Properties

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

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{2} - 1) q^{9} + ( - 3 \beta_{2} + \beta_1) q^{11} + (2 \beta_{3} + \beta_1 + 1) q^{13} + \beta_{2} q^{15} + ( - 2 \beta_{2} + 2) q^{17} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_{3} - 2) q^{21} + ( - 3 \beta_{2} + \beta_1) q^{23} + q^{25} - q^{27} + ( - \beta_{2} - 3 \beta_1) q^{29} + (2 \beta_{3} + 1) q^{31} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{33} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{35} + ( - 3 \beta_{2} + \beta_1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1) q^{39} + 2 \beta_{2} q^{41} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{43} + (\beta_{2} - 1) q^{45} + (\beta_{3} + 5) q^{47} + ( - \beta_{2} - 5 \beta_1) q^{49} + 2 q^{51} + (2 \beta_{3} - 2) q^{53} + ( - 3 \beta_{2} + \beta_1) q^{55} + (2 \beta_{3} + 2) q^{57} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 1) q^{59} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 6) q^{61} + ( - 2 \beta_{2} - \beta_1) q^{63} + (2 \beta_{3} + \beta_1 + 1) q^{65} + (4 \beta_{2} + \beta_1) q^{67} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{69} + ( - \beta_{3} + 6) q^{73} + \beta_{2} q^{75} + 2 q^{77} + (4 \beta_{3} - 7) q^{79} - \beta_{2} q^{81} + ( - 6 \beta_{3} - 2) q^{83} + ( - 2 \beta_{2} + 2) q^{85} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{87} - 2 \beta_{2} q^{89} + ( - 2 \beta_{3} - 6 \beta_{2} - 5 \beta_1 + 2) q^{91} + (\beta_{2} - 2 \beta_1) q^{93} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{95} + (\beta_{3} + 10 \beta_{2} + \beta_1 - 10) q^{97} + (\beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 5 q^{7} - 2 q^{9} - 5 q^{11} + q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{19} - 10 q^{21} - 5 q^{23} + 4 q^{25} - 4 q^{27} - 5 q^{29} + 5 q^{33} - 5 q^{35} - 5 q^{37} - q^{39} + 4 q^{41} + 4 q^{43} - 2 q^{45} + 18 q^{47} - 7 q^{49} + 8 q^{51} - 12 q^{53} - 5 q^{55} + 4 q^{57} - 5 q^{59} - 9 q^{61} - 5 q^{63} + q^{65} + 9 q^{67} + 5 q^{69} + 26 q^{73} + 2 q^{75} + 8 q^{77} - 36 q^{79} - 2 q^{81} + 4 q^{83} + 4 q^{85} + 5 q^{87} - 4 q^{89} - 5 q^{91} + 2 q^{95} - 21 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) 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
1.28078 + 2.21837i
−0.780776 1.35234i
1.28078 2.21837i
−0.780776 + 1.35234i
0 0.500000 + 0.866025i 0 1.00000 0 −2.28078 + 3.95042i 0 −0.500000 + 0.866025i 0
601.2 0 0.500000 + 0.866025i 0 1.00000 0 −0.219224 + 0.379706i 0 −0.500000 + 0.866025i 0
841.1 0 0.500000 0.866025i 0 1.00000 0 −2.28078 3.95042i 0 −0.500000 0.866025i 0
841.2 0 0.500000 0.866025i 0 1.00000 0 −0.219224 0.379706i 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.f 4
13.c even 3 1 inner 1560.2.bg.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.bg.f 4 1.a even 1 1 trivial
1560.2.bg.f 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 \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{4} + 5T_{7}^{3} + 23T_{7}^{2} + 10T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 5T_{11}^{3} + 23T_{11}^{2} + 10T_{11} + 4 \) 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
$7$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} - 12 T^{2} - 13 T + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$23$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( (T^{2} - 17)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + 29 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{4} + 9 T^{3} + 99 T^{2} - 162 T + 324 \) Copy content Toggle raw display
$67$ \( T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 13 T + 38)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 18 T + 13)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2 T - 152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 21 T^{3} + 335 T^{2} + \cdots + 11236 \) Copy content Toggle raw display
show more
show less