Properties

Label 4004.2.a.e
Level $4004$
Weight $2$
Character orbit 4004.a
Self dual yes
Analytic conductor $31.972$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4004,2,Mod(1,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.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: \(31.9721009693\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{7} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{7} + (\beta_1 - 1) q^{9} + q^{11} - q^{13} + (2 \beta_{3} - \beta_1 + 1) q^{15} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{17} + ( - 2 \beta_{3} + \beta_{2} - 2) q^{19} + (\beta_{3} + \beta_{2}) q^{21} - \beta_{2} q^{23} + ( - \beta_{3} - \beta_{2} - 3) q^{25} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{27} + 3 \beta_1 q^{29} + (2 \beta_1 + 1) q^{31} + ( - \beta_{3} - \beta_{2}) q^{33} + ( - \beta_{3} + \beta_1 - 1) q^{35} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{37} + (\beta_{3} + \beta_{2}) q^{39} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 1) q^{41} + (4 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{43} + (\beta_1 - 2) q^{45} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{47} + q^{49} + ( - 4 \beta_{3} + \beta_1 - 5) q^{51} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 5) q^{53} + (\beta_{3} - \beta_1 + 1) q^{55} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{57} + (6 \beta_{3} + 6 \beta_{2} - \beta_1 - 1) q^{59} + (3 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{61} + ( - \beta_1 + 1) q^{63} + ( - \beta_{3} + \beta_1 - 1) q^{65} + (4 \beta_1 - 5) q^{67} + (\beta_{3} + 2) q^{69} + ( - 4 \beta_{3} - 6) q^{71} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{73} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{75} - q^{77} + (7 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{79} + (\beta_{2} - 5 \beta_1 - 3) q^{81} + (5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{83} + ( - \beta_{3} + 3 \beta_{2} - 4) q^{85} + ( - 6 \beta_{3} - 3 \beta_{2} - 3) q^{87} + ( - 8 \beta_{3} - 3 \beta_{2} + \cdots - 10) q^{89}+ \cdots + (\beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.82709
−1.95630
−0.209057
1.33826
0 −1.95630 0 −0.209057 0 −1.00000 0 0.827091 0
1.2 0 −0.209057 0 1.33826 0 −1.00000 0 −2.95630 0
1.3 0 1.33826 0 1.82709 0 −1.00000 0 −1.20906 0
1.4 0 1.82709 0 −1.95630 0 −1.00000 0 0.338261 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(11\) \( -1 \)
\(13\) \( +1 \)

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 4004.2.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.e 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} - 4T_{3}^{2} + 4T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots - 59 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$23$ \( T^{4} + T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + \cdots - 29 \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$41$ \( T^{4} + 6 T^{3} + \cdots - 659 \) Copy content Toggle raw display
$43$ \( T^{4} + 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} + \cdots + 811 \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots - 6605 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + \cdots + 2311 \) Copy content Toggle raw display
$61$ \( T^{4} + 18 T^{3} + \cdots - 29 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 61 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 421 \) Copy content Toggle raw display
$79$ \( T^{4} - 9 T^{3} + \cdots - 2169 \) Copy content Toggle raw display
$83$ \( T^{4} + 14 T^{3} + \cdots + 31 \) Copy content Toggle raw display
$89$ \( T^{4} + 23 T^{3} + \cdots - 10679 \) Copy content Toggle raw display
$97$ \( T^{4} - 14 T^{3} + \cdots + 12211 \) Copy content Toggle raw display
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