Properties

Label 6762.2.a.cf
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3132.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 15x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + \beta_1 q^{11} + q^{12} + (\beta_{2} + \beta_1 + 1) q^{13} - q^{15} + q^{16} + (\beta_{2} - \beta_1 - 1) q^{17} - q^{18} + ( - \beta_{2} - 2) q^{19} - q^{20} - \beta_1 q^{22} - q^{23} - q^{24} - 4 q^{25} + ( - \beta_{2} - \beta_1 - 1) q^{26} + q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + q^{30} + ( - 2 \beta_{2} - \beta_1) q^{31} - q^{32} + \beta_1 q^{33} + ( - \beta_{2} + \beta_1 + 1) q^{34} + q^{36} + (\beta_{2} - 2 \beta_1 - 4) q^{37} + (\beta_{2} + 2) q^{38} + (\beta_{2} + \beta_1 + 1) q^{39} + q^{40} + ( - \beta_{2} + 6) q^{41} - 2 \beta_1 q^{43} + \beta_1 q^{44} - q^{45} + q^{46} + ( - 2 \beta_{2} - \beta_1 + 1) q^{47} + q^{48} + 4 q^{50} + (\beta_{2} - \beta_1 - 1) q^{51} + (\beta_{2} + \beta_1 + 1) q^{52} + ( - 2 \beta_1 - 1) q^{53} - q^{54} - \beta_1 q^{55} + ( - \beta_{2} - 2) q^{57} + (\beta_{2} + \beta_1 - 4) q^{58} + ( - \beta_{2} + \beta_1 + 4) q^{59} - q^{60} - 10 q^{61} + (2 \beta_{2} + \beta_1) q^{62} + q^{64} + ( - \beta_{2} - \beta_1 - 1) q^{65} - \beta_1 q^{66} + ( - 2 \beta_{2} - \beta_1 - 7) q^{67} + (\beta_{2} - \beta_1 - 1) q^{68} - q^{69} + (\beta_{2} + 3 \beta_1 - 1) q^{71} - q^{72} + ( - \beta_1 - 5) q^{73} + ( - \beta_{2} + 2 \beta_1 + 4) q^{74} - 4 q^{75} + ( - \beta_{2} - 2) q^{76} + ( - \beta_{2} - \beta_1 - 1) q^{78} + (\beta_{2} + 3 \beta_1 - 4) q^{79} - q^{80} + q^{81} + (\beta_{2} - 6) q^{82} + ( - \beta_1 + 4) q^{83} + ( - \beta_{2} + \beta_1 + 1) q^{85} + 2 \beta_1 q^{86} + ( - \beta_{2} - \beta_1 + 4) q^{87} - \beta_1 q^{88} + (\beta_{2} - 2 \beta_1 - 4) q^{89} + q^{90} - q^{92} + ( - 2 \beta_{2} - \beta_1) q^{93} + (2 \beta_{2} + \beta_1 - 1) q^{94} + (\beta_{2} + 2) q^{95} - q^{96} + (3 \beta_{2} + 3 \beta_1) q^{97} + \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 3 q^{10} + 3 q^{12} + 3 q^{13} - 3 q^{15} + 3 q^{16} - 3 q^{17} - 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{23} - 3 q^{24} - 12 q^{25} - 3 q^{26} + 3 q^{27} + 12 q^{29} + 3 q^{30} - 3 q^{32} + 3 q^{34} + 3 q^{36} - 12 q^{37} + 6 q^{38} + 3 q^{39} + 3 q^{40} + 18 q^{41} - 3 q^{45} + 3 q^{46} + 3 q^{47} + 3 q^{48} + 12 q^{50} - 3 q^{51} + 3 q^{52} - 3 q^{53} - 3 q^{54} - 6 q^{57} - 12 q^{58} + 12 q^{59} - 3 q^{60} - 30 q^{61} + 3 q^{64} - 3 q^{65} - 21 q^{67} - 3 q^{68} - 3 q^{69} - 3 q^{71} - 3 q^{72} - 15 q^{73} + 12 q^{74} - 12 q^{75} - 6 q^{76} - 3 q^{78} - 12 q^{79} - 3 q^{80} + 3 q^{81} - 18 q^{82} + 12 q^{83} + 3 q^{85} + 12 q^{87} - 12 q^{89} + 3 q^{90} - 3 q^{92} - 3 q^{94} + 6 q^{95} - 3 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 15x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 10 \) 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
−3.65491
−0.404409
4.05932
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.3 −1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)
\(23\) \( +1 \)

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 6762.2.a.cf 3
7.b odd 2 1 6762.2.a.ce 3
7.d odd 6 2 966.2.i.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.k 6 7.d odd 6 2
6762.2.a.ce 3 7.b odd 2 1
6762.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 15T_{11} - 6 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 24T_{13} + 22 \) Copy content Toggle raw display
\( T_{17}^{3} + 3T_{17}^{2} - 36T_{17} - 126 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 15T - 6 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} + \cdots + 22 \) Copy content Toggle raw display
$17$ \( T^{3} + 3 T^{2} + \cdots - 126 \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + \cdots - 48 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} + \cdots + 48 \) Copy content Toggle raw display
$31$ \( T^{3} - 75T - 214 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} + \cdots - 588 \) Copy content Toggle raw display
$41$ \( T^{3} - 18 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$43$ \( T^{3} - 60T + 48 \) Copy content Toggle raw display
$47$ \( T^{3} - 3 T^{2} + \cdots - 140 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} + \cdots - 11 \) Copy content Toggle raw display
$59$ \( T^{3} - 12 T^{2} + \cdots + 180 \) Copy content Toggle raw display
$61$ \( (T + 10)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + 21 T^{2} + \cdots - 396 \) Copy content Toggle raw display
$71$ \( T^{3} + 3 T^{2} + \cdots - 726 \) Copy content Toggle raw display
$73$ \( T^{3} + 15 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} + \cdots - 1068 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$89$ \( T^{3} + 12 T^{2} + \cdots - 588 \) Copy content Toggle raw display
$97$ \( T^{3} - 243T - 108 \) Copy content Toggle raw display
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