Properties

Label 1872.2.bf.f
Level $1872$
Weight $2$
Character orbit 1872.bf
Analytic conductor $14.948$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,Mod(1279,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.1279");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.bf (of order \(4\), degree \(2\), 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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_1 - 2) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_1 - 2) q^{5} - \beta_{2} q^{7} + 2 \beta_{2} q^{11} + (3 \beta_1 + 2) q^{13} + 4 \beta_1 q^{17} + \beta_{3} q^{19} + 3 \beta_1 q^{25} - 4 q^{29} + 3 \beta_{3} q^{31} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{35} + ( - 5 \beta_1 + 5) q^{37} + (6 \beta_1 + 6) q^{41} - 2 \beta_{2} q^{47} - \beta_1 q^{49} + 4 q^{53} + (4 \beta_{3} - 4 \beta_{2}) q^{55} - 6 \beta_{2} q^{59} + 4 q^{61} + ( - 10 \beta_1 + 2) q^{65} + 5 \beta_{3} q^{67} + 2 \beta_{3} q^{71} + (\beta_1 - 1) q^{73} - 12 \beta_1 q^{77} + (2 \beta_{3} - 2 \beta_{2}) q^{79} + 6 \beta_{3} q^{83} + ( - 8 \beta_1 + 8) q^{85} + (10 \beta_1 - 10) q^{89} + (3 \beta_{3} - 2 \beta_{2}) q^{91} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{95} + (13 \beta_1 + 13) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{5} + 8 q^{13} - 16 q^{29} + 20 q^{37} + 24 q^{41} + 16 q^{53} + 16 q^{61} + 8 q^{65} - 4 q^{73} + 32 q^{85} - 40 q^{89} + 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2 ) / 4 \) 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 \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 0 0 −2.00000 + 2.00000i 0 −1.73205 + 1.73205i 0 0 0
1279.2 0 0 0 −2.00000 + 2.00000i 0 1.73205 1.73205i 0 0 0
1711.1 0 0 0 −2.00000 2.00000i 0 −1.73205 1.73205i 0 0 0
1711.2 0 0 0 −2.00000 2.00000i 0 1.73205 + 1.73205i 0 0 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
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.bf.f 4
3.b odd 2 1 1872.2.bf.k yes 4
4.b odd 2 1 inner 1872.2.bf.f 4
12.b even 2 1 1872.2.bf.k yes 4
13.d odd 4 1 inner 1872.2.bf.f 4
39.f even 4 1 1872.2.bf.k yes 4
52.f even 4 1 inner 1872.2.bf.f 4
156.l odd 4 1 1872.2.bf.k yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.2.bf.f 4 1.a even 1 1 trivial
1872.2.bf.f 4 4.b odd 2 1 inner
1872.2.bf.f 4 13.d odd 4 1 inner
1872.2.bf.f 4 52.f even 4 1 inner
1872.2.bf.k yes 4 3.b odd 2 1
1872.2.bf.k yes 4 12.b even 2 1
1872.2.bf.k yes 4 39.f even 4 1
1872.2.bf.k yes 4 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{2} + 4T_{5} + 8 \) Copy content Toggle raw display
\( T_{7}^{4} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 36 \) Copy content Toggle raw display
$11$ \( T^{4} + 576 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 36 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2916 \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 576 \) Copy content Toggle raw display
$53$ \( (T - 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 46656 \) Copy content Toggle raw display
$61$ \( (T - 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 22500 \) Copy content Toggle raw display
$71$ \( T^{4} + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 46656 \) Copy content Toggle raw display
$89$ \( (T^{2} + 20 T + 200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 26 T + 338)^{2} \) Copy content Toggle raw display
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