Properties

Label 1560.2.g.h
Level $1560$
Weight $2$
Character orbit 1560.g
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.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9} + (\beta_{2} + \beta_1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} - \beta_{2} q^{15} + (\beta_{3} + 2) q^{17} + 2 \beta_1 q^{19} + ( - \beta_{2} + \beta_1) q^{21} - \beta_{3} q^{23} - q^{25} + q^{27} - 2 q^{29} - 2 \beta_{2} q^{31} + (\beta_{2} + \beta_1) q^{33} + (\beta_{3} - 2) q^{35} + ( - 3 \beta_{2} + \beta_1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{2} + 3 \beta_1) q^{41} + (2 \beta_{3} + 4) q^{43} - \beta_{2} q^{45} + (3 \beta_{3} - 1) q^{49} + (\beta_{3} + 2) q^{51} + (3 \beta_{3} - 2) q^{53} + \beta_{3} q^{55} + 2 \beta_1 q^{57} + ( - 4 \beta_{2} - 4 \beta_1) q^{59} + (3 \beta_{3} - 2) q^{61} + ( - \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{65} + 6 \beta_1 q^{67} - \beta_{3} q^{69} + (7 \beta_{2} - \beta_1) q^{71} + ( - 6 \beta_{2} - 2 \beta_1) q^{73} - q^{75} + (\beta_{3} - 4) q^{77} + (3 \beta_{3} - 4) q^{79} + q^{81} + (2 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{2} - \beta_1) q^{85} - 2 q^{87} + ( - 3 \beta_{2} - \beta_1) q^{89} + ( - 3 \beta_{3} + 2 \beta_{2} + 8) q^{91} - 2 \beta_{2} q^{93} + (2 \beta_{3} - 2) q^{95} + (\beta_{2} + \beta_1) q^{97} + (\beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 6 q^{13} + 10 q^{17} - 2 q^{23} - 4 q^{25} + 4 q^{27} - 8 q^{29} - 6 q^{35} + 6 q^{39} + 20 q^{43} + 2 q^{49} + 10 q^{51} - 2 q^{53} + 2 q^{55} - 2 q^{61} + 6 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{77} - 10 q^{79} + 4 q^{81} - 8 q^{87} + 26 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
2.56155i
1.56155i
1.56155i
2.56155i
0 1.00000 0 1.00000i 0 3.56155i 0 1.00000 0
961.2 0 1.00000 0 1.00000i 0 0.561553i 0 1.00000 0
961.3 0 1.00000 0 1.00000i 0 0.561553i 0 1.00000 0
961.4 0 1.00000 0 1.00000i 0 3.56155i 0 1.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.g.h 4
3.b odd 2 1 4680.2.g.g 4
4.b odd 2 1 3120.2.g.p 4
13.b even 2 1 inner 1560.2.g.h 4
39.d odd 2 1 4680.2.g.g 4
52.b odd 2 1 3120.2.g.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.g.h 4 1.a even 1 1 trivial
1560.2.g.h 4 13.b even 2 1 inner
3120.2.g.p 4 4.b odd 2 1
3120.2.g.p 4 52.b odd 2 1
4680.2.g.g 4 3.b odd 2 1
4680.2.g.g 4 39.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 \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{4} + 13T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 9T_{11}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + 18 T^{2} - 78 T + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$61$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 324 T^{2} + 20736 \) Copy content Toggle raw display
$71$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$73$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$89$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$97$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
show more
show less