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Properties

Label 500.4.i.b
Level $500$
Weight $4$
Character orbit 500.i
Analytic conductor $29.501$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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] = [500,4,Mod(49,500)] 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("500.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(500, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.i (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 26 q^{9} - 40 q^{11} - 4 q^{19} + 216 q^{21} - 384 q^{29} + 756 q^{31} - 1184 q^{39} + 596 q^{41} - 288 q^{49} - 3328 q^{51} - 288 q^{59} + 1032 q^{61} - 5992 q^{69} - 4052 q^{71} - 1792 q^{79} - 4406 q^{81}+ \cdots + 4200 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −8.18380 + 2.65908i 0 0 0 17.7526i 0 38.0604 − 27.6525i 0
49.2 0 −6.92876 + 2.25129i 0 0 0 6.47483i 0 21.0959 − 15.3271i 0
49.3 0 −6.44968 + 2.09563i 0 0 0 − 19.6435i 0 15.3633 − 11.1621i 0
49.4 0 −5.59338 + 1.81740i 0 0 0 − 11.1886i 0 6.13952 − 4.46063i 0
49.5 0 −3.20031 + 1.03984i 0 0 0 − 0.208499i 0 −12.6828 + 9.21457i 0
49.6 0 −2.14336 + 0.696421i 0 0 0 29.6874i 0 −17.7345 + 12.8848i 0
49.7 0 −0.0906021 + 0.0294384i 0 0 0 − 26.6009i 0 −21.8361 + 15.8649i 0
49.8 0 0.0906021 − 0.0294384i 0 0 0 26.6009i 0 −21.8361 + 15.8649i 0
49.9 0 2.14336 − 0.696421i 0 0 0 − 29.6874i 0 −17.7345 + 12.8848i 0
49.10 0 3.20031 − 1.03984i 0 0 0 0.208499i 0 −12.6828 + 9.21457i 0
49.11 0 5.59338 − 1.81740i 0 0 0 11.1886i 0 6.13952 − 4.46063i 0
49.12 0 6.44968 − 2.09563i 0 0 0 19.6435i 0 15.3633 − 11.1621i 0
49.13 0 6.92876 − 2.25129i 0 0 0 − 6.47483i 0 21.0959 − 15.3271i 0
49.14 0 8.18380 − 2.65908i 0 0 0 − 17.7526i 0 38.0604 − 27.6525i 0
149.1 0 −5.92322 + 8.15261i 0 0 0 − 17.4845i 0 −23.0371 − 70.9009i 0
149.2 0 −4.47641 + 6.16125i 0 0 0 − 17.5417i 0 −9.57931 − 29.4821i 0
149.3 0 −3.48863 + 4.80168i 0 0 0 21.1171i 0 −2.54218 − 7.82402i 0
149.4 0 −3.38546 + 4.65969i 0 0 0 12.5176i 0 −1.90790 − 5.87192i 0
149.5 0 −3.11890 + 4.29280i 0 0 0 − 5.90251i 0 −0.357125 − 1.09912i 0
149.6 0 −0.958173 + 1.31881i 0 0 0 34.0141i 0 7.52229 + 23.1512i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.4.i.b 56
5.b even 2 1 inner 500.4.i.b 56
5.c odd 4 1 100.4.g.a ✓ 28
5.c odd 4 1 500.4.g.a 28
25.d even 5 1 inner 500.4.i.b 56
25.e even 10 1 inner 500.4.i.b 56
25.f odd 20 1 100.4.g.a ✓ 28
25.f odd 20 1 500.4.g.a 28
25.f odd 20 1 2500.4.a.c 14
25.f odd 20 1 2500.4.a.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.g.a ✓ 28 5.c odd 4 1
100.4.g.a ✓ 28 25.f odd 20 1
500.4.g.a 28 5.c odd 4 1
500.4.g.a 28 25.f odd 20 1
500.4.i.b 56 1.a even 1 1 trivial
500.4.i.b 56 5.b even 2 1 inner
500.4.i.b 56 25.d even 5 1 inner
500.4.i.b 56 25.e even 10 1 inner
2500.4.a.c 14 25.f odd 20 1
2500.4.a.d 14 25.f odd 20 1