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Properties

Label 464.2.m.c
Level $464$
Weight $2$
Character orbit 464.m
Analytic conductor $3.705$
Analytic rank $0$
Dimension $112$
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] = [464,2,Mod(173,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.m (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: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 16 q^{6} - 16 q^{13} - 8 q^{16} - 40 q^{20} - 20 q^{22} - 20 q^{24} + 4 q^{28} - 24 q^{29} + 40 q^{30} - 44 q^{34} + 24 q^{35} - 8 q^{36} + 32 q^{38} + 36 q^{42} + 32 q^{45} - 112 q^{49} + 24 q^{51} - 108 q^{52} - 16 q^{53} + 96 q^{54} - 8 q^{58} - 32 q^{59} + 32 q^{62} + 16 q^{63} + 108 q^{64} + 72 q^{67} + 80 q^{74} + 60 q^{78} - 60 q^{80} - 112 q^{81} - 132 q^{82} + 16 q^{83} + 84 q^{86} - 64 q^{88} + 72 q^{91} - 68 q^{92} - 48 q^{93} - 36 q^{94} - 40 q^{96}+O(q^{100}) \) 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 −1.41310 − 0.0561855i 0.413822 − 0.413822i 1.99369 + 0.158791i −1.56513 + 1.56513i −0.608021 + 0.561519i − 2.37637i −2.80835 − 0.336404i 2.65750i 2.29962 − 2.12375i
173.2 −1.40389 + 0.170573i 2.23821 − 2.23821i 1.94181 − 0.478931i −2.03766 + 2.03766i −2.76042 + 3.52397i 3.68152i −2.64439 + 1.00359i − 7.01914i 2.51308 − 3.20822i
173.3 −1.40022 + 0.198438i 0.156525 − 0.156525i 1.92124 − 0.555716i −0.534205 + 0.534205i −0.188109 + 0.250230i 4.28982i −2.57989 + 1.15937i 2.95100i 0.641999 − 0.854012i
173.4 −1.39963 − 0.202567i −1.42187 + 1.42187i 1.91793 + 0.567037i 0.584479 − 0.584479i 2.27811 − 1.70206i 0.454315i −2.56954 − 1.18215i − 1.04341i −0.936450 + 0.699658i
173.5 −1.39056 + 0.257591i 1.55125 − 1.55125i 1.86729 − 0.716389i 1.32814 − 1.32814i −1.75751 + 2.55669i − 4.89123i −2.41204 + 1.47718i − 1.81276i −1.50474 + 2.18897i
173.6 −1.35771 + 0.395762i −1.75197 + 1.75197i 1.68675 − 1.07466i −1.87994 + 1.87994i 1.68530 − 3.07203i − 1.69017i −1.86480 + 2.12662i − 3.13881i 1.80840 − 3.29641i
173.7 −1.33853 − 0.456427i 1.76029 − 1.76029i 1.58335 + 1.22189i 1.85262 − 1.85262i −3.15966 + 1.55277i 1.34386i −1.56167 − 2.35822i − 3.19725i −3.32538 + 1.63421i
173.8 −1.30694 + 0.540281i 0.0678381 − 0.0678381i 1.41619 − 1.41223i 1.82655 − 1.82655i −0.0520088 + 0.125312i 0.155723i −1.08788 + 2.61085i 2.99080i −1.40035 + 3.37405i
173.9 −1.23810 − 0.683458i −1.38743 + 1.38743i 1.06577 + 1.69237i −2.62858 + 2.62858i 2.66602 − 0.769523i 2.47903i −0.162863 − 2.82373i − 0.849908i 5.05097 − 1.45792i
173.10 −1.22821 − 0.701074i −0.746128 + 0.746128i 1.01699 + 1.72213i 1.27515 − 1.27515i 1.43949 − 0.393310i − 3.84739i −0.0417361 − 2.82812i 1.88658i −2.46012 + 0.672174i
173.11 −1.15298 − 0.818930i 0.592554 − 0.592554i 0.658709 + 1.88841i −1.20296 + 1.20296i −1.16846 + 0.197941i 0.166330i 0.787002 − 2.71673i 2.29776i 2.37212 − 0.401845i
173.12 −1.07142 + 0.923075i −1.90973 + 1.90973i 0.295867 − 1.97799i 0.507761 − 0.507761i 0.283292 − 3.80893i 4.88924i 1.50884 + 2.39236i − 4.29410i −0.0753223 + 1.01273i
173.13 −1.04116 + 0.957068i 1.66414 − 1.66414i 0.168041 − 1.99293i −1.68723 + 1.68723i −0.139946 + 3.32534i − 3.40166i 1.73241 + 2.23579i − 2.53874i 0.141888 − 3.37148i
173.14 −0.996502 − 1.00349i 1.17260 − 1.17260i −0.0139671 + 1.99995i 0.506768 − 0.506768i −2.34519 − 0.00818898i 1.94624i 2.02084 − 1.97894i 0.250003i −1.01353 − 0.00353906i
173.15 −0.893574 − 1.09614i 2.18647 − 2.18647i −0.403051 + 1.95897i −2.42728 + 2.42728i −4.35045 − 0.442907i − 2.16728i 2.50746 − 1.30868i − 6.56130i 4.82959 + 0.491686i
173.16 −0.890198 − 1.09888i −1.16145 + 1.16145i −0.415094 + 1.95645i 2.67055 − 2.67055i 2.31021 + 0.242378i 4.43697i 2.51943 − 1.28549i 0.302083i −5.31194 − 0.557306i
173.17 −0.877481 + 1.10907i −0.588804 + 0.588804i −0.460054 − 1.94637i 2.66406 − 2.66406i −0.136358 − 1.16969i − 0.732496i 2.56234 + 1.19767i 2.30662i 0.616957 + 5.29229i
173.18 −0.786873 + 1.17509i 2.11954 − 2.11954i −0.761662 − 1.84929i 2.48644 − 2.48644i 0.822836 + 4.15845i 1.38922i 2.77241 + 0.560135i − 5.98487i 0.965272 + 4.87829i
173.19 −0.739692 − 1.20534i −2.38365 + 2.38365i −0.905713 + 1.78317i 0.304315 − 0.304315i 4.63629 + 1.10995i − 2.26455i 2.81928 − 0.227297i − 8.36357i −0.591904 − 0.141705i
173.20 −0.673229 + 1.24369i −1.85734 + 1.85734i −1.09352 − 1.67458i −0.263482 + 0.263482i −1.05954 − 3.56036i − 3.06253i 2.81884 − 0.232631i − 3.89940i −0.150306 − 0.505074i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
29.b even 2 1 inner
464.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.2.m.c ✓ 112
16.e even 4 1 inner 464.2.m.c ✓ 112
29.b even 2 1 inner 464.2.m.c ✓ 112
464.m even 4 1 inner 464.2.m.c ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
464.2.m.c ✓ 112 1.a even 1 1 trivial
464.2.m.c ✓ 112 16.e even 4 1 inner
464.2.m.c ✓ 112 29.b even 2 1 inner
464.2.m.c ✓ 112 464.m even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} + 784 T_{3}^{108} + 278500 T_{3}^{104} + 59480640 T_{3}^{100} + 8547183266 T_{3}^{96} + \cdots + 3564075791376 \) acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display