Properties

Label 525.2.b.f
Level 525
Weight 2
Character orbit 525.b
Analytic conductor 4.192
Analytic rank 0
Dimension 4
CM discriminant -35
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: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{-7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + 2 q^{4} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + 2 q^{4} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -1 + 2 \beta_{2} ) q^{11} + 2 \beta_{3} q^{12} + ( \beta_{1} - \beta_{3} ) q^{13} + 4 q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 4 - \beta_{2} ) q^{21} + ( -3 \beta_{1} - \beta_{3} ) q^{27} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{28} + ( 1 - 2 \beta_{2} ) q^{29} + ( -6 \beta_{1} - \beta_{3} ) q^{33} + ( -2 + 2 \beta_{2} ) q^{36} + ( 4 - \beta_{2} ) q^{39} + ( -2 + 4 \beta_{2} ) q^{44} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{47} + 4 \beta_{3} q^{48} -7 q^{49} + ( 2 + \beta_{2} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{52} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{63} + 8 q^{64} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 2 - 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{77} + q^{79} + ( -8 - \beta_{2} ) q^{81} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 8 - 2 \beta_{2} ) q^{84} + ( 6 \beta_{1} + \beta_{3} ) q^{87} -7 q^{91} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{97} + ( -17 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 2q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 2q^{9} + 16q^{16} + 14q^{21} - 4q^{36} + 14q^{39} - 28q^{49} + 10q^{51} + 32q^{64} + 4q^{79} - 34q^{81} + 28q^{84} - 28q^{91} - 70q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 7 \nu + 8 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - \nu^{2} + 7 \nu + 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} + 3 \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 9\)\()/2\)

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
0.809017 + 2.14046i
0.809017 2.14046i
−0.309017 0.817582i
−0.309017 + 0.817582i
0 −1.11803 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 + 2.95804i 0
251.2 0 −1.11803 + 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 2.95804i 0
251.3 0 1.11803 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 2.95804i 0
251.4 0 1.11803 + 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 + 2.95804i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
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
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.f 4
3.b odd 2 1 inner 525.2.b.f 4
5.b even 2 1 inner 525.2.b.f 4
5.c odd 4 2 105.2.g.b 4
7.b odd 2 1 inner 525.2.b.f 4
15.d odd 2 1 inner 525.2.b.f 4
15.e even 4 2 105.2.g.b 4
20.e even 4 2 1680.2.k.b 4
21.c even 2 1 inner 525.2.b.f 4
35.c odd 2 1 CM 525.2.b.f 4
35.f even 4 2 105.2.g.b 4
35.k even 12 4 735.2.p.b 8
35.l odd 12 4 735.2.p.b 8
60.l odd 4 2 1680.2.k.b 4
105.g even 2 1 inner 525.2.b.f 4
105.k odd 4 2 105.2.g.b 4
105.w odd 12 4 735.2.p.b 8
105.x even 12 4 735.2.p.b 8
140.j odd 4 2 1680.2.k.b 4
420.w even 4 2 1680.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.b 4 5.c odd 4 2
105.2.g.b 4 15.e even 4 2
105.2.g.b 4 35.f even 4 2
105.2.g.b 4 105.k odd 4 2
525.2.b.f 4 1.a even 1 1 trivial
525.2.b.f 4 3.b odd 2 1 inner
525.2.b.f 4 5.b even 2 1 inner
525.2.b.f 4 7.b odd 2 1 inner
525.2.b.f 4 15.d odd 2 1 inner
525.2.b.f 4 21.c even 2 1 inner
525.2.b.f 4 35.c odd 2 1 CM
525.2.b.f 4 105.g even 2 1 inner
735.2.p.b 8 35.k even 12 4
735.2.p.b 8 35.l odd 12 4
735.2.p.b 8 105.w odd 12 4
735.2.p.b 8 105.x even 12 4
1680.2.k.b 4 20.e even 4 2
1680.2.k.b 4 60.l odd 4 2
1680.2.k.b 4 140.j odd 4 2
1680.2.k.b 4 420.w even 4 2

Hecke kernels

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

\( T_{2} \)
\( T_{11}^{2} + 35 \)
\( T_{17}^{2} - 5 \)
\( T_{37} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{4} \)
$3$ \( 1 + T^{2} + 9 T^{4} \)
$5$ \( \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 19 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 29 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 - 9 T + 29 T^{2} )^{2}( 1 + 9 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 43 T^{2} )^{4} \)
$47$ \( ( 1 - 31 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 34 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 86 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 + 149 T^{2} + 9409 T^{4} )^{2} \)
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