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Properties

Label 500.3.b.c
Level $500$
Weight $3$
Character orbit 500.b
Analytic conductor $13.624$
Analytic rank $0$
Dimension $24$
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] = [500,3,Mod(251,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.251");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 500.b (of order \(2\), degree \(1\), 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: \(13.6240132180\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 5 q^{2} - 3 q^{4} - 9 q^{6} - 20 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 5 q^{2} - 3 q^{4} - 9 q^{6} - 20 q^{8} - 72 q^{9} + 50 q^{12} + 20 q^{13} - 4 q^{14} + 9 q^{16} + 80 q^{17} + 105 q^{18} - 8 q^{21} - 130 q^{22} - 19 q^{24} + 52 q^{26} - 170 q^{28} + 24 q^{29} + 125 q^{32} - 120 q^{33} - 57 q^{34} - 20 q^{36} - 100 q^{37} + 150 q^{38} - 12 q^{41} - 225 q^{42} + 40 q^{44} - 9 q^{46} - 205 q^{48} - 132 q^{49} + 265 q^{52} + 220 q^{53} + 217 q^{54} + 31 q^{56} + 320 q^{57} + 255 q^{58} + 48 q^{61} - 220 q^{62} + 252 q^{64} - 215 q^{66} - 345 q^{68} - 88 q^{69} + 525 q^{72} - 400 q^{73} + 313 q^{74} - 315 q^{76} - 400 q^{77} + 620 q^{78} + 256 q^{81} - 490 q^{82} + 312 q^{84} - 249 q^{86} - 595 q^{88} - 36 q^{89} + 595 q^{92} + 600 q^{93} - 9 q^{94} + 16 q^{96} + 360 q^{97} + 615 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −1.98396 − 0.252755i 3.08212i 3.87223 + 1.00292i 0 0.779022 − 6.11481i − 9.80821i −7.42887 − 2.96848i −0.499448 0
251.2 −1.98396 + 0.252755i − 3.08212i 3.87223 − 1.00292i 0 0.779022 + 6.11481i 9.80821i −7.42887 + 2.96848i −0.499448 0
251.3 −1.97160 − 0.335828i − 5.63820i 3.77444 + 1.32424i 0 −1.89346 + 11.1163i 5.62801i −6.99698 − 3.87844i −22.7892 0
251.4 −1.97160 + 0.335828i 5.63820i 3.77444 − 1.32424i 0 −1.89346 − 11.1163i − 5.62801i −6.99698 + 3.87844i −22.7892 0
251.5 −1.62698 − 1.16316i 0.628132i 1.29412 + 3.78487i 0 0.730618 − 1.02196i − 5.70922i 2.29689 − 7.66318i 8.60545 0
251.6 −1.62698 + 1.16316i − 0.628132i 1.29412 − 3.78487i 0 0.730618 + 1.02196i 5.70922i 2.29689 + 7.66318i 8.60545 0
251.7 −1.41456 − 1.41387i 2.66132i 0.00195879 + 4.00000i 0 3.76275 − 3.76459i 13.7883i 5.65270 − 5.66101i 1.91740 0
251.8 −1.41456 + 1.41387i − 2.66132i 0.00195879 − 4.00000i 0 3.76275 + 3.76459i − 13.7883i 5.65270 + 5.66101i 1.91740 0
251.9 −1.21298 − 1.59018i − 5.58157i −1.05735 + 3.85772i 0 −8.87571 + 6.77035i − 3.71470i 7.41702 − 2.99798i −22.1540 0
251.10 −1.21298 + 1.59018i 5.58157i −1.05735 − 3.85772i 0 −8.87571 − 6.77035i 3.71470i 7.41702 + 2.99798i −22.1540 0
251.11 −0.600134 − 1.90784i − 1.84848i −3.27968 + 2.28991i 0 −3.52660 + 1.10934i 10.0454i 6.33703 + 4.88283i 5.58311 0
251.12 −0.600134 + 1.90784i 1.84848i −3.27968 − 2.28991i 0 −3.52660 − 1.10934i − 10.0454i 6.33703 − 4.88283i 5.58311 0
251.13 0.167153 − 1.99300i − 1.30398i −3.94412 − 0.666273i 0 −2.59883 − 0.217964i 0.846062i −1.98716 + 7.74927i 7.29965 0
251.14 0.167153 + 1.99300i 1.30398i −3.94412 + 0.666273i 0 −2.59883 + 0.217964i − 0.846062i −1.98716 − 7.74927i 7.29965 0
251.15 0.688660 − 1.87770i 3.78208i −3.05149 − 2.58619i 0 7.10161 + 2.60457i − 1.63471i −6.95753 + 3.94878i −5.30414 0
251.16 0.688660 + 1.87770i − 3.78208i −3.05149 + 2.58619i 0 7.10161 − 2.60457i 1.63471i −6.95753 − 3.94878i −5.30414 0
251.17 0.850426 − 1.81019i 4.40247i −2.55355 − 3.07886i 0 7.96928 + 3.74397i − 5.32191i −7.74492 + 2.00406i −10.3817 0
251.18 0.850426 + 1.81019i − 4.40247i −2.55355 + 3.07886i 0 7.96928 − 3.74397i 5.32191i −7.74492 − 2.00406i −10.3817 0
251.19 0.879336 − 1.79632i − 4.13523i −2.45354 − 3.15914i 0 −7.42820 − 3.63626i − 10.0590i −7.83231 + 1.62939i −8.10014 0
251.20 0.879336 + 1.79632i 4.13523i −2.45354 + 3.15914i 0 −7.42820 + 3.63626i 10.0590i −7.83231 − 1.62939i −8.10014 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.3.b.c ✓ 24
4.b odd 2 1 inner 500.3.b.c ✓ 24
5.b even 2 1 500.3.b.e yes 24
5.c odd 4 2 500.3.d.e 48
20.d odd 2 1 500.3.b.e yes 24
20.e even 4 2 500.3.d.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.3.b.c ✓ 24 1.a even 1 1 trivial
500.3.b.c ✓ 24 4.b odd 2 1 inner
500.3.b.e yes 24 5.b even 2 1
500.3.b.e yes 24 20.d odd 2 1
500.3.d.e 48 5.c odd 4 2
500.3.d.e 48 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(500, [\chi])\):

\( T_{3}^{24} + 144 T_{3}^{22} + 8846 T_{3}^{20} + 304824 T_{3}^{18} + 6524471 T_{3}^{16} + \cdots + 11508998400 \) Copy content Toggle raw display
\( T_{13}^{12} - 10 T_{13}^{11} - 1059 T_{13}^{10} + 11130 T_{13}^{9} + 343526 T_{13}^{8} + \cdots + 282533536125 \) Copy content Toggle raw display