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Properties

Label 4000.2.a.p
Level $4000$
Weight $2$
Character orbit 4000.a
Self dual yes
Analytic conductor $31.940$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.26208800000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{6} q^{7} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} - \beta_{6} q^{7} + ( - \beta_{5} + 1) q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{4}) q^{11} + (\beta_{3} + 1) q^{13} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{17} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_1) q^{19} + (\beta_{5} - \beta_{2} - 2) q^{21} + ( - 2 \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{23}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{13} + 20 q^{17} - 12 q^{21} - 16 q^{29} + 36 q^{33} + 28 q^{37} + 8 q^{41} + 16 q^{49} + 48 q^{53} + 20 q^{57} - 28 q^{61} - 28 q^{69} + 44 q^{77} + 20 q^{81} + 20 q^{89} + 4 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 34x^{4} - 30x^{2} + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 12\nu^{4} - 37\nu^{2} + 13 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 18\nu^{4} - 32\nu^{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 24\nu^{5} + 83\nu^{3} - 71\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{6} - 66\nu^{4} + 151\nu^{2} - 64 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 21\nu^{5} - 56\nu^{3} + 26\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - 21\nu^{5} + 59\nu^{3} - 41\nu ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{5} + 6\beta_{3} + 13\beta_{2} + 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 8\beta_{6} - 3\beta_{4} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47\beta_{5} + 35\beta_{3} + 101\beta_{2} + 200 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 133\beta_{7} + 109\beta_{6} - 63\beta_{4} + 188\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−0.464606
1.81062
−1.05053
2.53025
−2.53025
1.05053
−1.81062
0.464606
0 −2.79703 0 0 0 2.30294 0 4.82335 0
1.2 0 −2.74204 0 0 0 1.42258 0 4.51880 0
1.3 0 −0.693685 0 0 0 −4.52199 0 −2.51880 0
1.4 0 −0.420293 0 0 0 −2.86781 0 −2.82335 0
1.5 0 0.420293 0 0 0 2.86781 0 −2.82335 0
1.6 0 0.693685 0 0 0 4.52199 0 −2.51880 0
1.7 0 2.74204 0 0 0 −1.42258 0 4.51880 0
1.8 0 2.79703 0 0 0 −2.30294 0 4.82335 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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
4.b 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.p yes 8
4.b odd 2 1 inner 4000.2.a.p yes 8
5.b even 2 1 4000.2.a.o ✓ 8
5.c odd 4 2 4000.2.c.i 16
8.b even 2 1 8000.2.a.by 8
8.d odd 2 1 8000.2.a.by 8
20.d odd 2 1 4000.2.a.o ✓ 8
20.e even 4 2 4000.2.c.i 16
40.e odd 2 1 8000.2.a.bz 8
40.f even 2 1 8000.2.a.bz 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.o ✓ 8 5.b even 2 1
4000.2.a.o ✓ 8 20.d odd 2 1
4000.2.a.p yes 8 1.a even 1 1 trivial
4000.2.a.p yes 8 4.b odd 2 1 inner
4000.2.c.i 16 5.c odd 4 2
4000.2.c.i 16 20.e even 4 2
8000.2.a.by 8 8.b even 2 1
8000.2.a.by 8 8.d odd 2 1
8000.2.a.bz 8 40.e odd 2 1
8000.2.a.bz 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):

\( T_{3}^{8} - 16T_{3}^{6} + 69T_{3}^{4} - 40T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{8} - 36T_{7}^{6} + 389T_{7}^{4} - 1540T_{7}^{2} + 1805 \) Copy content Toggle raw display
\( T_{11}^{8} - 64T_{11}^{6} + 1344T_{11}^{4} - 9280T_{11}^{2} + 1280 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} - 32T_{13}^{2} + 72T_{13} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 36 T^{6} + \cdots + 1805 \) Copy content Toggle raw display
$11$ \( T^{8} - 64 T^{6} + \cdots + 1280 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 10 T^{3} + \cdots - 304)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 140 T^{6} + \cdots + 800000 \) Copy content Toggle raw display
$23$ \( T^{8} - 116 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} - 19 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 224 T^{6} + \cdots + 8398080 \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots - 80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots - 695)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 104 T^{6} + \cdots + 10125 \) Copy content Toggle raw display
$47$ \( T^{8} - 176 T^{6} + \cdots + 96605 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 16)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} - 336 T^{6} + \cdots + 1280 \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots - 355)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 536 T^{6} + \cdots + 205825280 \) Copy content Toggle raw display
$71$ \( T^{8} - 364 T^{6} + \cdots + 2151680 \) Copy content Toggle raw display
$73$ \( (T^{4} - 192 T^{2} + \cdots - 2384)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 116 T^{6} + \cdots + 1280 \) Copy content Toggle raw display
$83$ \( T^{8} - 356 T^{6} + \cdots + 15753125 \) Copy content Toggle raw display
$89$ \( (T^{2} - 5 T - 5)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 8 T^{3} + \cdots + 4016)^{2} \) Copy content Toggle raw display
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