Properties

Label 4002.2.a.bb
Level $4002$
Weight $2$
Character orbit 4002.a
Self dual yes
Analytic conductor $31.956$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.23252.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 2 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 6\beta _1 + 3 \) 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
0.565882
−1.88474
2.95372
−0.634868
−1.00000 1.00000 1.00000 −3.96842 −1.00000 2.40254 −1.00000 1.00000 3.96842
1.2 −1.00000 1.00000 1.00000 −1.82358 −1.00000 2.70832 −1.00000 1.00000 1.82358
1.3 −1.00000 1.00000 1.00000 1.27661 −1.00000 −5.23034 −1.00000 1.00000 −1.27661
1.4 −1.00000 1.00000 1.00000 1.51539 −1.00000 −1.88053 −1.00000 1.00000 −1.51539
\(n\): 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.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\( T_{5}^{4} + 3T_{5}^{3} - 7T_{5}^{2} - 9T_{5} + 14 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} - 20T_{7}^{2} - 4T_{7} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} + 11T_{11}^{3} + 39T_{11}^{2} + 45T_{11} + 2 \) 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
$5$ \( T^{4} + 3 T^{3} + \cdots + 14 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{4} + 11 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + \cdots + 22 \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + \cdots + 232 \) Copy content Toggle raw display
$23$ \( (T - 1)^{4} \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 17 T^{3} + \cdots - 56 \) Copy content Toggle raw display
$37$ \( T^{4} - 15 T^{3} + \cdots - 2672 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} + \cdots + 434 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + \cdots + 10352 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + \cdots + 176 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + \cdots + 56 \) Copy content Toggle raw display
$59$ \( T^{4} + 13 T^{3} + \cdots - 2404 \) Copy content Toggle raw display
$61$ \( T^{4} + 13 T^{3} + \cdots + 56 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots - 9268 \) Copy content Toggle raw display
$71$ \( T^{4} + 15 T^{3} + \cdots - 3784 \) Copy content Toggle raw display
$73$ \( T^{4} + 22 T^{3} + \cdots - 472 \) Copy content Toggle raw display
$79$ \( T^{4} + 26 T^{3} + \cdots + 248 \) Copy content Toggle raw display
$83$ \( T^{4} + 22 T^{3} + \cdots - 10592 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots - 3776 \) Copy content Toggle raw display
$97$ \( T^{4} + 8 T^{3} + \cdots + 6224 \) Copy content Toggle raw display
show more
show less