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Properties

Label 501.2.e.a
Level $501$
Weight $2$
Character orbit 501.e
Analytic conductor $4.001$
Analytic rank $0$
Dimension $1148$
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] = [501,2,Mod(4,501)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(501, base_ring=CyclotomicField(166))
 
chi = DirichletCharacter(H, H._module([0, 80]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("501.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 501 = 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 501.e (of order \(83\), degree \(82\), 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: \(4.00050514127\)
Analytic rank: \(0\)
Dimension: \(1148\)
Relative dimension: \(14\) over \(\Q(\zeta_{83})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{83}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1148 q - 2 q^{2} - 14 q^{3} - 20 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1148 q - 2 q^{2} - 14 q^{3} - 20 q^{4} - 4 q^{5} - 2 q^{6} - 6 q^{8} - 14 q^{9} - 22 q^{10} - 10 q^{11} - 20 q^{12} - 14 q^{13} - 26 q^{14} - 4 q^{15} - 48 q^{16} - 18 q^{17} + 81 q^{18} - 16 q^{19} - 42 q^{20} - 26 q^{22} + 61 q^{23} - 6 q^{24} - 32 q^{25} - 52 q^{26} - 14 q^{27} - 38 q^{28} - 30 q^{29} - 22 q^{30} - 12 q^{31} - 64 q^{32} - 10 q^{33} - 42 q^{34} - 40 q^{35} - 20 q^{36} + 130 q^{37} - 80 q^{38} - 14 q^{39} - 100 q^{40} - 36 q^{41} + 140 q^{42} - 52 q^{43} - 82 q^{44} - 4 q^{45} - 80 q^{46} - 58 q^{47} - 48 q^{48} - 66 q^{49} - 92 q^{50} - 18 q^{51} - 56 q^{52} - 54 q^{53} - 2 q^{54} - 96 q^{55} - 114 q^{56} - 16 q^{57} - 42 q^{58} - 52 q^{59} - 42 q^{60} + 118 q^{61} - 80 q^{62} - 136 q^{64} - 76 q^{65} + 140 q^{66} - 68 q^{67} - 178 q^{68} - 22 q^{69} - 156 q^{70} - 98 q^{71} - 6 q^{72} - 48 q^{73} - 110 q^{74} - 32 q^{75} - 120 q^{76} - 72 q^{77} - 52 q^{78} - 102 q^{79} + 688 q^{80} - 14 q^{81} - 180 q^{82} - 112 q^{83} - 38 q^{84} + 558 q^{85} - 106 q^{86} - 30 q^{87} - 184 q^{88} - 64 q^{89} - 22 q^{90} - 80 q^{91} - 104 q^{92} - 12 q^{93} - 176 q^{94} - 166 q^{95} - 64 q^{96} - 58 q^{97} - 150 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.71477 − 2.03488i −0.584719 − 0.811236i −0.861281 + 5.00763i 0.181398 + 3.19156i −0.648107 + 2.58091i −1.07192 + 0.474014i 7.07461 − 4.14425i −0.316208 + 0.948690i 6.18337 − 5.84191i
4.2 −1.68553 − 2.00017i −0.584719 − 0.811236i −0.820692 + 4.77164i −0.220281 − 3.87567i −0.637055 + 2.53690i 2.93860 − 1.29947i 6.41353 − 3.75700i −0.316208 + 0.948690i −7.38073 + 6.97315i
4.3 −1.30283 − 1.54604i −0.584719 − 0.811236i −0.353862 + 2.05741i 0.0209672 + 0.368901i −0.492414 + 1.96091i 2.36318 − 1.04502i 0.152858 − 0.0895433i −0.316208 + 0.948690i 0.543021 − 0.513033i
4.4 −1.23206 − 1.46206i −0.584719 − 0.811236i −0.280634 + 1.63165i 0.0557298 + 0.980522i −0.465666 + 1.85439i −4.08270 + 1.80541i −0.568155 + 0.332821i −0.316208 + 0.948690i 1.36492 − 1.28955i
4.5 −1.03251 − 1.22525i −0.584719 − 0.811236i −0.0961622 + 0.559103i −0.146273 − 2.57356i −0.390242 + 1.55403i −1.05821 + 0.467951i −1.98074 + 1.16030i −0.316208 + 0.948690i −3.00223 + 2.83644i
4.6 −0.734896 − 0.872084i −0.584719 − 0.811236i 0.118551 − 0.689275i 0.171704 + 3.02100i −0.277759 + 1.10610i 2.23363 − 0.987731i −2.65630 + 1.55604i −0.316208 + 0.948690i 2.50838 − 2.36986i
