Properties

Label 525.2.t.b
Level 525
Weight 2
Character orbit 525.t
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.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 4 - 2 \zeta_{6} ) q^{12} + ( 4 - 8 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + ( 4 - 2 \zeta_{6} ) q^{19} + ( 5 - \zeta_{6} ) q^{21} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 - 6 \zeta_{6} ) q^{28} + ( -5 - 5 \zeta_{6} ) q^{31} -6 q^{36} -11 \zeta_{6} q^{37} + ( -12 + 12 \zeta_{6} ) q^{39} + 5 q^{43} + ( -4 + 8 \zeta_{6} ) q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 8 + 8 \zeta_{6} ) q^{52} -6 q^{57} + ( 18 - 9 \zeta_{6} ) q^{61} + ( -6 - 3 \zeta_{6} ) q^{63} + 8 q^{64} + ( -16 + 16 \zeta_{6} ) q^{67} + ( 1 + \zeta_{6} ) q^{73} + ( -4 + 8 \zeta_{6} ) q^{76} -17 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -8 + 10 \zeta_{6} ) q^{84} + ( 4 + 16 \zeta_{6} ) q^{91} + 15 \zeta_{6} q^{93} + ( -3 + 6 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 2q^{4} - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 2q^{4} - 4q^{7} + 3q^{9} + 6q^{12} - 4q^{16} + 6q^{19} + 9q^{21} - 2q^{28} - 15q^{31} - 12q^{36} - 11q^{37} - 12q^{39} + 10q^{43} + 2q^{49} + 24q^{52} - 12q^{57} + 27q^{61} - 15q^{63} + 16q^{64} - 16q^{67} + 3q^{73} - 17q^{79} - 9q^{81} - 6q^{84} + 24q^{91} + 15q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i −1.00000 1.73205i 0 0 −2.00000 1.73205i 0 1.50000 2.59808i 0
101.1 0 −1.50000 0.866025i −1.00000 + 1.73205i 0 0 −2.00000 + 1.73205i 0 1.50000 + 2.59808i 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.t.b 2
3.b odd 2 1 CM 525.2.t.b 2
5.b even 2 1 525.2.t.d yes 2
5.c odd 4 2 525.2.q.c 4
7.d odd 6 1 inner 525.2.t.b 2
15.d odd 2 1 525.2.t.d yes 2
15.e even 4 2 525.2.q.c 4
21.g even 6 1 inner 525.2.t.b 2
35.i odd 6 1 525.2.t.d yes 2
35.k even 12 2 525.2.q.c 4
105.p even 6 1 525.2.t.d yes 2
105.w odd 12 2 525.2.q.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.q.c 4 5.c odd 4 2
525.2.q.c 4 15.e even 4 2
525.2.q.c 4 35.k even 12 2
525.2.q.c 4 105.w odd 12 2
525.2.t.b 2 1.a even 1 1 trivial
525.2.t.b 2 3.b odd 2 1 CM
525.2.t.b 2 7.d odd 6 1 inner
525.2.t.b 2 21.g even 6 1 inner
525.2.t.d yes 2 5.b even 2 1
525.2.t.d yes 2 15.d odd 2 1
525.2.t.d yes 2 35.i odd 6 1
525.2.t.d yes 2 105.p even 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{13}^{2} + 48 \)
\( T_{37}^{2} + 11 T_{37} + 121 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( 1 + 23 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 53 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )( 1 - 13 T + 61 T^{2} ) \)
$67$ \( ( 1 + 5 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( ( 1 - 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \)
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