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Properties

Label 502.2.e.b
Level $502$
Weight $2$
Character orbit 502.e
Analytic conductor $4.008$
Analytic rank $0$
Dimension $220$
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] = [502,2,Mod(5,502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(502, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("502.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 502 = 2 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 502.e (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: \(4.00849018147\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 220 q - 55 q^{2} - 5 q^{3} - 55 q^{4} - 5 q^{5} - 5 q^{6} + 15 q^{7} - 55 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 220 q - 55 q^{2} - 5 q^{3} - 55 q^{4} - 5 q^{5} - 5 q^{6} + 15 q^{7} - 55 q^{8} - 5 q^{9} - 5 q^{10} - 25 q^{11} - 5 q^{12} - 10 q^{14} - 5 q^{15} - 55 q^{16} + 20 q^{18} - 20 q^{19} + 20 q^{20} + 50 q^{22} + 60 q^{23} + 20 q^{24} - 40 q^{25} - 50 q^{27} + 15 q^{28} - 30 q^{29} - 5 q^{30} + 5 q^{31} + 220 q^{32} - 55 q^{33} - 15 q^{35} - 5 q^{36} - 25 q^{37} - 20 q^{38} + 5 q^{39} - 5 q^{40} - 35 q^{41} - 55 q^{43} + 25 q^{44} + 125 q^{45} - 15 q^{46} - 5 q^{48} - 45 q^{49} - 90 q^{50} + 35 q^{51} - 30 q^{53} - 50 q^{54} - 30 q^{55} - 10 q^{56} - 10 q^{57} - 30 q^{58} - 60 q^{59} + 20 q^{60} - 35 q^{61} + 55 q^{62} - 40 q^{63} - 55 q^{64} - 35 q^{65} - 55 q^{66} + 135 q^{67} + 80 q^{69} + 35 q^{70} + 115 q^{71} - 5 q^{72} + 45 q^{73} - 25 q^{74} + 20 q^{75} - 20 q^{76} - 65 q^{77} + 30 q^{78} - 55 q^{79} - 5 q^{80} - 20 q^{81} + 15 q^{82} + 110 q^{83} + 10 q^{85} + 70 q^{86} + 55 q^{87} - 25 q^{88} + 25 q^{89} - 50 q^{90} - 125 q^{91} - 15 q^{92} + 65 q^{93} + 100 q^{95} - 5 q^{96} - 85 q^{97} + 80 q^{98} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.809017 − 0.587785i −1.32117 − 2.08183i 0.309017 + 0.951057i −0.271902 + 0.836827i −0.154821 + 2.46080i −4.10384 + 0.518436i 0.309017 − 0.951057i −1.31119 + 2.78643i 0.711848 − 0.517188i
5.2 −0.809017 − 0.587785i −1.25458 − 1.97690i 0.309017 + 0.951057i 0.320116 − 0.985216i −0.147017 + 2.33676i 1.77366 − 0.224065i 0.309017 − 0.951057i −1.05682 + 2.24585i −0.838075 + 0.608897i
5.3 −0.809017 − 0.587785i −0.932259 − 1.46900i 0.309017 + 0.951057i −0.420826 + 1.29517i −0.109246 + 1.73642i 2.01671 − 0.254770i 0.309017 − 0.951057i −0.0115301 + 0.0245027i 1.10174 − 0.800460i
5.4 −0.809017 − 0.587785i −0.688664 − 1.08516i 0.309017 + 0.951057i 0.699104 − 2.15162i −0.0807006 + 1.28270i −0.665662 + 0.0840927i 0.309017 − 0.951057i 0.574022 − 1.21986i −1.83028 + 1.32977i
5.5 −0.809017 − 0.587785i −0.00837639 − 0.0131991i 0.309017 + 0.951057i 1.14943 − 3.53757i −0.000981582 0.0156018i 3.56708 − 0.450627i 0.309017 − 0.951057i 1.27723 − 2.71426i −3.00924 + 2.18634i
5.6 −0.809017 − 0.587785i 0.0983019 + 0.154899i 0.309017 + 0.951057i −1.25092 + 3.84994i 0.0115194 − 0.183096i −1.85577 + 0.234438i 0.309017 − 0.951057i 1.26301 − 2.68403i 3.27495 − 2.37939i
5.7 −0.809017 − 0.587785i 0.464124 + 0.731342i 0.309017 + 0.951057i 0.151521 − 0.466335i 0.0543881 − 0.864474i −2.72741 + 0.344552i 0.309017 − 0.951057i 0.957887 − 2.03561i −0.396688 + 0.288211i
