Properties

Label 525.2.b.d
Level 525
Weight 2
Character orbit 525.b
Analytic conductor 4.192
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + 2 q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{12} + ( 3 - 6 \zeta_{6} ) q^{13} + 4 q^{16} + ( -5 + 10 \zeta_{6} ) q^{19} + ( 1 - 5 \zeta_{6} ) q^{21} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 2 \zeta_{6} ) q^{28} + ( 5 - 10 \zeta_{6} ) q^{31} -6 q^{36} -10 q^{37} -9 q^{39} + 5 q^{43} + ( 4 - 8 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 6 - 12 \zeta_{6} ) q^{52} + 15 q^{57} + ( -5 + 10 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} + 8 q^{64} -5 q^{67} + ( -8 + 16 \zeta_{6} ) q^{73} + ( -10 + 20 \zeta_{6} ) q^{76} -4 q^{79} + 9 q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( 3 - 15 \zeta_{6} ) q^{91} -15 q^{93} + ( -11 + 22 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 5q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 4q^{4} + 5q^{7} - 6q^{9} + 8q^{16} - 3q^{21} + 10q^{28} - 12q^{36} - 20q^{37} - 18q^{39} + 10q^{43} + 11q^{49} + 30q^{57} - 15q^{63} + 16q^{64} - 10q^{67} - 8q^{79} + 18q^{81} - 6q^{84} - 9q^{91} - 30q^{93} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 2.00000 0 0 2.50000 0.866025i 0 −3.00000 0
251.2 0 1.73205i 2.00000 0 0 2.50000 + 0.866025i 0 −3.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 inner
21.c 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.d yes 2
3.b odd 2 1 CM 525.2.b.d yes 2
5.b even 2 1 525.2.b.c 2
5.c odd 4 2 525.2.g.a 4
7.b odd 2 1 inner 525.2.b.d yes 2
15.d odd 2 1 525.2.b.c 2
15.e even 4 2 525.2.g.a 4
21.c even 2 1 inner 525.2.b.d yes 2
35.c odd 2 1 525.2.b.c 2
35.f even 4 2 525.2.g.a 4
105.g even 2 1 525.2.b.c 2
105.k odd 4 2 525.2.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.c 2 5.b even 2 1
525.2.b.c 2 15.d odd 2 1
525.2.b.c 2 35.c odd 2 1
525.2.b.c 2 105.g even 2 1
525.2.b.d yes 2 1.a even 1 1 trivial
525.2.b.d yes 2 3.b odd 2 1 CM
525.2.b.d yes 2 7.b odd 2 1 inner
525.2.b.d yes 2 21.c even 2 1 inner
525.2.g.a 4 5.c odd 4 2
525.2.g.a 4 15.e even 4 2
525.2.g.a 4 35.f even 4 2
525.2.g.a 4 105.k odd 4 2

Hecke kernels

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

\( T_{2} \)
\( T_{11} \)
\( T_{17} \)
\( T_{37} + 10 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ 1
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \)
$37$ \( ( 1 + 10 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \)
$67$ \( ( 1 + 5 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )( 1 + 5 T + 97 T^{2} ) \)
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