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Properties

Label 4000.2.a.r
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: 8.8.578340050000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 21x^{6} + 129x^{4} - 220x^{2} + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{7} - \beta_{5} + \beta_1) q^{11} + (\beta_{6} + 2) q^{13} + (\beta_{6} + \beta_{3} + 1) q^{17} + (\beta_{7} + \beta_{4} + 2 \beta_1) q^{19} + ( - \beta_{6} + \beta_{3} + \beta_{2} + 2) q^{21} + (\beta_{7} - \beta_{5} + \beta_{4}) q^{23} + (2 \beta_{5} - \beta_{4} - 3 \beta_1) q^{27} + (\beta_{6} - \beta_{3} + 3) q^{29} + \beta_{4} q^{31} + ( - \beta_{6} - \beta_{3} - \beta_{2} - 4) q^{33} + ( - \beta_{6} + \beta_{2} + 2) q^{37} + ( - 2 \beta_{7} - 3 \beta_{4} - 4 \beta_1) q^{39} + (\beta_{3} - \beta_{2} + 3) q^{41} + ( - 2 \beta_{5} + \beta_{4} - \beta_1) q^{43} + (\beta_{7} - \beta_{5} + \cdots - 2 \beta_1) q^{47}+ \cdots + ( - \beta_{7} + \beta_{5} + \cdots + 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 18 q^{9} + 12 q^{13} + 18 q^{21} + 24 q^{29} - 26 q^{33} + 22 q^{37} + 18 q^{41} + 26 q^{49} + 32 q^{53} - 50 q^{57} + 22 q^{61} + 32 q^{69} - 50 q^{73} + 36 q^{77} + 60 q^{81} + 10 q^{89} + 16 q^{93} - 46 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 21x^{6} + 129x^{4} - 220x^{2} + 80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 25\nu^{4} - 161\nu^{2} + 116 ) / 68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 25\nu^{5} - 161\nu^{3} + 116\nu ) / 68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 25\nu^{5} - 229\nu^{3} + 728\nu ) / 136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{6} - 157\nu^{4} + 701\nu^{2} - 568 ) / 68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 58\nu^{5} + 296\nu^{3} - 263\nu ) / 34 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} + \beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 9\beta_{3} + 11\beta_{2} + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 22\beta_{5} + 23\beta_{4} + 94\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25\beta_{6} + 157\beta_{3} + 114\beta_{2} + 511 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50\beta_{7} - 228\beta_{5} + 346\beta_{4} + 1017\beta_1 \) 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
3.33481
2.72684
1.39511
0.705030
−0.705030
−1.39511
−2.72684
−3.33481
0 −3.33481 0 0 0 −1.78602 0 8.12097 0
1.2 0 −2.72684 0 0 0 −2.84526 0 4.43565 0
1.3 0 −1.39511 0 0 0 4.73991 0 −1.05368 0
1.4 0 −0.705030 0 0 0 −2.69218 0 −2.50293 0
1.5 0 0.705030 0 0 0 2.69218 0 −2.50293 0
1.6 0 1.39511 0 0 0 −4.73991 0 −1.05368 0
1.7 0 2.72684 0 0 0 2.84526 0 4.43565 0
1.8 0 3.33481 0 0 0 1.78602 0 8.12097 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.r yes 8
4.b odd 2 1 inner 4000.2.a.r yes 8
5.b even 2 1 4000.2.a.q ✓ 8
5.c odd 4 2 4000.2.c.h 16
8.b even 2 1 8000.2.a.ca 8
8.d odd 2 1 8000.2.a.ca 8
20.d odd 2 1 4000.2.a.q ✓ 8
20.e even 4 2 4000.2.c.h 16
40.e odd 2 1 8000.2.a.cb 8
40.f even 2 1 8000.2.a.cb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.q ✓ 8 5.b even 2 1
4000.2.a.q ✓ 8 20.d odd 2 1
4000.2.a.r yes 8 1.a even 1 1 trivial
4000.2.a.r yes 8 4.b odd 2 1 inner
4000.2.c.h 16 5.c odd 4 2
4000.2.c.h 16 20.e even 4 2
8000.2.a.ca 8 8.b even 2 1
8000.2.a.ca 8 8.d odd 2 1
8000.2.a.cb 8 40.e odd 2 1
8000.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):

\( T_{3}^{8} - 21T_{3}^{6} + 129T_{3}^{4} - 220T_{3}^{2} + 80 \) Copy content Toggle raw display
\( T_{7}^{8} - 41T_{7}^{6} + 524T_{7}^{4} - 2605T_{7}^{2} + 4205 \) Copy content Toggle raw display
\( T_{11}^{8} - 49T_{11}^{6} + 684T_{11}^{4} - 2445T_{11}^{2} + 5 \) Copy content Toggle raw display
\( T_{13}^{4} - 6T_{13}^{3} - 17T_{13}^{2} + 78T_{13} + 149 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 21 T^{6} + \cdots + 80 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 41 T^{6} + \cdots + 4205 \) Copy content Toggle raw display
$11$ \( T^{8} - 49 T^{6} + \cdots + 5 \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + \cdots + 149)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 33 T^{2} + \cdots + 176)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 80 T^{6} + \cdots + 3125 \) Copy content Toggle raw display
$23$ \( T^{8} - 91 T^{6} + \cdots + 20480 \) Copy content Toggle raw display
$29$ \( (T^{4} - 12 T^{3} + \cdots + 20)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 29 T^{6} + \cdots + 80 \) Copy content Toggle raw display
$37$ \( (T^{4} - 11 T^{3} + \cdots - 320)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 9 T^{3} - 6 T^{2} + \cdots + 95)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 234 T^{6} + \cdots + 3872000 \) Copy content Toggle raw display
$47$ \( T^{8} - 176 T^{6} + \cdots + 17405 \) Copy content Toggle raw display
$53$ \( (T^{4} - 16 T^{3} + \cdots + 2501)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 196 T^{6} + \cdots + 85805 \) Copy content Toggle raw display
$61$ \( (T^{4} - 11 T^{3} + \cdots - 2620)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 81 T^{6} + \cdots + 80 \) Copy content Toggle raw display
$71$ \( T^{8} - 694 T^{6} + \cdots + 849686480 \) Copy content Toggle raw display
$73$ \( (T^{4} + 25 T^{3} + \cdots - 4124)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 546 T^{6} + \cdots + 3494480 \) Copy content Toggle raw display
$83$ \( T^{8} - 576 T^{6} + \cdots + 8192000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{3} + \cdots - 2000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 23 T^{3} + \cdots - 244)^{2} \) Copy content Toggle raw display
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