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Properties

Label 6534.2.a.cv
Level $6534$
Weight $2$
Character orbit 6534.a
Self dual yes
Analytic conductor $52.174$
Analytic rank $0$
Dimension $6$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.493090625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 20x^{4} + 15x^{3} + 115x^{2} + 73x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 594)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_1 q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_1 q^{5} - \beta_{3} q^{7} - q^{8} - \beta_1 q^{10} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{13} + \beta_{3} q^{14} + q^{16} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{17}+ \cdots + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 2 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 2 q^{5} + q^{7} - 6 q^{8} - 2 q^{10} - q^{13} - q^{14} + 6 q^{16} - 7 q^{19} + 2 q^{20} - 9 q^{23} + 14 q^{25} + q^{26} + q^{28} + q^{29} + 10 q^{31} - 6 q^{32} - 18 q^{35} + 12 q^{37} + 7 q^{38} - 2 q^{40} - 11 q^{41} - 5 q^{43} + 9 q^{46} - q^{47} + 29 q^{49} - 14 q^{50} - q^{52} + 6 q^{53} - q^{56} - q^{58} + 18 q^{59} + 17 q^{61} - 10 q^{62} + 6 q^{64} - 2 q^{65} + 14 q^{67} + 18 q^{70} + 10 q^{71} + 17 q^{73} - 12 q^{74} - 7 q^{76} + 22 q^{79} + 2 q^{80} + 11 q^{82} - 33 q^{83} - 35 q^{85} + 5 q^{86} - 12 q^{89} + 59 q^{91} - 9 q^{92} + q^{94} - 39 q^{95} + 29 q^{97} - 29 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 20x^{4} + 15x^{3} + 115x^{2} + 73x - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 12\nu^{3} - 36\nu^{2} - 26\nu - 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 27\nu^{3} + 33\nu^{2} - 14\nu - 13 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 5\nu^{4} + 12\nu^{3} + 75\nu^{2} + 56\nu - 17 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 8\nu^{4} - 51\nu^{3} - 141\nu^{2} - 71\nu + 11 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 3\beta_{2} + 12\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{5} + 12\beta_{4} + 13\beta_{3} - 14\beta_{2} + 23\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{5} + 24\beta_{4} + 22\beta_{3} + 31\beta_{2} + 160\beta _1 + 205 \) 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
−3.01515
−1.87736
−1.02098
0.125525
3.88962
3.89834
−1.00000 0 1.00000 −3.01515 0 −1.62220 −1.00000 0 3.01515
1.2 −1.00000 0 1.00000 −1.87736 0 4.94713 −1.00000 0 1.87736
1.3 −1.00000 0 1.00000 −1.02098 0 −2.69661 −1.00000 0 1.02098
1.4 −1.00000 0 1.00000 0.125525 0 4.72810 −1.00000 0 −0.125525
1.5 −1.00000 0 1.00000 3.88962 0 −3.72393 −1.00000 0 −3.88962
1.6 −1.00000 0 1.00000 3.89834 0 −0.632487 −1.00000 0 −3.89834
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cv 6
3.b odd 2 1 6534.2.a.cw 6
11.b odd 2 1 6534.2.a.cx 6
11.d odd 10 2 594.2.f.k ✓ 12
33.d even 2 1 6534.2.a.cu 6
33.f even 10 2 594.2.f.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
594.2.f.k ✓ 12 11.d odd 10 2
594.2.f.l yes 12 33.f even 10 2
6534.2.a.cu 6 33.d even 2 1
6534.2.a.cv 6 1.a even 1 1 trivial
6534.2.a.cw 6 3.b odd 2 1
6534.2.a.cx 6 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6534))\):

\( T_{5}^{6} - 2T_{5}^{5} - 20T_{5}^{4} + 15T_{5}^{3} + 115T_{5}^{2} + 73T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{5} - 35T_{7}^{4} - 15T_{7}^{3} + 325T_{7}^{2} + 584T_{7} + 241 \) Copy content Toggle raw display
\( T_{13}^{6} + T_{13}^{5} - 55T_{13}^{4} + 760T_{13}^{2} - 824T_{13} + 176 \) Copy content Toggle raw display
\( T_{17}^{6} - 85T_{17}^{4} - 40T_{17}^{3} + 1840T_{17}^{2} + 440T_{17} - 9680 \) Copy content Toggle raw display
\( T_{29}^{6} - T_{29}^{5} - 90T_{29}^{4} - 65T_{29}^{3} + 2130T_{29}^{2} + 3779T_{29} - 4799 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots - 11 \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} + \cdots + 241 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + T^{5} + \cdots + 176 \) Copy content Toggle raw display
$17$ \( T^{6} - 85 T^{4} + \cdots - 9680 \) Copy content Toggle raw display
$19$ \( T^{6} + 7 T^{5} + \cdots + 3344 \) Copy content Toggle raw display
$23$ \( T^{6} + 9 T^{5} + \cdots - 464 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + \cdots - 4799 \) Copy content Toggle raw display
$31$ \( T^{6} - 10 T^{5} + \cdots - 2705 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{6} + 11 T^{5} + \cdots + 45296 \) Copy content Toggle raw display
$43$ \( T^{6} + 5 T^{5} + \cdots + 720 \) Copy content Toggle raw display
$47$ \( T^{6} + T^{5} + \cdots + 976 \) Copy content Toggle raw display
$53$ \( T^{6} - 6 T^{5} + \cdots - 20929 \) Copy content Toggle raw display
$59$ \( T^{6} - 18 T^{5} + \cdots + 31124 \) Copy content Toggle raw display
$61$ \( T^{6} - 17 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$67$ \( T^{6} - 14 T^{5} + \cdots + 31936 \) Copy content Toggle raw display
$71$ \( T^{6} - 10 T^{5} + \cdots - 284400 \) Copy content Toggle raw display
$73$ \( T^{6} - 17 T^{5} + \cdots - 441 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} + \cdots - 441 \) Copy content Toggle raw display
$83$ \( T^{6} + 33 T^{5} + \cdots + 997744 \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + \cdots - 2872576 \) Copy content Toggle raw display
$97$ \( T^{6} - 29 T^{5} + \cdots + 561411 \) Copy content Toggle raw display
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