Properties

Label 525.2.bf.a
Level 525
Weight 2
Character orbit 525.bf
Analytic conductor 4.192
Analytic rank 0
Dimension 8
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.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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{3} + ( -2 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( -3 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{7} + ( 2 + 2 \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{3} + ( -2 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{6} + ( -3 \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{7} + ( 2 + 2 \zeta_{24}^{6} ) q^{8} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{9} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{11} + ( -5 \zeta_{24} + 5 \zeta_{24}^{5} ) q^{13} + ( -\zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{14} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{17} + ( 4 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{18} -7 \zeta_{24}^{2} q^{19} + ( 1 + 3 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{21} -6 \zeta_{24}^{3} q^{22} + ( -3 - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} ) q^{23} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{24} + ( -5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{26} + ( -1 + 5 \zeta_{24}^{3} + \zeta_{24}^{6} ) q^{27} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{29} + \zeta_{24}^{4} q^{31} + ( 3 + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{33} -2 \zeta_{24}^{6} q^{34} + ( \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{37} + ( 7 \zeta_{24}^{2} - 7 \zeta_{24}^{4} - 7 \zeta_{24}^{6} ) q^{38} + ( -5 \zeta_{24} + 5 \zeta_{24}^{2} - 5 \zeta_{24}^{6} - 5 \zeta_{24}^{7} ) q^{39} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{41} + ( -3 + \zeta_{24}^{2} + 6 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{42} + ( 5 \zeta_{24} - 5 \zeta_{24}^{5} ) q^{43} -6 \zeta_{24}^{4} q^{46} + ( 4 - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} ) q^{47} + ( -4 + 4 \zeta_{24} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{48} + ( 5 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{49} + ( 2 + \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{51} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{53} + ( -2 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{54} + ( 4 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{56} + ( 7 + 7 \zeta_{24}^{3} - 7 \zeta_{24}^{6} ) q^{57} + 10 \zeta_{24}^{5} q^{58} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{59} + ( -4 + 4 \zeta_{24}^{4} ) q^{61} + ( -1 + \zeta_{24}^{6} ) q^{62} + ( -4 + \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{63} + 8 \zeta_{24}^{6} q^{64} + ( 6 \zeta_{24} + 6 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{66} -\zeta_{24}^{7} q^{67} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{69} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{71} + ( 2 - 2 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 8 \zeta_{24}^{7} ) q^{72} -7 \zeta_{24} q^{73} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{74} + ( 3 + 6 \zeta_{24}^{2} + 6 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{77} + ( 5 + 10 \zeta_{24} - 10 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{78} -7 \zeta_{24}^{2} q^{79} + ( -4 \zeta_{24} - 7 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{81} + ( -10 \zeta_{24}^{3} + 10 \zeta_{24}^{7} ) q^{82} + ( -1 - \zeta_{24}^{6} ) q^{83} + ( 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{86} + ( -5 \zeta_{24}^{2} - 5 \zeta_{24}^{4} + 5 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{87} -12 \zeta_{24} q^{88} + ( -11 \zeta_{24}^{3} + 11 \zeta_{24}^{5} + 11 \zeta_{24}^{7} ) q^{89} + ( 15 - 5 \zeta_{24}^{4} ) q^{91} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{93} + ( 8 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{94} -10 \zeta_{24}^{3} q^{97} + ( -3 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{98} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} + 4q^{3} - 16q^{6} + 16q^{8} + O(q^{10}) \) \( 8q - 4q^{2} + 4q^{3} - 16q^{6} + 16q^{8} - 16q^{16} - 4q^{17} - 4q^{18} + 16q^{21} - 12q^{23} - 8q^{27} + 4q^{31} + 12q^{33} - 28q^{38} - 20q^{42} - 24q^{46} + 16q^{47} - 32q^{48} + 8q^{51} + 8q^{53} + 56q^{57} - 16q^{61} - 8q^{62} - 8q^{63} + 24q^{66} + 8q^{72} + 48q^{77} + 40q^{78} - 28q^{81} - 8q^{83} - 20q^{87} + 100q^{91} - 4q^{93} + 8q^{98} + O(q^{100}) \)

