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Properties

Label 500.2.m.a
Level $500$
Weight $2$
Character orbit 500.m
Analytic conductor $3.993$
Analytic rank $0$
Dimension $260$
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] = [500,2,Mod(21,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([0, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.m (of order \(25\), degree \(20\), 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: \(260\)
Relative dimension: \(13\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 260 q + 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 260 q + 5 q^{5} + 5 q^{11} - 20 q^{15} + 5 q^{17} + 10 q^{19} + 50 q^{23} - 25 q^{25} + 30 q^{27} + 10 q^{29} - 10 q^{31} + 35 q^{33} + 5 q^{35} - 15 q^{37} - 20 q^{39} + 15 q^{41} - 40 q^{43} + 80 q^{45} - 120 q^{47} - 85 q^{49} - 20 q^{51} - 15 q^{53} - 5 q^{55} - 80 q^{57} - 15 q^{59} + 20 q^{61} + 30 q^{63} + 20 q^{65} + 35 q^{67} + 60 q^{69} + 95 q^{71} - 20 q^{73} + 75 q^{75} + 60 q^{77} + 20 q^{79} - 5 q^{81} + 40 q^{83} + 10 q^{85} - 45 q^{87} + 10 q^{89} - 120 q^{91} + 70 q^{93} - 60 q^{95} + 5 q^{97} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 0 −0.571533 − 2.99608i 0 1.63304 + 1.52747i 0 −2.53423 − 1.84122i 0 −5.86052 + 2.32034i 0
21.2 0 −0.474947 − 2.48976i 0 −0.857999 + 2.06491i 0 2.47990 + 1.80175i 0 −3.18401 + 1.26064i 0
21.3 0 −0.467991 − 2.45329i 0 0.402131 − 2.19961i 0 −3.05752 − 2.22142i 0 −3.01030 + 1.19186i 0
21.4 0 −0.359937 − 1.88685i 0 −2.23289 + 0.119198i 0 0.357704 + 0.259887i 0 −0.641336 + 0.253923i 0
21.5 0 −0.289135 − 1.51570i 0 −0.615897 − 2.14957i 0 1.96345 + 1.42653i 0 0.575585 − 0.227890i 0
21.6 0 −0.0733308 − 0.384414i 0 2.15090 − 0.611264i 0 1.73779 + 1.26258i 0 2.64693 − 1.04799i 0
21.7 0 −0.0176213 − 0.0923742i 0 0.221900 + 2.22503i 0 −0.792758 − 0.575972i 0 2.78111 − 1.10112i 0
21.8 0 0.0273523 + 0.143386i 0 −1.79514 + 1.33321i 0 −3.34413 − 2.42965i 0 2.76952 − 1.09653i 0
21.9 0 0.214732 + 1.12567i 0 −1.62977 − 1.53096i 0 −0.653093 − 0.474500i 0 1.56832 − 0.620940i 0
21.10 0 0.247936 + 1.29973i 0 2.13511 − 0.664310i 0 −3.88410 − 2.82196i 0 1.16151 − 0.459874i 0
21.11 0 0.360596 + 1.89031i 0 0.902849 + 2.04569i 0 2.55780 + 1.85835i 0 −0.653918 + 0.258904i 0
21.12 0 0.549753 + 2.88191i 0 −2.03560 + 0.925385i 0 −0.451015 − 0.327681i 0 −5.21382 + 2.06430i 0
21.13 0 0.615242 + 3.22521i 0 1.37442 − 1.76379i 0 1.75680 + 1.27639i 0 −7.23415 + 2.86420i 0
41.1 0 −3.09246 − 0.794009i 0 2.23606 − 0.00447694i 0 −0.805918 − 2.48036i 0 6.30395 + 3.46563i 0
41.2 0 −2.16375 − 0.555557i 0 −0.0948148 − 2.23406i 0 0.464166 + 1.42856i 0 1.74426 + 0.958915i 0
41.3 0 −1.82968 − 0.469782i 0 −2.11225 − 0.733753i 0 0.379793 + 1.16888i 0 0.498107 + 0.273837i 0
41.4 0 −1.53683 − 0.394590i 0 −0.236557 + 2.22352i 0 −0.411791 − 1.26736i 0 −0.422782 − 0.232426i 0
41.5 0 −1.33567 − 0.342941i 0 1.49616 + 1.66178i 0 0.891131 + 2.74262i 0 −0.962521 − 0.529150i 0
41.6 0 0.0263610 + 0.00676836i 0 −1.71251 + 1.43781i 0 −0.640731 − 1.97197i 0 −2.62827 − 1.44490i 0
41.7 0 0.332246 + 0.0853062i 0 2.19743 − 0.413906i 0 −1.11255 − 3.42408i 0 −2.52581 − 1.38858i 0
See next 80 embeddings (of 260 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.g even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.m.a ✓ 260
125.g even 25 1 inner 500.2.m.a ✓ 260
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.2.m.a ✓ 260 1.a even 1 1 trivial
500.2.m.a ✓ 260 125.g even 25 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(500, [\chi])\).