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Properties

Label 459.2.ba.a
Level $459$
Weight $2$
Character orbit 459.ba
Analytic conductor $3.665$
Analytic rank $0$
Dimension $1248$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [459,2,Mod(25,459)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(459, base_ring=CyclotomicField(72))
 
chi = DirichletCharacter(H, H._module([40, 45]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("459.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 459 = 3^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 459.ba (of order \(72\), degree \(24\), 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: \(3.66513345278\)
Analytic rank: \(0\)
Dimension: \(1248\)
Relative dimension: \(52\) over \(\Q(\zeta_{72})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1248 q - 24 q^{2} - 24 q^{3} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9} - 12 q^{10} - 36 q^{11} + 12 q^{12} - 24 q^{14} - 24 q^{15} - 48 q^{16} - 12 q^{17} - 48 q^{18} - 12 q^{19} - 24 q^{20}+ \cdots - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −1.55139 − 2.21561i 0.627483 + 1.61439i −1.81809 + 4.99517i 0.510558 − 0.160978i 2.60340 − 3.89481i −2.15228 + 1.97220i 8.66274 − 2.32117i −2.21253 + 2.02601i −1.14874 − 0.881459i
25.2 −1.49539 − 2.13565i 0.344398 − 1.69747i −1.64073 + 4.50788i −0.145048 + 0.0457334i −4.14020 + 1.80287i 2.84495 − 2.60692i 7.04417 − 1.88748i −2.76278 − 1.16921i 0.314574 + 0.241381i
25.3 −1.45902 − 2.08370i 1.71956 + 0.207643i −1.52902 + 4.20095i −2.60696 + 0.821971i −2.07621 − 3.88600i 1.97919 − 1.81359i 6.07029 − 1.62653i 2.91377 + 0.714110i 5.51636 + 4.23285i
25.4 −1.45590 − 2.07925i −1.49626 − 0.872473i −1.51957 + 4.17497i −1.87764 + 0.592016i 0.364323 + 4.38133i −1.18104 + 1.08223i 5.98953 − 1.60489i 1.47758 + 2.61089i 3.96460 + 3.04215i
25.5 −1.41239 − 2.01710i 1.57022 − 0.731031i −1.38982 + 3.81850i 2.94630 − 0.928964i −3.69233 − 2.13480i −0.765334 + 0.701300i 4.90823 − 1.31516i 1.93119 − 2.29576i −6.03514 − 4.63093i
25.6 −1.26968 − 1.81329i −1.66712 + 0.469789i −0.991890 + 2.72520i −0.420160 + 0.132476i 2.96857 + 2.42649i 0.435898 − 0.399427i 1.92457 − 0.515686i 2.55860 − 1.56639i 0.773684 + 0.593668i
25.7 −1.24030 − 1.77133i −1.29422 + 1.15109i −0.915221 + 2.51455i 1.64865 − 0.519817i 3.64416 + 0.864790i −1.17396 + 1.07573i 1.41181 − 0.378295i 0.349993 − 2.97951i −2.96558 − 2.27557i
25.8 −1.22861 − 1.75463i 0.00134192 + 1.73205i −0.885220 + 2.43212i 2.58809 − 0.816022i 3.03746 − 2.13036i 2.95845 − 2.71092i 1.21702 − 0.326100i −3.00000 + 0.00464853i −4.61156 − 3.53858i
25.9 −1.11038 − 1.58579i 0.501377 − 1.65790i −0.597734 + 1.64226i −2.74308 + 0.864889i −3.18579 + 1.04582i −0.585095 + 0.536141i −0.471861 + 0.126435i −2.49724 − 1.66246i 4.41738 + 3.38958i
25.10 −1.11023 − 1.58557i 0.664068 + 1.59969i −0.597391 + 1.64132i −3.50205 + 1.10419i 1.79916 − 2.82895i 1.47118 − 1.34809i −0.473675 + 0.126921i −2.11803 + 2.12461i 5.63886 + 4.32685i
25.11 −1.08122 − 1.54414i 1.44943 − 0.948231i −0.531296 + 1.45972i −2.16213 + 0.681718i −3.03136 − 1.21288i −2.45463 + 2.24926i −0.813166 + 0.217887i 1.20171 − 2.74880i 3.39041 + 2.60155i
25.12 −0.903333 − 1.29009i 1.44992 + 0.947483i −0.164290 + 0.451383i 0.429370 − 0.135380i −0.0874213 − 2.72643i −3.73552 + 3.42297i −2.31176 + 0.619435i 1.20455 + 2.74755i −0.562517 − 0.431634i
25.13 −0.890643 − 1.27197i −1.43866 − 0.964506i −0.140623 + 0.386357i 3.34930 − 1.05603i 0.0545055 + 2.68896i 0.933498 − 0.855393i −2.38308 + 0.638544i 1.13946 + 2.77518i −4.32627 − 3.31967i
25.14 −0.870717 − 1.24351i 1.65409 + 0.513794i −0.104136 + 0.286110i 2.29872 − 0.724783i −0.801335 − 2.50425i 1.56020 − 1.42966i −2.48619 + 0.666173i 2.47203 + 1.69973i −2.90281 − 2.22740i
25.15 −0.808637 − 1.15485i 0.0715990 − 1.73057i 0.00424807 − 0.0116715i 1.94527 − 0.613343i −2.05645 + 1.31672i −0.521489 + 0.477857i −2.74047 + 0.734306i −2.98975 − 0.247814i −2.28134 − 1.75053i
25.16 −0.751830 − 1.07372i −1.65924 + 0.496913i 0.0964040 − 0.264868i −2.69307 + 0.849122i 1.78101 + 1.40797i 3.54742 − 3.25061i −2.88910 + 0.774132i 2.50616 − 1.64899i 2.93645 + 2.25322i
25.17 −0.623790 − 0.890865i −1.22090 − 1.22857i 0.279514 − 0.767960i −2.12009 + 0.668461i −0.332906 + 1.85403i −0.361337 + 0.331104i −2.95948 + 0.792991i −0.0187911 + 2.99994i 1.91800 + 1.47173i
25.18 −0.589138 − 0.841376i −1.73101 − 0.0600983i 0.323210 − 0.888013i 2.49663 − 0.787185i 0.969237 + 1.49183i −2.99699 + 2.74624i −2.92183 + 0.782902i 2.99278 + 0.208061i −2.13318 − 1.63685i
25.19 −0.541406 − 0.773208i −0.275133 + 1.71006i 0.379310 − 1.04215i −0.876235 + 0.276276i 1.47119 − 0.713101i 0.372163 − 0.341024i −2.83466 + 0.759544i −2.84860 − 0.940989i 0.688018 + 0.527935i
25.20 −0.442402 − 0.631815i 1.70567 − 0.301145i 0.480569 − 1.32035i −2.01144 + 0.634203i −0.944859 − 0.944442i 1.40032 − 1.28316i −2.53687 + 0.679752i 2.81862 − 1.02731i 1.29056 + 0.990283i
See next 80 embeddings (of 1248 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner
27.e even 9 1 inner
459.ba even 72 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 459.2.ba.a ✓ 1248
17.d even 8 1 inner 459.2.ba.a ✓ 1248
27.e even 9 1 inner 459.2.ba.a ✓ 1248
459.ba even 72 1 inner 459.2.ba.a ✓ 1248
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
459.2.ba.a ✓ 1248 1.a even 1 1 trivial
459.2.ba.a ✓ 1248 17.d even 8 1 inner
459.2.ba.a ✓ 1248 27.e even 9 1 inner
459.2.ba.a ✓ 1248 459.ba even 72 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(459, [\chi])\).