Properties

Label 525.2.g.a
Level 525
Weight 2
Character orbit 525.g
Analytic conductor 4.192
Analytic rank 0
Dimension 4
CM discriminant -3
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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 q^{4} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 q^{4} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{12} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{13} + 4 q^{16} + ( -5 + 10 \zeta_{12}^{2} ) q^{19} + ( -4 + 5 \zeta_{12}^{2} ) q^{21} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -2 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{28} + ( -5 + 10 \zeta_{12}^{2} ) q^{31} -6 q^{36} + 10 \zeta_{12}^{3} q^{37} + 9 q^{39} + 5 \zeta_{12}^{3} q^{43} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( -3 - 5 \zeta_{12}^{2} ) q^{49} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{52} -15 \zeta_{12}^{3} q^{57} + ( 5 - 10 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{63} -8 q^{64} + 5 \zeta_{12}^{3} q^{67} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{73} + ( 10 - 20 \zeta_{12}^{2} ) q^{76} + 4 q^{79} + 9 q^{81} + ( 8 - 10 \zeta_{12}^{2} ) q^{84} + ( -12 + 15 \zeta_{12}^{2} ) q^{91} -15 \zeta_{12}^{3} q^{93} + ( -22 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 12q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 12q^{9} + 16q^{16} - 6q^{21} - 24q^{36} + 36q^{39} - 22q^{49} - 32q^{64} + 16q^{79} + 36q^{81} + 12q^{84} - 18q^{91} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 −2.00000 0 0 0.866025 2.50000i 0 3.00000 0
524.2 0 −1.73205 −2.00000 0 0 0.866025 + 2.50000i 0 3.00000 0
524.3 0 1.73205 −2.00000 0 0 −0.866025 2.50000i 0 3.00000 0
524.4 0 1.73205 −2.00000 0 0 −0.866025 + 2.50000i 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 \(\Q(\sqrt{-3}) \)
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.g.a 4
3.b odd 2 1 CM 525.2.g.a 4
5.b even 2 1 inner 525.2.g.a 4
5.c odd 4 1 525.2.b.c 2
5.c odd 4 1 525.2.b.d yes 2
7.b odd 2 1 inner 525.2.g.a 4
15.d odd 2 1 inner 525.2.g.a 4
15.e even 4 1 525.2.b.c 2
15.e even 4 1 525.2.b.d yes 2
21.c even 2 1 inner 525.2.g.a 4
35.c odd 2 1 inner 525.2.g.a 4
35.f even 4 1 525.2.b.c 2
35.f even 4 1 525.2.b.d yes 2
105.g even 2 1 inner 525.2.g.a 4
105.k odd 4 1 525.2.b.c 2
105.k odd 4 1 525.2.b.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.b.c 2 5.c odd 4 1
525.2.b.c 2 15.e even 4 1
525.2.b.c 2 35.f even 4 1
525.2.b.c 2 105.k odd 4 1
525.2.b.d yes 2 5.c odd 4 1
525.2.b.d yes 2 15.e even 4 1
525.2.b.d yes 2 35.f even 4 1
525.2.b.d yes 2 105.k odd 4 1
525.2.g.a 4 1.a even 1 1 trivial
525.2.g.a 4 3.b odd 2 1 CM
525.2.g.a 4 5.b even 2 1 inner
525.2.g.a 4 7.b odd 2 1 inner
525.2.g.a 4 15.d odd 2 1 inner
525.2.g.a 4 21.c even 2 1 inner
525.2.g.a 4 35.c odd 2 1 inner
525.2.g.a 4 105.g even 2 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{41} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{4} \)
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 17 T^{2} )^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 26 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + 13 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 109 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 46 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \)
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