Properties

Label 6762.2.a.bz
Level $6762$
Weight $2$
Character orbit 6762.a
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{12} - \beta q^{13} + 2 q^{15} + q^{16} + \beta q^{17} + q^{18} + (\beta + 2) q^{19} - 2 q^{20} - q^{23} - q^{24} - q^{25} - \beta q^{26} - q^{27} - 2 q^{29} + 2 q^{30} + ( - \beta - 2) q^{31} + q^{32} + \beta q^{34} + q^{36} + ( - 2 \beta - 2) q^{37} + (\beta + 2) q^{38} + \beta q^{39} - 2 q^{40} + 6 q^{41} + (2 \beta + 4) q^{43} - 2 q^{45} - q^{46} + ( - \beta - 2) q^{47} - q^{48} - q^{50} - \beta q^{51} - \beta q^{52} + (2 \beta - 2) q^{53} - q^{54} + ( - \beta - 2) q^{57} - 2 q^{58} - 4 q^{59} + 2 q^{60} + 6 q^{61} + ( - \beta - 2) q^{62} + q^{64} + 2 \beta q^{65} + ( - 2 \beta - 4) q^{67} + \beta q^{68} + q^{69} + ( - 2 \beta - 4) q^{71} + q^{72} + (2 \beta - 6) q^{73} + ( - 2 \beta - 2) q^{74} + q^{75} + (\beta + 2) q^{76} + \beta q^{78} + (2 \beta + 4) q^{79} - 2 q^{80} + q^{81} + 6 q^{82} + (\beta + 2) q^{83} - 2 \beta q^{85} + (2 \beta + 4) q^{86} + 2 q^{87} + (3 \beta + 4) q^{89} - 2 q^{90} - q^{92} + (\beta + 2) q^{93} + ( - \beta - 2) q^{94} + ( - 2 \beta - 4) q^{95} - q^{96} + ( - \beta - 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{12} + 4 q^{15} + 2 q^{16} + 2 q^{18} + 4 q^{19} - 4 q^{20} - 2 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{36} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} + 8 q^{43} - 4 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{50} - 4 q^{53} - 2 q^{54} - 4 q^{57} - 4 q^{58} - 8 q^{59} + 4 q^{60} + 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 2 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 2 q^{75} + 4 q^{76} + 8 q^{79} - 4 q^{80} + 2 q^{81} + 12 q^{82} + 4 q^{83} + 8 q^{86} + 4 q^{87} + 8 q^{89} - 4 q^{90} - 2 q^{92} + 4 q^{93} - 4 q^{94} - 8 q^{95} - 2 q^{96} - 8 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
\(n\): 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.bz 2
7.b odd 2 1 966.2.a.p 2
21.c even 2 1 2898.2.a.v 2
28.d even 2 1 7728.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.p 2 7.b odd 2 1
2898.2.a.v 2 21.c even 2 1
6762.2.a.bz 2 1.a even 1 1 trivial
7728.2.a.bd 2 28.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6762))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 164 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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