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Properties

Label 483.2.y.a
Level $483$
Weight $2$
Character orbit 483.y
Analytic conductor $3.857$
Analytic rank $0$
Dimension $320$
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] = [483,2,Mod(4,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 44, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.y (of order \(33\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(16\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9} + 8 q^{10} - 6 q^{11} - 18 q^{12} - 10 q^{14} + 18 q^{15} + 8 q^{16} + 4 q^{17} + 2 q^{18} + 18 q^{20} + 4 q^{21} - 176 q^{22} - 18 q^{23} - 6 q^{24} + 8 q^{25} - 14 q^{26} + 32 q^{27} + 46 q^{28} + 34 q^{29} - 8 q^{30} + 52 q^{31} - 8 q^{32} - 5 q^{33} - 24 q^{34} - 12 q^{35} - 14 q^{36} - 30 q^{37} - 157 q^{38} + 88 q^{40} - 28 q^{41} - 45 q^{42} + 64 q^{43} - 71 q^{44} - 2 q^{45} + 4 q^{46} + 36 q^{47} + 60 q^{48} + 28 q^{49} + 210 q^{50} - 26 q^{51} - 198 q^{52} - 10 q^{53} - 2 q^{54} - 4 q^{55} + 44 q^{57} + 31 q^{58} + 10 q^{59} - 2 q^{60} - 34 q^{61} - 8 q^{62} + 2 q^{63} + 84 q^{64} + 38 q^{65} - 12 q^{67} - 22 q^{68} + 8 q^{69} - 336 q^{70} - 144 q^{71} + 6 q^{72} - 16 q^{73} - 68 q^{74} - 30 q^{75} + 8 q^{76} + 98 q^{77} + 16 q^{78} + 26 q^{79} + 225 q^{80} + 16 q^{81} - 122 q^{82} + 44 q^{84} - 240 q^{85} - 26 q^{86} - 16 q^{87} + 43 q^{88} + 68 q^{89} - 16 q^{90} + 40 q^{91} - 222 q^{92} - 8 q^{93} + 137 q^{94} - 49 q^{95} + 30 q^{96} - 8 q^{97} + 137 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.59717 + 2.24291i −0.235759 + 0.971812i −1.82557 − 5.27464i −0.128956 + 2.70712i −1.80314 − 2.08094i 2.64469 + 0.0748341i 9.46243 + 2.77842i −0.888835 − 0.458227i −5.86587 − 4.61297i
4.2 −1.42823 + 2.00566i −0.235759 + 0.971812i −1.32872 − 3.83909i 0.0678834 − 1.42505i −1.61241 − 1.86082i −0.624376 + 2.57102i 4.87268 + 1.43075i −0.888835 − 0.458227i 2.76121 + 2.17144i
4.3 −1.09070 + 1.53167i −0.235759 + 0.971812i −0.502260 − 1.45118i −0.0927520 + 1.94710i −1.23135 − 1.42106i 0.850242 − 2.50541i −0.837777 − 0.245993i −0.888835 − 0.458227i −2.88116 − 2.26577i
4.4 −0.971087 + 1.36370i −0.235759 + 0.971812i −0.262535 − 0.758544i −0.194859 + 4.09059i −1.09632 − 1.26522i −2.20100 + 1.46819i −1.92325 − 0.564717i −0.888835 − 0.458227i −5.38912 − 4.23805i
4.5 −0.957403 + 1.34448i −0.235759 + 0.971812i −0.236882 − 0.684426i 0.115570 − 2.42610i −1.08087 − 1.24739i 2.21964 − 1.43986i −2.02036 − 0.593231i −0.888835 − 0.458227i 3.15121 + 2.47814i
4.6 −0.851765 + 1.19614i −0.235759 + 0.971812i −0.0511029 − 0.147652i 0.117190 − 2.46013i −0.961608 − 1.10975i −2.08452 − 1.62935i −2.59773 − 0.762762i −0.888835 − 0.458227i 2.84283 + 2.23562i
4.7 −0.364327 + 0.511625i −0.235759 + 0.971812i 0.525109 + 1.51720i −0.00931300 + 0.195504i −0.411310 − 0.474677i 0.694215 + 2.55305i −2.17284 − 0.638004i −0.888835 − 0.458227i −0.0966318 − 0.0759921i
4.8 −0.264798 + 0.371856i −0.235759 + 0.971812i 0.585977 + 1.69307i 0.162988 − 3.42155i −0.298946 − 0.345002i −2.64103 − 0.158003i −1.66077 − 0.487645i −0.888835 − 0.458227i 1.22916 + 0.966626i
