Properties

Label 5625.2.a.l
Level $5625$
Weight $2$
Character orbit 5625.a
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q - 2 q^{4} + ( - 2 \beta_{3} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} + ( - 2 \beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + 3 \beta_1 + 1) q^{13} + 4 q^{16} + ( - 2 \beta_{3} + 5 \beta_1 - 2) q^{19} + (4 \beta_{3} - 2 \beta_{2}) q^{28} + ( - \beta_{3} + 5 \beta_{2}) q^{31} + ( - 7 \beta_{3} + 4 \beta_1 - 7) q^{37} + (6 \beta_{3} - 7 \beta_1 + 6) q^{43} + ( - 3 \beta_{3} - 5 \beta_1 + 4) q^{49} + ( - 2 \beta_{3} - 6 \beta_1 - 2) q^{52} + ( - 9 \beta_{3} - 5 \beta_{2}) q^{61} - 8 q^{64} + ( - 7 \beta_{3} - 9 \beta_{2}) q^{67} + (9 \beta_{3} + 8 \beta_{2}) q^{73} + (4 \beta_{3} - 10 \beta_1 + 4) q^{76} + (3 \beta_{3} + 10 \beta_{2}) q^{79} + ( - 3 \beta_{3} - 5 \beta_{2} + 10 \beta_1 - 9) q^{91} + (3 \beta_{3} + 11 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 5 q^{7} + 5 q^{13} + 16 q^{16} + q^{19} - 10 q^{28} + 7 q^{31} - 10 q^{37} + 5 q^{43} + 17 q^{49} - 10 q^{52} + 13 q^{61} - 32 q^{64} + 5 q^{67} - 10 q^{73} - 2 q^{76} + 4 q^{79} - 25 q^{91} + 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.209057
1.82709
1.33826
−1.95630
0 0 −2.00000 0 0 −3.19236 0 0 0
1.2 0 0 −2.00000 0 0 0.102193 0 0 0
1.3 0 0 −2.00000 0 0 3.02701 0 0 0
1.4 0 0 −2.00000 0 0 5.06316 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.l yes 4
3.b odd 2 1 CM 5625.2.a.l yes 4
5.b even 2 1 5625.2.a.k 4
15.d odd 2 1 5625.2.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5625.2.a.k 4 5.b even 2 1
5625.2.a.k 4 15.d odd 2 1
5625.2.a.l yes 4 1.a even 1 1 trivial
5625.2.a.l yes 4 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} - 10T_{7}^{2} + 50T_{7} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 5 T^{3} - 10 T^{2} + 50 T - 5 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} - 40 T^{2} + 200 T - 155 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} - 94 T^{2} + 94 T + 1711 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 7 T^{3} - 106 T^{2} + \cdots - 389 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} - 85 T^{2} + \cdots - 1655 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 5 T^{3} - 190 T^{2} + \cdots + 4495 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 13 T^{3} - 136 T^{2} + \cdots - 4379 \) Copy content Toggle raw display
$67$ \( T^{4} - 5 T^{3} - 310 T^{2} + \cdots + 14695 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} - 265 T^{2} + \cdots + 145 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} - 379 T^{2} + \cdots + 25141 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 5 T^{3} - 460 T^{2} + \cdots + 35545 \) Copy content Toggle raw display
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