Properties

Label 525.2.b.j
Level 525
Weight 2
Character orbit 525.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} - q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 1 - 2 \zeta_{24}^{4} ) q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{24}^{4} ) q^{2} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} - q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{6} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 1 - 2 \zeta_{24}^{4} ) q^{8} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{9} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{11} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{12} -4 \zeta_{24}^{6} q^{13} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{14} -5 q^{16} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{17} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{18} + ( -1 - 2 \zeta_{24} - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{21} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{22} + ( -2 + 4 \zeta_{24}^{4} ) q^{23} + ( \zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{24} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{26} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{27} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{28} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{29} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{31} + ( -3 + 6 \zeta_{24}^{4} ) q^{32} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{33} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{34} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{36} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{39} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{41} + ( -1 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{42} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{44} + 6 q^{46} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{48} + ( 5 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 4 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{51} + 4 \zeta_{24}^{6} q^{52} + ( -\zeta_{24} - 10 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{54} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{56} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{58} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{59} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{61} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} ) q^{62} + ( -4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} - q^{64} + ( 2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{66} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{67} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{68} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{69} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{71} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{72} + 8 \zeta_{24}^{6} q^{73} + ( 4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{77} + ( -4 + 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{78} -8 q^{79} + ( -7 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{81} -6 \zeta_{24}^{6} q^{82} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{83} + ( 1 + 2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{84} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{86} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} ) q^{87} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{88} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{89} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{91} + ( 2 - 4 \zeta_{24}^{4} ) q^{92} + ( 8 + 4 \zeta_{24} + 4 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{93} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{94} + ( -3 \zeta_{24} + 6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{96} -8 \zeta_{24}^{6} q^{97} + ( 5 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{98} + ( 8 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{4} + 8q^{9} - 40q^{16} - 8q^{21} - 8q^{36} - 32q^{39} + 48q^{46} + 40q^{49} + 32q^{51} - 8q^{64} - 64q^{79} - 56q^{81} + 8q^{84} - 32q^{91} + 64q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
1.73205i −1.41421 1.00000i −1.00000 0 −1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.2 1.73205i −1.41421 + 1.00000i −1.00000 0 1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.3 1.73205i 1.41421 1.00000i −1.00000 0 −1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.4 1.73205i 1.41421 + 1.00000i −1.00000 0 1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.5 1.73205i −1.41421 1.00000i −1.00000 0 1.73205 2.44949i −2.44949 1.00000i 1.73205i 1.00000 + 2.82843i 0
251.6 1.73205i −1.41421 + 1.00000i −1.00000 0 −1.73205 2.44949i 2.44949 + 1.00000i 1.73205i 1.00000 2.82843i 0
251.7 1.73205i 1.41421 1.00000i −1.00000 0 1.73205 + 2.44949i 2.44949 1.00000i 1.73205i 1.00000 2.82843i 0
251.8 1.73205i 1.41421 + 1.00000i −1.00000 0 −1.73205 + 2.44949i −2.44949 + 1.00000i 1.73205i 1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.b.j 8
3.b odd 2 1 inner 525.2.b.j 8
5.b even 2 1 inner 525.2.b.j 8
5.c odd 4 1 105.2.g.a 4
5.c odd 4 1 105.2.g.c yes 4
7.b odd 2 1 inner 525.2.b.j 8
15.d odd 2 1 inner 525.2.b.j 8
15.e even 4 1 105.2.g.a 4
15.e even 4 1 105.2.g.c yes 4
20.e even 4 1 1680.2.k.a 4
20.e even 4 1 1680.2.k.c 4
21.c even 2 1 inner 525.2.b.j 8
35.c odd 2 1 inner 525.2.b.j 8
35.f even 4 1 105.2.g.a 4
35.f even 4 1 105.2.g.c yes 4
35.k even 12 2 735.2.p.a 8
35.k even 12 2 735.2.p.c 8
35.l odd 12 2 735.2.p.a 8
35.l odd 12 2 735.2.p.c 8
60.l odd 4 1 1680.2.k.a 4
60.l odd 4 1 1680.2.k.c 4
105.g even 2 1 inner 525.2.b.j 8
105.k odd 4 1 105.2.g.a 4
105.k odd 4 1 105.2.g.c yes 4
105.w odd 12 2 735.2.p.a 8
105.w odd 12 2 735.2.p.c 8
105.x even 12 2 735.2.p.a 8
105.x even 12 2 735.2.p.c 8
140.j odd 4 1 1680.2.k.a 4
140.j odd 4 1 1680.2.k.c 4
420.w even 4 1 1680.2.k.a 4
420.w even 4 1 1680.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 5.c odd 4 1
105.2.g.a 4 15.e even 4 1
105.2.g.a 4 35.f even 4 1
105.2.g.a 4 105.k odd 4 1
105.2.g.c yes 4 5.c odd 4 1
105.2.g.c yes 4 15.e even 4 1
105.2.g.c yes 4 35.f even 4 1
105.2.g.c yes 4 105.k odd 4 1
525.2.b.j 8 1.a even 1 1 trivial
525.2.b.j 8 3.b odd 2 1 inner
525.2.b.j 8 5.b even 2 1 inner
525.2.b.j 8 7.b odd 2 1 inner
525.2.b.j 8 15.d odd 2 1 inner
525.2.b.j 8 21.c even 2 1 inner
525.2.b.j 8 35.c odd 2 1 inner
525.2.b.j 8 105.g even 2 1 inner
735.2.p.a 8 35.k even 12 2
735.2.p.a 8 35.l odd 12 2
735.2.p.a 8 105.w odd 12 2
735.2.p.a 8 105.x even 12 2
735.2.p.c 8 35.k even 12 2
735.2.p.c 8 35.l odd 12 2
735.2.p.c 8 105.w odd 12 2
735.2.p.c 8 105.x even 12 2
1680.2.k.a 4 20.e even 4 1
1680.2.k.a 4 60.l odd 4 1
1680.2.k.a 4 140.j odd 4 1
1680.2.k.a 4 420.w even 4 1
1680.2.k.c 4 20.e even 4 1
1680.2.k.c 4 60.l odd 4 1
1680.2.k.c 4 140.j odd 4 1
1680.2.k.c 4 420.w even 4 1

Hecke kernels

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

\( T_{2}^{2} + 3 \)
\( T_{11}^{2} + 8 \)
\( T_{17}^{2} - 8 \)
\( T_{37} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + 4 T^{4} )^{4} \)
$3$ \( ( 1 - 2 T^{2} + 9 T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{4}( 1 + 6 T + 11 T^{2} )^{4} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{4}( 1 + 6 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 26 T^{2} + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{8} \)
$23$ \( ( 1 - 34 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 - 26 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 34 T^{2} + 961 T^{4} )^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{8} \)
$41$ \( ( 1 + 70 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 + 62 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{8} \)
$59$ \( ( 1 + 70 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 - 26 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 + 110 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 82 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{8} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )^{4}( 1 + 18 T + 97 T^{2} )^{4} \)
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