Properties

Label 1520.2.a.p
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: 3.3.148.1
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 95)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + q^{5} + (\beta_{2} - \beta_1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_{2} + \beta_1 + 2) q^{11} + ( - \beta_{2} + 3) q^{13} + (\beta_1 - 1) q^{15} + 2 \beta_{2} q^{17} + q^{19} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{21} + ( - \beta_{2} - \beta_1 + 2) q^{23} + q^{25} + ( - 2 \beta_{2} - 2) q^{27} + (2 \beta_1 - 4) q^{29} + ( - 2 \beta_{2} - 2 \beta_1) q^{31} + 4 \beta_1 q^{33} + (\beta_{2} - \beta_1) q^{35} + ( - \beta_{2} + 7) q^{37} + (\beta_{2} + \beta_1 - 2) q^{39} - 2 \beta_{2} q^{41} + ( - \beta_{2} + 5 \beta_1) q^{43} + (\beta_{2} - \beta_1 + 1) q^{45} + (\beta_{2} - \beta_1) q^{47} + ( - 4 \beta_1 + 5) q^{49} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{51} + (\beta_{2} + 5) q^{53} + (\beta_{2} + \beta_1 + 2) q^{55} + (\beta_1 - 1) q^{57} + (2 \beta_1 + 6) q^{59} + (3 \beta_{2} + \beta_1 - 2) q^{61} + (\beta_{2} - 5 \beta_1 + 12) q^{63} + ( - \beta_{2} + 3) q^{65} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{67} - 4 q^{69} + 4 \beta_1 q^{71} + 2 \beta_1 q^{73} + (\beta_1 - 1) q^{75} + ( - 4 \beta_1 + 4) q^{77} + ( - 6 \beta_1 + 2) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (\beta_{2} + \beta_1 + 10) q^{83} + 2 \beta_{2} q^{85} + (2 \beta_{2} - 4 \beta_1 + 10) q^{87} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} + (4 \beta_{2} - 8) q^{91} + ( - 4 \beta_1 - 4) q^{93} + q^{95} + ( - \beta_{2} - 6 \beta_1 + 9) q^{97} + (\beta_{2} - 3 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{9} + 8 q^{11} + 8 q^{13} - 2 q^{15} + 2 q^{17} + 3 q^{19} - 12 q^{21} + 4 q^{23} + 3 q^{25} - 8 q^{27} - 10 q^{29} - 4 q^{31} + 4 q^{33} + 20 q^{37} - 4 q^{39} - 2 q^{41} + 4 q^{43} + 3 q^{45} + 11 q^{49} - 4 q^{51} + 16 q^{53} + 8 q^{55} - 2 q^{57} + 20 q^{59} - 2 q^{61} + 32 q^{63} + 8 q^{65} - 2 q^{67} - 12 q^{69} + 4 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{77} - q^{81} + 32 q^{83} + 2 q^{85} + 28 q^{87} + 2 q^{89} - 20 q^{91} - 16 q^{93} + 3 q^{95} + 20 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
0.311108
−1.48119
2.17009
0 −2.90321 0 1.00000 0 4.42864 0 5.42864 0
1.2 0 −0.806063 0 1.00000 0 −3.35026 0 −2.35026 0
1.3 0 1.70928 0 1.00000 0 −1.07838 0 −0.0783777 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.p 3
4.b odd 2 1 95.2.a.a 3
5.b even 2 1 7600.2.a.bx 3
8.b even 2 1 6080.2.a.by 3
8.d odd 2 1 6080.2.a.bo 3
12.b even 2 1 855.2.a.i 3
20.d odd 2 1 475.2.a.f 3
20.e even 4 2 475.2.b.d 6
28.d even 2 1 4655.2.a.u 3
60.h even 2 1 4275.2.a.bk 3
76.d even 2 1 1805.2.a.f 3
380.d even 2 1 9025.2.a.bb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 4.b odd 2 1
475.2.a.f 3 20.d odd 2 1
475.2.b.d 6 20.e even 4 2
855.2.a.i 3 12.b even 2 1
1520.2.a.p 3 1.a even 1 1 trivial
1805.2.a.f 3 76.d even 2 1
4275.2.a.bk 3 60.h even 2 1
4655.2.a.u 3 28.d even 2 1
6080.2.a.bo 3 8.d odd 2 1
6080.2.a.by 3 8.b even 2 1
7600.2.a.bx 3 5.b even 2 1
9025.2.a.bb 3 380.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 12 T - 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 36 T + 104 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$29$ \( T^{3} + 10 T^{2} + 12 T - 40 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$37$ \( T^{3} - 20 T^{2} + 124 T - 244 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 36 T - 104 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} - 144 T + 592 \) Copy content Toggle raw display
$47$ \( T^{3} - 16T - 16 \) Copy content Toggle raw display
$53$ \( T^{3} - 16 T^{2} + 76 T - 92 \) Copy content Toggle raw display
$59$ \( T^{3} - 20 T^{2} + 112 T - 160 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 84 T + 232 \) Copy content Toggle raw display
$67$ \( T^{3} + 2 T^{2} - 76 T + 116 \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 80 T + 64 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} - 20 T + 8 \) Copy content Toggle raw display
$79$ \( T^{3} - 192T + 160 \) Copy content Toggle raw display
$83$ \( T^{3} - 32 T^{2} + 328 T - 1072 \) Copy content Toggle raw display
$89$ \( T^{3} - 2 T^{2} - 132 T + 680 \) Copy content Toggle raw display
$97$ \( T^{3} - 20 T^{2} - 60 T + 1748 \) Copy content Toggle raw display
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