Properties

Label 4002.2.a.z
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - \beta_{2} q^{5} - q^{6} + (\beta_{2} - 2 \beta_1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - \beta_{2} q^{5} - q^{6} + (\beta_{2} - 2 \beta_1) q^{7} + q^{8} + q^{9} - \beta_{2} q^{10} + ( - \beta_{2} + \beta_1 - 1) q^{11} - q^{12} + (\beta_{2} + \beta_1 - 3) q^{13} + (\beta_{2} - 2 \beta_1) q^{14} + \beta_{2} q^{15} + q^{16} + \beta_{2} q^{17} + q^{18} + (\beta_{2} + 2 \beta_1) q^{19} - \beta_{2} q^{20} + ( - \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{2} + \beta_1 - 1) q^{22} - q^{23} - q^{24} + ( - \beta_{2} + 2 \beta_1 - 1) q^{25} + (\beta_{2} + \beta_1 - 3) q^{26} - q^{27} + (\beta_{2} - 2 \beta_1) q^{28} + q^{29} + \beta_{2} q^{30} + ( - 3 \beta_{2} + \beta_1 - 1) q^{31} + q^{32} + (\beta_{2} - \beta_1 + 1) q^{33} + \beta_{2} q^{34} + (3 \beta_{2} - 2) q^{35} + q^{36} + (\beta_1 - 5) q^{37} + (\beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{2} - \beta_1 + 3) q^{39} - \beta_{2} q^{40} + ( - \beta_1 - 3) q^{41} + ( - \beta_{2} + 2 \beta_1) q^{42} + (\beta_{2} - 2) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{44} - \beta_{2} q^{45} - q^{46} + ( - \beta_{2} + 4 \beta_1 - 6) q^{47} - q^{48} + ( - \beta_{2} - 2 \beta_1 + 5) q^{49} + ( - \beta_{2} + 2 \beta_1 - 1) q^{50} - \beta_{2} q^{51} + (\beta_{2} + \beta_1 - 3) q^{52} + (4 \beta_{2} + 2 \beta_1 - 4) q^{53} - q^{54} + ( - \beta_{2} + \beta_1 + 3) q^{55} + (\beta_{2} - 2 \beta_1) q^{56} + ( - \beta_{2} - 2 \beta_1) q^{57} + q^{58} + (5 \beta_{2} - 2 \beta_1 + 4) q^{59} + \beta_{2} q^{60} + (2 \beta_{2} - 6 \beta_1 - 4) q^{61} + ( - 3 \beta_{2} + \beta_1 - 1) q^{62} + (\beta_{2} - 2 \beta_1) q^{63} + q^{64} + (3 \beta_{2} - 3 \beta_1 - 5) q^{65} + (\beta_{2} - \beta_1 + 1) q^{66} + ( - 2 \beta_{2} + 2 \beta_1 - 10) q^{67} + \beta_{2} q^{68} + q^{69} + (3 \beta_{2} - 2) q^{70} + ( - \beta_{2} + \beta_1 + 11) q^{71} + q^{72} + 2 q^{73} + (\beta_1 - 5) q^{74} + (\beta_{2} - 2 \beta_1 + 1) q^{75} + (\beta_{2} + 2 \beta_1) q^{76} + (\beta_{2} + 3 \beta_1 - 7) q^{77} + ( - \beta_{2} - \beta_1 + 3) q^{78} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{79} - \beta_{2} q^{80} + q^{81} + ( - \beta_1 - 3) q^{82} + ( - 2 \beta_1 - 2) q^{83} + ( - \beta_{2} + 2 \beta_1) q^{84} + (\beta_{2} - 2 \beta_1 - 4) q^{85} + (\beta_{2} - 2) q^{86} - q^{87} + ( - \beta_{2} + \beta_1 - 1) q^{88} + ( - 2 \beta_1 - 4) q^{89} - \beta_{2} q^{90} + ( - 7 \beta_{2} + 7 \beta_1 - 3) q^{91} - q^{92} + (3 \beta_{2} - \beta_1 + 1) q^{93} + ( - \beta_{2} + 4 \beta_1 - 6) q^{94} + ( - \beta_{2} - 4 \beta_1 - 6) q^{95} - q^{96} + ( - 6 \beta_{2} + 4 \beta_1 - 2) q^{97} + ( - \beta_{2} - 2 \beta_1 + 5) q^{98} + ( - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 3 q^{9} - 2 q^{11} - 3 q^{12} - 8 q^{13} - 2 q^{14} + 3 q^{16} + 3 q^{18} + 2 q^{19} + 2 q^{21} - 2 q^{22} - 3 q^{23} - 3 q^{24} - q^{25} - 8 q^{26} - 3 q^{27} - 2 q^{28} + 3 q^{29} - 2 q^{31} + 3 q^{32} + 2 q^{33} - 6 q^{35} + 3 q^{36} - 14 q^{37} + 2 q^{38} + 8 q^{39} - 10 q^{41} + 2 q^{42} - 6 q^{43} - 2 q^{44} - 3 q^{46} - 14 q^{47} - 3 q^{48} + 13 q^{49} - q^{50} - 8 q^{52} - 10 q^{53} - 3 q^{54} + 10 q^{55} - 2 q^{56} - 2 q^{57} + 3 q^{58} + 10 q^{59} - 18 q^{61} - 2 q^{62} - 2 q^{63} + 3 q^{64} - 18 q^{65} + 2 q^{66} - 28 q^{67} + 3 q^{69} - 6 q^{70} + 34 q^{71} + 3 q^{72} + 6 q^{73} - 14 q^{74} + q^{75} + 2 q^{76} - 18 q^{77} + 8 q^{78} - 6 q^{79} + 3 q^{81} - 10 q^{82} - 8 q^{83} + 2 q^{84} - 14 q^{85} - 6 q^{86} - 3 q^{87} - 2 q^{88} - 14 q^{89} - 2 q^{91} - 3 q^{92} + 2 q^{93} - 14 q^{94} - 22 q^{95} - 3 q^{96} - 2 q^{97} + 13 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
2.34292
−1.81361
0.470683
1.00000 −1.00000 1.00000 −2.48929 −1.00000 −2.19656 1.00000 1.00000 −2.48929
1.2 1.00000 −1.00000 1.00000 −0.289169 −1.00000 3.91638 1.00000 1.00000 −0.289169
1.3 1.00000 −1.00000 1.00000 2.77846 −1.00000 −3.71982 1.00000 1.00000 2.77846
\(n\): 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.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{3} - 7T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 15T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 6T_{11} - 8 \) 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
$5$ \( T^{3} - 7T - 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + \cdots - 44 \) Copy content Toggle raw display
$17$ \( T^{3} - 7T + 2 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} + \cdots - 44 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} + \cdots - 176 \) Copy content Toggle raw display
$37$ \( T^{3} + 14 T^{2} + \cdots + 82 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + \cdots + 22 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$47$ \( T^{3} + 14 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} + \cdots - 976 \) Copy content Toggle raw display
$59$ \( T^{3} - 10 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + \cdots - 1208 \) Copy content Toggle raw display
$67$ \( T^{3} + 28 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$71$ \( T^{3} - 34 T^{2} + \cdots - 1376 \) Copy content Toggle raw display
$73$ \( (T - 2)^{3} \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} + \cdots + 328 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( T^{3} + 2 T^{2} + \cdots - 1376 \) Copy content Toggle raw display
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