Properties

Label 4000.2.a.h
Level $4000$
Weight $2$
Character orbit 4000.a
Self dual yes
Analytic conductor $31.940$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: 4.4.16400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 13x^{2} + 41 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - 2) q^{9} - \beta_1 q^{11} + \beta_1 q^{13} - \beta_{3} q^{17} + \beta_{3} q^{19} - q^{21} + (\beta_{2} - 3) q^{23} + (4 \beta_{2} + 1) q^{27} + (5 \beta_{2} - 2) q^{29} + \beta_1 q^{31} + \beta_{3} q^{33} + (\beta_{3} + \beta_1) q^{37} - \beta_{3} q^{39} + (3 \beta_{2} + 4) q^{41} + ( - 7 \beta_{2} - 8) q^{43} + (5 \beta_{2} - 2) q^{47} + (\beta_{2} - 5) q^{49} + ( - \beta_{3} + \beta_1) q^{51} - 2 \beta_1 q^{53} + (\beta_{3} - \beta_1) q^{57} + ( - 2 \beta_{3} - \beta_1) q^{59} + ( - 3 \beta_{2} + 3) q^{61} + ( - 2 \beta_{2} - 3) q^{63} + ( - 4 \beta_{2} - 8) q^{67} + (4 \beta_{2} - 1) q^{69} + \beta_{3} q^{71} + ( - 2 \beta_{3} - \beta_1) q^{73} + ( - \beta_{3} - \beta_1) q^{77} + (\beta_{3} + \beta_1) q^{79} + (6 \beta_{2} + 2) q^{81} + ( - 9 \beta_{2} - 7) q^{83} + (7 \beta_{2} - 5) q^{87} + ( - 3 \beta_{2} + 5) q^{89} + (\beta_{3} + \beta_1) q^{91} - \beta_{3} q^{93} + (2 \beta_{3} + \beta_1) q^{97} + (\beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{7} - 6 q^{9} - 4 q^{21} - 14 q^{23} - 4 q^{27} - 18 q^{29} + 10 q^{41} - 18 q^{43} - 18 q^{47} - 22 q^{49} + 18 q^{61} - 8 q^{63} - 24 q^{67} - 12 q^{69} - 4 q^{81} - 10 q^{83} - 34 q^{87} + 26 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 41 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - 14\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 2 \) 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
2.76008
−2.76008
2.31991
−2.31991
0 −0.618034 0 0 0 1.61803 0 −2.61803 0
1.2 0 −0.618034 0 0 0 1.61803 0 −2.61803 0
1.3 0 1.61803 0 0 0 −0.618034 0 −0.381966 0
1.4 0 1.61803 0 0 0 −0.618034 0 −0.381966 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.a.h yes 4
4.b odd 2 1 4000.2.a.c 4
5.b even 2 1 4000.2.a.c 4
5.c odd 4 2 4000.2.c.e 8
8.b even 2 1 8000.2.a.bc 4
8.d odd 2 1 8000.2.a.bp 4
20.d odd 2 1 inner 4000.2.a.h yes 4
20.e even 4 2 4000.2.c.e 8
40.e odd 2 1 8000.2.a.bc 4
40.f even 2 1 8000.2.a.bp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.c 4 4.b odd 2 1
4000.2.a.c 4 5.b even 2 1
4000.2.a.h yes 4 1.a even 1 1 trivial
4000.2.a.h yes 4 20.d odd 2 1 inner
4000.2.c.e 8 5.c odd 4 2
4000.2.c.e 8 20.e even 4 2
8000.2.a.bc 4 8.b even 2 1
8000.2.a.bc 4 40.e odd 2 1
8000.2.a.bp 4 8.d odd 2 1
8000.2.a.bp 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):

\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 52T_{11}^{2} + 656 \) Copy content Toggle raw display
\( T_{13}^{4} - 52T_{13}^{2} + 656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 52T^{2} + 656 \) Copy content Toggle raw display
$13$ \( T^{4} - 52T^{2} + 656 \) Copy content Toggle raw display
$17$ \( T^{4} - 68T^{2} + 656 \) Copy content Toggle raw display
$19$ \( T^{4} - 68T^{2} + 656 \) Copy content Toggle raw display
$23$ \( (T^{2} + 7 T + 11)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T - 11)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 52T^{2} + 656 \) Copy content Toggle raw display
$37$ \( T^{4} - 88T^{2} + 656 \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T - 5)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 9 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9 T - 11)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 208 T^{2} + 10496 \) Copy content Toggle raw display
$59$ \( T^{4} - 260 T^{2} + 16400 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 68T^{2} + 656 \) Copy content Toggle raw display
$73$ \( T^{4} - 260 T^{2} + 16400 \) Copy content Toggle raw display
$79$ \( T^{4} - 88T^{2} + 656 \) Copy content Toggle raw display
$83$ \( (T^{2} + 5 T - 95)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 13 T + 31)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 260 T^{2} + 16400 \) Copy content Toggle raw display
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