Properties

Label 1521.4.a.n
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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 \(\beta = 18\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{4} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{4} + \beta q^{7} + 64 q^{16} - 5 \beta q^{19} - 125 q^{25} - 8 \beta q^{28} + 5 \beta q^{31} - 14 \beta q^{37} + 520 q^{43} + 629 q^{49} - 182 q^{61} - 512 q^{64} + 21 \beta q^{67} - 12 \beta q^{73} + 40 \beta q^{76} - 884 q^{79} + 44 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} + 128 q^{16} - 250 q^{25} + 1040 q^{43} + 1258 q^{49} - 364 q^{61} - 1024 q^{64} - 1768 q^{79}+O(q^{100}) \) 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
−1.73205
1.73205
0 0 −8.00000 0 0 −31.1769 0 0 0
1.2 0 0 −8.00000 0 0 31.1769 0 0 0
\(n\): 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.n 2
3.b odd 2 1 CM 1521.4.a.n 2
13.b even 2 1 inner 1521.4.a.n 2
13.d odd 4 2 117.4.b.b 2
39.d odd 2 1 inner 1521.4.a.n 2
39.f even 4 2 117.4.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.b.b 2 13.d odd 4 2
117.4.b.b 2 39.f even 4 2
1521.4.a.n 2 1.a even 1 1 trivial
1521.4.a.n 2 3.b odd 2 1 CM
1521.4.a.n 2 13.b even 2 1 inner
1521.4.a.n 2 39.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{2} - 972 \) 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
$7$ \( T^{2} - 972 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 24300 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 24300 \) Copy content Toggle raw display
$37$ \( T^{2} - 190512 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 520)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 428652 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 139968 \) Copy content Toggle raw display
$79$ \( (T + 884)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1881792 \) Copy content Toggle raw display
show more
show less