Properties

Label 4998.2.a.cj
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
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] = [4998,2,Mod(1,4998)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4998, 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("4998.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.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: \(39.9092309302\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + q^{8} + q^{9} + \beta_{2} q^{10} + (\beta_1 + 1) q^{11} + q^{12} + ( - \beta_1 + 1) q^{13} + \beta_{2} q^{15} + q^{16} - q^{17} + q^{18} + (\beta_{2} + \beta_1 - 1) q^{19} + \beta_{2} q^{20} + (\beta_1 + 1) q^{22} + 2 q^{23} + q^{24} + (\beta_{2} - 2 \beta_1 + 3) q^{25} + ( - \beta_1 + 1) q^{26} + q^{27} + ( - \beta_{2} + \beta_1 + 1) q^{29} + \beta_{2} q^{30} + ( - 2 \beta_1 + 2) q^{31} + q^{32} + (\beta_1 + 1) q^{33} - q^{34} + q^{36} + ( - 2 \beta_{2} + \beta_1 + 1) q^{37} + (\beta_{2} + \beta_1 - 1) q^{38} + ( - \beta_1 + 1) q^{39} + \beta_{2} q^{40} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{41} + ( - \beta_{2} + 2) q^{43} + (\beta_1 + 1) q^{44} + \beta_{2} q^{45} + 2 q^{46} + 2 \beta_{2} q^{47} + q^{48} + (\beta_{2} - 2 \beta_1 + 3) q^{50} - q^{51} + ( - \beta_1 + 1) q^{52} + ( - \beta_{2} + 4) q^{53} + q^{54} + ( - \beta_{2} + 2) q^{55} + (\beta_{2} + \beta_1 - 1) q^{57} + ( - \beta_{2} + \beta_1 + 1) q^{58} + ( - \beta_{2} + \beta_1 - 3) q^{59} + \beta_{2} q^{60} + (2 \beta_{2} - 2 \beta_1) q^{61} + ( - 2 \beta_1 + 2) q^{62} + q^{64} + (3 \beta_{2} - 2) q^{65} + (\beta_1 + 1) q^{66} + ( - \beta_{2} + 2 \beta_1 + 8) q^{67} - q^{68} + 2 q^{69} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + q^{72} + ( - \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - 2 \beta_{2} + \beta_1 + 1) q^{74} + (\beta_{2} - 2 \beta_1 + 3) q^{75} + (\beta_{2} + \beta_1 - 1) q^{76} + ( - \beta_1 + 1) q^{78} + ( - \beta_{2} - 4 \beta_1 + 2) q^{79} + \beta_{2} q^{80} + q^{81} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{83} - \beta_{2} q^{85} + ( - \beta_{2} + 2) q^{86} + ( - \beta_{2} + \beta_1 + 1) q^{87} + (\beta_1 + 1) q^{88} + ( - 3 \beta_{2} + 4) q^{89} + \beta_{2} q^{90} + 2 q^{92} + ( - 2 \beta_1 + 2) q^{93} + 2 \beta_{2} q^{94} + ( - 2 \beta_{2} - 2 \beta_1 + 10) q^{95} + q^{96} + ( - \beta_{2} - 2 \beta_1 + 2) q^{97} + (\beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + q^{5} + 3 q^{6} + 3 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} + 3 q^{12} + 3 q^{13} + q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{18} - 2 q^{19} + q^{20} + 3 q^{22} + 6 q^{23} + 3 q^{24} + 10 q^{25} + 3 q^{26} + 3 q^{27} + 2 q^{29} + q^{30} + 6 q^{31} + 3 q^{32} + 3 q^{33} - 3 q^{34} + 3 q^{36} + q^{37} - 2 q^{38} + 3 q^{39} + q^{40} + 10 q^{41} + 5 q^{43} + 3 q^{44} + q^{45} + 6 q^{46} + 2 q^{47} + 3 q^{48} + 10 q^{50} - 3 q^{51} + 3 q^{52} + 11 q^{53} + 3 q^{54} + 5 q^{55} - 2 q^{57} + 2 q^{58} - 10 q^{59} + q^{60} + 2 q^{61} + 6 q^{62} + 3 q^{64} - 3 q^{65} + 3 q^{66} + 23 q^{67} - 3 q^{68} + 6 q^{69} + 10 q^{71} + 3 q^{72} - 7 q^{73} + q^{74} + 10 q^{75} - 2 q^{76} + 3 q^{78} + 5 q^{79} + q^{80} + 3 q^{81} + 10 q^{82} + q^{83} - q^{85} + 5 q^{86} + 2 q^{87} + 3 q^{88} + 9 q^{89} + q^{90} + 6 q^{92} + 6 q^{93} + 2 q^{94} + 28 q^{95} + 3 q^{96} + 5 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 7 ) / 2 \) 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
0.772866
2.39138
−2.16425
1.00000 1.00000 1.00000 −3.17554 1.00000 0 1.00000 1.00000 −3.17554
1.2 1.00000 1.00000 1.00000 0.327327 1.00000 0 1.00000 1.00000 0.327327
1.3 1.00000 1.00000 1.00000 3.84822 1.00000 0 1.00000 1.00000 3.84822
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(17\) \(1\)

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 4998.2.a.cj yes 3
7.b odd 2 1 4998.2.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.ci 3 7.b odd 2 1
4998.2.a.cj yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{3} - T_{5}^{2} - 12T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 10T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 10T_{13} + 28 \) Copy content Toggle raw display
\( T_{23} - 2 \) 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
$5$ \( T^{3} - T^{2} - 12T + 4 \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} - 10 T - 4 \) Copy content Toggle raw display
$13$ \( T^{3} - 3 T^{2} - 10 T + 28 \) Copy content Toggle raw display
$17$ \( (T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 28 T + 32 \) Copy content Toggle raw display
$23$ \( (T - 2)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} - 20 T + 32 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 40 T + 224 \) Copy content Toggle raw display
$37$ \( T^{3} - T^{2} - 54 T + 172 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} - 52 T + 392 \) Copy content Toggle raw display
$43$ \( T^{3} - 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 48 T + 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 11 T^{2} + 28 T - 4 \) Copy content Toggle raw display
$59$ \( T^{3} + 10 T^{2} + 12 T - 16 \) Copy content Toggle raw display
$61$ \( T^{3} - 2 T^{2} - 84 T - 88 \) Copy content Toggle raw display
$67$ \( T^{3} - 23 T^{2} + 120 T - 112 \) Copy content Toggle raw display
$71$ \( T^{3} - 10 T^{2} - 84 T - 56 \) Copy content Toggle raw display
$73$ \( T^{3} + 7 T^{2} - 40 T - 212 \) Copy content Toggle raw display
$79$ \( T^{3} - 5 T^{2} - 228 T + 944 \) Copy content Toggle raw display
$83$ \( T^{3} - T^{2} - 142 T - 76 \) Copy content Toggle raw display
$89$ \( T^{3} - 9 T^{2} - 84 T + 308 \) Copy content Toggle raw display
$97$ \( T^{3} - 5 T^{2} - 64 T + 76 \) Copy content Toggle raw display
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