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Properties

Label 9072.2.a.by
Level $9072$
Weight $2$
Character orbit 9072.a
Self dual yes
Analytic conductor $72.440$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9072,2,Mod(1,9072)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9072, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9072.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

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: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{2} + 2 \beta_1 + 1) q^{13} + ( - \beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} - \beta_1 - 4) q^{23} + (\beta_{2} + \beta_1 - 2) q^{25} + ( - \beta_{2} - 3 \beta_1 + 1) q^{29} + 3 \beta_1 q^{31} + \beta_1 q^{35} + ( - 2 \beta_{2} + \beta_1 - 2) q^{37} + ( - 2 \beta_{2} + \beta_1 - 1) q^{41} + (4 \beta_{2} + \beta_1) q^{43} + ( - \beta_{2} + 2 \beta_1 - 8) q^{47} + q^{49} + (\beta_{2} + \beta_1 + 2) q^{53} + ( - \beta_{2} - 4 \beta_1 - 1) q^{55} + ( - \beta_{2} + \beta_1 - 11) q^{59} + ( - \beta_{2} - 7 \beta_1 + 4) q^{61} + (2 \beta_{2} + 5 \beta_1 + 4) q^{65} + ( - \beta_{2} + 2 \beta_1 - 3) q^{67} + (3 \beta_{2} - 5 \beta_1 - 3) q^{71} + (\beta_{2} - 2 \beta_1) q^{73} + ( - 2 \beta_{2} - \beta_1 - 1) q^{77} + ( - \beta_{2} - 4 \beta_1 + 4) q^{79} + (\beta_{2} + 2 \beta_1 - 7) q^{83} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{85} + (4 \beta_{2} + \beta_1 + 5) q^{89} + (2 \beta_{2} + 2 \beta_1 + 1) q^{91} + ( - 2 \beta_{2} - \beta_1 - 7) q^{95} + (5 \beta_{2} - \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{5} + 3 q^{7} - 2 q^{11} + 3 q^{13} + 2 q^{17} - 3 q^{19} - 14 q^{23} - 6 q^{25} + q^{29} + 3 q^{31} + q^{35} - 3 q^{37} - 3 q^{43} - 21 q^{47} + 3 q^{49} + 6 q^{53} - 6 q^{55} - 31 q^{59} + 6 q^{61} + 15 q^{65} - 6 q^{67} - 17 q^{71} - 3 q^{73} - 2 q^{77} + 9 q^{79} - 20 q^{83} - 15 q^{85} + 12 q^{89} + 3 q^{91} - 20 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4x + 1 \) : Copy content Toggle raw display

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.69963
0.239123
2.46050
0 0 0 −1.69963 0 1.00000 0 0 0
1.2 0 0 0 0.239123 0 1.00000 0 0 0
1.3 0 0 0 2.46050 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9072.2.a.by 3
3.b odd 2 1 9072.2.a.bv 3
4.b odd 2 1 2268.2.a.i 3
9.c even 3 2 1008.2.r.j 6
9.d odd 6 2 3024.2.r.j 6
12.b even 2 1 2268.2.a.h 3
36.f odd 6 2 252.2.j.a ✓ 6
36.h even 6 2 756.2.j.b 6
252.n even 6 2 1764.2.l.f 6
252.o even 6 2 5292.2.l.e 6
252.r odd 6 2 5292.2.i.e 6
252.s odd 6 2 5292.2.j.d 6
252.u odd 6 2 1764.2.i.g 6
252.bb even 6 2 5292.2.i.f 6
252.bi even 6 2 1764.2.j.e 6
252.bj even 6 2 1764.2.i.d 6
252.bl odd 6 2 1764.2.l.e 6
252.bn odd 6 2 5292.2.l.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a ✓ 6 36.f odd 6 2
756.2.j.b 6 36.h even 6 2
1008.2.r.j 6 9.c even 3 2
1764.2.i.d 6 252.bj even 6 2
1764.2.i.g 6 252.u odd 6 2
1764.2.j.e 6 252.bi even 6 2
1764.2.l.e 6 252.bl odd 6 2
1764.2.l.f 6 252.n even 6 2
2268.2.a.h 3 12.b even 2 1
2268.2.a.i 3 4.b odd 2 1
3024.2.r.j 6 9.d odd 6 2
5292.2.i.e 6 252.r odd 6 2
5292.2.i.f 6 252.bb even 6 2
5292.2.j.d 6 252.s odd 6 2
5292.2.l.e 6 252.o even 6 2
5292.2.l.f 6 252.bn odd 6 2
9072.2.a.bv 3 3.b odd 2 1
9072.2.a.by 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{3} - T_{5}^{2} - 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 25T_{11} - 59 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 33T_{13} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 4T + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} + \cdots - 59 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots + 47 \) Copy content Toggle raw display
$19$ \( T^{3} + 3 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$23$ \( T^{3} + 14 T^{2} + \cdots + 31 \) Copy content Toggle raw display
$29$ \( T^{3} - T^{2} + \cdots + 109 \) Copy content Toggle raw display
$31$ \( T^{3} - 3 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$37$ \( T^{3} + 3 T^{2} + \cdots - 23 \) Copy content Toggle raw display
$41$ \( T^{3} - 33T + 9 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} + \cdots + 281 \) Copy content Toggle raw display
$47$ \( T^{3} + 21 T^{2} + \cdots + 207 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$59$ \( T^{3} + 31 T^{2} + \cdots + 989 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} + \cdots + 7 \) Copy content Toggle raw display
$71$ \( T^{3} + 17 T^{2} + \cdots - 1907 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$79$ \( T^{3} - 9 T^{2} + \cdots + 369 \) Copy content Toggle raw display
$83$ \( T^{3} + 20 T^{2} + \cdots + 127 \) Copy content Toggle raw display
$89$ \( T^{3} - 12 T^{2} + \cdots + 711 \) Copy content Toggle raw display
$97$ \( T^{3} + 9 T^{2} + \cdots - 73 \) Copy content Toggle raw display
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