Properties

Label 1520.2.g.a
Level $1520$
Weight $2$
Character orbit 1520.g
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \(x^{2} - x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-19})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} -3 q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta q^{5} -3 q^{7} + 3 q^{9} + ( 1 - 2 \beta ) q^{11} + ( -1 + 2 \beta ) q^{17} + ( 1 - 2 \beta ) q^{19} -4 q^{23} + ( -5 + \beta ) q^{25} + 3 \beta q^{35} + q^{43} -3 \beta q^{45} -13 q^{47} + 2 q^{49} + ( -10 + \beta ) q^{55} -15 q^{61} -9 q^{63} + ( 3 - 6 \beta ) q^{73} + ( -3 + 6 \beta ) q^{77} + 9 q^{81} -16 q^{83} + ( 10 - \beta ) q^{85} + ( -10 + \beta ) q^{95} + ( 3 - 6 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 6 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q - q^{5} - 6 q^{7} + 6 q^{9} - 8 q^{23} - 9 q^{25} + 3 q^{35} + 2 q^{43} - 3 q^{45} - 26 q^{47} + 4 q^{49} - 19 q^{55} - 30 q^{61} - 18 q^{63} + 18 q^{81} - 32 q^{83} + 19 q^{85} - 19 q^{95} + O(q^{100}) \)

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
0.500000 + 2.17945i
0.500000 2.17945i
0 0 0 −0.500000 2.17945i 0 −3.00000 0 3.00000 0
1519.2 0 0 0 −0.500000 + 2.17945i 0 −3.00000 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.a 2
4.b odd 2 1 1520.2.g.b yes 2
5.b even 2 1 1520.2.g.b yes 2
19.b odd 2 1 CM 1520.2.g.a 2
20.d odd 2 1 inner 1520.2.g.a 2
76.d even 2 1 1520.2.g.b yes 2
95.d odd 2 1 1520.2.g.b yes 2
380.d even 2 1 inner 1520.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1520.2.g.a 2 1.a even 1 1 trivial
1520.2.g.a 2 19.b odd 2 1 CM
1520.2.g.a 2 20.d odd 2 1 inner
1520.2.g.a 2 380.d even 2 1 inner
1520.2.g.b yes 2 4.b odd 2 1
1520.2.g.b yes 2 5.b even 2 1
1520.2.g.b yes 2 76.d even 2 1
1520.2.g.b yes 2 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 \(S_{2}^{\mathrm{new}}(1520, [\chi])\):

\( T_{3} \)
\( T_{7} + 3 \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + T + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( 19 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 19 + T^{2} \)
$19$ \( 19 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( ( 13 + T )^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 15 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 171 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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