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Properties

Label 403.2.y.a
Level $403$
Weight $2$
Character orbit 403.y
Analytic conductor $3.218$
Analytic rank $0$
Dimension $136$
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] = [403,2,Mod(64,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.y (of order \(10\), degree \(4\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9} + 6 q^{10} + 48 q^{12} - 16 q^{13} - 4 q^{14} - 46 q^{16} + 12 q^{17} - 44 q^{22} + 16 q^{23} - 92 q^{25} - 48 q^{26} - 52 q^{27} - 10 q^{29} + 28 q^{35} - 62 q^{38} - 31 q^{39} + 12 q^{40} + 22 q^{42} - 66 q^{43} + 64 q^{48} + 58 q^{49} + 28 q^{51} + 27 q^{52} + 44 q^{55} - 208 q^{56} + 4 q^{61} + 72 q^{62} + 10 q^{64} + 7 q^{65} + 38 q^{66} + 44 q^{68} + 46 q^{69} + 44 q^{74} + 34 q^{75} + 48 q^{77} + 5 q^{78} + 48 q^{79} - 12 q^{81} + 8 q^{82} - 60 q^{87} - 160 q^{88} + 142 q^{90} - 80 q^{91} + 28 q^{92} - 108 q^{94} + 60 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.59762 + 2.19894i −0.583936 + 0.424254i −1.66490 − 5.12402i 2.47143i − 1.96183i −0.903099 + 0.293435i 8.75727 + 2.84541i −0.766062 + 2.35770i −5.43453 − 3.94842i
64.2 −1.47859 + 2.03510i −2.57815 + 1.87313i −1.33738 − 4.11602i − 1.88262i − 8.01636i −2.52898 + 0.821714i 5.56913 + 1.80952i 2.21116 − 6.80524i 3.83131 + 2.78361i
64.3 −1.44841 + 1.99357i 1.15476 − 0.838982i −1.25838 − 3.87289i − 1.37198i 3.51728i −4.14268 + 1.34604i 4.85636 + 1.57793i −0.297472 + 0.915526i 2.73514 + 1.98720i
64.4 −1.37219 + 1.88866i 1.37165 − 0.996564i −1.06609 − 3.28109i 0.910017i 3.95806i 1.80401 − 0.586159i 3.21924 + 1.04600i −0.0387606 + 0.119293i −1.71871 − 1.24872i
64.5 −1.33163 + 1.83284i −1.79188 + 1.30188i −0.968010 − 2.97923i 0.557599i − 5.01786i 4.29621 − 1.39592i 2.44022 + 0.792875i 0.588906 − 1.81247i −1.02199 − 0.742519i
64.6 −1.20132 + 1.65347i −0.773570 + 0.562032i −0.672771 − 2.07058i − 3.18878i − 1.95426i 3.58968 − 1.16636i 0.344303 + 0.111871i −0.644520 + 1.98363i 5.27256 + 3.83074i
64.7 −1.07240 + 1.47603i −0.526166 + 0.382282i −0.410588 − 1.26366i − 2.39403i − 1.18659i −1.79656 + 0.583739i −1.16484 − 0.378479i −0.796340 + 2.45088i 3.53365 + 2.56735i
64.8 −0.987540 + 1.35923i −1.39528 + 1.01373i −0.254242 − 0.782477i 3.09070i − 2.89760i −3.01663 + 0.980163i −1.88110 − 0.611207i −0.00790000 + 0.0243137i −4.20097 − 3.05219i
64.9 −0.918537 + 1.26426i 1.35395 − 0.983703i −0.136603 − 0.420420i 3.21606i 2.61531i 1.56427 − 0.508261i −2.31545 − 0.752336i −0.0615391 + 0.189398i −4.06592 − 2.95406i
64.10 −0.801121 + 1.10265i 1.77414 − 1.28899i 0.0439947 + 0.135402i − 3.51161i 2.98889i 0.360511 − 0.117137i −2.77703 − 0.902311i 0.559035 − 1.72053i 3.87208 + 2.81323i
64.11 −0.675748 + 0.930087i −0.384489 + 0.279347i 0.209607 + 0.645103i 1.11176i − 0.546376i 0.340635 − 0.110679i −2.92841 − 0.951498i −0.857254 + 2.63836i −1.03404 − 0.751273i
64.12 −0.618027 + 0.850641i 2.69208 − 1.95591i 0.276401 + 0.850674i 1.05598i 3.49880i 0.656905 − 0.213441i −2.89442 − 0.940454i 2.49466 − 7.67776i −0.898257 − 0.652622i
64.13 −0.493698 + 0.679518i −2.07909 + 1.51055i 0.400028 + 1.23116i 0.0344567i − 2.15854i −1.65083 + 0.536388i −2.63173 − 0.855101i 1.11382 − 3.42798i −0.0234140 − 0.0170112i
64.14 −0.336818 + 0.463590i −2.27344 + 1.65175i 0.516564 + 1.58982i − 3.22494i − 1.61029i 1.88146 − 0.611323i −2.00098 − 0.650158i 1.51320 − 4.65716i 1.49505 + 1.08622i
64.15 −0.169061 + 0.232692i −0.823965 + 0.598646i 0.592470 + 1.82343i 4.12859i − 0.292938i 4.35350 − 1.41454i −1.07156 − 0.348169i −0.606509 + 1.86664i −0.960691 − 0.697983i
64.16 −0.160682 + 0.221160i 0.324832 − 0.236004i 0.594941 + 1.83104i 0.569297i 0.109762i −3.20061 + 1.03994i −1.02053 − 0.331590i −0.877233 + 2.69985i −0.125906 − 0.0914759i
64.17 −0.0184596 + 0.0254075i 1.61150 − 1.17082i 0.617729 + 1.90118i − 2.11185i 0.0625570i 3.19880 − 1.03935i −0.119444 − 0.0388096i 0.299052 − 0.920387i 0.0536567 + 0.0389839i
64.18 0.0184596 − 0.0254075i 1.61150 − 1.17082i 0.617729 + 1.90118i 2.11185i − 0.0625570i −3.19880 + 1.03935i 0.119444 + 0.0388096i 0.299052 − 0.920387i 0.0536567 + 0.0389839i
64.19 0.160682 − 0.221160i 0.324832 − 0.236004i 0.594941 + 1.83104i − 0.569297i − 0.109762i 3.20061 − 1.03994i 1.02053 + 0.331590i −0.877233 + 2.69985i −0.125906 − 0.0914759i
64.20 0.169061 − 0.232692i −0.823965 + 0.598646i 0.592470 + 1.82343i − 4.12859i 0.292938i −4.35350 + 1.41454i 1.07156 + 0.348169i −0.606509 + 1.86664i −0.960691 − 0.697983i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
31.d even 5 1 inner
403.y even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.y.a ✓ 136
13.b even 2 1 inner 403.2.y.a ✓ 136
31.d even 5 1 inner 403.2.y.a ✓ 136
403.y even 10 1 inner 403.2.y.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.y.a ✓ 136 1.a even 1 1 trivial
403.2.y.a ✓ 136 13.b even 2 1 inner
403.2.y.a ✓ 136 31.d even 5 1 inner
403.2.y.a ✓ 136 403.y even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).