Properties

Label 9025.2.a.p
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{4} - \beta q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} - \beta q^{7} - 3 q^{9} + 5 q^{11} + 4 q^{16} - \beta q^{17} - 2 \beta q^{23} + 2 \beta q^{28} + 6 q^{36} - 3 \beta q^{43} - 10 q^{44} - \beta q^{47} + 12 q^{49} - 15 q^{61} + 3 \beta q^{63} - 8 q^{64} + 2 \beta q^{68} - 3 \beta q^{73} - 5 \beta q^{77} + 9 q^{81} + 2 \beta q^{83} + 4 \beta q^{92} - 15 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 6 q^{9} + 10 q^{11} + 8 q^{16} + 12 q^{36} - 20 q^{44} + 24 q^{49} - 30 q^{61} - 16 q^{64} + 18 q^{81} - 30 q^{99}+O(q^{100}) \) 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
4.35890
−4.35890
0 0 −2.00000 0 0 −4.35890 0 −3.00000 0
1.2 0 0 −2.00000 0 0 4.35890 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(19\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9025.2.a.p 2
5.b even 2 1 inner 9025.2.a.p 2
5.c odd 4 2 1805.2.b.d 2
19.b odd 2 1 CM 9025.2.a.p 2
95.d odd 2 1 inner 9025.2.a.p 2
95.g even 4 2 1805.2.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.d 2 5.c odd 4 2
1805.2.b.d 2 95.g even 4 2
9025.2.a.p 2 1.a even 1 1 trivial
9025.2.a.p 2 5.b even 2 1 inner
9025.2.a.p 2 19.b odd 2 1 CM
9025.2.a.p 2 95.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 19 \) Copy content Toggle raw display
\( T_{11} - 5 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 19 \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 19 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 76 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 171 \) Copy content Toggle raw display
$47$ \( T^{2} - 19 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 171 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 76 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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