor 4.192
Analytic rank 0
Dimension 24
CM no
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 24q^{11} + 48q^{16} - 24q^{21} - 144q^{26} - 36q^{31} - 48q^{36} + 48q^{46} + 24q^{51} + 168q^{56} + 144q^{61} - 72q^{66} + 96q^{71} + 12q^{81} - 168q^{86} + 12q^{91} + 144q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −2.60685 + 0.698503i 0.258819 0.965926i 4.57570 2.64178i 0 2.69881i −0.543835 2.58926i −6.26618 + 6.26618i −0.866025 0.500000i 0
82.2 −2.01147 + 0.538972i −0.258819 + 0.965926i 2.02348 1.16825i 0 2.08243i −1.99060 + 1.74285i −0.495509 + 0.495509i −0.866025 0.500000i 0
82.3 −0.595377 + 0.159531i −0.258819 + 0.965926i −1.40303 + 0.810038i 0 0.616380i 2.22322 1.43432i 1.57780 1.57780i −0.866025 0.500000i 0
82.4 0.595377 0.159531i 0.258819 0.965926i −1.40303 + 0.810038i 0 0.616380i −2.22322 + 1.43432i −1.57780 + 1.57780i −0.866025 0.500000i 0
82.5 2.01147 0.538972i 0.258819 0.965926i 2.02348 1.16825i 0 2.08243i 1.99060 1.74285i 0.495509 0.495509i −0.866025 0.500000i 0
82.6 2.60685 0.698503i −0.258819 + 0.965926i 4.57570 2.64178i 0 2.69881i 0.543835 + 2.58926i 6.26618 6.26618i −0.866025 0.500000i 0
157.1 −0.698503 + 2.60685i −0.965926 + 0.258819i −4.57570 2.64178i 0 2.69881i 2.58926 + 0.543835i 6.26618 6.26618i 0.866025 0.500000i 0
157.2 −0.538972 + 2.01147i 0.965926 0.258819i −2.02348 1.16825i 0 2.08243i −1.74285 + 1.99060i 0.495509 0.495509i 0.866025 0.500000i 0
157.3 −0.159531 + 0.595377i 0.965926 0.258819i 1.40303 + 0.810038i 0 0.616380i 1.43432 2.22322i −1.57780 + 1.57780i 0.866025 0.500000i 0
157.4 0.159531 0.595377i −0.965926 + 0.258819i 1.40303 + 0.810038i 0 0.616380i −1.43432 + 2.22322i 1.57780 1.57780i 0.866025 0.500000i 0
157.5 0.538972 2.01147i −0.965926 + 0.258819i −2.02348 1.16825i 0 2.08243i 1.74285 1.99060i −0.495509 + 0.495509i 0.866025 0.500000i 0
157.6 0.698503 2.60685i 0.965926 0.258819i −4.57570 2.64178i 0 2.69881i −2.58926 0.543835i −6.26618 + 6.26618i 0.866025 0.500000i 0
418.1 −0.698503 2.60685i −0.965926 0.258819i −4.57570 + 2.64178i 0 2.69881i 2.58926 0.543835i 6.26618 + 6.26618i 0.866025 + 0.500000i 0
418.2 −0.538972 2.01147i 0.965926 + 0.258819i −2.02348 + 1.16825i 0 2.08243i −1.74285 1.99060i 0.495509 + 0.495509i 0.866025 + 0.500000i 0
418.3 −0.159531 0.595377i 0.965926 + 0.258819i 1.40303 0.810038i 0 0.616380i 1.43432 + 2.22322i −1.57780 1.57780i 0.866025 + 0.500000i 0
418.4 0.159531 + 0.595377i −0.965926 0.258819i 1.40303 0.810038i 0 0.616380i −1.43432 2.22322i 1.57780 + 1.57780i 0.866025 + 0.500000i 0
418.5 0.538972 + 2.01147i −0.965926 0.258819i −2.02348 + 1.16825i 0 2.08243i 1.74285 + 1.99060i −0.495509 0.495509i 0.866025 + 0.500000i 0
418.6 0.698503 + 2.60685i 0.965926 + 0.258819i −4.57570 + 2.64178i 0 2.69881i −2.58926 + 0.543835i −6.26618 6.26618i 0.866025 + 0.500000i 0
493.1 −2.60685 0.698503i 0.258819 + 0.965926i 4.57570 + 2.64178i 0 2.69881i −0.543835 + 2.58926i −6.26618 6.26618i −0.866025 + 0.500000i 0
493.2 −2.01147 0.538972i −0.258819 0.965926i 2.02348 + 1.16825i 0 2.08243i −1.99060 1.74285i −0.495509 0.495509i −0.866025 + 0.500000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bc.d 24
5.b even 2 1 inner 525.2.bc.d 24
5.c odd 4 2 inner 525.2.bc.d 24
7.d odd 6 1 inner 525.2.bc.d 24
35.i odd 6 1 inner 525.2.bc.d 24
35.k even 12 2 inner 525.2.bc.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bc.d 24 1.a even 1 1 trivial
525.2.bc.d 24 5.b even 2 1 inner
525.2.bc.d 24 5.c odd 4 2 inner
525.2.bc.d 24 7.d odd 6 1 inner
525.2.bc.d 24 35.i odd 6 1 inner
525.2.bc.d 24 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 72 T_{2}^{20} + 4176 T_{2}^{16} - 72288 T_{2}^{12} + 1005696 T_{2}^{8} - 145152 T_{2}^{4} + 20736 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database