Properties

Label 538.2.e.a
Level $538$
Weight $2$
Character orbit 538.e
Analytic conductor $4.296$
Analytic rank $0$
Dimension $1452$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(9,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(134))
 
chi = DirichletCharacter(H, H._module([109]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.e (of order \(134\), degree \(66\), 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: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(1452\)
Relative dimension: \(22\) over \(\Q(\zeta_{134})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{134}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1452 q + 22 q^{4} - 2 q^{5} + 2 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1452 q + 22 q^{4} - 2 q^{5} + 2 q^{6} + 20 q^{9} + 2 q^{11} - 10 q^{13} + 12 q^{14} - 22 q^{16} + 2 q^{20} + 12 q^{21} - 4 q^{23} - 2 q^{24} - 24 q^{25} - 8 q^{30} - 12 q^{34} - 20 q^{36} - 396 q^{37} - 2 q^{38} + 24 q^{41} + 14 q^{43} - 2 q^{44} - 42 q^{45} - 402 q^{47} + 22 q^{49} + 12 q^{51} + 10 q^{52} - 10 q^{53} - 8 q^{54} - 4 q^{55} - 12 q^{56} + 16 q^{57} - 18 q^{58} - 268 q^{60} - 2 q^{61} + 4 q^{62} + 22 q^{64} + 16 q^{65} + 40 q^{66} + 6 q^{67} - 106 q^{73} + 40 q^{78} + 40 q^{79} - 2 q^{80} - 14 q^{81} - 402 q^{83} - 12 q^{84} - 100 q^{87} + 4 q^{92} - 340 q^{93} + 2 q^{96} - 64 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.833094 + 0.553131i −1.43549 + 2.53183i 0.388092 0.921621i 1.94372 + 1.35724i −0.204537 2.90327i 0.225409 + 1.59183i 0.186460 + 0.982463i −2.80913 4.69454i −2.37004 0.0555750i
9.2 −0.833094 + 0.553131i −1.05597 + 1.86246i 0.388092 0.921621i −1.04785 0.731677i −0.150460 2.13569i −0.574974 4.06045i 0.186460 + 0.982463i −0.813253 1.35909i 1.27767 + 0.0299601i
9.3 −0.833094 + 0.553131i −0.805229 + 1.42021i 0.388092 0.921621i −1.22081 0.852450i −0.114733 1.62857i 0.0206347 + 0.145722i 0.186460 + 0.982463i 0.171807 + 0.287119i 1.48856 + 0.0349053i
9.4 −0.833094 + 0.553131i −0.552092 + 0.973747i 0.388092 0.921621i 1.05205 + 0.734610i −0.0786651 1.11660i 0.586003 + 4.13834i 0.186460 + 0.982463i 0.897045 + 1.49912i −1.28279 0.0300801i
9.5 −0.833094 + 0.553131i 0.0651030 0.114825i 0.388092 0.921621i 2.50671 + 1.75036i 0.00927622 + 0.131670i −0.310567 2.19321i 0.186460 + 0.982463i 1.53148 + 2.55936i −3.05651 0.0716721i
9.6 −0.833094 + 0.553131i 0.283306 0.499678i 0.388092 0.921621i −0.723479 0.505183i 0.0403669 + 0.572984i −0.341899 2.41448i 0.186460 + 0.982463i 1.37101 + 2.29119i 0.882159 + 0.0206858i
9.7 −0.833094 + 0.553131i 0.602356 1.06240i 0.388092 0.921621i −2.46979 1.72457i 0.0858269 + 1.21826i 0.527223 + 3.72323i 0.186460 + 0.982463i 0.774562 + 1.29443i 3.01148 + 0.0706163i
9.8 −0.833094 + 0.553131i 0.865942 1.52730i 0.388092 0.921621i 0.921477 + 0.643438i 0.123384 + 1.75136i 0.350952 + 2.47841i 0.186460 + 0.982463i −0.0423564 0.0707850i −1.12358 0.0263469i
9.9 −0.833094 + 0.553131i 1.30591 2.30329i 0.388092 0.921621i −3.05509 2.13327i 0.186074 + 2.64120i −0.565573 3.99406i 0.186460 + 0.982463i −2.05933 3.44150i 3.72515 + 0.0873512i
9.10 −0.833094 + 0.553131i 1.52425 2.68838i 0.388092 0.921621i −0.943666 0.658932i 0.217183 + 3.08279i 0.465524 + 3.28751i 0.186460 + 0.982463i −3.36364 5.62122i 1.15064 + 0.0269813i
9.11 −0.833094 + 0.553131i 1.61279 2.84454i 0.388092 0.921621i 2.64265 + 1.84528i 0.229799 + 3.26186i −0.192859 1.36197i 0.186460 + 0.982463i −3.94990 6.60098i −3.22226 0.0755588i
9.12 0.833094 0.553131i −1.66856 + 2.94291i 0.388092 0.921621i −0.524262 0.366075i 0.237745 + 3.37465i −0.443772 3.13391i −0.186460 0.982463i −4.33618 7.24652i −0.639247 0.0149897i
9.13 0.833094 0.553131i −1.12161 + 1.97822i 0.388092 0.921621i 3.58960 + 2.50650i 0.159813 + 2.26844i 0.304181 + 2.14812i −0.186460 0.982463i −1.11495 1.86327i 4.37690 + 0.102634i
9.14 0.833094 0.553131i −0.952778 + 1.68045i 0.388092 0.921621i 0.475910 + 0.332313i 0.135757 + 1.92699i −0.0793537 0.560394i −0.186460 0.982463i −0.375712 0.627880i 0.580291 + 0.0136072i
9.15 0.833094 0.553131i −0.745603 + 1.31505i 0.388092 0.921621i −3.35521 2.34284i 0.106238 + 1.50798i 0.255112 + 1.80159i −0.186460 0.982463i 0.366990 + 0.613304i −4.09110 0.0959323i
9.16 0.833094 0.553131i −0.559957 + 0.987619i 0.388092 0.921621i −2.31476 1.61632i 0.0797857 + 1.13251i −0.0304501 0.215038i −0.186460 0.982463i 0.878584 + 1.46827i −2.82245 0.0661836i
9.17 0.833094 0.553131i 0.115924 0.204459i 0.388092 0.921621i −1.66371 1.16171i −0.0165174 0.234455i −0.662220 4.67658i −0.186460 0.982463i 1.51206 + 2.52691i −2.02860 0.0475687i
9.18 0.833094 0.553131i 0.261679 0.461534i 0.388092 0.921621i 1.06696 + 0.745027i −0.0372855 0.529245i 0.525099 + 3.70823i −0.186460 0.982463i 1.39588 + 2.33277i 1.30098 + 0.0305067i
9.19 0.833094 0.553131i 0.453043 0.799051i 0.388092 0.921621i 0.923295 + 0.644708i −0.0645520 0.916277i −0.343905 2.42865i −0.186460 0.982463i 1.10719 + 1.85031i 1.12580 + 0.0263989i
9.20 0.833094 0.553131i 1.10088 1.94167i 0.388092 0.921621i −0.0185458 0.0129499i −0.156860 2.22653i −0.0718859 0.507656i −0.186460 0.982463i −1.01772 1.70078i −0.0226134 0.000530262i
See next 80 embeddings (of 1452 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
269.e even 134 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.2.e.a 1452
269.e even 134 1 inner 538.2.e.a 1452
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.e.a 1452 1.a even 1 1 trivial
538.2.e.a 1452 269.e even 134 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(538, [\chi])\).