Properties

Label 1560.2.g.g
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

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,2,Mod(961,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_{2} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{3} - 4) q^{17} + (\beta_{2} + 4 \beta_1) q^{19} + \beta_{2} q^{21} + ( - \beta_{3} - 6) q^{23} - q^{25} + q^{27} + \beta_{3} q^{29} + 2 \beta_1 q^{31} + \beta_{3} q^{35} - 8 \beta_1 q^{37} + (\beta_{2} + 1) q^{39} + 2 \beta_1 q^{41} + 2 \beta_{3} q^{43} - \beta_1 q^{45} - 4 \beta_1 q^{47} - 5 q^{49} + ( - \beta_{3} - 4) q^{51} + 2 q^{53} + (\beta_{2} + 4 \beta_1) q^{57} + (2 \beta_{2} + 8 \beta_1) q^{59} + ( - 2 \beta_{3} - 2) q^{61} + \beta_{2} q^{63} + (\beta_{3} - \beta_1) q^{65} - 2 \beta_1 q^{67} + ( - \beta_{3} - 6) q^{69} + (2 \beta_{2} - 4 \beta_1) q^{71} + (3 \beta_{2} + 2 \beta_1) q^{73} - q^{75} + (2 \beta_{3} + 4) q^{79} + q^{81} - 2 \beta_{2} q^{83} + (\beta_{2} + 4 \beta_1) q^{85} + \beta_{3} q^{87} + ( - 2 \beta_{2} + 6 \beta_1) q^{89} + (\beta_{2} - 12) q^{91} + 2 \beta_1 q^{93} + (\beta_{3} + 4) q^{95} + (\beta_{2} - 10 \beta_1) q^{97}+O(q^{100}) \) 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 \( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{13} - 16 q^{17} - 24 q^{23} - 4 q^{25} + 4 q^{27} + 4 q^{39} - 20 q^{49} - 16 q^{51} + 8 q^{53} - 8 q^{61} - 24 q^{69} - 4 q^{75} + 16 q^{79} + 4 q^{81} - 48 q^{91} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{12}^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0 1.00000 0 1.00000i 0 3.46410i 0 1.00000 0
961.2 0 1.00000 0 1.00000i 0 3.46410i 0 1.00000 0
961.3 0 1.00000 0 1.00000i 0 3.46410i 0 1.00000 0
961.4 0 1.00000 0 1.00000i 0 3.46410i 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.g 4
3.b odd 2 1 4680.2.g.f 4
4.b odd 2 1 3120.2.g.m 4
13.b even 2 1 inner 1560.2.g.g 4
39.d odd 2 1 4680.2.g.f 4
52.b odd 2 1 3120.2.g.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.g.g 4 1.a even 1 1 trivial
1560.2.g.g 4 13.b even 2 1 inner
3120.2.g.m 4 4.b odd 2 1
3120.2.g.m 4 52.b odd 2 1
4680.2.g.f 4 3.b odd 2 1
4680.2.g.f 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 \(S_{2}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{2} + 12 \) Copy content Toggle raw display
\( T_{11} \) 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
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 224T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$97$ \( T^{4} + 224T^{2} + 7744 \) Copy content Toggle raw display
show more
show less