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Properties

Label 6498.2.a.bx
Level $6498$
Weight $2$
Character orbit 6498.a
Self dual yes
Analytic conductor $51.887$
Analytic rank $0$
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] = [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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) 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 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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1) q^{7} - q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{10} + ( - 3 \beta_{2} + \beta_1 - 2) q^{11} + (\beta_{3} - \beta_{2} + \beta_1 + 4) q^{13} + ( - \beta_{2} - \beta_1) q^{14} + q^{16} + (\beta_{2} + 2 \beta_1 - 1) q^{17} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{20} + (3 \beta_{2} - \beta_1 + 2) q^{22} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots + 1) q^{23}+ \cdots + ( - 2 \beta_{3} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{8} - 2 q^{10} - 2 q^{11} + 18 q^{13} + 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{20} + 2 q^{22} + 10 q^{23} + 6 q^{25} - 18 q^{26} - 2 q^{28} + 2 q^{29} + 26 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{37} - 2 q^{40} + 12 q^{41} - 10 q^{43} - 2 q^{44} - 10 q^{46} + 12 q^{47} - 12 q^{49} - 6 q^{50} + 18 q^{52} - 8 q^{53} - 26 q^{55} + 2 q^{56} - 2 q^{58} + 8 q^{59} - 26 q^{62} + 4 q^{64} + 4 q^{65} + 10 q^{67} - 6 q^{68} + 6 q^{70} - 14 q^{73} - 4 q^{74} - 4 q^{77} + 22 q^{79} + 2 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{85} + 10 q^{86} + 2 q^{88} + 16 q^{89} - 4 q^{91} + 10 q^{92} - 12 q^{94} + 28 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.17557
1.90211
−1.90211
−1.17557
−1.00000 0 1.00000 −3.69572 0 −0.442463 −1.00000 0 3.69572
1.2 −1.00000 0 1.00000 0.891491 0 2.52015 −1.00000 0 −0.891491
1.3 −1.00000 0 1.00000 2.34458 0 −1.28408 −1.00000 0 −2.34458
1.4 −1.00000 0 1.00000 2.45965 0 −2.79360 −1.00000 0 −2.45965
\(n\): 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.bx 4
3.b odd 2 1 722.2.a.n yes 4
12.b even 2 1 5776.2.a.bt 4
19.b odd 2 1 6498.2.a.ca 4
57.d even 2 1 722.2.a.m ✓ 4
57.f even 6 2 722.2.c.n 8
57.h odd 6 2 722.2.c.m 8
57.j even 18 6 722.2.e.s 24
57.l odd 18 6 722.2.e.r 24
228.b odd 2 1 5776.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m ✓ 4 57.d even 2 1
722.2.a.n yes 4 3.b odd 2 1
722.2.c.m 8 57.h odd 6 2
722.2.c.n 8 57.f even 6 2
722.2.e.r 24 57.l odd 18 6
722.2.e.s 24 57.j even 18 6
5776.2.a.bt 4 12.b even 2 1
5776.2.a.bv 4 228.b odd 2 1
6498.2.a.bx 4 1.a even 1 1 trivial
6498.2.a.ca 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6498))\):

\( T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 32T_{5} - 19 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} - 6T_{7}^{2} - 12T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 26T_{11}^{2} - 12T_{11} + 76 \) Copy content Toggle raw display
\( T_{13}^{4} - 18T_{13}^{3} + 109T_{13}^{2} - 232T_{13} + 61 \) Copy content Toggle raw display
\( T_{29}^{4} - 2T_{29}^{3} - 31T_{29}^{2} + 92T_{29} - 19 \) 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
$5$ \( T^{4} - 2 T^{3} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 76 \) Copy content Toggle raw display
$13$ \( T^{4} - 18 T^{3} + \cdots + 61 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots - 19 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + \cdots - 1220 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots - 19 \) Copy content Toggle raw display
$31$ \( T^{4} - 26 T^{3} + \cdots + 1436 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + \cdots - 19 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots - 359 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots - 100 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots + 76 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + \cdots + 181 \) Copy content Toggle raw display
$59$ \( T^{4} - 8 T^{3} + \cdots - 164 \) Copy content Toggle raw display
$61$ \( T^{4} - 115 T^{2} + \cdots + 1025 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + \cdots - 20 \) Copy content Toggle raw display
$71$ \( T^{4} - 100 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots - 139 \) Copy content Toggle raw display
$79$ \( T^{4} - 22 T^{3} + \cdots - 4724 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + \cdots - 6884 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots - 1159 \) Copy content Toggle raw display
$97$ \( T^{4} - 28 T^{3} + \cdots - 7979 \) Copy content Toggle raw display
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