Properties

Label 1560.2.bg.e
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: \(\Q(\sqrt{-3}, \sqrt{10})\)
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - q^{5} + ( - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{2} - 1) q^{9} - 2 \beta_{2} q^{11} + ( - \beta_{2} - \beta_1 + 1) q^{13} + \beta_{2} q^{15} + (4 \beta_{2} + \beta_1 + 4) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + ( - \beta_{3} - 1) q^{21} - 2 \beta_{2} q^{23} + q^{25} - q^{27} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{3} - 3) q^{31} + ( - 2 \beta_{2} - 2) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{37} + (\beta_{3} - 2 \beta_{2} - 1) q^{39} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{41} + ( - 7 \beta_{2} + \beta_1 - 7) q^{43} + (\beta_{2} + 1) q^{45} + ( - 3 \beta_{3} + 2) q^{47} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{49} + ( - \beta_{3} + 4) q^{51} - 2 \beta_{3} q^{53} + 2 \beta_{2} q^{55} - 2 q^{57} + (4 \beta_{2} - \beta_1 + 4) q^{59} + ( - 13 \beta_{2} - 13) q^{61} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{63} + (\beta_{2} + \beta_1 - 1) q^{65} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{2} - 2) q^{69} + (4 \beta_{2} + 3 \beta_1 + 4) q^{71} + ( - \beta_{3} - 3) q^{73} - \beta_{2} q^{75} + ( - 2 \beta_{3} - 2) q^{77} + 11 q^{79} + \beta_{2} q^{81} + ( - 4 \beta_{2} - \beta_1 - 4) q^{85} + ( - 4 \beta_{2} + \beta_1 - 4) q^{87} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{89} + ( - 11 \beta_{2} + 2 \beta_1 - 2) q^{91} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{93} + (2 \beta_{2} + 2) q^{95} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{97} - 2 q^{99}+O(q^{100}) \) 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 \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} + 8 q^{17} - 4 q^{19} - 4 q^{21} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 8 q^{29} - 12 q^{31} - 4 q^{33} + 2 q^{35} - 4 q^{37} - 4 q^{41} - 14 q^{43} + 2 q^{45} + 8 q^{47} - 8 q^{49} + 16 q^{51} - 4 q^{55} - 8 q^{57} + 8 q^{59} - 26 q^{61} - 2 q^{63} - 6 q^{65} - 14 q^{67} - 4 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} + 44 q^{79} - 2 q^{81} - 8 q^{85} - 8 q^{87} + 12 q^{89} + 14 q^{91} - 6 q^{93} + 4 q^{95} - 6 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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
−1.58114 + 2.73861i
1.58114 2.73861i
−1.58114 2.73861i
1.58114 + 2.73861i
0 0.500000 + 0.866025i 0 −1.00000 0 −2.08114 + 3.60464i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 −1.00000 0 1.08114 1.87259i 0 −0.500000 + 0.866025i 0
841.1 0 0.500000 0.866025i 0 −1.00000 0 −2.08114 3.60464i 0 −0.500000 0.866025i 0
841.2 0 0.500000 0.866025i 0 −1.00000 0 1.08114 + 1.87259i 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.e 4
13.c even 3 1 inner 1560.2.bg.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bg.e 4 1.a even 1 1 trivial
1560.2.bg.e 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 \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{4} + 2T_{7}^{3} + 13T_{7}^{2} - 18T_{7} + 81 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) 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
$7$ \( T^{4} + 2 T^{3} + 13 T^{2} - 18 T + 81 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 25 T^{2} - 78 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 52 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + 22 T^{2} - 24 T + 36 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} + 157 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T - 86)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + 58 T^{2} - 48 T + 36 \) Copy content Toggle raw display
$61$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + 157 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + 138 T^{2} + \cdots + 5476 \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$79$ \( (T - 11)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 198 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} + 277 T^{2} + \cdots + 58081 \) Copy content Toggle raw display
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