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Properties

Label 500.2.r.b
Level $500$
Weight $2$
Character orbit 500.r
Analytic conductor $3.993$
Analytic rank $0$
Dimension $2880$
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(3,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(100))
 
chi = DirichletCharacter(H, H._module([50, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.r (of order \(100\), degree \(40\), 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: \(2880\)
Relative dimension: \(72\) over \(\Q(\zeta_{100})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{100}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2880 q - 40 q^{2} - 40 q^{4} - 80 q^{5} - 40 q^{6} - 40 q^{8} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2880 q - 40 q^{2} - 40 q^{4} - 80 q^{5} - 40 q^{6} - 40 q^{8} - 80 q^{9} - 40 q^{10} - 40 q^{12} - 80 q^{13} - 40 q^{14} - 40 q^{16} - 80 q^{17} - 70 q^{18} - 60 q^{20} - 80 q^{21} - 40 q^{22} - 50 q^{24} - 80 q^{25} - 30 q^{26} - 40 q^{28} - 80 q^{29} - 40 q^{30} - 80 q^{33} + 10 q^{34} - 40 q^{36} - 140 q^{37} - 70 q^{38} - 90 q^{40} - 80 q^{41} - 140 q^{42} - 110 q^{44} - 80 q^{45} - 40 q^{46} - 170 q^{48} - 100 q^{49} - 130 q^{50} - 130 q^{52} - 60 q^{53} - 170 q^{54} - 40 q^{56} - 80 q^{57} - 110 q^{58} - 140 q^{60} - 80 q^{61} - 90 q^{62} - 70 q^{64} - 40 q^{66} - 40 q^{68} - 80 q^{69} - 40 q^{70} - 10 q^{72} - 80 q^{73} - 50 q^{74} - 30 q^{76} - 80 q^{77} - 70 q^{78} - 40 q^{80} - 80 q^{81} + 40 q^{84} - 200 q^{85} - 40 q^{86} + 80 q^{88} - 260 q^{89} + 110 q^{90} + 60 q^{92} - 160 q^{93} + 120 q^{94} - 40 q^{96} - 280 q^{97} + 80 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.40787 − 0.133844i −1.26205 + 0.0396616i 1.96417 + 0.376869i −1.24974 − 1.85422i 1.78211 + 0.113080i −0.427374 − 2.69833i −2.71485 − 0.793474i −1.40288 + 0.0882615i 1.51129 + 2.77777i
3.2 −1.38816 + 0.270196i 1.38481 − 0.0435194i 1.85399 − 0.750153i 1.38951 − 1.75193i −1.91058 + 0.434583i −0.0362075 − 0.228605i −2.37095 + 1.54227i −1.07827 + 0.0678391i −1.45550 + 2.80740i
3.3 −1.38189 − 0.300645i 2.57363 − 0.0808795i 1.81922 + 0.830916i 1.81547 + 1.30540i −3.58078 − 0.661983i 0.164448 + 1.03828i −2.26415 − 1.69517i 3.62293 − 0.227935i −2.11631 − 2.34973i
3.4 −1.37838 − 0.316347i −2.57363 + 0.0808795i 1.79985 + 0.872091i 1.81547 + 1.30540i 3.57301 + 0.702676i −0.164448 − 1.03828i −2.20499 − 1.77145i 3.62293 − 0.227935i −2.08945 − 2.37365i
3.5 −1.37808 + 0.317638i −2.46610 + 0.0775002i 1.79821 − 0.875462i 0.852914 − 2.06701i 3.37386 − 0.890127i 0.366323 + 2.31287i −2.20000 + 1.77764i 3.08154 − 0.193874i −0.518822 + 3.11943i
3.6 −1.37460 + 0.332365i 2.93713 − 0.0923029i 1.77907 − 0.913740i −2.07132 − 0.842393i −4.00671 + 1.10308i −0.763860 − 4.82282i −2.14182 + 1.84733i 5.62412 − 0.353839i 3.12723 + 0.469520i
3.7 −1.37371 + 0.336022i 2.40409 − 0.0755516i 1.77418 − 0.923196i −1.91270 + 1.15826i −3.27715 + 0.911614i 0.667937 + 4.21719i −2.12700 + 1.86437i 2.77987 − 0.174895i 2.23831 − 2.23383i
