Properties

Label 1521.2.a.t
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $4$
CM discriminant -39
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.8112.1
Defining polynomial: \( x^{4} - 5x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + 2) q^{4} + \beta_1 q^{5} + ( - 3 \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} + 2) q^{4} + \beta_1 q^{5} + ( - 3 \beta_{2} + \beta_1) q^{8} + ( - \beta_{3} + 1) q^{10} + (2 \beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{3} + 9) q^{16} + ( - 4 \beta_{2} - \beta_1) q^{20} + (\beta_{3} - 7) q^{22} + (2 \beta_{3} + 5) q^{25} - 9 \beta_{2} q^{32} + ( - \beta_{3} + 13) q^{40} + (4 \beta_{2} - \beta_1) q^{41} + 4 q^{43} + (6 \beta_{2} - 3 \beta_1) q^{44} + ( - 2 \beta_{2} - 3 \beta_1) q^{47} - 7 q^{49} + (\beta_{2} - 2 \beta_1) q^{50} + (4 \beta_{3} + 8) q^{55} + ( - 2 \beta_{2} + 3 \beta_1) q^{59} - 2 \beta_{3} q^{61} + ( - 5 \beta_{3} + 18) q^{64} + (6 \beta_{2} + \beta_1) q^{71} + 4 \beta_{3} q^{79} + ( - 8 \beta_{2} + 3 \beta_1) q^{80} + (5 \beta_{3} - 17) q^{82} + ( - 6 \beta_{2} + \beta_1) q^{83} - 4 \beta_{2} q^{86} + (7 \beta_{3} - 13) q^{88} + (4 \beta_{2} - 3 \beta_1) q^{89} + (\beta_{3} + 5) q^{94} + 7 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 4 q^{10} + 36 q^{16} - 28 q^{22} + 20 q^{25} + 52 q^{40} + 16 q^{43} - 28 q^{49} + 32 q^{55} + 72 q^{64} - 68 q^{82} - 52 q^{88} + 20 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta_1 \) 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
−0.835000
2.07431
−2.07431
0.835000
−2.75782 0 5.60555 −1.67000 0 0 −9.94345 0 4.60555
1.2 −0.628052 0 −1.60555 4.14863 0 0 2.26447 0 −2.60555
1.3 0.628052 0 −1.60555 −4.14863 0 0 −2.26447 0 −2.60555
1.4 2.75782 0 5.60555 1.67000 0 0 9.94345 0 4.60555
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.a.t 4
3.b odd 2 1 inner 1521.2.a.t 4
13.b even 2 1 inner 1521.2.a.t 4
13.d odd 4 2 117.2.b.b 4
39.d odd 2 1 CM 1521.2.a.t 4
39.f even 4 2 117.2.b.b 4
52.f even 4 2 1872.2.c.k 4
156.l odd 4 2 1872.2.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.b.b 4 13.d odd 4 2
117.2.b.b 4 39.f even 4 2
1521.2.a.t 4 1.a even 1 1 trivial
1521.2.a.t 4 3.b odd 2 1 inner
1521.2.a.t 4 13.b even 2 1 inner
1521.2.a.t 4 39.d odd 2 1 CM
1872.2.c.k 4 52.f even 4 2
1872.2.c.k 4 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{4} - 8T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{4} - 20T_{5}^{2} + 48 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 8T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 20T^{2} + 48 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 44T^{2} + 432 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 164T^{2} + 432 \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 188T^{2} + 48 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 236 T^{2} + 13872 \) Copy content Toggle raw display
$61$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 284 T^{2} + 13872 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 332T^{2} + 48 \) Copy content Toggle raw display
$89$ \( T^{4} - 356 T^{2} + 25392 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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