5.8 −0.809017 − 0.587785i 0.693110 + 1.09217i 0.309017 + 0.951057i −0.569075 + 1.75143i 0.0812216 − 1.29098i 3.30218 − 0.417162i 0.309017 − 0.951057i 0.564912 − 1.20050i 1.48986 − 1.08245i
5.9 −0.809017 − 0.587785i 0.948775 + 1.49503i 0.309017 + 0.951057i 0.522321 − 1.60754i 0.111182 − 1.76718i −4.27362 + 0.539884i 0.309017 − 0.951057i −0.0576026 + 0.122412i −1.36746 + 0.993514i
5.10 −0.809017 − 0.587785i 1.67017 + 2.63176i 0.309017 + 0.951057i 1.34710 − 4.14596i 0.195717 − 3.11084i −0.387656 + 0.0489724i 0.309017 − 0.951057i −2.85937 + 6.07648i −3.52676 + 2.56234i
5.11 −0.809017 − 0.587785i 1.70357 + 2.68440i 0.309017 + 0.951057i −0.427991 + 1.31722i 0.199632 − 3.17306i 2.26164 − 0.285711i 0.309017 − 0.951057i −3.02651 + 6.43167i 1.12050 − 0.814088i
25.1 0.309017 + 0.951057i −1.15606 + 2.45675i −0.809017 + 0.587785i −2.66934 − 1.93939i −2.69375 − 0.340300i 4.57610 − 1.17494i −0.809017 − 0.587785i −2.78689 − 3.36877i 1.01960 − 3.13799i
25.2 0.309017 + 0.951057i −1.14277 + 2.42850i −0.809017 + 0.587785i 2.32243 + 1.68734i −2.66277 − 0.336387i −3.18451 + 0.817642i −0.809017 − 0.587785i −2.67943 − 3.23887i −0.887089 + 2.73018i
25.3 0.309017 + 0.951057i −0.978929 + 2.08033i −0.809017 + 0.587785i −1.29836 − 0.943317i −2.28102 − 0.288160i −2.72796 + 0.700422i −0.809017 − 0.587785i −1.45720 − 1.76145i 0.495931 − 1.52632i
25.4 0.309017 + 0.951057i −0.908517 + 1.93070i −0.809017 + 0.587785i 1.63552 + 1.18828i −2.11695 − 0.267433i 4.05628 − 1.04147i −0.809017 − 0.587785i −0.989917 − 1.19660i −0.624715 + 1.92267i
25.5 0.309017 + 0.951057i −0.325564 + 0.691858i −0.809017 + 0.587785i −0.680281 − 0.494253i −0.758601 − 0.0958336i 0.381394 − 0.0979253i −0.809017 − 0.587785i 1.53960 + 1.86105i 0.259844 − 0.799718i
25.6 0.309017 + 0.951057i 0.0849275 − 0.180480i −0.809017 + 0.587785i 2.79335 + 2.02949i 0.197891 + 0.0249994i 2.48312 − 0.637557i −0.809017 − 0.587785i 1.88691 + 2.28088i −1.06696 + 3.28378i
25.7 0.309017 + 0.951057i 0.160832 − 0.341785i −0.809017 + 0.587785i −1.85500 − 1.34774i 0.374756 + 0.0473427i 1.46483 − 0.376104i −0.809017 − 0.587785i 1.82132 + 2.20160i 0.708546 − 2.18068i
25.8 0.309017 + 0.951057i 0.447025 − 0.949976i −0.809017 + 0.587785i 2.00087 + 1.45371i 1.04162 + 0.131587i −2.96441 + 0.761131i −0.809017 − 0.587785i 1.20965 + 1.46221i −0.764263 + 2.35216i
25.9 0.309017 + 0.951057i 0.885840 − 1.88251i −0.809017 + 0.587785i −2.75552 − 2.00200i 2.06411 + 0.260758i 1.33519 − 0.342818i −0.809017 − 0.587785i −0.846846 − 1.02366i 1.05251 − 3.23930i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
251.e even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 502.2.e.b ✓ 220
251.e even 25 1 inner 502.2.e.b ✓ 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
502.2.e.b ✓ 220 1.a even 1 1 trivial
502.2.e.b ✓ 220 251.e even 25 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{220} + 5 T_{3}^{219} + 15 T_{3}^{218} + 60 T_{3}^{217} + 210 T_{3}^{216} + 525 T_{3}^{215} + \cdots + 65\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(502, [\chi])\). Copy content Toggle raw display