Properties

Label 1920.2.d.a
Level $1920$
Weight $2$
Character orbit 1920.d
Analytic conductor $15.331$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(1729,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1920.1729");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.d (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: \(15.3312771881\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_1 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_1 q^{5} + q^{9} + ( - \beta_{2} - \beta_1) q^{13} + \beta_1 q^{15} + \beta_{3} q^{17} - \beta_{3} q^{19} + (\beta_{2} - \beta_1) q^{23} + (\beta_{3} + 1) q^{25} - q^{27} + ( - \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{2} - 2 \beta_1) q^{31} + (\beta_{2} + \beta_1) q^{37} + (\beta_{2} + \beta_1) q^{39} + 6 q^{41} - 4 q^{43} - \beta_1 q^{45} + ( - \beta_{2} + \beta_1) q^{47} + 7 q^{49} - \beta_{3} q^{51} + (\beta_{2} + \beta_1) q^{53} + \beta_{3} q^{57} + 2 \beta_{3} q^{59} + (\beta_{3} + 6) q^{65} - 4 q^{67} + ( - \beta_{2} + \beta_1) q^{69} + (4 \beta_{2} + 4 \beta_1) q^{71} + 2 \beta_{3} q^{73} + ( - \beta_{3} - 1) q^{75} + ( - 2 \beta_{2} - 2 \beta_1) q^{79} + q^{81} + 12 q^{83} + (5 \beta_{2} - \beta_1) q^{85} + (\beta_{2} - \beta_1) q^{87} + 6 q^{89} + (2 \beta_{2} + 2 \beta_1) q^{93} + ( - 5 \beta_{2} + \beta_1) q^{95} + 2 \beta_{3} q^{97}+O(q^{100}) \) 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 \( 4 q - 4 q^{3} + 4 q^{9} + 4 q^{25} - 4 q^{27} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 24 q^{65} - 16 q^{67} - 4 q^{75} + 4 q^{81} + 48 q^{83} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + \nu^{2} + 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + \nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
0.517638i
0.517638i
1.93185i
1.93185i
0 −1.00000 0 −1.73205 1.41421i 0 0 0 1.00000 0
1729.2 0 −1.00000 0 −1.73205 + 1.41421i 0 0 0 1.00000 0
1729.3 0 −1.00000 0 1.73205 1.41421i 0 0 0 1.00000 0
1729.4 0 −1.00000 0 1.73205 + 1.41421i 0 0 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
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.2.d.a 4
4.b odd 2 1 1920.2.d.b yes 4
5.b even 2 1 1920.2.d.b yes 4
8.b even 2 1 1920.2.d.b yes 4
8.d odd 2 1 inner 1920.2.d.a 4
16.e even 4 2 3840.2.f.j 8
16.f odd 4 2 3840.2.f.j 8
20.d odd 2 1 inner 1920.2.d.a 4
40.e odd 2 1 1920.2.d.b yes 4
40.f even 2 1 inner 1920.2.d.a 4
80.k odd 4 2 3840.2.f.j 8
80.q even 4 2 3840.2.f.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.2.d.a 4 1.a even 1 1 trivial
1920.2.d.a 4 8.d odd 2 1 inner
1920.2.d.a 4 20.d odd 2 1 inner
1920.2.d.a 4 40.f even 2 1 inner
1920.2.d.b yes 4 4.b odd 2 1
1920.2.d.b yes 4 5.b even 2 1
1920.2.d.b yes 4 8.b even 2 1
1920.2.d.b yes 4 40.e odd 2 1
3840.2.f.j 8 16.e even 4 2
3840.2.f.j 8 16.f odd 4 2
3840.2.f.j 8 80.k odd 4 2
3840.2.f.j 8 80.q even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1920, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13}^{2} - 12 \) Copy content Toggle raw display
\( T_{31}^{2} - 48 \) Copy content Toggle raw display
\( T_{43} + 4 \) 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
$5$ \( T^{4} - 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T + 4)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{4} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
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