Properties

Label 1520.2.a.s
Level $1520$
Weight $2$
Character orbit 1520.a
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9} + (\beta_{2} + 2) q^{13} - \beta_{2} q^{15} + (\beta_1 + 2) q^{17} - q^{19} + ( - 2 \beta_{2} - \beta_1) q^{21} + ( - 2 \beta_{2} + \beta_1) q^{23} + q^{25} - \beta_1 q^{27} + (\beta_1 + 6) q^{29} + 2 \beta_1 q^{31} - \beta_1 q^{35} + ( - \beta_{2} - \beta_1 + 2) q^{37} + ( - 2 \beta_{2} + \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 4) q^{43} + ( - \beta_1 + 1) q^{45} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 + 1) q^{49} + \beta_1 q^{51} + ( - \beta_{2} + 2 \beta_1 + 2) q^{53} + \beta_{2} q^{57} + (4 \beta_{2} - \beta_1 - 4) q^{59} + ( - 2 \beta_{2} + 2) q^{61} + ( - 2 \beta_{2} + 8) q^{63} + (\beta_{2} + 2) q^{65} + (5 \beta_{2} - 2 \beta_1) q^{67} + (2 \beta_{2} - \beta_1 + 8) q^{69} - 4 \beta_1 q^{71} + (4 \beta_{2} - \beta_1 + 2) q^{73} - \beta_{2} q^{75} + 2 \beta_{2} q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{81} + (2 \beta_1 + 4) q^{83} + (\beta_1 + 2) q^{85} + ( - 4 \beta_{2} + \beta_1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 2) q^{89} + (2 \beta_{2} - \beta_1) q^{91} + (4 \beta_{2} + 2 \beta_1) q^{93} - q^{95} + (\beta_{2} - 3 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
−1.86081
2.11491
−0.254102
0 −2.32340 0 1.00000 0 1.39821 0 2.39821 0
1.2 0 0.642074 0 1.00000 0 −3.58774 0 −2.58774 0
1.3 0 2.68133 0 1.00000 0 3.18953 0 4.18953 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.a.s 3
4.b odd 2 1 760.2.a.j 3
5.b even 2 1 7600.2.a.bq 3
8.b even 2 1 6080.2.a.bq 3
8.d odd 2 1 6080.2.a.bv 3
12.b even 2 1 6840.2.a.bg 3
20.d odd 2 1 3800.2.a.x 3
20.e even 4 2 3800.2.d.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.j 3 4.b odd 2 1
1520.2.a.s 3 1.a even 1 1 trivial
3800.2.a.x 3 20.d odd 2 1
3800.2.d.l 6 20.e even 4 2
6080.2.a.bq 3 8.b even 2 1
6080.2.a.bv 3 8.d odd 2 1
6840.2.a.bg 3 12.b even 2 1
7600.2.a.bq 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\):

\( T_{3}^{3} - T_{3}^{2} - 6T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 12T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 6T + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 12 T + 16 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 5 T^{2} + 2 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 5 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} - 32 T + 64 \) Copy content Toggle raw display
$29$ \( T^{3} - 17 T^{2} + 84 T - 124 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 48 T - 128 \) Copy content Toggle raw display
$37$ \( T^{3} - 8T^{2} + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 6 T^{2} - 52 T + 56 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + 8 T + 32 \) Copy content Toggle raw display
$47$ \( T^{3} + 4 T^{2} - 80 T + 128 \) Copy content Toggle raw display
$53$ \( T^{3} - 5 T^{2} - 42 T - 52 \) Copy content Toggle raw display
$59$ \( T^{3} + 15 T^{2} - 28 T - 784 \) Copy content Toggle raw display
$61$ \( T^{3} - 8 T^{2} - 4 T + 64 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} - 178 T - 1052 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 192 T + 1024 \) Copy content Toggle raw display
$73$ \( T^{3} - 3 T^{2} - 100 T - 292 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} - 24 T - 32 \) Copy content Toggle raw display
$83$ \( T^{3} - 10 T^{2} - 16 T + 32 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 52 T + 184 \) Copy content Toggle raw display
$97$ \( T^{3} + 4 T^{2} - 104 T + 296 \) Copy content Toggle raw display
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