Properties

Label 4002.2.a.bd
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.19796.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - \beta_{3} q^{5} + q^{6} + (\beta_{3} + \beta_1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} - \beta_{3} q^{5} + q^{6} + (\beta_{3} + \beta_1) q^{7} + q^{8} + q^{9} - \beta_{3} q^{10} + ( - \beta_1 + 4) q^{11} + q^{12} + ( - \beta_{2} + \beta_1 + 2) q^{13} + (\beta_{3} + \beta_1) q^{14} - \beta_{3} q^{15} + q^{16} + (\beta_{3} + \beta_1) q^{17} + q^{18} + ( - \beta_{3} - \beta_1) q^{19} - \beta_{3} q^{20} + (\beta_{3} + \beta_1) q^{21} + ( - \beta_1 + 4) q^{22} - q^{23} + q^{24} + (2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{25} + ( - \beta_{2} + \beta_1 + 2) q^{26} + q^{27} + (\beta_{3} + \beta_1) q^{28} + q^{29} - \beta_{3} q^{30} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{31} + q^{32} + ( - \beta_1 + 4) q^{33} + (\beta_{3} + \beta_1) q^{34} + ( - \beta_{3} + \beta_1 - 4) q^{35} + q^{36} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{37}+ \cdots + ( - \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - q^{5} + 4 q^{6} + 2 q^{7} + 4 q^{8} + 4 q^{9} - q^{10} + 15 q^{11} + 4 q^{12} + 11 q^{13} + 2 q^{14} - q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} - 2 q^{19} - q^{20} + 2 q^{21} + 15 q^{22} - 4 q^{23} + 4 q^{24} - q^{25} + 11 q^{26} + 4 q^{27} + 2 q^{28} + 4 q^{29} - q^{30} - 5 q^{31} + 4 q^{32} + 15 q^{33} + 2 q^{34} - 16 q^{35} + 4 q^{36} + 3 q^{37} - 2 q^{38} + 11 q^{39} - q^{40} + 11 q^{41} + 2 q^{42} + 10 q^{43} + 15 q^{44} - q^{45} - 4 q^{46} + 10 q^{47} + 4 q^{48} - q^{50} + 2 q^{51} + 11 q^{52} + 24 q^{53} + 4 q^{54} - 7 q^{55} + 2 q^{56} - 2 q^{57} + 4 q^{58} + q^{59} - q^{60} + 3 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 3 q^{65} + 15 q^{66} + 7 q^{67} + 2 q^{68} - 4 q^{69} - 16 q^{70} - 13 q^{71} + 4 q^{72} - 2 q^{73} + 3 q^{74} - q^{75} - 2 q^{76} - 4 q^{77} + 11 q^{78} - 22 q^{79} - q^{80} + 4 q^{81} + 11 q^{82} - 16 q^{83} + 2 q^{84} - 16 q^{85} + 10 q^{86} + 4 q^{87} + 15 q^{88} - 8 q^{89} - q^{90} + 14 q^{91} - 4 q^{92} - 5 q^{93} + 10 q^{94} + 16 q^{95} + 4 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 4 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 6\beta _1 + 4 \) 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
−1.28586
2.94741
−1.82082
1.15927
1.00000 1.00000 1.00000 −3.71045 1.00000 2.42459 1.00000 1.00000 −3.71045
1.2 1.00000 1.00000 1.00000 −0.440752 1.00000 3.38816 1.00000 1.00000 −0.440752
1.3 1.00000 1.00000 1.00000 1.38424 1.00000 −3.20506 1.00000 1.00000 1.38424
1.4 1.00000 1.00000 1.00000 1.76696 1.00000 −0.607688 1.00000 1.00000 1.76696
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(23\) \( +1 \)
\(29\) \( -1 \)

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 4002.2.a.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{4} + T_{5}^{3} - 9T_{5}^{2} + 5T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 12T_{7}^{2} + 20T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 15T_{11}^{3} + 77T_{11}^{2} - 153T_{11} + 92 \) 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
$5$ \( T^{4} + T^{3} - 9 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{4} - 15 T^{3} + \cdots + 92 \) Copy content Toggle raw display
$13$ \( T^{4} - 11 T^{3} + \cdots - 38 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots - 124 \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + \cdots + 326 \) Copy content Toggle raw display
$41$ \( T^{4} - 11 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots - 416 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + \cdots - 2336 \) Copy content Toggle raw display
$53$ \( T^{4} - 24 T^{3} + \cdots - 14416 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{3} + \cdots + 596 \) Copy content Toggle raw display
$61$ \( T^{4} - 3 T^{3} + \cdots + 5210 \) Copy content Toggle raw display
$67$ \( T^{4} - 7 T^{3} + \cdots - 4322 \) Copy content Toggle raw display
$71$ \( T^{4} + 13 T^{3} + \cdots - 976 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} + \cdots + 3208 \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + \cdots - 5080 \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + \cdots - 2224 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} + \cdots - 5120 \) Copy content Toggle raw display
$97$ \( T^{4} + 4 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
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