Properties

Label 10000.2.a.bc
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10000,2,Mod(1,10000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10000, 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("10000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.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: \(79.8504020213\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.103238125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 14x^{4} + 3x^{3} + 49x^{2} + 34x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{4} - \beta_{3} + 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{4} - \beta_{3} + 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{9} + (\beta_{5} - \beta_{2} - \beta_1 - 1) q^{11} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{13} + ( - \beta_{3} - \beta_{2} - 2) q^{17} + (\beta_{4} + \beta_{3} - \beta_1 - 1) q^{19} + (\beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{21} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 3) q^{27} + (\beta_{5} - \beta_{3} + \beta_{2} + 3) q^{29} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{31} + (\beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 7) q^{33} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - 3) q^{37} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{39} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{41} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 5) q^{43} + (\beta_{5} - \beta_1 - 1) q^{47} + ( - \beta_{5} - 2 \beta_{3} - 2 \beta_1 + 4) q^{49} + ( - \beta_{5} - 4 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{51} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 4) q^{53} + (2 \beta_{4} - \beta_{2} + \beta_1 + 6) q^{57} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{59} + (2 \beta_{5} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{61} + ( - 2 \beta_{5} - 6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 10) q^{63} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_1) q^{67} + ( - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{69} + ( - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_1 + 1) q^{71} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{73} + ( - 2 \beta_{5} + 8 \beta_{3} - 4 \beta_1 - 2) q^{77} + (2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 1) q^{79} + ( - \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 2) q^{81} + (2 \beta_{4} + 6 \beta_{3} + \beta_1 - 2) q^{83} + (2 \beta_{5} + 6 \beta_{3} - 3 \beta_{2} - \beta_1) q^{87} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 7) q^{89} + (3 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{91} + ( - 2 \beta_{5} + \beta_{4} - 9 \beta_{3} + \beta_{2} - \beta_1 + 7) q^{93} + (2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - 2 \beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_{2} - 6 \beta_1 - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + q^{7} + 11 q^{9} - 5 q^{11} - 11 q^{13} - 12 q^{17} - 6 q^{19} + 11 q^{21} + 3 q^{23} - 28 q^{27} + 11 q^{29} - q^{31} + 30 q^{33} - 16 q^{37} - q^{39} + 16 q^{41} - 25 q^{43} - 8 q^{47} + 17 q^{49} - 7 q^{51} - 22 q^{53} + 36 q^{57} - 12 q^{59} + 12 q^{61} + 36 q^{63} + 21 q^{67} + 28 q^{69} - 8 q^{71} - 16 q^{73} + 10 q^{77} + 2 q^{79} + 34 q^{81} + 3 q^{83} + 24 q^{87} + 34 q^{89} - 9 q^{91} + 11 q^{93} + 9 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 14x^{4} + 3x^{3} + 49x^{2} + 34x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - 12\nu^{3} + 29\nu^{2} + 42\nu + 8 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 3\nu^{4} - 24\nu^{3} + 18\nu^{2} + 69\nu + 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 7\nu^{4} - 36\nu^{3} + 57\nu^{2} + 101\nu - 16 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} + 46\nu^{3} - 7\nu^{2} - 176\nu - 74 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{3} - \beta_{2} + 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{4} - 6\beta_{3} - 12\beta_{2} + 11\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + 3\beta_{4} + 29\beta_{3} - 21\beta_{2} + 69\beta _1 + 35 \) 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
3.31243
2.78801
−0.149258
−0.691973
−1.54514
−2.71407
0 −3.31243 0 0 0 −1.69984 0 7.97222 0
1.2 0 −2.78801 0 0 0 2.70809 0 4.77301 0
1.3 0 0.149258 0 0 0 −3.58696 0 −2.97772 0
1.4 0 0.691973 0 0 0 −3.25686 0 −2.52117 0
1.5 0 1.54514 0 0 0 2.43270 0 −0.612535 0
1.6 0 2.71407 0 0 0 4.40288 0 4.36620 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 10000.2.a.bc 6
4.b odd 2 1 2500.2.a.d 6
5.b even 2 1 10000.2.a.bd 6
20.d odd 2 1 2500.2.a.c 6
20.e even 4 2 2500.2.c.c 12
25.d even 5 2 400.2.u.f 12
100.h odd 10 2 500.2.g.a 12
100.j odd 10 2 100.2.g.a 12
100.l even 20 4 500.2.i.b 24
300.n even 10 2 900.2.n.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a 12 100.j odd 10 2
400.2.u.f 12 25.d even 5 2
500.2.g.a 12 100.h odd 10 2
500.2.i.b 24 100.l even 20 4
900.2.n.c 12 300.n even 10 2
2500.2.a.c 6 20.d odd 2 1
2500.2.a.d 6 4.b odd 2 1
2500.2.c.c 12 20.e even 4 2
10000.2.a.bc 6 1.a even 1 1 trivial
10000.2.a.bd 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):

\( T_{3}^{6} + T_{3}^{5} - 14T_{3}^{4} - 3T_{3}^{3} + 49T_{3}^{2} - 34T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{5} - 29T_{7}^{4} + 18T_{7}^{3} + 244T_{7}^{2} - 96T_{7} - 576 \) Copy content Toggle raw display
\( T_{11}^{6} + 5T_{11}^{5} - 35T_{11}^{4} - 90T_{11}^{3} + 480T_{11}^{2} - 200T_{11} - 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 14 T^{4} - 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} - 29 T^{4} + 18 T^{3} + \cdots - 576 \) Copy content Toggle raw display
$11$ \( T^{6} + 5 T^{5} - 35 T^{4} - 90 T^{3} + \cdots - 400 \) Copy content Toggle raw display
$13$ \( T^{6} + 11 T^{5} + 16 T^{4} + \cdots - 181 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 39 T^{4} - 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} - 35 T^{4} + \cdots + 3236 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} - 46 T^{4} - 59 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 11 T^{5} - 30 T^{4} + \cdots + 1021 \) Copy content Toggle raw display
$31$ \( T^{6} + T^{5} - 80 T^{4} - 85 T^{3} + \cdots - 10124 \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} + 36 T^{4} + \cdots + 1459 \) Copy content Toggle raw display
$41$ \( T^{6} - 16 T^{5} + 35 T^{4} + \cdots + 6436 \) Copy content Toggle raw display
$43$ \( T^{6} + 25 T^{5} + 185 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} - 31 T^{4} - 316 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{6} + 22 T^{5} + 114 T^{4} + \cdots + 1321 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} - 85 T^{4} + \cdots + 11404 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} - 140 T^{4} + \cdots + 40759 \) Copy content Toggle raw display
$67$ \( T^{6} - 21 T^{5} - 29 T^{4} + \cdots + 241424 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} - 135 T^{4} + \cdots + 74484 \) Copy content Toggle raw display
$73$ \( T^{6} + 16 T^{5} - 44 T^{4} + \cdots - 65851 \) Copy content Toggle raw display
$79$ \( T^{6} - 2 T^{5} - 245 T^{4} + \cdots - 237196 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} - 246 T^{4} + \cdots - 6004 \) Copy content Toggle raw display
$89$ \( T^{6} - 34 T^{5} + 355 T^{4} + \cdots - 7984 \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} - 214 T^{4} + \cdots + 349009 \) Copy content Toggle raw display
show more
show less