Properties

Label 4002.2.a.y
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
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] = [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: \(3\)
Coefficient field: 3.3.1772.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 12x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + \beta _1 + 8 \) 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
3.67370
0.654334
−3.32803
−1.00000 −1.00000 1.00000 −3.67370 1.00000 2.91118 −1.00000 1.00000 3.67370
1.2 −1.00000 −1.00000 1.00000 −0.654334 1.00000 −2.11309 −1.00000 1.00000 0.654334
1.3 −1.00000 −1.00000 1.00000 3.32803 1.00000 5.20191 −1.00000 1.00000 −3.32803
\(n\): 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.y 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{3} + T_{5}^{2} - 12T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 6T_{7}^{2} - 2T_{7} + 32 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 12T_{11} - 8 \) 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
$5$ \( T^{3} + T^{2} - 12T - 8 \) Copy content Toggle raw display
$7$ \( T^{3} - 6 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 12T - 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 7 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$17$ \( (T - 4)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( (T + 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 7 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$37$ \( T^{3} + 7 T^{2} + \cdots - 212 \) Copy content Toggle raw display
$41$ \( T^{3} + 11 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots - 180 \) Copy content Toggle raw display
$47$ \( T^{3} + 10 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$53$ \( T^{3} + 2 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$59$ \( T^{3} - 23 T^{2} + \cdots - 360 \) Copy content Toggle raw display
$61$ \( T^{3} + 7 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$67$ \( T^{3} + 7 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$71$ \( T^{3} + 15 T^{2} + \cdots - 804 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} + \cdots - 144 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} + \cdots - 200 \) Copy content Toggle raw display
$83$ \( T^{3} - 126T - 324 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$97$ \( T^{3} - 28 T^{2} + \cdots - 300 \) Copy content Toggle raw display
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