Properties

Label 10000.2.a.ba
Level $10000$
Weight $2$
Character orbit 10000.a
Self dual yes
Analytic conductor $79.850$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Coefficient field: 4.4.7625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5000)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{2} q^{7} + (\beta_{3} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{2} q^{7} + (\beta_{3} - \beta_1 + 3) q^{9} + ( - \beta_1 + 3) q^{11} + q^{13} + (\beta_{2} - \beta_1 + 2) q^{17} + (2 \beta_{2} - \beta_1 + 4) q^{19} + ( - \beta_{3} + \beta_{2} - 1) q^{21} + ( - \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{3} - 4 \beta_{2} + 5) q^{27} + ( - \beta_{3} + 2 \beta_{2}) q^{29} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{31} + (\beta_{3} - 3 \beta_1 + 8) q^{33} + (3 \beta_{2} - 3 \beta_1 + 2) q^{37} + ( - \beta_1 + 1) q^{39} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + ( - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{43} + (3 \beta_{3} + \beta_1 + 4) q^{47} + ( - \beta_{2} - 6) q^{49} + (\beta_{2} - 2 \beta_1 + 6) q^{51} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{53} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 7) q^{57} + ( - 2 \beta_{3} - 2 \beta_{2} - 5) q^{59} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{61} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{63} + (2 \beta_1 + 1) q^{67} + ( - 4 \beta_{3} + 5 \beta_{2} - 11) q^{69} + ( - 2 \beta_{3} + 7 \beta_{2} - 2 \beta_1 + 5) q^{71} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{73} + ( - \beta_{3} + 3 \beta_{2} - 1) q^{77} + ( - 4 \beta_{3} - 3) q^{79} + (3 \beta_{3} - 12 \beta_{2} - 2 \beta_1) q^{81} + (3 \beta_{3} - 9 \beta_{2} + \beta_1 - 5) q^{83} + ( - 3 \beta_{3} + 6 \beta_{2} - 2) q^{87} + (8 \beta_{2} - 3 \beta_1 + 2) q^{89} + \beta_{2} q^{91} + (3 \beta_{3} - 14 \beta_{2} - 8) q^{93} + ( - 2 \beta_{2} + \beta_1 + 9) q^{97} + (4 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} - 2 q^{7} + 9 q^{9} + 11 q^{11} + 4 q^{13} + 5 q^{17} + 11 q^{19} - 4 q^{21} + 2 q^{23} + 24 q^{27} - 2 q^{29} + 27 q^{33} - q^{37} + 3 q^{39} - 2 q^{41} + 11 q^{43} + 11 q^{47} - 22 q^{49} + 20 q^{51} + 16 q^{53} + 22 q^{57} - 12 q^{59} - 15 q^{61} - 7 q^{63} + 6 q^{67} - 46 q^{69} + 8 q^{71} + 3 q^{73} - 8 q^{77} - 4 q^{79} + 16 q^{81} - 7 q^{83} - 14 q^{87} - 11 q^{89} - 2 q^{91} - 10 q^{93} + 41 q^{97} + 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 9x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \) 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
2.96645
1.71472
−1.34841
−2.33275
0 −1.96645 0 0 0 0.618034 0 0.866918 0
1.2 0 −0.714715 0 0 0 −1.61803 0 −2.48918 0
1.3 0 2.34841 0 0 0 0.618034 0 2.51505 0
1.4 0 3.33275 0 0 0 −1.61803 0 8.10722 0
\(n\): e.g. 2-40 or 990-1000
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.ba 4
4.b odd 2 1 5000.2.a.e 4
5.b even 2 1 10000.2.a.p 4
20.d odd 2 1 5000.2.a.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5000.2.a.e 4 4.b odd 2 1
5000.2.a.j yes 4 20.d odd 2 1
10000.2.a.p 4 5.b even 2 1
10000.2.a.ba 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{4} - 3T_{3}^{3} - 6T_{3}^{2} + 13T_{3} + 11 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 11T_{11}^{3} + 36T_{11}^{2} - 31T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} - 6 T^{2} + 13 T + 11 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + 36 T^{2} - 31 T + 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 5 T^{3} + 15 T + 5 \) Copy content Toggle raw display
$19$ \( T^{4} - 11 T^{3} + 31 T^{2} - 6 T - 44 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} - 71 T^{2} + 272 T + 71 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 21 T^{2} + 18 T - 4 \) Copy content Toggle raw display
$31$ \( T^{4} - 105 T^{2} + 80 T + 2180 \) Copy content Toggle raw display
$37$ \( T^{4} + T^{3} - 84 T^{2} - 59 T + 1321 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 76 T^{2} + 208 T - 64 \) Copy content Toggle raw display
$43$ \( T^{4} - 11 T^{3} + 31 T^{2} - 16 T - 4 \) Copy content Toggle raw display
$47$ \( T^{4} - 11 T^{3} - 54 T^{2} + \cdots + 1511 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} - 9 T^{2} + 664 T + 436 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} - 6 T^{2} - 412 T - 859 \) Copy content Toggle raw display
$61$ \( T^{4} + 15 T^{3} - 70 T^{2} + \cdots - 4895 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} - 24 T^{2} + 94 T + 191 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} - 121 T^{2} + \cdots + 431 \) Copy content Toggle raw display
$73$ \( T^{4} - 3 T^{3} - 121 T^{2} + \cdots + 1516 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} - 194 T^{2} + \cdots + 4681 \) Copy content Toggle raw display
$83$ \( T^{4} + 7 T^{3} - 261 T^{2} - 1392 T - 64 \) Copy content Toggle raw display
$89$ \( T^{4} + 11 T^{3} - 139 T^{2} + \cdots - 1364 \) Copy content Toggle raw display
$97$ \( T^{4} - 41 T^{3} + 616 T^{2} + \cdots + 9511 \) Copy content Toggle raw display
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