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Properties

Label 720.2.cq.a
Level $720$
Weight $2$
Character orbit 720.cq
Analytic conductor $5.749$
Analytic rank $0$
Dimension $560$
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(173,720)]
 
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, 2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.cq (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: \(560\)
Relative dimension: \(140\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 6 q^{2} - 6 q^{5} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 6 q^{2} - 6 q^{5} - 8 q^{6} - 12 q^{11} - 10 q^{12} - 8 q^{15} - 4 q^{16} + 20 q^{18} - 6 q^{20} - 8 q^{21} + 6 q^{22} - 24 q^{24} + 8 q^{28} + 28 q^{30} - 8 q^{31} + 54 q^{32} - 8 q^{33} - 12 q^{34} + 8 q^{36} - 18 q^{38} - 24 q^{39} + 8 q^{40} - 46 q^{42} - 4 q^{43} - 14 q^{45} - 40 q^{46} - 12 q^{47} - 22 q^{48} - 6 q^{50} - 8 q^{51} - 18 q^{52} + 24 q^{54} - 12 q^{56} + 24 q^{57} - 10 q^{58} - 88 q^{60} - 4 q^{61} + 20 q^{63} - 24 q^{64} - 12 q^{65} + 24 q^{66} - 4 q^{67} - 30 q^{68} + 12 q^{69} + 2 q^{70} - 22 q^{72} + 24 q^{74} - 66 q^{75} + 8 q^{76} - 12 q^{77} - 122 q^{78} - 16 q^{81} - 12 q^{84} - 12 q^{85} - 12 q^{86} + 12 q^{87} - 10 q^{88} - 94 q^{90} - 16 q^{91} - 120 q^{92} - 12 q^{95} + 48 q^{96} - 4 q^{97} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −1.41421 + 0.00351547i −1.59675 − 0.671116i 1.99998 − 0.00994323i 0.792144 + 2.09105i 2.26049 + 0.943485i 4.41987 − 1.18430i −2.82835 + 0.0210927i 2.09921 + 2.14321i −1.12761 − 2.95440i
173.2 −1.41219 − 0.0755346i −1.07340 + 1.35934i 1.98859 + 0.213339i −1.44309 + 1.70806i 1.61853 − 1.83857i −0.233709 + 0.0626222i −2.79216 − 0.451484i −0.695608 − 2.91824i 2.16695 − 2.30311i
173.3 −1.41125 − 0.0915137i −1.21160 − 1.23775i 1.98325 + 0.258297i 0.142911 − 2.23150i 1.59659 + 1.85766i 1.80078 − 0.482518i −2.77522 − 0.546017i −0.0640682 + 2.99932i −0.405896 + 3.13612i
173.4 −1.41100 − 0.0952149i −0.823073 − 1.52399i 1.98187 + 0.268697i 2.07542 + 0.832256i 1.01625 + 2.22873i −3.99488 + 1.07043i −2.77084 − 0.567837i −1.64510 + 2.50871i −2.84918 − 1.37193i
173.5 −1.40690 + 0.143627i 1.70215 − 0.320465i 1.95874 − 0.404137i −0.747438 − 2.10745i −2.34872 + 0.695336i 0.426653 − 0.114321i −2.69771 + 0.849908i 2.79460 − 1.09096i 1.35426 + 2.85762i
173.6 −1.40664 − 0.146199i 1.16660 + 1.28025i 1.95725 + 0.411298i −1.86599 + 1.23211i −1.45381 − 1.97140i −2.96377 + 0.794141i −2.69301 − 0.864695i −0.278082 + 2.98708i 2.80490 − 1.46032i
173.7 −1.40201 + 0.185396i 0.925575 − 1.46401i 1.93126 − 0.519854i 2.11985 − 0.711505i −1.02624 + 2.22415i 3.06030 − 0.820005i −2.61126 + 1.08689i −1.28662 − 2.71009i −2.84014 + 1.39055i
173.8 −1.40148 + 0.189382i −1.35263 + 1.08184i 1.92827 − 0.530828i 2.05945 + 0.871024i 1.69080 − 1.77234i −2.15913 + 0.578537i −2.60189 + 1.10912i 0.659238 − 2.92667i −3.05122 − 0.830698i
