Properties

Label 6498.2.a.be
Level $6498$
Weight $2$
Character orbit 6498.a
Self dual yes
Analytic conductor $51.887$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6498,2,Mod(1,6498)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6498, 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("6498.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6498.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: \(51.8867912334\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 722)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta + 3) q^{5} + (2 \beta - 2) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta + 3) q^{5} + (2 \beta - 2) q^{7} - q^{8} + (\beta - 3) q^{10} + ( - 2 \beta + 2) q^{11} + ( - \beta - 2) q^{13} + ( - 2 \beta + 2) q^{14} + q^{16} + (\beta + 4) q^{17} + ( - \beta + 3) q^{20} + (2 \beta - 2) q^{22} + (2 \beta - 4) q^{23} + ( - 5 \beta + 5) q^{25} + (\beta + 2) q^{26} + (2 \beta - 2) q^{28} + (5 \beta - 6) q^{29} - 2 \beta q^{31} - q^{32} + ( - \beta - 4) q^{34} + (6 \beta - 8) q^{35} + ( - \beta - 9) q^{37} + (\beta - 3) q^{40} + ( - 3 \beta - 1) q^{41} + (2 \beta - 8) q^{43} + ( - 2 \beta + 2) q^{44} + ( - 2 \beta + 4) q^{46} + (4 \beta - 2) q^{47} + ( - 4 \beta + 1) q^{49} + (5 \beta - 5) q^{50} + ( - \beta - 2) q^{52} + (5 \beta - 3) q^{53} + ( - 6 \beta + 8) q^{55} + ( - 2 \beta + 2) q^{56} + ( - 5 \beta + 6) q^{58} + (4 \beta + 2) q^{59} + (\beta + 2) q^{61} + 2 \beta q^{62} + q^{64} - 5 q^{65} + (6 \beta - 8) q^{67} + (\beta + 4) q^{68} + ( - 6 \beta + 8) q^{70} + ( - 8 \beta - 2) q^{71} + (\beta - 5) q^{73} + (\beta + 9) q^{74} + (4 \beta - 8) q^{77} + (2 \beta + 4) q^{79} + ( - \beta + 3) q^{80} + (3 \beta + 1) q^{82} + ( - 4 \beta - 2) q^{83} + ( - 2 \beta + 11) q^{85} + ( - 2 \beta + 8) q^{86} + (2 \beta - 2) q^{88} + ( - 3 \beta + 7) q^{89} + ( - 4 \beta + 2) q^{91} + (2 \beta - 4) q^{92} + ( - 4 \beta + 2) q^{94} + (\beta - 8) q^{97} + (4 \beta - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} - 2 q^{8} - 5 q^{10} + 2 q^{11} - 5 q^{13} + 2 q^{14} + 2 q^{16} + 9 q^{17} + 5 q^{20} - 2 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - 2 q^{28} - 7 q^{29} - 2 q^{31} - 2 q^{32} - 9 q^{34} - 10 q^{35} - 19 q^{37} - 5 q^{40} - 5 q^{41} - 14 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{49} - 5 q^{50} - 5 q^{52} - q^{53} + 10 q^{55} + 2 q^{56} + 7 q^{58} + 8 q^{59} + 5 q^{61} + 2 q^{62} + 2 q^{64} - 10 q^{65} - 10 q^{67} + 9 q^{68} + 10 q^{70} - 12 q^{71} - 9 q^{73} + 19 q^{74} - 12 q^{77} + 10 q^{79} + 5 q^{80} + 5 q^{82} - 8 q^{83} + 20 q^{85} + 14 q^{86} - 2 q^{88} + 11 q^{89} - 6 q^{92} - 15 q^{97} + 2 q^{98}+O(q^{100}) \) 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 0 1.00000 1.38197 0 1.23607 −1.00000 0 −1.38197
1.2 −1.00000 0 1.00000 3.61803 0 −3.23607 −1.00000 0 −3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(-1\)

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 6498.2.a.be 2
3.b odd 2 1 722.2.a.i yes 2
12.b even 2 1 5776.2.a.be 2
19.b odd 2 1 6498.2.a.bk 2
57.d even 2 1 722.2.a.h 2
57.f even 6 2 722.2.c.i 4
57.h odd 6 2 722.2.c.h 4
57.j even 18 6 722.2.e.q 12
57.l odd 18 6 722.2.e.p 12
228.b odd 2 1 5776.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.h 2 57.d even 2 1
722.2.a.i yes 2 3.b odd 2 1
722.2.c.h 4 57.h odd 6 2
722.2.c.i 4 57.f even 6 2
722.2.e.p 12 57.l odd 18 6
722.2.e.q 12 57.j even 18 6
5776.2.a.t 2 228.b odd 2 1
5776.2.a.be 2 12.b even 2 1
6498.2.a.be 2 1.a even 1 1 trivial
6498.2.a.bk 2 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6498))\):

\( T_{5}^{2} - 5T_{5} + 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 5T_{13} + 5 \) Copy content Toggle raw display
\( T_{29}^{2} + 7T_{29} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$17$ \( T^{2} - 9T + 19 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} + 7T - 19 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 19T + 89 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$43$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 20 \) Copy content Toggle raw display
$53$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$97$ \( T^{2} + 15T + 55 \) Copy content Toggle raw display
show more
show less