Properties

Label 4998.2.a.cp
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
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] = [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: \(4\)
Coefficient field: 4.4.16448.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + ( - \beta_{2} + \beta_1) q^{5} + q^{6} + q^{8} + q^{9} + ( - \beta_{2} + \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{11} + q^{12} + ( - \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_{2} + \beta_1) q^{15} + q^{16} + q^{17} + q^{18} - \beta_{3} q^{19} + ( - \beta_{2} + \beta_1) q^{20} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{22} + (\beta_{3} - \beta_{2}) q^{23} + q^{24} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{25} + ( - \beta_{2} - \beta_1 + 2) q^{26} + q^{27} + (\beta_{3} + 2 \beta_{2} + 2) q^{29} + ( - \beta_{2} + \beta_1) q^{30} + (\beta_{3} - 2) q^{31} + q^{32} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{33} + q^{34} + q^{36} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{37} - \beta_{3} q^{38} + ( - \beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{2} + \beta_1) q^{40} + (\beta_{3} + 2 \beta_1 - 2) q^{41} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{43} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{44} + ( - \beta_{2} + \beta_1) q^{45} + (\beta_{3} - \beta_{2}) q^{46} + (3 \beta_{3} - 2) q^{47} + q^{48} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{50} + q^{51} + ( - \beta_{2} - \beta_1 + 2) q^{52} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 6) q^{53} + q^{54} + ( - \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{55} - \beta_{3} q^{57} + (\beta_{3} + 2 \beta_{2} + 2) q^{58} + ( - \beta_{3} + 3 \beta_{2} + 6) q^{59} + ( - \beta_{2} + \beta_1) q^{60} + (2 \beta_{3} - 2 \beta_1) q^{61} + (\beta_{3} - 2) q^{62} + q^{64} + ( - 3 \beta_{2} + \beta_1 - 2) q^{65} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{66} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 4) q^{67} + q^{68} + (\beta_{3} - \beta_{2}) q^{69} + (\beta_{2} + 2 \beta_1 + 2) q^{71} + q^{72} + (5 \beta_{2} - \beta_1 - 4) q^{73} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{74} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{75} - \beta_{3} q^{76} + ( - \beta_{2} - \beta_1 + 2) q^{78} + (3 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{79} + ( - \beta_{2} + \beta_1) q^{80} + q^{81} + (\beta_{3} + 2 \beta_1 - 2) q^{82} + ( - 4 \beta_{2} - 3 \beta_1 - 2) q^{83} + ( - \beta_{2} + \beta_1) q^{85} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 4) q^{86} + (\beta_{3} + 2 \beta_{2} + 2) q^{87} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{88} + ( - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 8) q^{89} + ( - \beta_{2} + \beta_1) q^{90} + (\beta_{3} - \beta_{2}) q^{92} + (\beta_{3} - 2) q^{93} + (3 \beta_{3} - 2) q^{94} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{95} + q^{96} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{97} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 10 q^{11} + 4 q^{12} + 6 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 2 q^{20} + 10 q^{22} + 4 q^{24} + 6 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 2 q^{30} - 8 q^{31} + 4 q^{32} + 10 q^{33} + 4 q^{34} + 4 q^{36} + 6 q^{37} + 6 q^{39} + 2 q^{40} - 4 q^{41} + 6 q^{43} + 10 q^{44} + 2 q^{45} - 8 q^{47} + 4 q^{48} + 6 q^{50} + 4 q^{51} + 6 q^{52} + 18 q^{53} + 4 q^{54} + 10 q^{55} + 8 q^{58} + 24 q^{59} + 2 q^{60} - 4 q^{61} - 8 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{66} + 14 q^{67} + 4 q^{68} + 12 q^{71} + 4 q^{72} - 18 q^{73} + 6 q^{74} + 6 q^{75} + 6 q^{78} + 10 q^{79} + 2 q^{80} + 4 q^{81} - 4 q^{82} - 14 q^{83} + 2 q^{85} + 6 q^{86} + 8 q^{87} + 10 q^{88} - 34 q^{89} + 2 q^{90} - 8 q^{93} - 8 q^{94} - 4 q^{95} + 4 q^{96} - 6 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 4 \) 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
−1.87996
−1.18398
2.87996
2.18398
1.00000 1.00000 1.00000 −3.29417 1.00000 0 1.00000 1.00000 −3.29417
1.2 1.00000 1.00000 1.00000 0.230234 1.00000 0 1.00000 1.00000 0.230234
1.3 1.00000 1.00000 1.00000 1.46575 1.00000 0 1.00000 1.00000 1.46575
1.4 1.00000 1.00000 1.00000 3.59819 1.00000 0 1.00000 1.00000 3.59819
\(n\): 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.cp yes 4
7.b odd 2 1 4998.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.co 4 7.b odd 2 1
4998.2.a.cp yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 20T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 10T_{11}^{3} + 19T_{11}^{2} + 52T_{11} - 124 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} + T_{13}^{2} + 32T_{13} - 32 \) Copy content Toggle raw display
\( T_{23}^{4} - 18T_{23}^{2} + 8T_{23} + 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - 11 T^{2} + 20 T - 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 10 T^{3} + 19 T^{2} + \cdots - 124 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + T^{2} + 32 T - 32 \) Copy content Toggle raw display
$17$ \( (T - 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 18 T^{2} + 8 T + 56 \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{2} + 8 T + 56 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} - 18 T^{2} + 96 T - 64 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 6 T^{2} - 48 T - 16 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} - 13 T^{2} + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} - 62 T^{2} - 224 T - 128 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} - 185 T^{2} + \cdots + 6944 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} - 138 T^{2} + \cdots + 3472 \) Copy content Toggle raw display
$53$ \( T^{4} - 18 T^{3} + 25 T^{2} + \cdots - 1528 \) Copy content Toggle raw display
$59$ \( T^{4} - 24 T^{3} + 174 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} - 68 T^{2} + 32 T + 224 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + 15 T^{2} + 8 T - 8 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 16 T^{2} + 88 T + 28 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 13 T^{2} + \cdots + 124 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} - 145 T^{2} + \cdots - 4088 \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} - 67 T^{2} + \cdots - 686 \) Copy content Toggle raw display
$89$ \( T^{4} + 34 T^{3} + 325 T^{2} + \cdots - 6044 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} - 195 T^{2} + \cdots - 3556 \) Copy content Toggle raw display
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