Properties

Label 4000.2.a.k
Level $4000$
Weight $2$
Character orbit 4000.a
Self dual yes
Analytic conductor $31.940$
Analytic rank $1$
Dimension $6$
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] = [4000,2,Mod(1,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, 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("4000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.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: \(31.9401608085\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.30040000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{5} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{5} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + ( - \beta_{4} - \beta_{2} - \beta_1 - 2) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{17} + (\beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{19} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{23} + (\beta_{2} + 2) q^{27} + (\beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{31} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 2) q^{33} + (\beta_{5} + 3 \beta_{2} + 3) q^{37} + ( - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{39} + (\beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{41} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{43} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2} - 2) q^{47} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{49} + (2 \beta_{5} + 4 \beta_{3} - 3 \beta_1 - 2) q^{51} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{53} + ( - \beta_{4} + \beta_{3} - 3 \beta_1 - 4) q^{57} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 5) q^{59} + (3 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{61} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{63} + ( - 2 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{67} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{69} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{71} + (2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{73} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 1) q^{77} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 4) q^{79} + (\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1 - 3) q^{81} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{83} + (2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{87} + ( - 3 \beta_{4} - 4 \beta_{2} - \beta_1) q^{89} + (\beta_{5} + \beta_{4} + 8 \beta_{3} - 3 \beta_{2} - 1) q^{91} + (3 \beta_{5} + \beta_{4} + 6 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{93} + (2 \beta_{5} + \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{97} + ( - \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{7} + 4 q^{9} - 13 q^{11} + 7 q^{13} - 4 q^{17} - 9 q^{19} - 7 q^{23} + 12 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} + 21 q^{37} - 26 q^{39} - 5 q^{41} + 8 q^{43} - 19 q^{47} - 3 q^{49} - 18 q^{51} + 11 q^{53} - 28 q^{57} - 25 q^{59} - 6 q^{61} + 3 q^{63} - 12 q^{69} - 34 q^{71} - 20 q^{73} + 14 q^{77} - 16 q^{79} - 18 q^{81} + 4 q^{83} + 26 q^{87} - 3 q^{89} - 26 q^{91} + 16 q^{93} - 20 q^{97} - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 6\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 11\nu^{3} - 2\nu^{2} + 29\nu + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 11\nu^{3} + 4\nu^{2} - 31\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} + 31\nu^{3} - 6\nu^{2} - 79\nu - 26 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 6\beta_{4} + 9\beta_{3} + \beta_{2} + 8\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{4} + 4\beta_{3} + 11\beta_{2} + 39\beta _1 + 20 \) 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
−2.40554
−1.81030
−0.481965
−0.306558
2.29226
2.71210
0 −2.40554 0 0 0 2.20893 0 2.78662 0
1.2 0 −1.81030 0 0 0 −4.37760 0 0.277175 0
1.3 0 −0.481965 0 0 0 2.69841 0 −2.76771 0
1.4 0 −0.306558 0 0 0 −2.60656 0 −2.90602 0
1.5 0 2.29226 0 0 0 −0.938845 0 2.25447 0
1.6 0 2.71210 0 0 0 0.0156607 0 4.35547 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 4000.2.a.k 6
4.b odd 2 1 4000.2.a.n yes 6
5.b even 2 1 4000.2.a.m yes 6
5.c odd 4 2 4000.2.c.f 12
8.b even 2 1 8000.2.a.bv 6
8.d odd 2 1 8000.2.a.bw 6
20.d odd 2 1 4000.2.a.l yes 6
20.e even 4 2 4000.2.c.g 12
40.e odd 2 1 8000.2.a.bu 6
40.f even 2 1 8000.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.k 6 1.a even 1 1 trivial
4000.2.a.l yes 6 20.d odd 2 1
4000.2.a.m yes 6 5.b even 2 1
4000.2.a.n yes 6 4.b odd 2 1
4000.2.c.f 12 5.c odd 4 2
4000.2.c.g 12 20.e even 4 2
8000.2.a.bu 6 40.e odd 2 1
8000.2.a.bv 6 8.b even 2 1
8000.2.a.bw 6 8.d odd 2 1
8000.2.a.bx 6 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):

\( T_{3}^{6} - 11T_{3}^{4} - 4T_{3}^{3} + 29T_{3}^{2} + 22T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} + 3T_{7}^{5} - 15T_{7}^{4} - 30T_{7}^{3} + 55T_{7}^{2} + 63T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 13T_{11}^{5} + 41T_{11}^{4} - 50T_{11}^{3} - 325T_{11}^{2} - 375T_{11} - 125 \) Copy content Toggle raw display
\( T_{13}^{6} - 7T_{13}^{5} - 34T_{13}^{4} + 245T_{13}^{3} + 150T_{13}^{2} - 1375T_{13} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 11 T^{4} - 4 T^{3} + 29 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} - 15 T^{4} - 30 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 13 T^{5} + 41 T^{4} + \cdots - 125 \) Copy content Toggle raw display
$13$ \( T^{6} - 7 T^{5} - 34 T^{4} + 245 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} - 66 T^{4} + \cdots + 2000 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} - 16 T^{4} - 285 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} - 11 T^{4} - 104 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} - 54 T^{4} + \cdots - 2636 \) Copy content Toggle raw display
$31$ \( T^{6} + 12 T^{5} - 59 T^{4} + \cdots - 29500 \) Copy content Toggle raw display
$37$ \( T^{6} - 21 T^{5} + 69 T^{4} + \cdots - 8000 \) Copy content Toggle raw display
$41$ \( T^{6} + 5 T^{5} - 75 T^{4} + \cdots + 10475 \) Copy content Toggle raw display
$43$ \( T^{6} - 8 T^{5} - 74 T^{4} + \cdots - 24256 \) Copy content Toggle raw display
$47$ \( T^{6} + 19 T^{5} + 56 T^{4} + \cdots + 2549 \) Copy content Toggle raw display
$53$ \( T^{6} - 11 T^{5} - 106 T^{4} + \cdots - 9875 \) Copy content Toggle raw display
$59$ \( T^{6} + 25 T^{5} + 80 T^{4} + \cdots - 59375 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} - 181 T^{4} + \cdots - 32220 \) Copy content Toggle raw display
$67$ \( T^{6} - 159 T^{4} - 68 T^{3} + \cdots - 5364 \) Copy content Toggle raw display
$71$ \( T^{6} + 34 T^{5} + 274 T^{4} + \cdots - 372500 \) Copy content Toggle raw display
$73$ \( T^{6} + 20 T^{5} - 45 T^{4} + \cdots + 2500 \) Copy content Toggle raw display
$79$ \( T^{6} + 16 T^{5} - 206 T^{4} + \cdots + 174500 \) Copy content Toggle raw display
$83$ \( T^{6} - 4 T^{5} - 144 T^{4} + \cdots - 46080 \) Copy content Toggle raw display
$89$ \( T^{6} + 3 T^{5} - 361 T^{4} + \cdots - 142144 \) Copy content Toggle raw display
$97$ \( T^{6} + 20 T^{5} - 45 T^{4} + \cdots + 177500 \) Copy content Toggle raw display
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