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Properties

Label 500.4.g.a
Level $500$
Weight $4$
Character orbit 500.g
Analytic conductor $29.501$
Analytic rank $0$
Dimension $28$
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] = [500,4,Mod(101,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.101");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.g (of order \(5\), degree \(4\), not minimal)

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: no
Analytic conductor: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{3} - 16 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{3} - 16 q^{7} - 13 q^{9} - 20 q^{11} + 86 q^{13} + 178 q^{17} + 2 q^{19} + 108 q^{21} - 102 q^{23} - 92 q^{27} + 192 q^{29} + 378 q^{31} - 560 q^{33} + 399 q^{37} + 592 q^{39} + 298 q^{41} + 180 q^{43} + 78 q^{47} + 144 q^{49} - 1664 q^{51} + 657 q^{53} - 384 q^{57} + 144 q^{59} + 516 q^{61} - 584 q^{63} + 134 q^{67} + 2996 q^{69} - 2026 q^{71} + 1346 q^{73} - 2320 q^{77} + 896 q^{79} - 2203 q^{81} - 2082 q^{83} + 4316 q^{87} - 167 q^{89} + 2212 q^{91} + 5664 q^{93} - 1156 q^{97} - 2100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −8.15261 − 5.92322i 0 0 0 −17.4845 0 23.0371 + 70.9009i 0
101.2 0 −4.29280 − 3.11890i 0 0 0 −5.90251 0 0.357125 + 1.09912i 0
101.3 0 −1.31881 − 0.958173i 0 0 0 34.0141 0 −7.52229 − 23.1512i 0
101.4 0 −0.858402 − 0.623665i 0 0 0 −3.00616 0 −7.99556 − 24.6078i 0
101.5 0 4.65969 + 3.38546i 0 0 0 −12.5176 0 1.90790 + 5.87192i 0
101.6 0 4.80168 + 3.48863i 0 0 0 −21.1171 0 2.54218 + 7.82402i 0
101.7 0 6.16125 + 4.47641i 0 0 0 17.5417 0 9.57931 + 29.4821i 0
201.1 0 −2.25129 − 6.92876i 0 0 0 6.47483 0 −21.0959 + 15.3271i 0
201.2 0 −1.81740 − 5.59338i 0 0 0 −11.1886 0 −6.13952 + 4.46063i 0
201.3 0 −0.696421 − 2.14336i 0 0 0 29.6874 0 17.7345 − 12.8848i 0
201.4 0 −0.0294384 − 0.0906021i 0 0 0 −26.6009 0 21.8361 − 15.8649i 0
201.5 0 1.03984 + 3.20031i 0 0 0 0.208499 0 12.6828 − 9.21457i 0
201.6 0 2.09563 + 6.44968i 0 0 0 19.6435 0 −15.3633 + 11.1621i 0
201.7 0 2.65908 + 8.18380i 0 0 0 −17.7526 0 −38.0604 + 27.6525i 0
301.1 0 −2.25129 + 6.92876i 0 0 0 6.47483 0 −21.0959 − 15.3271i 0
301.2 0 −1.81740 + 5.59338i 0 0 0 −11.1886 0 −6.13952 − 4.46063i 0
301.3 0 −0.696421 + 2.14336i 0 0 0 29.6874 0 17.7345 + 12.8848i 0
301.4 0 −0.0294384 + 0.0906021i 0 0 0 −26.6009 0 21.8361 + 15.8649i 0
301.5 0 1.03984 − 3.20031i 0 0 0 0.208499 0 12.6828 + 9.21457i 0
301.6 0 2.09563 − 6.44968i 0 0 0 19.6435 0 −15.3633 − 11.1621i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

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