Properties

Label 9072.2.a.bd
Level $9072$
Weight $2$
Character orbit 9072.a
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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 \(\beta = \sqrt{6}\). 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 \( q + (\beta - 1) q^{5} + q^{7} + 2 q^{11} + 2 \beta q^{13} - 2 q^{17} + (\beta - 5) q^{19} - q^{23} + ( - 2 \beta + 2) q^{25} + (2 \beta + 2) q^{29} - 6 q^{31} + (\beta - 1) q^{35} + (4 \beta + 2) q^{37} - 4 \beta q^{41} + ( - 2 \beta - 2) q^{43} + 4 \beta q^{47} + q^{49} + (2 \beta + 6) q^{53} + (2 \beta - 2) q^{55} - 2 q^{59} + ( - \beta + 9) q^{61} + ( - 2 \beta + 12) q^{65} + (2 \beta + 8) q^{67} + ( - 2 \beta + 5) q^{71} + ( - 2 \beta - 2) q^{73} + 2 q^{77} + (2 \beta - 3) q^{79} + 2 q^{83} + ( - 2 \beta + 2) q^{85} + (2 \beta + 12) q^{89} + 2 \beta q^{91} + ( - 6 \beta + 11) q^{95} + (2 \beta - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{17} - 10 q^{19} - 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{35} + 4 q^{37} - 4 q^{43} + 2 q^{49} + 12 q^{53} - 4 q^{55} - 4 q^{59} + 18 q^{61} + 24 q^{65} + 16 q^{67} + 10 q^{71} - 4 q^{73} + 4 q^{77} - 6 q^{79} + 4 q^{83} + 4 q^{85} + 24 q^{89} + 22 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−2.44949
2.44949
0 0 0 −3.44949 0 1.00000 0 0 0
1.2 0 0 0 1.44949 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.bd 2
3.b odd 2 1 9072.2.a.bk 2
4.b odd 2 1 1134.2.a.i 2
9.c even 3 2 3024.2.r.e 4
9.d odd 6 2 1008.2.r.e 4
12.b even 2 1 1134.2.a.p 2
28.d even 2 1 7938.2.a.bm 2
36.f odd 6 2 378.2.f.d 4
36.h even 6 2 126.2.f.c 4
84.h odd 2 1 7938.2.a.bn 2
252.n even 6 2 2646.2.h.n 4
252.o even 6 2 882.2.h.k 4
252.r odd 6 2 882.2.e.n 4
252.s odd 6 2 882.2.f.j 4
252.u odd 6 2 2646.2.e.l 4
252.bb even 6 2 882.2.e.m 4
252.bi even 6 2 2646.2.f.k 4
252.bj even 6 2 2646.2.e.k 4
252.bl odd 6 2 2646.2.h.m 4
252.bn odd 6 2 882.2.h.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 36.h even 6 2
378.2.f.d 4 36.f odd 6 2
882.2.e.m 4 252.bb even 6 2
882.2.e.n 4 252.r odd 6 2
882.2.f.j 4 252.s odd 6 2
882.2.h.k 4 252.o even 6 2
882.2.h.l 4 252.bn odd 6 2
1008.2.r.e 4 9.d odd 6 2
1134.2.a.i 2 4.b odd 2 1
1134.2.a.p 2 12.b even 2 1
2646.2.e.k 4 252.bj even 6 2
2646.2.e.l 4 252.u odd 6 2
2646.2.f.k 4 252.bi even 6 2
2646.2.h.m 4 252.bl odd 6 2
2646.2.h.n 4 252.n even 6 2
3024.2.r.e 4 9.c even 3 2
7938.2.a.bm 2 28.d even 2 1
7938.2.a.bn 2 84.h odd 2 1
9072.2.a.bd 2 1.a even 1 1 trivial
9072.2.a.bk 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 2T_{5} - 5 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 24 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 19 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 92 \) Copy content Toggle raw display
$41$ \( T^{2} - 96 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$47$ \( T^{2} - 96 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 12 \) Copy content Toggle raw display
$59$ \( (T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 18T + 75 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 40 \) Copy content Toggle raw display
$71$ \( T^{2} - 10T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T - 15 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 24T + 120 \) Copy content Toggle raw display
$97$ \( T^{2} + 4T - 20 \) Copy content Toggle raw display
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