Properties

Label 4002.2.a.ba
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: 4.4.16448.2
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: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + \beta_{3} q^{5} + q^{6} + ( - \beta_{2} + 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + \beta_{3} q^{5} + q^{6} + ( - \beta_{2} + 1) q^{7} - q^{8} + q^{9} - \beta_{3} q^{10} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} - q^{12} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{2} - 1) q^{14} - \beta_{3} q^{15} + q^{16} + ( - \beta_{3} + 2 \beta_{2}) q^{17} - q^{18} + (\beta_{2} - 2 \beta_1 + 1) q^{19} + \beta_{3} q^{20} + (\beta_{2} - 1) q^{21} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{22} + q^{23} + q^{24} + (\beta_{2} - 2 \beta_1 + 2) q^{25} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{26} - q^{27} + ( - \beta_{2} + 1) q^{28} - q^{29} + \beta_{3} q^{30} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{31} - q^{32} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{33} + (\beta_{3} - 2 \beta_{2}) q^{34} + (2 \beta_{3} - \beta_{2} - 3) q^{35} + q^{36} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{37} + ( - \beta_{2} + 2 \beta_1 - 1) q^{38} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{39} - \beta_{3} q^{40} + ( - \beta_{3} - \beta_1 + 2) q^{41} + ( - \beta_{2} + 1) q^{42} + ( - 2 \beta_{3} + \beta_{2} + 5) q^{43} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{44} + \beta_{3} q^{45} - q^{46} + (\beta_{2} + 3) q^{47} - q^{48} + ( - 3 \beta_{2} + 2 \beta_1) q^{49} + ( - \beta_{2} + 2 \beta_1 - 2) q^{50} + (\beta_{3} - 2 \beta_{2}) q^{51} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{52} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{53} + q^{54} + (3 \beta_{3} - \beta_{2} + \beta_1 - 7) q^{55} + (\beta_{2} - 1) q^{56} + ( - \beta_{2} + 2 \beta_1 - 1) q^{57} + q^{58} + (2 \beta_{3} - \beta_{2} - 7) q^{59} - \beta_{3} q^{60} + ( - 2 \beta_{3} + 2 \beta_1 + 8) q^{61} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{62} + ( - \beta_{2} + 1) q^{63} + q^{64} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{65} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{66} + (3 \beta_{3} + \beta_{2} + 5) q^{67} + ( - \beta_{3} + 2 \beta_{2}) q^{68} - q^{69} + ( - 2 \beta_{3} + \beta_{2} + 3) q^{70} + (3 \beta_{3} - \beta_{2} + \beta_1 + 5) q^{71} - q^{72} + (4 \beta_{2} + 2) q^{73} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{74} + ( - \beta_{2} + 2 \beta_1 - 2) q^{75} + (\beta_{2} - 2 \beta_1 + 1) q^{76} + ( - 3 \beta_{3} + 5 \beta_{2} + \cdots + 3) q^{77}+ \cdots + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + 2\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + \beta _1 + 7 \) 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 2.46575 −1.00000 1.00000 3.29417
1.2 −1.00000 −1.00000 1.00000 0.230234 1.00000 4.59819 −1.00000 1.00000 −0.230234
1.3 −1.00000 −1.00000 1.00000 1.46575 1.00000 −2.29417 −1.00000 1.00000 −1.46575
1.4 −1.00000 −1.00000 1.00000 3.59819 1.00000 1.23023 −1.00000 1.00000 −3.59819
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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

\( T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 20T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + T_{7}^{2} + 32T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{4} - 26T_{11}^{2} - 24T_{11} + 8 \) 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
$5$ \( T^{4} - 2 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{4} - 26 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{4} - 42 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots + 56 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 772 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + \cdots - 1568 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} + \cdots - 124 \) Copy content Toggle raw display
$47$ \( T^{4} - 10 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$59$ \( T^{4} + 22 T^{3} + \cdots - 112 \) Copy content Toggle raw display
$61$ \( T^{4} - 28 T^{3} + \cdots - 7168 \) Copy content Toggle raw display
$67$ \( T^{4} - 24 T^{3} + \cdots - 2624 \) Copy content Toggle raw display
$71$ \( T^{4} - 28 T^{3} + \cdots - 8768 \) Copy content Toggle raw display
$73$ \( T^{4} - 200 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 1736 \) Copy content Toggle raw display
$83$ \( T^{4} - 20 T^{3} + \cdots - 224 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots - 21376 \) Copy content Toggle raw display
$97$ \( T^{4} + 32 T^{3} + \cdots + 896 \) Copy content Toggle raw display
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