Properties

Label 1944.3.e.g
Level $1944$
Weight $3$
Character orbit 1944.e
Analytic conductor $52.970$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1944,3,Mod(1457,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1944.1457");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1944.e (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: \(52.9701633916\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.102302208.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 16x^{4} + 61x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{4} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{4} + 1) q^{7} + \beta_{2} q^{11} + (\beta_{5} + \beta_{4} - 2) q^{13} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{17} + (\beta_{5} + 4 \beta_{4} - 4) q^{19} + ( - \beta_{3} - \beta_1) q^{23} + (\beta_{5} - 2 \beta_{4} - 1) q^{25} + (3 \beta_{3} - 2 \beta_{2} - \beta_1) q^{29} + (3 \beta_{5} - 4 \beta_{4} + 9) q^{31} + (3 \beta_{3} - \beta_{2} - 2 \beta_1) q^{35} + ( - 3 \beta_{5} - 2 \beta_{4} + 5) q^{37} + ( - \beta_{2} + 4 \beta_1) q^{41} + (\beta_{5} + 8 \beta_{4} - 3) q^{43} + (8 \beta_{3} + \beta_1) q^{47} + ( - \beta_{5} - 6 \beta_{4} - 14) q^{49} + (2 \beta_{3} + 11 \beta_1) q^{53} + ( - 3 \beta_{5} - 7 \beta_{4} - 4) q^{55} + ( - 2 \beta_{3} - 3 \beta_{2} - 6 \beta_1) q^{59} + (2 \beta_{5} - 7 \beta_{4} - 11) q^{61} + (7 \beta_{3} + 3 \beta_{2} - 6 \beta_1) q^{65} + ( - 2 \beta_{5} + 2 \beta_{4} + 19) q^{67} + (5 \beta_{3} + 6 \beta_{2} - 15 \beta_1) q^{71} + (4 \beta_{5} - 6 \beta_{4} + 11) q^{73} + ( - 11 \beta_{3} + 4 \beta_{2} - 8 \beta_1) q^{77} + ( - 2 \beta_{4} + 30) q^{79} + (13 \beta_{3} - \beta_{2} + 8 \beta_1) q^{83} + (3 \beta_{5} + 5 \beta_{4} - 48) q^{85} + (11 \beta_{3} + 4 \beta_{2} - 6 \beta_1) q^{89} + ( - 2 \beta_{5} + 8 \beta_{4} - 22) q^{91} + ( - 2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{95} + ( - 9 \beta_{5} - 2 \beta_{4} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} - 12 q^{13} - 24 q^{19} - 6 q^{25} + 54 q^{31} + 30 q^{37} - 18 q^{43} - 84 q^{49} - 24 q^{55} - 66 q^{61} + 114 q^{67} + 66 q^{73} + 180 q^{79} - 288 q^{85} - 132 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 16x^{4} + 61x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 19\nu^{3} + 82\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 13\nu^{3} - 2\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 13\nu^{3} - 34\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 11\nu^{2} + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\nu^{2} - 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} - 2\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} + 8\beta_{2} + 6\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} + 6\beta_{4} + 134 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -46\beta_{3} - 35\beta_{2} - 39\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
0.567166i
3.19116i
2.34411i
2.34411i
3.19116i
0.567166i
0 0 0 7.18331i 0 −4.28251 0 0 0
1457.2 0 0 0 4.13939i 0 −1.84248 0 0 0
1457.3 0 0 0 3.04393i 0 9.12499 0 0 0
1457.4 0 0 0 3.04393i 0 9.12499 0 0 0
1457.5 0 0 0 4.13939i 0 −1.84248 0 0 0
1457.6 0 0 0 7.18331i 0 −4.28251 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.3.e.g 6
3.b odd 2 1 inner 1944.3.e.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.3.e.g 6 1.a even 1 1 trivial
1944.3.e.g 6 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 78T_{5}^{4} + 1521T_{5}^{2} + 8192 \) acting on \(S_{3}^{\mathrm{new}}(1944, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 78 T^{4} + 1521 T^{2} + \cdots + 8192 \) Copy content Toggle raw display
$7$ \( (T^{3} - 3 T^{2} - 48 T - 72)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 552 T^{4} + 76176 T^{2} + \cdots + 25088 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} - 216 T - 1744)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 912 T^{4} + 75840 T^{2} + \cdots + 401408 \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} - 819 T + 1654)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 174 T^{4} + 7665 T^{2} + \cdots + 16928 \) Copy content Toggle raw display
$29$ \( T^{6} + 3198 T^{4} + \cdots + 472596768 \) Copy content Toggle raw display
$31$ \( (T^{3} - 27 T^{2} - 3048 T + 101488)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 15 T^{2} - 1848 T + 34704)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 1704 T^{4} + 725904 T^{2} + \cdots + 41472 \) Copy content Toggle raw display
$43$ \( (T^{3} + 9 T^{2} - 3120 T + 56084)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 6222 T^{4} + \cdots + 8249444352 \) Copy content Toggle raw display
$53$ \( T^{6} + 9822 T^{4} + \cdots + 29725071488 \) Copy content Toggle raw display
$59$ \( T^{6} + 8592 T^{4} + \cdots + 2914813952 \) Copy content Toggle raw display
$61$ \( (T^{3} + 33 T^{2} - 3600 T - 6568)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 57 T^{2} - 165 T + 469)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 37662 T^{4} + \cdots + 1127076957728 \) Copy content Toggle raw display
$73$ \( (T^{3} - 33 T^{2} - 5985 T + 260033)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 90 T^{2} + 2496 T - 21856)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 21576 T^{4} + \cdots + 25824462848 \) Copy content Toggle raw display
$89$ \( T^{6} + 22680 T^{4} + \cdots + 68053934592 \) Copy content Toggle raw display
$97$ \( (T^{3} - 12 T^{2} - 17139 T + 271342)^{2} \) Copy content Toggle raw display
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