Properties

Label 1520.2.a.t
Level $1520$
Weight $2$
Character orbit 1520.a
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.11344.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9} + ( - \beta_1 - 1) q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (2 \beta_{2} + 2) q^{17} - q^{19} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{21} + (2 \beta_{2} + \beta_1 + 3) q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{27} + (2 \beta_{2} + 2) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{31} + ( - \beta_{3} + \beta_1 + 2) q^{33} + ( - \beta_1 + 1) q^{35} + (\beta_{3} + \beta_{2} - 1) q^{37} + ( - \beta_1 + 3) q^{39} + (\beta_{3} - \beta_1 + 4) q^{41} + (\beta_1 - 1) q^{43} + (\beta_{3} - 2) q^{45} + (2 \beta_{3} - \beta_1 + 3) q^{47} + (2 \beta_{3} - 2 \beta_1 + 5) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2} + 10) q^{51} + ( - \beta_{3} - \beta_{2} - 3) q^{53} + (\beta_1 + 1) q^{55} - \beta_{2} q^{57} + ( - \beta_{3} - \beta_1) q^{59} + ( - \beta_{3} + 2 \beta_1 + 5) q^{61} + ( - 4 \beta_{2} - \beta_1 - 7) q^{63} + ( - \beta_{2} - \beta_1 - 1) q^{65} + ( - \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{67} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 8) q^{69} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 6) q^{73} + \beta_{2} q^{75} + ( - 2 \beta_{3} - 10) q^{77} + ( - \beta_{3} - \beta_1 + 4) q^{79} + ( - \beta_{3} + 2 \beta_1 + 4) q^{81} + ( - 2 \beta_{2} + \beta_1 - 1) q^{83} + ( - 2 \beta_{2} - 2) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} + 10) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{89} + (3 \beta_{3} - 2 \beta_{2} - \beta_1 + 8) q^{91} + (4 \beta_{3} - 2 \beta_1 - 10) q^{93} + q^{95} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{97} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} - 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{33} + 4 q^{35} - 6 q^{37} + 12 q^{39} + 16 q^{41} - 4 q^{43} - 8 q^{45} + 12 q^{47} + 20 q^{49} + 36 q^{51} - 10 q^{53} + 4 q^{55} + 2 q^{57} + 20 q^{61} - 20 q^{63} - 2 q^{65} + 18 q^{67} + 28 q^{69} + 20 q^{71} + 28 q^{73} - 2 q^{75} - 40 q^{77} + 16 q^{79} + 16 q^{81} - 4 q^{85} + 36 q^{87} + 4 q^{89} + 36 q^{91} - 40 q^{93} + 4 q^{95} + 30 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 11 ) / 2 \) 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
1.28734
−1.51658
2.78165
−0.552409
0 −3.04306 0 −1.00000 0 0.574672 0 6.26020 0
1.2 0 −1.53844 0 −1.00000 0 −5.03316 0 −0.633188 0
1.3 0 −0.296842 0 −1.00000 0 3.56331 0 −2.91188 0
1.4 0 2.87834 0 −1.00000 0 −3.10482 0 5.28487 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.t 4
4.b odd 2 1 95.2.a.b 4
5.b even 2 1 7600.2.a.cf 4
8.b even 2 1 6080.2.a.ch 4
8.d odd 2 1 6080.2.a.cc 4
12.b even 2 1 855.2.a.m 4
20.d odd 2 1 475.2.a.i 4
20.e even 4 2 475.2.b.e 8
28.d even 2 1 4655.2.a.y 4
60.h even 2 1 4275.2.a.bo 4
76.d even 2 1 1805.2.a.p 4
380.d even 2 1 9025.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 4.b odd 2 1
475.2.a.i 4 20.d odd 2 1
475.2.b.e 8 20.e even 4 2
855.2.a.m 4 12.b even 2 1
1520.2.a.t 4 1.a even 1 1 trivial
1805.2.a.p 4 76.d even 2 1
4275.2.a.bo 4 60.h even 2 1
4655.2.a.y 4 28.d even 2 1
6080.2.a.cc 4 8.d odd 2 1
6080.2.a.ch 4 8.b even 2 1
7600.2.a.cf 4 5.b even 2 1
9025.2.a.bf 4 380.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\):

\( T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 16T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 48T_{7} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 8 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} - 16 T^{2} - 32 T + 48 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} - 24 T^{2} + 32 T + 20 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 24 T^{2} + 176 T + 288 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 80 T^{2} - 512 T - 640 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} - 24 T^{2} - 40 T + 4 \) Copy content Toggle raw display
$41$ \( T^{4} - 16 T^{3} + 56 T^{2} + \cdots - 240 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 1056 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} - 184 T - 348 \) Copy content Toggle raw display
$59$ \( T^{4} - 64 T^{2} + 224 T - 192 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + 56 T^{2} + \cdots - 2656 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 8 T^{2} + \cdots - 1076 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 4224 \) Copy content Toggle raw display
$73$ \( T^{4} - 28 T^{3} + 256 T^{2} + \cdots + 176 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 1856 \) Copy content Toggle raw display
$83$ \( T^{4} - 72 T^{2} + 112 T + 480 \) Copy content Toggle raw display
$89$ \( T^{4} - 4 T^{3} - 144 T^{2} + \cdots + 240 \) Copy content Toggle raw display
$97$ \( T^{4} - 30 T^{3} + 224 T^{2} + \cdots - 1388 \) Copy content Toggle raw display
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