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Properties

Label 2500.2.a.d
Level $2500$
Weight $2$
Character orbit 2500.a
Self dual yes
Analytic conductor $19.963$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2500,2,Mod(1,2500)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2500.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2500, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2500.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.9626005053\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.103238125.1
Copy content comment:defining polynomial
 
Copy content 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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} + \beta_{3} - 1) q^{7} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 2) q^{9} + ( - \beta_{5} + \beta_{2} + \beta_1 + 1) q^{11} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{13}+ \cdots + (2 \beta_{5} + \beta_{4} - 7 \beta_{3} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} - q^{7} + 11 q^{9} + 5 q^{11} - 11 q^{13} - 12 q^{17} + 6 q^{19} + 11 q^{21} - 3 q^{23} + 28 q^{27} + 11 q^{29} + q^{31} + 30 q^{33} - 16 q^{37} + q^{39} + 16 q^{41} + 25 q^{43} + 8 q^{47}+ \cdots + 20 q^{99}+O(q^{100}) \) 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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.71407
−1.54514
−0.691973
−0.149258
2.78801
3.31243
0 −2.71407 0 0 0 −4.40288 0 4.36620 0
1.2 0 −1.54514 0 0 0 −2.43270 0 −0.612535 0
1.3 0 −0.691973 0 0 0 3.25686 0 −2.52117 0
1.4 0 −0.149258 0 0 0 3.58696 0 −2.97772 0
1.5 0 2.78801 0 0 0 −2.70809 0 4.77301 0
1.6 0 3.31243 0 0 0 1.69984 0 7.97222 0
\(n\): e.g. 2-40 or 80-90
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 2500.2.a.d 6
4.b odd 2 1 10000.2.a.bc 6
5.b even 2 1 2500.2.a.c 6
5.c odd 4 2 2500.2.c.c 12
20.d odd 2 1 10000.2.a.bd 6
25.d even 5 2 100.2.g.a ✓ 12
25.e even 10 2 500.2.g.a 12
25.f odd 20 4 500.2.i.b 24
75.j odd 10 2 900.2.n.c 12
100.j odd 10 2 400.2.u.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a ✓ 12 25.d even 5 2
400.2.u.f 12 100.j odd 10 2
500.2.g.a 12 25.e even 10 2
500.2.i.b 24 25.f odd 20 4
900.2.n.c 12 75.j odd 10 2
2500.2.a.c 6 5.b even 2 1
2500.2.a.d 6 1.a even 1 1 trivial
2500.2.c.c 12 5.c odd 4 2
10000.2.a.bc 6 4.b odd 2 1
10000.2.a.bd 6 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - T_{3}^{5} - 14T_{3}^{4} + 3T_{3}^{3} + 49T_{3}^{2} + 34T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} - 14 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + T^{5} + \cdots - 576 \) Copy content Toggle raw display
$11$ \( T^{6} - 5 T^{5} + \cdots - 400 \) Copy content Toggle raw display
$13$ \( T^{6} + 11 T^{5} + \cdots - 181 \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{6} - 6 T^{5} + \cdots + 3236 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$29$ \( T^{6} - 11 T^{5} + \cdots + 1021 \) Copy content Toggle raw display
$31$ \( T^{6} - T^{5} + \cdots - 10124 \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} + \cdots + 1459 \) Copy content Toggle raw display
$41$ \( T^{6} - 16 T^{5} + \cdots + 6436 \) Copy content Toggle raw display
$43$ \( T^{6} - 25 T^{5} + \cdots + 6400 \) Copy content Toggle raw display
$47$ \( T^{6} - 8 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{6} + 22 T^{5} + \cdots + 1321 \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 11404 \) Copy content Toggle raw display
$61$ \( T^{6} - 12 T^{5} + \cdots + 40759 \) Copy content Toggle raw display
$67$ \( T^{6} + 21 T^{5} + \cdots + 241424 \) Copy content Toggle raw display
$71$ \( T^{6} - 8 T^{5} + \cdots + 74484 \) Copy content Toggle raw display
$73$ \( T^{6} + 16 T^{5} + \cdots - 65851 \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + \cdots - 237196 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{5} + \cdots - 6004 \) Copy content Toggle raw display
$89$ \( T^{6} - 34 T^{5} + \cdots - 7984 \) Copy content Toggle raw display
$97$ \( T^{6} - 9 T^{5} + \cdots + 349009 \) Copy content Toggle raw display
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