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Properties

Label 720.2.db.a
Level $720$
Weight $2$
Character orbit 720.db
Analytic conductor $5.749$
Analytic rank $0$
Dimension $384$
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] = [720,2,Mod(61,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.db (of order \(12\), degree \(4\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(96\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 384 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q + 12 q^{6} - 24 q^{12} + 20 q^{18} + 56 q^{24} + 32 q^{26} - 24 q^{27} + 40 q^{32} - 12 q^{36} - 88 q^{44} - 24 q^{46} + 80 q^{47} + 40 q^{48} + 192 q^{49} + 40 q^{51} - 100 q^{54} - 80 q^{56} + 36 q^{58} - 24 q^{59} + 28 q^{60} - 200 q^{62} + 72 q^{64} + 72 q^{66} - 52 q^{68} - 112 q^{72} - 56 q^{74} - 12 q^{76} - 116 q^{78} - 32 q^{80} + 72 q^{82} + 40 q^{83} - 116 q^{84} - 104 q^{86} - 36 q^{90} + 48 q^{93} - 64 q^{95} - 156 q^{96} - 136 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 −1.41415 + 0.0136511i −0.724175 − 1.57339i 1.99963 − 0.0386093i 0.965926 + 0.258819i 1.04557 + 2.21513i −0.516190 − 0.298022i −2.82724 + 0.0818963i −1.95114 + 2.27883i −1.36949 − 0.352822i
61.2 −1.41385 − 0.0320594i 0.585913 − 1.62994i 1.99794 + 0.0906545i −0.965926 − 0.258819i −0.880648 + 2.28571i 2.85514 + 1.64842i −2.82189 − 0.192225i −2.31341 − 1.91001i 1.35738 + 0.396898i
61.3 −1.40940 − 0.116602i −1.12598 + 1.31612i 1.97281 + 0.328677i 0.965926 + 0.258819i 1.74042 − 1.72364i −0.466455 − 0.269308i −2.74215 − 0.693270i −0.464325 − 2.96385i −1.33120 − 0.477408i
61.4 −1.40860 + 0.125839i 1.65652 − 0.505903i 1.96833 − 0.354513i −0.965926 − 0.258819i −2.26972 + 0.921071i 1.15324 + 0.665824i −2.72798 + 0.747061i 2.48812 − 1.67608i 1.39318 + 0.243023i
61.5 −1.40618 − 0.150570i −0.576907 − 1.63315i 1.95466 + 0.423456i −0.965926 − 0.258819i 0.565329 + 2.38336i −4.27106 − 2.46590i −2.68483 − 0.889766i −2.33436 + 1.88435i 1.31929 + 0.509384i
61.6 −1.40314 − 0.176641i 1.71750 + 0.224076i 1.93760 + 0.495703i 0.965926 + 0.258819i −2.37030 − 0.617790i 0.0154882 + 0.00894211i −2.63115 − 1.03780i 2.89958 + 0.769700i −1.30961 − 0.533781i
61.7 −1.37257 + 0.340654i −1.70187 + 0.321939i 1.76791 − 0.935145i −0.965926 − 0.258819i 2.22627 − 1.02163i −3.41244 − 1.97017i −2.10802 + 1.88580i 2.79271 − 1.09580i 1.41397 + 0.0262011i
61.8 −1.37008 + 0.350563i −1.66005 − 0.494198i 1.75421 − 0.960594i 0.965926 + 0.258819i 2.44764 + 0.0951372i 4.02022 + 2.32107i −2.06665 + 1.93105i 2.51154 + 1.64079i −1.41412 − 0.0159841i
61.9 −1.34510 + 0.436695i 0.781208 + 1.54587i 1.61860 − 1.17480i −0.965926 − 0.258819i −1.72588 − 1.73820i 0.148094 + 0.0855022i −1.66415 + 2.28705i −1.77943 + 2.41529i 1.41229 − 0.0736769i
