Properties

Label 1520.2.g.c
Level $1520$
Weight $2$
Character orbit 1520.g
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + ( -2 - \beta_{1} - \beta_{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{5} + ( -2 - \beta_{1} - \beta_{3} ) q^{7} + 3 q^{9} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{17} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{19} + 4 q^{23} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{25} + ( 7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{35} + ( 2 + 3 \beta_{1} + 3 \beta_{3} ) q^{43} -3 \beta_{3} q^{45} + ( -6 + \beta_{1} + \beta_{3} ) q^{47} + ( 11 + 3 \beta_{1} + 3 \beta_{3} ) q^{49} + ( 5 + 5 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 8 + \beta_{1} + \beta_{3} ) q^{61} + ( -6 - 3 \beta_{1} - 3 \beta_{3} ) q^{63} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 9 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} ) q^{77} + 9 q^{81} + 16 q^{83} + ( -1 - 7 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -5 + 5 \beta_{2} - \beta_{3} ) q^{95} + ( -3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{5} - 6q^{7} + 12q^{9} + O(q^{10}) \) \( 4q + q^{5} - 6q^{7} + 12q^{9} + 16q^{23} + 9q^{25} + 27q^{35} + 2q^{43} + 3q^{45} - 26q^{47} + 38q^{49} + 17q^{55} + 30q^{61} - 18q^{63} + 36q^{81} + 64q^{83} - q^{85} - 19q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} + 16 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 4 \nu^{2} + 4 \nu + 15 \)\()/10\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} - 5 \beta_{2} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1519.1
2.13746 0.656712i
2.13746 + 0.656712i
−1.63746 1.52274i
−1.63746 + 1.52274i
0 0 0 −1.63746 1.52274i 0 −5.27492 0 3.00000 0
1519.2 0 0 0 −1.63746 + 1.52274i 0 −5.27492 0 3.00000 0
1519.3 0 0 0 2.13746 0.656712i 0 2.27492 0 3.00000 0
1519.4 0 0 0 2.13746 + 0.656712i 0 2.27492 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
20.d odd 2 1 inner
380.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.g.c 4
4.b odd 2 1 1520.2.g.d yes 4
5.b even 2 1 1520.2.g.d yes 4
19.b odd 2 1 CM 1520.2.g.c 4
20.d odd 2 1 inner 1520.2.g.c 4
76.d even 2 1 1520.2.g.d yes 4
95.d odd 2 1 1520.2.g.d yes 4
380.d even 2 1 inner 1520.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.c 4 1.a even 1 1 trivial
1520.2.g.c 4 19.b odd 2 1 CM
1520.2.g.c 4 20.d odd 2 1 inner
1520.2.g.c 4 380.d even 2 1 inner
1520.2.g.d yes 4 4.b odd 2 1
1520.2.g.d yes 4 5.b even 2 1
1520.2.g.d yes 4 76.d even 2 1
1520.2.g.d yes 4 95.d odd 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{7}^{2} + 3 T_{7} - 12 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 5 T - 4 T^{2} - T^{3} + T^{4} \)
$7$ \( ( -12 + 3 T + T^{2} )^{2} \)
$11$ \( 196 + 47 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 1024 + 83 T^{2} + T^{4} \)
$19$ \( ( 19 + T^{2} )^{2} \)
$23$ \( ( -4 + T )^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( -128 - T + T^{2} )^{2} \)
$47$ \( ( 28 + 13 T + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 42 - 15 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 2304 + 267 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -16 + T )^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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