173.9 −1.40067 + 0.195247i 0.492103 − 1.66067i 1.92376 − 0.546952i −1.75055 − 1.39125i −0.365034 + 2.42214i −4.31147 + 1.15526i −2.58776 + 1.14171i −2.51567 − 1.63445i 2.72358 + 1.60689i
173.10 −1.38181 − 0.300999i 0.0466910 + 1.73142i 1.81880 + 0.831846i 0.784513 − 2.09393i 0.456637 − 2.40655i −0.988857 + 0.264963i −2.26285 − 1.69691i −2.99564 + 0.161684i −1.71432 + 2.65728i
173.11 −1.36843 − 0.356938i 0.114404 + 1.72827i 1.74519 + 0.976889i 0.267661 + 2.21999i 0.460331 − 2.40585i 3.21344 − 0.861040i −2.03948 − 1.95973i −2.97382 + 0.395443i 0.426125 − 3.13344i
173.12 −1.36664 + 0.363715i 1.70669 − 0.295293i 1.73542 − 0.994137i −0.118386 + 2.23293i −2.22504 + 1.02431i −2.12221 + 0.568645i −2.01012 + 1.98983i 2.82560 − 1.00795i −0.650359 − 3.09468i
173.13 −1.36072 − 0.385272i 1.60971 − 0.639410i 1.70313 + 1.04850i −2.03912 + 0.917596i −2.43671 + 0.249885i 3.92493 − 1.05168i −1.91353 − 2.08288i 2.18231 − 2.05853i 3.12820 − 0.462976i
173.14 −1.33591 + 0.464066i −1.69698 + 0.346801i 1.56929 − 1.23990i −2.22855 + 0.183196i 2.10606 − 1.25080i 1.37032 − 0.367175i −1.52102 + 2.38464i 2.75946 − 1.17703i 2.89212 − 1.27893i
173.15 −1.32706 + 0.488794i −0.132011 + 1.72701i 1.52216 − 1.29732i −1.20259 − 1.88514i −0.668967 − 2.35637i 1.44560 − 0.387347i −1.38587 + 2.46563i −2.96515 − 0.455971i 2.51736 + 1.91387i
173.16 −1.31959 + 0.508601i 0.346998 − 1.69694i 1.48265 − 1.34229i −0.826514 + 2.07771i 0.405167 + 2.41575i 0.910045 − 0.243846i −1.27380 + 2.52536i −2.75918 − 1.17767i 0.0339376 − 3.16210i
173.17 −1.30498 − 0.545000i −1.69430 − 0.359639i 1.40595 + 1.42243i −1.85901 − 1.24261i 2.01503 + 1.39272i −1.49150 + 0.399646i −1.05952 − 2.62248i 2.74132 + 1.21867i 1.74875 + 2.63474i
173.18 −1.29878 − 0.559606i 1.72857 + 0.109827i 1.37368 + 1.45362i 1.89741 + 1.18314i −2.18357 − 1.10996i −0.405417 + 0.108631i −0.970663 − 2.65665i 2.97588 + 0.379688i −1.80224 − 2.59845i
173.19 −1.29466 + 0.569091i −1.73119 − 0.0547502i 1.35227 − 1.47356i 0.776792 − 2.09681i 2.27245 − 0.914320i −2.64311 + 0.708218i −0.912138 + 2.67731i 2.99400 + 0.189565i 0.187596 + 3.15671i
173.20 −1.28655 + 0.587188i 1.16745 + 1.27948i 1.31042 − 1.51089i 1.28573 + 1.82946i −2.25328 − 0.960598i 1.09060 − 0.292224i −0.798745 + 2.71330i −0.274120 + 2.98745i −2.72839 − 1.59872i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
80.i odd 4 1 inner
720.cq 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.cq.a yes 560
5.c odd 4 1 720.2.cm.a ✓ 560
9.d odd 6 1 inner 720.2.cq.a yes 560
16.e even 4 1 720.2.cm.a ✓ 560
45.l even 12 1 720.2.cm.a ✓ 560
80.i odd 4 1 inner 720.2.cq.a yes 560
144.w odd 12 1 720.2.cm.a ✓ 560
720.cq even 12 1 inner 720.2.cq.a yes 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.cm.a ✓ 560 5.c odd 4 1
720.2.cm.a ✓ 560 16.e even 4 1
720.2.cm.a ✓ 560 45.l even 12 1
720.2.cm.a ✓ 560 144.w odd 12 1
720.2.cq.a yes 560 1.a even 1 1 trivial
720.2.cq.a yes 560 9.d odd 6 1 inner
720.2.cq.a yes 560 80.i odd 4 1 inner
720.2.cq.a yes 560 720.cq even 12 1 inner

Hecke kernels

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