Properties

Label 525.2.bf.b
Level 525
Weight 2
Character orbit 525.bf
Analytic conductor 4.192
Analytic rank 0
Dimension 8
CM discriminant -3
Inner twists 16

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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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{2} q^{4} + ( -2 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{2} q^{4} + ( -2 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{7} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{9} + ( 4 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{12} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{13} + 4 \zeta_{24}^{4} q^{16} + ( 7 \zeta_{24}^{2} - 7 \zeta_{24}^{6} ) q^{19} + ( -5 + 4 \zeta_{24}^{4} ) q^{21} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{27} + ( -6 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{28} + ( -7 + 7 \zeta_{24}^{4} ) q^{31} + 6 q^{36} + ( -14 \zeta_{24}^{3} + 7 \zeta_{24}^{7} ) q^{37} + 3 \zeta_{24}^{2} q^{39} + ( -7 \zeta_{24} - 7 \zeta_{24}^{5} ) q^{43} + ( 4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{48} + ( 3 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{49} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{52} + ( 7 \zeta_{24}^{3} - 14 \zeta_{24}^{7} ) q^{57} -14 \zeta_{24}^{4} q^{61} + ( -6 \zeta_{24} + 9 \zeta_{24}^{5} ) q^{63} + 8 \zeta_{24}^{6} q^{64} + ( 7 \zeta_{24}^{3} + 7 \zeta_{24}^{7} ) q^{67} + ( 18 \zeta_{24} - 9 \zeta_{24}^{5} ) q^{73} + 14 q^{76} + ( -13 \zeta_{24}^{2} + 13 \zeta_{24}^{6} ) q^{79} + ( 9 - 9 \zeta_{24}^{4} ) q^{81} + ( -10 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{84} + ( -4 - \zeta_{24}^{4} ) q^{91} + ( -7 \zeta_{24} + 14 \zeta_{24}^{5} ) q^{93} + ( -8 \zeta_{24}^{3} + 16 \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{16} - 24q^{21} - 28q^{31} + 48q^{36} - 56q^{61} + 112q^{76} + 36q^{81} - 36q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-\zeta_{24}^{6}\) \(-1\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0 −0.448288 1.67303i −1.73205 1.00000i 0 0 −1.48356 2.19067i 0 −2.59808 + 1.50000i 0
32.2 0 0.448288 + 1.67303i −1.73205 1.00000i 0 0 1.48356 + 2.19067i 0 −2.59808 + 1.50000i 0
107.1 0 −1.67303 0.448288i 1.73205 1.00000i 0 0 2.19067 + 1.48356i 0 2.59808 + 1.50000i 0
107.2 0 1.67303 + 0.448288i 1.73205 1.00000i 0 0 −2.19067 1.48356i 0 2.59808 + 1.50000i 0
368.1 0 −1.67303 + 0.448288i 1.73205 + 1.00000i 0 0 2.19067 1.48356i 0 2.59808 1.50000i 0
368.2 0 1.67303 0.448288i 1.73205 + 1.00000i 0 0 −2.19067 + 1.48356i 0 2.59808 1.50000i 0
443.1 0 −0.448288 + 1.67303i −1.73205 + 1.00000i 0 0 −1.48356 + 2.19067i 0 −2.59808 1.50000i 0
443.2 0 0.448288 1.67303i −1.73205 + 1.00000i 0 0 1.48356 2.19067i 0 −2.59808 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
21.h odd 6 1 inner
35.j even 6 1 inner
35.l odd 12 2 inner
105.o odd 6 1 inner
105.x 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.bf.b 8
3.b odd 2 1 CM 525.2.bf.b 8
5.b even 2 1 inner 525.2.bf.b 8
5.c odd 4 2 inner 525.2.bf.b 8
7.c even 3 1 inner 525.2.bf.b 8
15.d odd 2 1 inner 525.2.bf.b 8
15.e even 4 2 inner 525.2.bf.b 8
21.h odd 6 1 inner 525.2.bf.b 8
35.j even 6 1 inner 525.2.bf.b 8
35.l odd 12 2 inner 525.2.bf.b 8
105.o odd 6 1 inner 525.2.bf.b 8
105.x even 12 2 inner 525.2.bf.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.b 8 1.a even 1 1 trivial
525.2.bf.b 8 3.b odd 2 1 CM
525.2.bf.b 8 5.b even 2 1 inner
525.2.bf.b 8 5.c odd 4 2 inner
525.2.bf.b 8 7.c even 3 1 inner
525.2.bf.b 8 15.d odd 2 1 inner
525.2.bf.b 8 15.e even 4 2 inner
525.2.bf.b 8 21.h odd 6 1 inner
525.2.bf.b 8 35.j even 6 1 inner
525.2.bf.b 8 35.l odd 12 2 inner
525.2.bf.b 8 105.o odd 6 1 inner
525.2.bf.b 8 105.x even 12 2 inner

Hecke kernels

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

\( T_{2} \)
\( T_{13}^{4} + 9 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{2} \)
$3$ \( 1 - 9 T^{4} + 81 T^{8} \)
$5$ 1
$7$ \( 1 + 71 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 11 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 191 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 289 T^{4} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 529 T^{4} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{8} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{4}( 1 + 11 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} ) \)
$41$ \( ( 1 - 41 T^{2} )^{8} \)
$43$ \( ( 1 + 23 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 2209 T^{4} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 2809 T^{4} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 59 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + T + 61 T^{2} )^{4}( 1 + 13 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 2903 T^{4} + 20151121 T^{8} )( 1 + 5906 T^{4} + 20151121 T^{8} ) \)
$71$ \( ( 1 - 71 T^{2} )^{8} \)
$73$ \( ( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} ) \)
$79$ \( ( 1 - 142 T^{2} + 6241 T^{4} )^{2}( 1 + 131 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 6889 T^{4} )^{4} \)
$89$ \( ( 1 - 89 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 18814 T^{4} + 88529281 T^{8} )^{2} \)
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