Character values

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

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

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
−1.36603 0.366025i 1.10721 1.33195i 0 0 −2.00000 + 1.41421i 1.15539 + 2.38014i 2.00000 + 2.00000i −0.548188 2.94949i 0
32.2 −1.36603 0.366025i 1.62484 + 0.599900i 0 0 −2.00000 1.41421i −1.15539 2.38014i 2.00000 + 2.00000i 2.28024 + 1.94949i 0
107.1 0.366025 + 1.36603i −1.33195 + 1.10721i 0 0 −2.00000 1.41421i −2.38014 1.15539i 2.00000 + 2.00000i 0.548188 2.94949i 0
107.2 0.366025 + 1.36603i 0.599900 + 1.62484i 0 0 −2.00000 + 1.41421i 2.38014 + 1.15539i 2.00000 + 2.00000i −2.28024 + 1.94949i 0
368.1 0.366025 1.36603i −1.33195 1.10721i 0 0 −2.00000 + 1.41421i −2.38014 + 1.15539i 2.00000 2.00000i 0.548188 + 2.94949i 0
368.2 0.366025 1.36603i 0.599900 1.62484i 0 0 −2.00000 1.41421i 2.38014 1.15539i 2.00000 2.00000i −2.28024 1.94949i 0
443.1 −1.36603 + 0.366025i 1.10721 + 1.33195i 0 0 −2.00000 1.41421i 1.15539 2.38014i 2.00000 2.00000i −0.548188 + 2.94949i 0
443.2 −1.36603 + 0.366025i 1.62484 0.599900i 0 0 −2.00000 + 1.41421i −1.15539 + 2.38014i 2.00000 2.00000i 2.28024 1.94949i 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
5.c odd 4 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.l odd 12 1 inner
105.o odd 6 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.a 8
3.b odd 2 1 525.2.bf.d yes 8
5.b even 2 1 525.2.bf.d yes 8
5.c odd 4 1 inner 525.2.bf.a 8
5.c odd 4 1 525.2.bf.d yes 8
7.c even 3 1 inner 525.2.bf.a 8
15.d odd 2 1 inner 525.2.bf.a 8
15.e even 4 1 inner 525.2.bf.a 8
15.e even 4 1 525.2.bf.d yes 8
21.h odd 6 1 525.2.bf.d yes 8
35.j even 6 1 525.2.bf.d yes 8
35.l odd 12 1 inner 525.2.bf.a 8
35.l odd 12 1 525.2.bf.d yes 8
105.o odd 6 1 inner 525.2.bf.a 8
105.x even 12 1 inner 525.2.bf.a 8
105.x even 12 1 525.2.bf.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bf.a 8 1.a even 1 1 trivial
525.2.bf.a 8 5.c odd 4 1 inner
525.2.bf.a 8 7.c even 3 1 inner
525.2.bf.a 8 15.d odd 2 1 inner
525.2.bf.a 8 15.e even 4 1 inner
525.2.bf.a 8 35.l odd 12 1 inner
525.2.bf.a 8 105.o odd 6 1 inner
525.2.bf.a 8 105.x even 12 1 inner
525.2.bf.d yes 8 3.b odd 2 1
525.2.bf.d yes 8 5.b even 2 1
525.2.bf.d yes 8 5.c odd 4 1
525.2.bf.d yes 8 15.e even 4 1
525.2.bf.d yes 8 21.h odd 6 1
525.2.bf.d yes 8 35.j even 6 1
525.2.bf.d yes 8 35.l odd 12 1
525.2.bf.d yes 8 105.x even 12 1

Hecke kernels

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

\( T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} + 4 T_{2} + 4 \)
\( T_{13}^{4} + 625 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T + 2 T^{2} )^{4}( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \)
$3$ \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 24 T^{5} + 72 T^{6} - 108 T^{7} + 81 T^{8} \)
$5$ \( \)
$7$ \( 1 + 23 T^{4} + 2401 T^{8} \)
$11$ \( ( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 337 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 2 T + 2 T^{2} - 64 T^{3} - 353 T^{4} - 1088 T^{5} + 578 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )^{2}( 1 + 26 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 6 T + 18 T^{2} - 168 T^{3} - 1033 T^{4} - 3864 T^{5} + 9522 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 + 8 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4} \)
$37$ \( ( 1 - 2062 T^{4} + 1874161 T^{8} )( 1 - 529 T^{4} + 1874161 T^{8} ) \)
$41$ \( ( 1 - 32 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 23 T^{4} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 8 T + 32 T^{2} + 496 T^{3} - 4193 T^{4} + 23312 T^{5} + 70688 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 4 T + 8 T^{2} + 392 T^{3} - 3593 T^{4} + 20776 T^{5} + 22472 T^{6} - 595508 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 116 T^{2} + 9975 T^{4} - 403796 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 - 8711 T^{4} + 55730400 T^{8} - 175536415031 T^{12} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 92 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 8542 T^{4} + 28398241 T^{8} )( 1 + 9791 T^{4} + 28398241 T^{8} ) \)
$79$ \( ( 1 + 109 T^{2} + 5640 T^{4} + 680269 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 2 T + 2 T^{2} + 166 T^{3} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 64 T^{2} - 3825 T^{4} + 506944 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 24 T + 288 T^{2} - 2328 T^{3} + 9409 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 2328 T^{3} + 9409 T^{4} )^{2} \)
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