Properties

Label 1560.2.bg.g
Level $1560$
Weight $2$
Character orbit 1560.bg
Analytic conductor $12.457$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bg (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + q^{5} + \beta_1 q^{7} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + q^{5} + \beta_1 q^{7} + \beta_1 q^{9} + ( - \beta_{3} + \beta_1 + 1) q^{11} + (3 \beta_1 + 4) q^{13} + (\beta_1 + 1) q^{15} + (\beta_{3} - \beta_{2} + \beta_1) q^{19} - q^{21} + 2 \beta_{3} q^{23} + q^{25} - q^{27} + ( - 2 \beta_{3} + 4 \beta_1 + 4) q^{29} + (\beta_{2} - 4) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{33} + \beta_1 q^{35} + (\beta_{3} + 3 \beta_1 + 3) q^{37} + (4 \beta_1 + 1) q^{39} + (2 \beta_{3} - 2 \beta_1 - 2) q^{41} + ( - \beta_{3} + \beta_{2} + 4 \beta_1) q^{43} + \beta_1 q^{45} + ( - \beta_{2} + 1) q^{47} + (6 \beta_1 + 6) q^{49} + (\beta_{2} + 5) q^{53} + ( - \beta_{3} + \beta_1 + 1) q^{55} + ( - \beta_{2} - 1) q^{57} + ( - 2 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{59} + (\beta_{3} - \beta_{2}) q^{61} + ( - \beta_1 - 1) q^{63} + (3 \beta_1 + 4) q^{65} + ( - \beta_{3} + 2 \beta_1 + 2) q^{67} + (2 \beta_{3} - 2 \beta_{2}) q^{69} - 2 \beta_1 q^{71} + ( - 3 \beta_{2} + 2) q^{73} + (\beta_1 + 1) q^{75} + (\beta_{2} - 1) q^{77} + ( - \beta_{2} - 4) q^{79} + ( - \beta_1 - 1) q^{81} + ( - 4 \beta_{2} + 4) q^{83} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{87} + (\beta_{3} + 9 \beta_1 + 9) q^{89} + (\beta_1 - 3) q^{91} + (\beta_{3} - 4 \beta_1 - 4) q^{93} + (\beta_{3} - \beta_{2} + \beta_1) q^{95} + ( - 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 2 q^{7} - 2 q^{9} + q^{11} + 10 q^{13} + 2 q^{15} - 3 q^{19} - 4 q^{21} + 2 q^{23} + 4 q^{25} - 4 q^{27} + 6 q^{29} - 14 q^{31} - q^{33} - 2 q^{35} + 7 q^{37} - 4 q^{39} - 2 q^{41} - 7 q^{43} - 2 q^{45} + 2 q^{47} + 12 q^{49} + 22 q^{53} + q^{55} - 6 q^{57} - 14 q^{59} - q^{61} - 2 q^{63} + 10 q^{65} + 3 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} - 2 q^{77} - 18 q^{79} - 2 q^{81} + 8 q^{83} - 6 q^{87} + 19 q^{89} - 14 q^{91} - 7 q^{93} - 3 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
2.13746 + 0.656712i
−1.63746 1.52274i
2.13746 0.656712i
−1.63746 + 1.52274i
0 0.500000 + 0.866025i 0 1.00000 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 1.00000 0 −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
841.1 0 0.500000 0.866025i 0 1.00000 0 −0.500000 0.866025i 0 −0.500000 0.866025i 0
841.2 0 0.500000 0.866025i 0 1.00000 0 −0.500000 0.866025i 0 −0.500000 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.bg.g 4
13.c even 3 1 inner 1560.2.bg.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bg.g 4 1.a even 1 1 trivial
1560.2.bg.g 4 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} + 15T_{11}^{2} + 14T_{11} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + 21 T^{2} - 36 T + 144 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 60 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 84 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 7 T^{3} + 51 T^{2} + 14 T + 4 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 60 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$43$ \( T^{4} + 7 T^{3} + 51 T^{2} - 14 T + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} - T - 14)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 11 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + 204 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + T^{3} + 15 T^{2} - 14 T + 196 \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} + 21 T^{2} + 36 T + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - T - 128)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9 T + 6)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 224)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 19 T^{3} + 285 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
$97$ \( T^{4} - 7 T^{3} + 165 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
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