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Properties

Label 688.2.k.a
Level $688$
Weight $2$
Character orbit 688.k
Analytic conductor $5.494$
Analytic rank $0$
Dimension $168$
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] = [688,2,Mod(173,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.k (of order \(4\), degree \(2\), 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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(84\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 12 q^{6} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 12 q^{6} - 12 q^{8} - 4 q^{10} - 12 q^{12} - 16 q^{15} - 8 q^{16} - 20 q^{18} - 16 q^{19} - 24 q^{22} - 4 q^{24} - 12 q^{26} + 24 q^{27} - 4 q^{28} + 24 q^{31} - 20 q^{32} - 20 q^{34} + 24 q^{35} - 20 q^{36} + 20 q^{38} + 52 q^{40} + 60 q^{42} + 44 q^{44} - 20 q^{46} - 40 q^{47} - 72 q^{48} - 168 q^{49} - 8 q^{50} - 24 q^{51} + 56 q^{52} + 48 q^{54} + 40 q^{56} + 68 q^{58} - 24 q^{59} + 20 q^{60} + 32 q^{61} - 32 q^{62} + 24 q^{64} - 16 q^{65} - 20 q^{66} - 24 q^{67} - 20 q^{68} + 32 q^{69} - 12 q^{70} + 96 q^{72} + 24 q^{74} - 60 q^{76} + 64 q^{78} - 8 q^{79} - 60 q^{80} - 168 q^{81} - 36 q^{82} + 60 q^{84} - 32 q^{85} + 24 q^{88} + 60 q^{90} + 48 q^{91} + 76 q^{92} - 48 q^{93} + 72 q^{94} + 16 q^{95} + 32 q^{96} - 68 q^{98} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −1.41363 + 0.0407094i 1.16520 + 1.16520i 1.99669 − 0.115096i 2.83091 − 2.83091i −1.69459 − 1.59972i − 0.201491i −2.81788 + 0.243986i − 0.284615i −3.88661 + 4.11710i
173.2 −1.41093 − 0.0963031i 1.96476 + 1.96476i 1.98145 + 0.271754i −0.104002 + 0.104002i −2.58293 − 2.96136i 3.83856i −2.76952 − 0.574246i 4.72060i 0.156755 − 0.136724i
173.3 −1.40832 − 0.128985i −0.0399061 − 0.0399061i 1.96673 + 0.363304i −2.06330 + 2.06330i 0.0510533 + 0.0613478i 2.97607i −2.72292 − 0.765326i − 2.99682i 3.17192 − 2.63965i
173.4 −1.40815 + 0.130865i −1.64198 − 1.64198i 1.96575 − 0.368555i −2.60209 + 2.60209i 2.52702 + 2.09726i − 2.45776i −2.71983 + 0.776228i 2.39217i 3.32360 − 4.00465i
173.5 −1.40484 − 0.162549i 1.27455 + 1.27455i 1.94716 + 0.456711i −2.14036 + 2.14036i −1.58337 − 1.99772i − 2.47629i −2.66121 − 0.958114i 0.248966i 3.35478 − 2.65896i
173.6 −1.38538 − 0.284116i −1.82996 − 1.82996i 1.83856 + 0.787217i 1.71838 − 1.71838i 2.01527 + 3.05511i − 1.69237i −2.32344 − 1.61296i 3.69749i −2.86883 + 1.89239i
173.7 −1.37921 + 0.312708i −2.04806 − 2.04806i 1.80443 − 0.862578i −0.604239 + 0.604239i 3.46515 + 2.18426i 3.25848i −2.21895 + 1.75393i 5.38913i 0.644420 − 1.02232i
173.8 −1.37546 − 0.328813i −0.675700 − 0.675700i 1.78376 + 0.904535i 2.02963 − 2.02963i 0.707218 + 1.15158i 4.39013i −2.15607 − 1.83067i − 2.08686i −3.45904 + 2.12430i
173.9 −1.37288 + 0.339430i −0.0437988 − 0.0437988i 1.76957 − 0.931990i −0.587631 + 0.587631i 0.0749969 + 0.0452637i 0.131298i −2.11306 + 1.88015i − 2.99616i 0.607285 − 1.00620i
173.10 −1.32881 + 0.484009i −0.880208 − 0.880208i 1.53147 − 1.28631i 1.45115 − 1.45115i 1.59566 + 0.743600i − 4.08001i −1.41244 + 2.45051i − 1.45047i −1.22593 + 2.63067i
173.11 −1.32619 − 0.491138i 1.68328 + 1.68328i 1.51757 + 1.30269i −0.695103 + 0.695103i −1.40563 − 3.05907i − 4.17872i −1.37278 − 2.47295i 2.66686i 1.26323 − 0.580448i
173.12 −1.28237 + 0.596257i 1.73583 + 1.73583i 1.28896 − 1.52925i 0.0304711 − 0.0304711i −3.26097 − 1.19098i − 0.334244i −0.741096 + 2.72961i 3.02618i −0.0209067 + 0.0572439i
173.13 −1.25884 − 0.644452i 0.212252 + 0.212252i 1.16936 + 1.62253i 1.08351 − 1.08351i −0.130405 − 0.403978i − 2.10195i −0.426402 − 2.79610i − 2.90990i −2.06223 + 0.665694i
173.14 −1.23765 + 0.684259i −0.973378 − 0.973378i 1.06358 − 1.69375i 1.70751 − 1.70751i 1.87075 + 0.538663i 3.73828i −0.157375 + 2.82405i − 1.10507i −0.944930 + 3.28169i
173.15 −1.22865 − 0.700306i −1.20036 − 1.20036i 1.01914 + 1.72086i −1.11417 + 1.11417i 0.634200 + 2.31544i 0.812252i −0.0470430 − 2.82804i − 0.118262i 2.14918 − 0.588662i
173.16 −1.19630 + 0.754235i 0.828338 + 0.828338i 0.862258 − 1.80458i −2.19041 + 2.19041i −1.61570 − 0.366177i − 4.02114i 0.329561 + 2.80916i − 1.62771i 0.968299 − 4.27247i
173.17 −1.15760 − 0.812384i −1.59242 − 1.59242i 0.680066 + 1.88083i −1.64531 + 1.64531i 0.549726 + 3.13703i − 4.56444i 0.740710 − 2.72972i 2.07159i 3.24123 − 0.567984i
173.18 −1.11674 − 0.867697i 2.37121 + 2.37121i 0.494202 + 1.93798i 2.18197 − 2.18197i −0.590525 − 4.70551i − 0.427614i 1.12969 − 2.59303i 8.24529i −4.32997 + 0.543396i
173.19 −1.09984 − 0.889016i 0.646392 + 0.646392i 0.419300 + 1.95555i −2.24814 + 2.24814i −0.136275 − 1.28558i 3.19763i 1.27736 − 2.52356i − 2.16435i 4.47122 − 0.473962i
173.20 −1.03125 − 0.967744i 0.702591 + 0.702591i 0.126944 + 1.99597i 0.867214 − 0.867214i −0.0446175 − 1.40447i 1.84300i 1.80067 − 2.18119i − 2.01273i −1.73355 + 0.0550717i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.k.a ✓ 168
16.e even 4 1 inner 688.2.k.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.k.a ✓ 168 1.a even 1 1 trivial
688.2.k.a ✓ 168 16.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(688, [\chi])\).