Properties

Label 1521.4.a.y
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM discriminant -39
Inner twists $4$

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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: \(4\)
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: \( 2^{2}\cdot 3^{2} \)
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 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + 8) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + 8) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{8} + (17 \beta_{2} - 29) q^{10} + (3 \beta_{3} - 8 \beta_1) q^{11} + ( - 8 \beta_{2} - 51) q^{16} + ( - 9 \beta_{3} + 30 \beta_1) q^{20} + ( - 53 \beta_{2} + 119) q^{22} + ( - 46 \beta_{2} + 125) q^{25} + 51 \beta_1 q^{32} + (29 \beta_{2} - 221) q^{40} + ( - 23 \beta_{3} + 50 \beta_1) q^{41} - 452 q^{43} + (29 \beta_{3} - 108 \beta_1) q^{44} + (23 \beta_{3} + 76 \beta_1) q^{47} - 343 q^{49} + (46 \beta_{3} - 171 \beta_1) q^{50} + (172 \beta_{2} - 808) q^{55} + (\beta_{3} + 160 \beta_1) q^{59} + 262 \beta_{2} q^{61} + (115 \beta_{2} - 408) q^{64} + (43 \beta_{3} + 140 \beta_1) q^{71} - 116 \beta_{2} q^{79} + (43 \beta_{3} + 10 \beta_1) q^{80} + (395 \beta_{2} - 731) q^{82} + (51 \beta_{3} - 208 \beta_1) q^{83} + 452 \beta_1 q^{86} + ( - 119 \beta_{2} + 689) q^{88} + ( - 61 \beta_{3} - 194 \beta_1) q^{89} + ( - 269 \beta_{2} - 1285) q^{94} + 343 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{4} - 116 q^{10} - 204 q^{16} + 476 q^{22} + 500 q^{25} - 884 q^{40} - 1808 q^{43} - 1372 q^{49} - 3232 q^{55} - 1632 q^{64} - 2924 q^{82} + 2756 q^{88} - 5140 q^{94}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( -\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{3} - 18\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 6\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 3 \) 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
−4.42782 0 11.6056 20.3925 0 0 −15.9647 0 −90.2944
1.2 −3.52058 0 4.39445 −9.17304 0 0 12.6936 0 32.2944
1.3 3.52058 0 4.39445 9.17304 0 0 −12.6936 0 32.2944
1.4 4.42782 0 11.6056 −20.3925 0 0 15.9647 0 −90.2944
\(n\): 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 1521.4.a.y 4
3.b odd 2 1 inner 1521.4.a.y 4
13.b even 2 1 inner 1521.4.a.y 4
13.d odd 4 2 117.4.b.c 4
39.d odd 2 1 CM 1521.4.a.y 4
39.f even 4 2 117.4.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.b.c 4 13.d odd 4 2
117.4.b.c 4 39.f even 4 2
1521.4.a.y 4 1.a even 1 1 trivial
1521.4.a.y 4 3.b odd 2 1 inner
1521.4.a.y 4 13.b even 2 1 inner
1521.4.a.y 4 39.d 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{4} - 32T_{2}^{2} + 243 \) Copy content Toggle raw display
\( T_{5}^{4} - 500T_{5}^{2} + 34992 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 32T^{2} + 243 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 500 T^{2} + 34992 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 5324 T^{2} + \cdots + 2056752 \) 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
$41$ \( T^{4} - 275684 T^{2} + \cdots + 9184890672 \) Copy content Toggle raw display
$43$ \( (T + 452)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 415292 T^{2} + \cdots + 2077595568 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 821516 T^{2} + \cdots + 163107544752 \) Copy content Toggle raw display
$61$ \( (T^{2} - 892372)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 1431644 T^{2} + \cdots + 21709693872 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 174928)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 2287148 T^{2} + \cdots + 20406871728 \) Copy content Toggle raw display
$89$ \( T^{4} - 2819876 T^{2} + \cdots + 67381252272 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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