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Properties

Label 500.2.l.f
Level $500$
Weight $2$
Character orbit 500.l
Analytic conductor $3.993$
Analytic rank $0$
Dimension $96$
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(7,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.l (of order \(20\), degree \(8\), 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: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 10 q^{2} - 10 q^{4} - 6 q^{6} + 10 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 10 q^{2} - 10 q^{4} - 6 q^{6} + 10 q^{8} - 20 q^{9} + 10 q^{12} + 20 q^{13} - 10 q^{14} - 14 q^{16} + 20 q^{17} - 20 q^{18} - 12 q^{21} + 10 q^{22} - 12 q^{26} + 10 q^{28} - 20 q^{29} + 50 q^{32} + 20 q^{33} - 60 q^{34} - 10 q^{36} - 40 q^{37} - 20 q^{38} - 28 q^{41} - 90 q^{42} + 60 q^{44} - 6 q^{46} - 120 q^{48} - 80 q^{52} + 40 q^{53} + 120 q^{54} - 6 q^{56} + 20 q^{57} - 60 q^{58} + 12 q^{61} - 40 q^{62} + 20 q^{64} - 30 q^{66} + 10 q^{68} - 20 q^{69} + 40 q^{72} + 20 q^{73} + 20 q^{77} - 20 q^{78} - 36 q^{81} + 50 q^{82} - 90 q^{84} - 6 q^{86} + 130 q^{88} + 160 q^{89} + 110 q^{92} - 60 q^{93} - 170 q^{94} + 14 q^{96} - 180 q^{97} + 130 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.38120 + 0.303793i 0.372429 + 2.35142i 1.81542 − 0.839197i 0 −1.22875 − 3.13464i 2.24334 − 2.24334i −2.25251 + 1.71061i −2.53732 + 0.824426i 0
7.2 −1.29270 − 0.573516i −0.0944343 − 0.596234i 1.34216 + 1.48277i 0 −0.219875 + 0.824913i 0.340930 − 0.340930i −0.884618 − 2.68653i 2.50659 − 0.814441i 0
7.3 −1.05762 + 0.938848i −0.372429 − 2.35142i 0.237129 − 1.98589i 0 2.60152 + 2.13726i −2.24334 + 2.24334i 1.61366 + 2.32295i −2.53732 + 0.824426i 0
7.4 −0.808309 − 1.16045i 0.282384 + 1.78290i −0.693273 + 1.87600i 0 1.84071 − 1.76883i 0.0608994 − 0.0608994i 2.73738 − 0.711881i −0.245830 + 0.0798750i 0
7.5 −0.295847 + 1.38292i 0.0944343 + 0.596234i −1.82495 − 0.818267i 0 −0.852484 − 0.0457989i −0.340930 + 0.340930i 1.67151 − 2.28168i 2.50659 − 0.814441i 0
7.6 −0.106562 − 1.41019i −0.458400 − 2.89422i −1.97729 + 0.300545i 0 −4.03257 + 0.954846i −1.61403 + 1.61403i 0.634530 + 2.75633i −5.31324 + 1.72637i 0
7.7 0.381403 − 1.36181i 0.427120 + 2.69673i −1.70906 − 1.03880i 0 3.83535 + 0.446882i −2.39694 + 2.39694i −2.06649 + 1.93122i −4.23676 + 1.37661i 0
7.8 0.463709 + 1.33603i −0.282384 − 1.78290i −1.56995 + 1.23906i 0 2.25107 − 1.20402i −0.0608994 + 0.0608994i −2.38342 − 1.52293i −0.245830 + 0.0798750i 0
7.9 0.626683 − 1.26778i −0.118091 − 0.745599i −1.21454 − 1.58899i 0 −1.01926 − 0.317541i 2.75590 − 2.75590i −2.77563 + 0.543971i 2.31120 − 0.750953i 0
7.10 1.07823 + 0.915101i 0.458400 + 2.89422i 0.325181 + 1.97339i 0 −2.15424 + 3.54014i 1.61403 − 1.61403i −1.45523 + 2.42535i −5.31324 + 1.72637i 0
7.11 1.32591 + 0.491892i −0.427120 − 2.69673i 1.51608 + 1.30441i 0 0.760176 − 3.78572i 2.39694 − 2.39694i 1.36857 + 2.47528i −4.23676 + 1.37661i 0
