Properties

Label 1872.1.cm.a
Level $1872$
Weight $1$
Character orbit 1872.cm
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
RM discriminant 13
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,1,Mod(545,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.545");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1872.cm (of order \(6\), degree \(2\), not 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: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 468)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.691896816.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{3} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{3} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{13} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} + ( - \zeta_{6}^{2} + 1) q^{23} - \zeta_{6}^{2} q^{25} + q^{27} - \zeta_{6}^{2} q^{39} + \zeta_{6}^{2} q^{43} - \zeta_{6} q^{49} + (\zeta_{6}^{2} - 1) q^{51} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{53} + \zeta_{6}^{2} q^{61} + ( - \zeta_{6} - 1) q^{69} - q^{75} - \zeta_{6}^{2} q^{79} - \zeta_{6} q^{81} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - q^{9} + q^{13} + 3 q^{23} + q^{25} + 2 q^{27} + q^{39} - q^{43} - q^{49} - 3 q^{51} - q^{61} - 3 q^{69} - 2 q^{75} + q^{79} - q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
1793.1 0 −0.500000 0.866025i 0 0 0 0 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.b even 2 1 RM by \(\Q(\sqrt{13}) \)
9.d odd 6 1 inner
117.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.1.cm.a 2
4.b odd 2 1 468.1.z.a 2
9.d odd 6 1 inner 1872.1.cm.a 2
12.b even 2 1 1404.1.z.a 2
13.b even 2 1 RM 1872.1.cm.a 2
36.f odd 6 1 1404.1.z.a 2
36.h even 6 1 468.1.z.a 2
52.b odd 2 1 468.1.z.a 2
117.n odd 6 1 inner 1872.1.cm.a 2
156.h even 2 1 1404.1.z.a 2
468.x even 6 1 468.1.z.a 2
468.bg odd 6 1 1404.1.z.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.1.z.a 2 4.b odd 2 1
468.1.z.a 2 36.h even 6 1
468.1.z.a 2 52.b odd 2 1
468.1.z.a 2 468.x even 6 1
1404.1.z.a 2 12.b even 2 1
1404.1.z.a 2 36.f odd 6 1
1404.1.z.a 2 156.h even 2 1
1404.1.z.a 2 468.bg odd 6 1
1872.1.cm.a 2 1.a even 1 1 trivial
1872.1.cm.a 2 9.d odd 6 1 inner
1872.1.cm.a 2 13.b even 2 1 RM
1872.1.cm.a 2 117.n odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1872, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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