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Properties

Label 2500.2.a.f
Level $2500$
Weight $2$
Character orbit 2500.a
Self dual yes
Analytic conductor $19.963$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2500,2,Mod(1,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2500.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: \(19.9626005053\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{6} - \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{7} + 2 \beta_{5}) q^{13} + ( - \beta_{7} + 2 \beta_{6} + \cdots - \beta_1) q^{17}+ \cdots + (4 \beta_{4} - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{9} - 10 q^{11} - 12 q^{19} - 22 q^{21} - 32 q^{29} - 2 q^{31} - 2 q^{39} - 42 q^{41} - 14 q^{49} - 14 q^{51} - 24 q^{59} - 34 q^{61} - 36 q^{69} + 4 q^{71} + 4 q^{79} - 28 q^{81} - 58 q^{89} - 18 q^{91} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1 \) 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
−2.30927
−2.08529
−1.13370
−0.183172
0.183172
1.13370
2.08529
2.30927
0 −2.30927 0 0 0 4.21139 0 2.33275 0
1.2 0 −2.08529 0 0 0 0.909715 0 1.34841 0
1.3 0 −1.13370 0 0 0 −0.768409 0 −1.71472 0
1.4 0 −0.183172 0 0 0 1.35874 0 −2.96645 0
1.5 0 0.183172 0 0 0 −1.35874 0 −2.96645 0
1.6 0 1.13370 0 0 0 0.768409 0 −1.71472 0
1.7 0 2.08529 0 0 0 −0.909715 0 1.34841 0
1.8 0 2.30927 0 0 0 −4.21139 0 2.33275 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.2.a.f 8
4.b odd 2 1 10000.2.a.bi 8
5.b even 2 1 inner 2500.2.a.f 8
5.c odd 4 2 2500.2.c.b 8
20.d odd 2 1 10000.2.a.bi 8
25.d even 5 2 500.2.g.b 16
25.e even 10 2 500.2.g.b 16
25.f odd 20 2 100.2.i.a ✓ 8
25.f odd 20 2 500.2.i.a 8
75.l even 20 2 900.2.w.a 8
100.l even 20 2 400.2.y.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.i.a ✓ 8 25.f odd 20 2
400.2.y.b 8 100.l even 20 2
500.2.g.b 16 25.d even 5 2
500.2.g.b 16 25.e even 10 2
500.2.i.a 8 25.f odd 20 2
900.2.w.a 8 75.l even 20 2
2500.2.a.f 8 1.a even 1 1 trivial
2500.2.a.f 8 5.b even 2 1 inner
2500.2.c.b 8 5.c odd 4 2
10000.2.a.bi 8 4.b odd 2 1
10000.2.a.bi 8 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 11T_{3}^{6} + 36T_{3}^{4} - 31T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 5 T^{3} - 5 T^{2} + \cdots - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 46 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} - 89 T^{6} + \cdots + 190096 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 39 T^{2} + \cdots + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 119 T^{6} + \cdots + 7921 \) Copy content Toggle raw display
$29$ \( (T^{4} + 16 T^{3} + 61 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} - 84 T^{2} + \cdots - 79)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 151 T^{6} + \cdots + 57121 \) Copy content Toggle raw display
$41$ \( (T^{4} + 21 T^{3} + \cdots + 316)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 125 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$47$ \( T^{8} - 234 T^{6} + \cdots + 73441 \) Copy content Toggle raw display
$53$ \( T^{8} - 199 T^{6} + \cdots + 116281 \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + \cdots - 409)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 17 T^{3} + \cdots + 61)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 281 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} - 191 T^{2} + \cdots - 29)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 511 T^{6} + \cdots + 20693401 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots - 3529)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 259 T^{6} + \cdots + 2627641 \) Copy content Toggle raw display
$89$ \( (T^{4} + 29 T^{3} + \cdots - 14324)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 506 T^{6} + \cdots + 9554281 \) Copy content Toggle raw display
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