7.12 1.39401 + 0.238186i 0.118091 + 0.745599i 1.88654 + 0.664067i 0 −0.0129704 + 1.06750i −2.75590 + 2.75590i 2.47168 + 1.37506i 2.31120 − 0.750953i 0
43.1 −1.08354 − 0.908815i −1.31518 − 0.670120i 0.348110 + 1.96947i 0 0.816037 + 1.92136i 2.85931 + 2.85931i 1.41269 − 2.45037i −0.482708 − 0.664390i 0
43.2 −0.882822 − 1.10482i 1.00304 + 0.511075i −0.441252 + 1.95072i 0 −0.320861 − 1.55937i −0.797631 − 0.797631i 2.54474 − 1.23463i −1.01846 − 1.40179i 0
43.3 −0.749667 + 1.19917i 1.31518 + 0.670120i −0.875999 − 1.79795i 0 −1.78953 + 1.07476i −2.85931 − 2.85931i 2.81275 + 0.297395i −0.482708 − 0.664390i 0
43.4 −0.498205 + 1.32355i −1.00304 − 0.511075i −1.50358 − 1.31880i 0 1.17616 − 1.07296i 0.797631 + 0.797631i 2.49460 − 1.33304i −1.01846 − 1.40179i 0
43.5 0.124137 − 1.40875i −1.47876 − 0.753467i −1.96918 − 0.349758i 0 −1.24502 + 1.98968i −2.36078 − 2.36078i −0.737172 + 2.73067i −0.144331 − 0.198654i 0
43.6 0.153683 − 1.40584i 2.37743 + 1.21136i −1.95276 − 0.432108i 0 2.06835 − 3.15612i 1.01154 + 1.01154i −0.907581 + 2.67886i 2.42143 + 3.33282i 0
43.7 0.553391 + 1.30144i 1.47876 + 0.753467i −1.38752 + 1.44041i 0 −0.162262 + 2.34149i 2.36078 + 2.36078i −2.64246 − 1.00867i −0.144331 − 0.198654i 0
43.8 0.580589 + 1.28954i −2.37743 − 1.21136i −1.32583 + 1.49739i 0 0.181789 − 3.76910i −1.01154 − 1.01154i −2.70071 − 0.840347i 2.42143 + 3.33282i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.l.f 96
4.b odd 2 1 inner 500.2.l.f 96
5.b even 2 1 100.2.l.b ✓ 96
5.c odd 4 1 500.2.l.d 96
5.c odd 4 1 500.2.l.e 96
15.d odd 2 1 900.2.bj.d 96
20.d odd 2 1 100.2.l.b ✓ 96
20.e even 4 1 500.2.l.d 96
20.e even 4 1 500.2.l.e 96
25.d even 5 1 500.2.l.d 96
25.e even 10 1 500.2.l.e 96
25.f odd 20 1 100.2.l.b ✓ 96
25.f odd 20 1 inner 500.2.l.f 96
60.h even 2 1 900.2.bj.d 96
75.l even 20 1 900.2.bj.d 96
100.h odd 10 1 500.2.l.e 96
100.j odd 10 1 500.2.l.d 96
100.l even 20 1 100.2.l.b ✓ 96
100.l even 20 1 inner 500.2.l.f 96
300.u odd 20 1 900.2.bj.d 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.l.b ✓ 96 5.b even 2 1
100.2.l.b ✓ 96 20.d odd 2 1
100.2.l.b ✓ 96 25.f odd 20 1
100.2.l.b ✓ 96 100.l even 20 1
500.2.l.d 96 5.c odd 4 1
500.2.l.d 96 20.e even 4 1
500.2.l.d 96 25.d even 5 1
500.2.l.d 96 100.j odd 10 1
500.2.l.e 96 5.c odd 4 1
500.2.l.e 96 20.e even 4 1
500.2.l.e 96 25.e even 10 1
500.2.l.e 96 100.h odd 10 1
500.2.l.f 96 1.a even 1 1 trivial
500.2.l.f 96 4.b odd 2 1 inner
500.2.l.f 96 25.f odd 20 1 inner
500.2.l.f 96 100.l even 20 1 inner
900.2.bj.d 96 15.d odd 2 1
900.2.bj.d 96 60.h even 2 1
900.2.bj.d 96 75.l even 20 1
900.2.bj.d 96 300.u odd 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\):

\( T_{3}^{96} + 10 T_{3}^{94} - 103 T_{3}^{92} - 1600 T_{3}^{90} + 7218 T_{3}^{88} + 167890 T_{3}^{86} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{48} - 10 T_{13}^{47} + 55 T_{13}^{46} - 190 T_{13}^{45} - 218 T_{13}^{44} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display