Properties

Label 6534.2.a.cs
Level $6534$
Weight $2$
Character orbit 6534.a
Self dual yes
Analytic conductor $52.174$
Analytic rank $1$
Dimension $4$
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] = [6534,2,Mod(1,6534)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6534, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6534.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6534.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: \(52.1742526807\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{5} + ( - \beta_{3} - \beta_{2} - 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{5} + ( - \beta_{3} - \beta_{2} - 1) q^{7} + q^{8} + \beta_1 q^{10} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{3} - \beta_{2} - 1) q^{14} + q^{16} + (\beta_{3} - \beta_1 - 2) q^{19} + \beta_1 q^{20} + ( - \beta_1 - 3) q^{23} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{25} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{26} + ( - \beta_{3} - \beta_{2} - 1) q^{28} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - \beta_{3} - \beta_1) q^{31} + q^{32} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{35} + ( - 3 \beta_{2} - 2) q^{37} + (\beta_{3} - \beta_1 - 2) q^{38} + \beta_1 q^{40} + (\beta_{3} - 2 \beta_1 - 2) q^{41} + ( - \beta_{3} + 2 \beta_1 - 4) q^{43} + ( - \beta_1 - 3) q^{46} + (3 \beta_{2} - 3 \beta_1 - 3) q^{47} + (2 \beta_{3} + 2 \beta_1 + 4) q^{49} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{50} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{52} + 3 \beta_{3} q^{53} + ( - \beta_{3} - \beta_{2} - 1) q^{56} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{58} + ( - 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{59} + (2 \beta_1 - 3) q^{61} + ( - \beta_{3} - \beta_1) q^{62} + q^{64} + ( - \beta_{3} - \beta_1 - 4) q^{65} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{67} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{70} + (2 \beta_1 - 6) q^{71} + (4 \beta_{2} - \beta_1 - 3) q^{73} + ( - 3 \beta_{2} - 2) q^{74} + (\beta_{3} - \beta_1 - 2) q^{76} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 7) q^{79} + \beta_1 q^{80} + (\beta_{3} - 2 \beta_1 - 2) q^{82} + ( - 2 \beta_{3} + \beta_1 - 5) q^{83} + ( - \beta_{3} + 2 \beta_1 - 4) q^{86} + (3 \beta_{2} + \beta_1 - 6) q^{89} + (\beta_{3} + 6 \beta_{2} + \beta_1 - 5) q^{91} + ( - \beta_1 - 3) q^{92} + (3 \beta_{2} - 3 \beta_1 - 3) q^{94} + ( - 2 \beta_{3} - 2 \beta_1 - 5) q^{95} + (2 \beta_{3} + 2 \beta_1 - 2) q^{97} + (2 \beta_{3} + 2 \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8} - 6 q^{13} - 6 q^{14} + 4 q^{16} - 6 q^{19} - 12 q^{23} + 10 q^{25} - 6 q^{26} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 4 q^{32} - 12 q^{35} - 8 q^{37} - 6 q^{38} - 6 q^{41} - 18 q^{43} - 12 q^{46} - 12 q^{47} + 20 q^{49} + 10 q^{50} - 6 q^{52} + 6 q^{53} - 6 q^{56} + 6 q^{58} + 6 q^{59} - 12 q^{61} - 2 q^{62} + 4 q^{64} - 18 q^{65} - 2 q^{67} - 12 q^{70} - 24 q^{71} - 12 q^{73} - 8 q^{74} - 6 q^{76} - 30 q^{79} - 6 q^{82} - 24 q^{83} - 18 q^{86} - 24 q^{89} - 18 q^{91} - 12 q^{92} - 12 q^{94} - 24 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} + \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 3 \) 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
−1.49551
1.21969
−0.219687
2.49551
1.00000 0 1.00000 −2.59030 0 0.164177 1.00000 0 −2.59030
1.2 1.00000 0 1.00000 −2.11256 0 −4.03957 1.00000 0 −2.11256
1.3 1.00000 0 1.00000 0.380509 0 2.77162 1.00000 0 0.380509
1.4 1.00000 0 1.00000 4.32235 0 −4.89623 1.00000 0 4.32235
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)

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 6534.2.a.cs yes 4
3.b odd 2 1 6534.2.a.cq 4
11.b odd 2 1 6534.2.a.cr yes 4
33.d even 2 1 6534.2.a.ct yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6534.2.a.cq 4 3.b odd 2 1
6534.2.a.cr yes 4 11.b odd 2 1
6534.2.a.cs yes 4 1.a even 1 1 trivial
6534.2.a.ct yes 4 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6534))\):

\( T_{5}^{4} - 15T_{5}^{2} - 18T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 6T_{7}^{3} - 6T_{7}^{2} - 54T_{7} + 9 \) Copy content Toggle raw display
\( T_{13}^{4} + 6T_{13}^{3} - 9T_{13}^{2} - 54T_{13} - 27 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{29}^{4} - 6T_{29}^{3} - 54T_{29}^{2} + 270T_{29} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 15 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} - 6 T^{2} - 54 T + 9 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} - 9 T^{2} - 54 T - 27 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} - 15 T^{2} - 144 T - 207 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 39 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} - 54 T^{2} + 270 T + 81 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} - 39 T^{2} - 40 T + 157 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 23)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} - 54 T^{2} - 270 T + 81 \) Copy content Toggle raw display
$43$ \( T^{4} + 18 T^{3} + 54 T^{2} + \cdots - 243 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} - 81 T^{2} + \cdots - 1863 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} - 162 T^{2} + \cdots + 4941 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 141 T^{2} + \cdots + 2061 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} - 6 T^{2} - 396 T - 747 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} - 66 T^{2} + 14 T + 373 \) Copy content Toggle raw display
$71$ \( T^{4} + 24 T^{3} + 156 T^{2} + \cdots - 1584 \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} - 33 T^{2} + \cdots - 207 \) Copy content Toggle raw display
$79$ \( T^{4} + 30 T^{3} + 309 T^{2} + \cdots + 981 \) Copy content Toggle raw display
$83$ \( T^{4} + 24 T^{3} + 135 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 129 T^{2} + \cdots - 1719 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} - 156 T^{2} + \cdots + 2512 \) Copy content Toggle raw display
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