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Properties

Label 272.3.x.a
Level $272$
Weight $3$
Character orbit 272.x
Analytic conductor $7.411$
Analytic rank $0$
Dimension $280$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [272,3,Mod(155,272)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("272.155"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 2, 7])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 272.x (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.41146319060\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(70\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 4 q^{2} - 4 q^{3} - 4 q^{5} + 12 q^{6} - 8 q^{7} - 4 q^{8} + 12 q^{10} - 4 q^{11} - 24 q^{12} + 40 q^{14} - 8 q^{16} - 8 q^{17} - 8 q^{18} - 4 q^{20} + 4 q^{22} - 8 q^{23} - 324 q^{24} - 132 q^{26}+ \cdots + 284 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1 −1.99991 + 0.0193634i −4.02433 − 1.66693i 3.99925 − 0.0774501i 1.08514 − 2.61976i 8.08056 + 3.25578i −1.46792 + 3.54388i −7.99663 + 0.232332i 7.05262 + 7.05262i −2.11945 + 5.26028i
155.2 −1.99664 + 0.115876i −3.90579 − 1.61783i 3.97315 − 0.462724i −3.74650 + 9.04486i 7.98593 + 2.77764i −0.137900 + 0.332921i −7.87932 + 1.38428i 6.27388 + 6.27388i 6.43234 − 18.4935i
155.3 −1.99137 + 0.185641i −2.62102 − 1.08566i 3.93107 − 0.739358i 3.16911 − 7.65091i 5.42096 + 1.67538i 4.43061 − 10.6964i −7.69095 + 2.20210i −0.672866 − 0.672866i −4.89054 + 15.8241i
155.4 −1.97415 + 0.320494i 0.567746 + 0.235168i 3.79457 − 1.26541i −0.802763 + 1.93804i −1.19619 − 0.282299i −4.28348 + 10.3412i −7.08550 + 3.71425i −6.09693 − 6.09693i 0.963647 − 4.08327i
155.5 −1.96547 − 0.370012i 0.834919 + 0.345835i 3.72618 + 1.45450i −1.80797 + 4.36483i −1.51305 − 0.988659i 1.40007 − 3.38006i −6.78554 − 4.23751i −5.78647 − 5.78647i 5.16857 − 7.91000i
155.6 −1.95557 + 0.419209i 4.50783 + 1.86720i 3.64853 − 1.63959i 1.59844 − 3.85899i −9.59814 − 1.76173i 1.19737 − 2.89070i −6.44763 + 4.73583i 10.4701 + 10.4701i −1.50815 + 8.21661i
155.7 −1.94797 − 0.453215i 1.30483 + 0.540479i 3.58919 + 1.76570i −0.932568 + 2.25142i −2.29682 − 1.64421i 3.05803 − 7.38274i −6.19140 − 5.06621i −4.95349 − 4.95349i 2.83699 − 3.96305i
155.8 −1.90154 − 0.619800i 4.22836 + 1.75144i 3.23170 + 2.35715i 0.733287 − 1.77031i −6.95484 − 5.95117i −3.63309 + 8.77105i −4.68424 − 6.48521i 8.44750 + 8.44750i −2.49161 + 2.91182i
155.9 −1.85168 + 0.755822i 4.12789 + 1.70983i 2.85746 − 2.79909i −3.32759 + 8.03351i −8.93588 − 0.0461087i −0.508046 + 1.22653i −3.17551 + 7.34276i 7.75201 + 7.75201i 0.0897346 − 17.3906i
155.10 −1.80399 + 0.863498i −0.384827 − 0.159401i 2.50874 − 3.11548i 1.50866 − 3.64224i 0.831866 − 0.0447408i −3.31711 + 8.00822i −1.83553 + 7.78658i −6.24128 − 6.24128i 0.423454 + 7.87328i
155.11 −1.74970 − 0.968791i −2.53528 − 1.05015i 2.12289 + 3.39018i 0.387682 − 0.935947i 3.41860 + 4.29359i −0.0463828 + 0.111978i −0.430039 − 7.98843i −1.03913 − 1.03913i −1.58506 + 1.26204i
155.12 −1.72798 + 1.00702i 0.244535 + 0.101290i 1.97183 − 3.48021i 0.0867710 − 0.209484i −0.524552 + 0.0712246i 3.71684 − 8.97324i 0.0973550 + 7.99941i −6.31442 − 6.31442i 0.0610153 + 0.449364i
155.13 −1.70774 − 1.04097i 0.459249 + 0.190227i 1.83277 + 3.55541i 3.27855 − 7.91511i −0.586259 − 0.802922i −1.99967 + 4.82763i 0.571157 − 7.97959i −6.18924 − 6.18924i −13.8383 + 10.1041i
155.14 −1.56595 − 1.24411i 3.79886 + 1.57354i 0.904403 + 3.89642i −2.85209 + 6.88556i −3.99118 − 7.19027i −1.34336 + 3.24315i 3.43130 − 7.22677i 5.59138 + 5.59138i 13.0326 − 7.23414i
155.15 −1.54844 + 1.26583i −2.28710 − 0.947346i 0.795363 − 3.92013i −1.67918 + 4.05391i 4.74062 − 1.42815i 0.739896 − 1.78627i 3.73063 + 7.07689i −2.03062 − 2.03062i −2.53142 − 8.40281i
155.16 −1.52269 + 1.29670i −4.96980 − 2.05856i 0.637156 − 3.94893i 0.212700 − 0.513504i 10.2368 − 3.30978i −0.864761 + 2.08772i 4.15037 + 6.83918i 14.0973 + 14.0973i 0.341983 + 1.05771i
155.17 −1.51503 − 1.30564i −5.10521 − 2.11465i 0.590634 + 3.95615i −1.23196 + 2.97420i 4.97359 + 9.86931i 5.08037 − 12.2651i 4.27047 − 6.76485i 15.2275 + 15.2275i 5.74968 − 2.89752i
155.18 −1.41018 + 1.41824i 2.02567 + 0.839058i −0.0228089 − 3.99993i 3.32204 − 8.02012i −4.04653 + 1.68966i −0.771065 + 1.86152i 5.70503 + 5.60826i −2.96466 − 2.96466i 6.68979 + 16.0212i
155.19 −1.40894 − 1.41947i 4.07727 + 1.68886i −0.0297944 + 3.99989i 1.55509 − 3.75431i −3.34732 − 8.16705i 4.18755 − 10.1096i 5.71970 − 5.59330i 7.40789 + 7.40789i −7.52015 + 3.08219i
155.20 −1.37735 − 1.45014i −2.31294 − 0.958051i −0.205791 + 3.99470i −2.57391 + 6.21396i 1.79643 + 4.67366i −3.91405 + 9.44936i 6.07631 − 5.20370i −1.93213 − 1.93213i 12.5563 − 4.82631i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.x odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.3.x.a ✓ 280
16.f odd 4 1 272.3.z.a yes 280
17.d even 8 1 272.3.z.a yes 280
272.x odd 8 1 inner 272.3.x.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.3.x.a ✓ 280 1.a even 1 1 trivial
272.3.x.a ✓ 280 272.x odd 8 1 inner
272.3.z.a yes 280 16.f odd 4 1
272.3.z.a yes 280 17.d even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(272, [\chi])\).