Properties

Label 1944.1.m.a
Level $1944$
Weight $1$
Character orbit 1944.m
Analytic conductor $0.970$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1944,1,Mod(161,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1944.161");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1944.m (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: \(0.970182384559\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.3888.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{5} - \beta_1 q^{11} - \beta_{2} q^{13} - \beta_{3} q^{17} - q^{19} + ( - \beta_{3} + \beta_1) q^{23} + ( - \beta_{2} + 1) q^{25} + \beta_{2} q^{31} + (\beta_{2} - 1) q^{43} + \beta_{2} q^{49} + \beta_{3} q^{53} - 2 q^{55} + ( - \beta_{3} + \beta_1) q^{59} + (\beta_{2} - 1) q^{61} - \beta_1 q^{65} + \beta_{2} q^{67} - \beta_{3} q^{71} + q^{73} + (\beta_{2} - 1) q^{79} + \beta_1 q^{83} - 2 \beta_{2} q^{85} + (\beta_{3} - \beta_1) q^{95} + ( - \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{13} - 4 q^{19} + 2 q^{25} + 2 q^{31} - 2 q^{43} + 2 q^{49} - 8 q^{55} - 2 q^{61} + 2 q^{67} + 4 q^{73} - 2 q^{79} - 4 q^{85} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 −1.22474 0.707107i 0 0 0 0 0
161.2 0 0 0 1.22474 + 0.707107i 0 0 0 0 0
809.1 0 0 0 −1.22474 + 0.707107i 0 0 0 0 0
809.2 0 0 0 1.22474 0.707107i 0 0 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.1.m.a 4
3.b odd 2 1 inner 1944.1.m.a 4
4.b odd 2 1 3888.1.q.d 4
9.c even 3 1 1944.1.e.a 2
9.c even 3 1 inner 1944.1.m.a 4
9.d odd 6 1 1944.1.e.a 2
9.d odd 6 1 inner 1944.1.m.a 4
12.b even 2 1 3888.1.q.d 4
36.f odd 6 1 3888.1.e.d 2
36.f odd 6 1 3888.1.q.d 4
36.h even 6 1 3888.1.e.d 2
36.h even 6 1 3888.1.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.1.e.a 2 9.c even 3 1
1944.1.e.a 2 9.d odd 6 1
1944.1.m.a 4 1.a even 1 1 trivial
1944.1.m.a 4 3.b odd 2 1 inner
1944.1.m.a 4 9.c even 3 1 inner
1944.1.m.a 4 9.d odd 6 1 inner
3888.1.e.d 2 36.f odd 6 1
3888.1.e.d 2 36.h even 6 1
3888.1.q.d 4 4.b odd 2 1
3888.1.q.d 4 12.b even 2 1
3888.1.q.d 4 36.f odd 6 1
3888.1.q.d 4 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1944, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T - 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
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