61.10 −1.34207 − 0.445925i −1.45492 − 0.939796i 1.60230 + 1.19692i −0.965926 − 0.258819i 1.53352 + 1.91006i 1.80069 + 1.03963i −1.61666 − 2.32086i 1.23357 + 2.73465i 1.18093 + 0.778084i
61.11 −1.31321 + 0.524866i −0.855555 + 1.50600i 1.44903 − 1.37852i −0.965926 − 0.258819i 0.333076 − 2.42674i 4.49718 + 2.59645i −1.17934 + 2.57083i −1.53605 − 2.57693i 1.40431 − 0.167099i
61.12 −1.28594 − 0.588516i 1.20923 + 1.24006i 1.30730 + 1.51360i 0.965926 + 0.258819i −0.825205 − 2.30630i −4.04284 − 2.33414i −0.790336 − 2.71576i −0.0755201 + 2.99905i −1.08981 − 0.901289i
61.13 −1.27299 − 0.616042i −0.900966 + 1.47928i 1.24098 + 1.56842i −0.965926 − 0.258819i 2.05821 − 1.32807i 1.40713 + 0.812407i −0.613541 − 2.76108i −1.37652 − 2.66556i 1.07017 + 0.924524i
61.14 −1.26766 + 0.626935i −0.138615 + 1.72650i 1.21391 − 1.58948i 0.965926 + 0.258819i −0.906684 − 2.27551i −2.42799 − 1.40180i −0.542316 + 2.77595i −2.96157 − 0.478637i −1.38672 + 0.277479i
61.15 −1.26443 + 0.633413i 0.751358 − 1.56060i 1.19758 − 1.60182i 0.965926 + 0.258819i 0.0384613 + 2.44919i 2.25502 + 1.30194i −0.499642 + 2.78395i −1.87092 − 2.34513i −1.38529 + 0.284571i
61.16 −1.26192 − 0.638409i 1.63714 − 0.565481i 1.18487 + 1.61124i 0.965926 + 0.258819i −2.42694 − 0.331577i 3.50132 + 2.02149i −0.466573 − 2.78968i 2.36046 − 1.85154i −1.05369 − 0.943264i
61.17 −1.25077 + 0.659977i 1.43203 + 0.974309i 1.12886 − 1.65096i 0.965926 + 0.258819i −2.43417 − 0.273529i 1.05702 + 0.610273i −0.322349 + 2.81000i 1.10144 + 2.79049i −1.37897 + 0.313766i
61.18 −1.20129 − 0.746258i −1.68371 − 0.406338i 0.886198 + 1.79295i 0.965926 + 0.258819i 1.71940 + 1.74461i −1.88214 − 1.08666i 0.273418 − 2.81518i 2.66978 + 1.36831i −0.967212 − 1.03175i
61.19 −1.13501 + 0.843648i −1.50905 − 0.850156i 0.576518 − 1.91511i −0.965926 − 0.258819i 2.43003 − 0.308168i 0.136135 + 0.0785978i 0.961318 + 2.66005i 1.55447 + 2.56586i 1.31469 − 0.521138i
61.20 −1.10657 − 0.880621i 1.69010 + 0.378888i 0.449014 + 1.94895i −0.965926 − 0.258819i −1.53657 − 1.90761i −2.53408 − 1.46305i 1.21941 − 2.55206i 2.71289 + 1.28072i 0.840947 + 1.13702i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.96
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
16.e even 4 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.db.a ✓ 384
9.c even 3 1 inner 720.2.db.a ✓ 384
16.e even 4 1 inner 720.2.db.a ✓ 384
144.x even 12 1 inner 720.2.db.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.db.a ✓ 384 1.a even 1 1 trivial
720.2.db.a ✓ 384 9.c even 3 1 inner
720.2.db.a ✓ 384 16.e even 4 1 inner
720.2.db.a ✓ 384 144.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).