4.9 0.282147 − 0.396220i −0.235759 + 0.971812i 0.576752 + 1.66642i −0.0698936 + 1.46725i 0.318533 + 0.367606i −2.64518 + 0.0550715i 1.75642 + 0.515730i −0.888835 − 0.458227i 0.561633 + 0.441673i
4.10 0.327609 − 0.460062i −0.235759 + 0.971812i 0.549806 + 1.58856i −0.198810 + 4.17354i 0.369857 + 0.426838i 2.54045 + 0.738979i 1.99478 + 0.585719i −0.888835 − 0.458227i 1.85495 + 1.45875i
4.11 0.466737 − 0.655441i −0.235759 + 0.971812i 0.442377 + 1.27816i −0.00881404 + 0.185029i 0.526928 + 0.608107i 1.59903 − 2.10787i 2.58833 + 0.760002i −0.888835 − 0.458227i 0.117162 + 0.0921373i
4.12 0.618045 − 0.867923i −0.235759 + 0.971812i 0.282825 + 0.817169i 0.184708 − 3.87751i 0.697748 + 0.805244i 0.447567 + 2.60762i 2.92870 + 0.859944i −0.888835 − 0.458227i −3.25122 − 2.55679i
4.13 0.910508 − 1.27863i −0.235759 + 0.971812i −0.151733 − 0.438405i 0.0858188 − 1.80156i 1.02793 + 1.18629i −0.649635 − 2.56476i 2.31350 + 0.679304i −0.888835 − 0.458227i −2.22539 − 1.75007i
4.14 1.13327 − 1.59145i −0.235759 + 0.971812i −0.594283 − 1.71707i −0.0776676 + 1.63044i 1.27941 + 1.47652i −1.33962 + 2.28154i 0.343047 + 0.100728i −0.888835 − 0.458227i 2.50675 + 1.97133i
4.15 1.47255 − 2.06790i −0.235759 + 0.971812i −1.45369 − 4.20016i 0.132912 − 2.79016i 1.66245 + 1.91856i 2.51171 − 0.831465i −5.95457 − 1.74842i −0.888835 − 0.458227i −5.57407 − 4.38350i
4.16 1.55491 − 2.18356i −0.235759 + 0.971812i −1.69606 − 4.90045i 0.00530352 − 0.111335i 1.75543 + 2.02587i −2.64473 − 0.0735467i −8.19360 − 2.40586i −0.888835 − 0.458227i −0.234859 − 0.184695i
16.1 −0.843018 − 2.43574i 0.888835 + 0.458227i −3.65004 + 2.87043i −0.918397 − 0.0876963i 0.366817 − 2.55126i −0.232163 − 2.63555i 5.73200 + 3.68373i 0.580057 + 0.814576i 0.560620 + 2.31091i
16.2 −0.783498 − 2.26377i 0.888835 + 0.458227i −2.93867 + 2.31100i −2.65564 − 0.253583i 0.340918 − 2.37114i −1.79847 + 1.94049i 3.50353 + 2.25158i 0.580057 + 0.814576i 1.50664 + 6.21044i
16.3 −0.741922 − 2.14364i 0.888835 + 0.458227i −2.47265 + 1.94451i 1.61087 + 0.153820i 0.322827 − 2.24531i 2.64162 − 0.147800i 2.18625 + 1.40502i 0.580057 + 0.814576i −0.865407 − 3.56726i
16.4 −0.461078 − 1.33220i 0.888835 + 0.458227i 0.00994599 − 0.00782161i 3.91624 + 0.373956i 0.200626 − 1.39538i −1.00683 − 2.44669i −2.38689 − 1.53396i 0.580057 + 0.814576i −1.30751 − 5.38963i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.y.a ✓ 320
7.c even 3 1 inner 483.2.y.a ✓ 320
23.c even 11 1 inner 483.2.y.a ✓ 320
161.m even 33 1 inner 483.2.y.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.y.a ✓ 320 1.a even 1 1 trivial
483.2.y.a ✓ 320 7.c even 3 1 inner
483.2.y.a ✓ 320 23.c even 11 1 inner
483.2.y.a ✓ 320 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{320} - 2 T_{2}^{319} - 23 T_{2}^{318} + 58 T_{2}^{317} + 200 T_{2}^{316} - 716 T_{2}^{315} + \cdots + 15\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\). Copy content Toggle raw display