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Properties

Label 1944.2.i.n
Level $1944$
Weight $2$
Character orbit 1944.i
Analytic conductor $15.523$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1944,2,Mod(649,1944)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1944.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1944.i (of order \(3\), 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: \(15.5229181529\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} + 1) q^{17} + ( - 2 \beta_{3} - 1) q^{19} + (\beta_{2} + 7 \beta_1) q^{23} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{25} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 6) q^{29}+ \cdots + ( - 4 \beta_{3} + 4 \beta_{2} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 14 q^{23} + 2 q^{25} + 12 q^{29} + 2 q^{31} + 24 q^{35} + 24 q^{37} + 12 q^{41} - 2 q^{43} - 10 q^{49} - 4 q^{53} - 32 q^{55} + 14 q^{59} + 10 q^{61} + 14 q^{65} + 14 q^{67} + 4 q^{71} + 4 q^{73} + 36 q^{77} - 14 q^{79} + 22 q^{83} + 8 q^{85} + 48 q^{89} - 48 q^{91} + 10 q^{95} - 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 0 0 −0.366025 − 0.633975i 0 −1.73205 + 3.00000i 0 0 0
649.2 0 0 0 1.36603 + 2.36603i 0 1.73205 − 3.00000i 0 0 0
1297.1 0 0 0 −0.366025 + 0.633975i 0 −1.73205 − 3.00000i 0 0 0
1297.2 0 0 0 1.36603 − 2.36603i 0 1.73205 + 3.00000i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.2.i.n 4
3.b odd 2 1 1944.2.i.k 4
9.c even 3 1 1944.2.a.k ✓ 2
9.c even 3 1 inner 1944.2.i.n 4
9.d odd 6 1 1944.2.a.n yes 2
9.d odd 6 1 1944.2.i.k 4
36.f odd 6 1 3888.2.a.w 2
36.h even 6 1 3888.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.2.a.k ✓ 2 9.c even 3 1
1944.2.a.n yes 2 9.d odd 6 1
1944.2.i.k 4 3.b odd 2 1
1944.2.i.k 4 9.d odd 6 1
1944.2.i.n 4 1.a even 1 1 trivial
1944.2.i.n 4 9.c even 3 1 inner
3888.2.a.w 2 36.f odd 6 1
3888.2.a.bc 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1944, [\chi])\):

\( T_{5}^{4} - 2T_{5}^{3} + 6T_{5}^{2} + 4T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 12T_{7}^{2} + 144 \) 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
$5$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 14 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 74)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 14 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 107)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 14 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{4} - 22 T^{3} + \cdots + 13924 \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + \cdots + 529 \) Copy content Toggle raw display
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