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Properties

Label 500.2.i.b
Level $500$
Weight $2$
Character orbit 500.i
Analytic conductor $3.993$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,2,Mod(49,500)]
 
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, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.i (of order \(10\), 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: \(3.99252010106\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{9} - 10 q^{11} + 16 q^{19} + 4 q^{21} + 36 q^{29} + 24 q^{31} - 44 q^{39} - 46 q^{41} - 68 q^{49} + 28 q^{51} - 18 q^{59} - 52 q^{61} + 88 q^{69} + 42 q^{71} + 48 q^{79} + 36 q^{81} + 4 q^{89} - 64 q^{91} - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −3.15031 + 1.02360i 0 0 0 − 1.69984i 0 6.44966 − 4.68596i 0
49.2 0 −1.46952 + 0.477475i 0 0 0 − 2.43270i 0 −0.495552 + 0.360039i 0
49.3 0 −0.141953 + 0.0461234i 0 0 0 3.58696i 0 −2.40903 + 1.75026i 0
49.4 0 0.141953 − 0.0461234i 0 0 0 − 3.58696i 0 −2.40903 + 1.75026i 0
49.5 0 1.46952 − 0.477475i 0 0 0 2.43270i 0 −0.495552 + 0.360039i 0
49.6 0 3.15031 − 1.02360i 0 0 0 1.69984i 0 6.44966 − 4.68596i 0
149.1 0 −1.63875 + 2.25555i 0 0 0 − 2.70809i 0 −1.47494 − 4.53940i 0
149.2 0 −1.59529 + 2.19573i 0 0 0 4.40288i 0 −1.34923 − 4.15250i 0
149.3 0 −0.406731 + 0.559818i 0 0 0 − 3.25686i 0 0.779086 + 2.39778i 0
149.4 0 0.406731 − 0.559818i 0 0 0 3.25686i 0 0.779086 + 2.39778i 0
149.5 0 1.59529 − 2.19573i 0 0 0 − 4.40288i 0 −1.34923 − 4.15250i 0
149.6 0 1.63875 − 2.25555i 0 0 0 2.70809i 0 −1.47494 − 4.53940i 0
349.1 0 −1.63875 − 2.25555i 0 0 0 2.70809i 0 −1.47494 + 4.53940i 0
349.2 0 −1.59529 − 2.19573i 0 0 0 − 4.40288i 0 −1.34923 + 4.15250i 0
349.3 0 −0.406731 − 0.559818i 0 0 0 3.25686i 0 0.779086 − 2.39778i 0
349.4 0 0.406731 + 0.559818i 0 0 0 − 3.25686i 0 0.779086 − 2.39778i 0
349.5 0 1.59529 + 2.19573i 0 0 0 4.40288i 0 −1.34923 + 4.15250i 0
349.6 0 1.63875 + 2.25555i 0 0 0 − 2.70809i 0 −1.47494 + 4.53940i 0
449.1 0 −3.15031 − 1.02360i 0 0 0 1.69984i 0 6.44966 + 4.68596i 0
449.2 0 −1.46952 − 0.477475i 0 0 0 2.43270i 0 −0.495552 − 0.360039i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

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 500.2.i.b 24
5.b even 2 1 inner 500.2.i.b 24
5.c odd 4 1 100.2.g.a ✓ 12
5.c odd 4 1 500.2.g.a 12
15.e even 4 1 900.2.n.c 12
20.e even 4 1 400.2.u.f 12
25.d even 5 1 inner 500.2.i.b 24
25.d even 5 1 2500.2.c.c 12
25.e even 10 1 inner 500.2.i.b 24
25.e even 10 1 2500.2.c.c 12
25.f odd 20 1 100.2.g.a ✓ 12
25.f odd 20 1 500.2.g.a 12
25.f odd 20 1 2500.2.a.c 6
25.f odd 20 1 2500.2.a.d 6
75.l even 20 1 900.2.n.c 12
100.l even 20 1 400.2.u.f 12
100.l even 20 1 10000.2.a.bc 6
100.l even 20 1 10000.2.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a ✓ 12 5.c odd 4 1
100.2.g.a ✓ 12 25.f odd 20 1
400.2.u.f 12 20.e even 4 1
400.2.u.f 12 100.l even 20 1
500.2.g.a 12 5.c odd 4 1
500.2.g.a 12 25.f odd 20 1
500.2.i.b 24 1.a even 1 1 trivial
500.2.i.b 24 5.b even 2 1 inner
500.2.i.b 24 25.d even 5 1 inner
500.2.i.b 24 25.e even 10 1 inner
900.2.n.c 12 15.e even 4 1
900.2.n.c 12 75.l even 20 1
2500.2.a.c 6 25.f odd 20 1
2500.2.a.d 6 25.f odd 20 1
2500.2.c.c 12 25.d even 5 1
2500.2.c.c 12 25.e even 10 1
10000.2.a.bc 6 100.l even 20 1
10000.2.a.bd 6 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 12 T_{3}^{22} + 126 T_{3}^{20} - 1130 T_{3}^{18} + 13395 T_{3}^{16} - 34182 T_{3}^{14} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display