4.7 −0.147141 − 0.174609i −0.584719 − 0.811236i 0.330172 − 1.91967i 0.131937 + 2.32133i −0.0556129 + 0.221463i −2.63513 + 1.16528i −0.777821 + 0.455642i −0.316208 + 0.948690i 0.385912 − 0.364600i
4.8 0.0835653 + 0.0991650i −0.584719 − 0.811236i 0.336159 − 1.95449i −0.131706 − 2.31726i 0.0315840 − 0.125775i 4.42696 − 1.95764i 0.445697 − 0.261086i −0.316208 + 0.948690i 0.218785 − 0.206703i
4.9 0.430138 + 0.510435i −0.584719 − 0.811236i 0.263485 − 1.53194i −0.219431 − 3.86072i 0.162573 − 0.647404i −4.55328 + 2.01350i 2.04721 − 1.19924i −0.316208 + 0.948690i 1.87626 − 1.77265i
4.10 0.584518 + 0.693634i −0.584719 − 0.811236i 0.199543 − 1.16018i 0.0222854 + 0.392095i 0.220922 − 0.879762i 0.104917 − 0.0463951i 2.48672 − 1.45670i −0.316208 + 0.948690i −0.258944 + 0.244644i
4.11 1.06293 + 1.26136i −0.584719 − 0.811236i −0.122187 + 0.710413i 0.117131 + 2.06082i 0.401742 − 1.59983i −0.763193 + 0.337491i 1.82059 − 1.06649i −0.316208 + 0.948690i −2.47493 + 2.33825i
4.12 1.18963 + 1.41171i −0.584719 − 0.811236i −0.238685 + 1.38776i −0.0772593 − 1.35932i 0.449628 − 1.79052i 0.456952 − 0.202068i 0.942802 − 0.552286i −0.316208 + 0.948690i 1.82704 − 1.72615i
4.13 1.58415 + 1.87987i −0.584719 − 0.811236i −0.685383 + 3.98493i 0.212272 + 3.73477i 0.598738 − 2.38431i −2.40028 + 1.06142i −4.33453 + 2.53913i −0.316208 + 0.948690i −6.68461 + 6.31547i
4.14 1.71724 + 2.03781i −0.584719 − 0.811236i −0.864738 + 5.02773i 0.0268028 + 0.471575i 0.649040 − 2.58463i 4.04050 − 1.78674i −7.13170 + 4.17769i −0.316208 + 0.948690i −0.914950 + 0.864424i
7.1 −0.566491 − 2.68176i −0.843109 + 0.537743i −5.04180 + 2.22953i −0.158464 + 0.203177i 1.91971 + 1.95639i 2.74280 − 1.09076i 5.62983 + 7.81080i 0.421665 − 0.906752i 0.634640 + 0.309865i
7.2 −0.473589 − 2.24197i −0.843109 + 0.537743i −2.97299 + 1.31468i −2.12099 + 2.71946i 1.60489 + 1.63555i −1.15020 + 0.457411i 1.67575 + 2.32493i 0.421665 − 0.906752i 7.10141 + 3.46729i
7.3 −0.398609 − 1.88701i −0.843109 + 0.537743i −1.57279 + 0.695500i −1.12854 + 1.44697i 1.35080 + 1.37661i 1.86378 − 0.741188i −0.316096 − 0.438550i 0.421665 − 0.906752i 3.18030 + 1.55279i
7.4 −0.351350 − 1.66329i −0.843109 + 0.537743i −0.813941 + 0.359932i 2.15929 − 2.76856i 1.19065 + 1.21340i 1.47251 − 0.585589i −1.10339 − 1.53083i 0.421665 − 0.906752i −5.36357 − 2.61878i
7.5 −0.221599 − 1.04905i −0.843109 + 0.537743i 0.777745 − 0.343925i 0.749563 − 0.961061i 0.750949 + 0.765297i −2.94659 + 1.17180i −1.78701 − 2.47929i 0.421665 − 0.906752i −1.17430 − 0.573356i
7.6 −0.0919744 − 0.435406i −0.843109 + 0.537743i 1.64802 − 0.728768i −1.01597 + 1.30264i 0.311681 + 0.317636i −2.75543 + 1.09578i −0.989302 − 1.37255i 0.421665 − 0.906752i 0.660618 + 0.322549i
See next 80 embeddings (of 1148 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.c even 83 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 501.2.e.a ✓ 1148
167.c even 83 1 inner 501.2.e.a ✓ 1148
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.e.a ✓ 1148 1.a even 1 1 trivial
501.2.e.a ✓ 1148 167.c even 83 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1148} + 2 T_{2}^{1147} + 26 T_{2}^{1146} + 54 T_{2}^{1145} + 416 T_{2}^{1144} + \cdots + 13\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(501, [\chi])\). Copy content Toggle raw display