3.8 −1.35622 + 0.400833i −0.334582 + 0.0105147i 1.67867 − 1.08724i −0.309000 + 2.21461i 0.449553 − 0.148372i −0.349033 − 2.20371i −1.84084 + 2.14740i −2.88225 + 0.181336i −0.468619 − 3.12736i
3.9 −1.33086 − 0.478334i 1.26205 − 0.0396616i 1.54239 + 1.27319i −1.24974 − 1.85422i −1.69859 − 0.550899i 0.427374 + 2.69833i −1.44370 − 2.43223i −1.40288 + 0.0882615i 0.776299 + 3.06551i
3.10 −1.33014 + 0.480341i 0.383911 − 0.0120649i 1.53855 − 1.27784i 2.09380 + 0.784854i −0.504860 + 0.200456i 0.0189128 + 0.119411i −1.43268 + 2.43873i −2.84684 + 0.179108i −3.16205 − 0.0382278i
3.11 −1.21979 + 0.715612i −3.32560 + 0.104511i 0.975799 − 1.74580i −1.53924 + 1.62196i 3.98176 − 2.50732i −0.134723 − 0.850605i 0.0590398 + 2.82781i 8.05459 − 0.506752i 0.716861 − 3.07995i
3.12 −1.14100 − 0.835532i −1.38481 + 0.0435194i 0.603773 + 1.90669i 1.38951 − 1.75193i 1.61644 + 1.10740i 0.0362075 + 0.228605i 0.904191 − 2.68001i −1.07827 + 0.0678391i −3.04923 + 0.837978i
3.13 −1.11168 − 0.874166i 2.46610 − 0.0775002i 0.471669 + 1.94359i 0.852914 − 2.06701i −2.80926 − 2.06962i −0.366323 − 2.31287i 1.17467 − 2.57296i 3.08154 − 0.193874i −2.75508 + 1.55227i
3.14 −1.10226 − 0.886010i −2.93713 + 0.0923029i 0.429971 + 1.95323i −2.07132 − 0.842393i 3.31927 + 2.50058i 0.763860 + 4.82282i 1.25664 − 2.53394i 5.62412 − 0.353839i 1.53677 + 2.76375i
3.15 −1.09990 − 0.888941i −2.40409 + 0.0755516i 0.419569 + 1.95550i −1.91270 + 1.15826i 2.71143 + 2.05400i −0.667937 − 4.21719i 1.27683 − 2.52383i 2.77987 − 0.174895i 3.13341 + 0.426307i
3.16 −1.06519 + 0.930253i −0.931938 + 0.0292873i 0.269258 − 1.98179i −2.08752 − 0.801409i 0.965446 − 0.898135i 0.370793 + 2.34109i 1.55676 + 2.36146i −2.12643 + 0.133784i 2.96912 − 1.08827i
3.17 −1.05648 − 0.940135i 0.334582 − 0.0105147i 0.232292 + 1.98646i −0.309000 + 2.21461i −0.363364 − 0.303444i 0.349033 + 2.20371i 1.62213 − 2.31704i −2.88225 + 0.181336i 2.40849 − 2.04919i
3.18 −1.04117 + 0.957063i −1.05768 + 0.0332390i 0.168061 − 1.99293i 1.80936 + 1.31385i 1.06941 − 1.04687i 0.711041 + 4.48934i 1.73238 + 2.23582i −1.87650 + 0.118059i −3.14129 + 0.363734i
3.19 −0.999027 − 1.00097i −0.383911 + 0.0120649i −0.00388824 + 2.00000i 2.09380 + 0.784854i 0.395614 + 0.372231i −0.0189128 − 0.119411i 2.00582 − 1.99416i −2.84684 + 0.179108i −1.30615 − 2.87993i
3.20 −0.995379 + 1.00460i −1.65112 + 0.0518885i −0.0184429 − 1.99991i −0.303175 − 2.21542i 1.59136 − 1.71036i −0.819982 − 5.17717i 2.02747 + 1.97214i −0.270583 + 0.0170236i 2.52738 + 1.90061i
See next 80 embeddings (of 2880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
125.i odd 100 1 inner
500.r even 100 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.r.b ✓ 2880
4.b odd 2 1 inner 500.2.r.b ✓ 2880
125.i odd 100 1 inner 500.2.r.b ✓ 2880
500.r even 100 1 inner 500.2.r.b ✓ 2880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.2.r.b ✓ 2880 1.a even 1 1 trivial
500.2.r.b ✓ 2880 4.b odd 2 1 inner
500.2.r.b ✓ 2880 125.i odd 100 1 inner
500.2.r.b ✓ 2880 500.r even 100 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2880} + 40 T_{3}^{2878} + 820 T_{3}^{2876} + 11720 T_{3}^{2874} + 133260 T_{3}^{2872} + \cdots + 13\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display