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Properties

Label 729.2.i.a
Level $729$
Weight $2$
Character orbit 729.i
Analytic conductor $5.821$
Analytic rank $0$
Dimension $1404$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,2,Mod(10,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(162))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.i (of order \(81\), degree \(54\), not 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: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(1404\)
Relative dimension: \(26\) over \(\Q(\zeta_{81})\)
Twist minimal: no (minimal twist has level 243)
Sato-Tate group: $\mathrm{SU}(2)[C_{81}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1404 q + 54 q^{2} - 54 q^{4} + 54 q^{5} - 54 q^{7} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1404 q + 54 q^{2} - 54 q^{4} + 54 q^{5} - 54 q^{7} + 54 q^{8} - 54 q^{10} + 54 q^{11} - 54 q^{13} + 54 q^{14} - 54 q^{16} + 54 q^{17} - 54 q^{19} + 54 q^{20} - 54 q^{22} + 54 q^{23} - 54 q^{25} + 54 q^{26} - 54 q^{28} + 54 q^{29} - 54 q^{31} + 54 q^{32} - 54 q^{34} + 54 q^{35} - 54 q^{37} + 54 q^{38} - 54 q^{40} + 54 q^{41} - 54 q^{43} + 54 q^{44} - 54 q^{46} + 54 q^{47} - 54 q^{49} + 54 q^{50} - 54 q^{52} + 54 q^{53} - 54 q^{55} + 54 q^{56} - 54 q^{58} + 54 q^{59} - 54 q^{61} + 54 q^{62} - 54 q^{64} - 54 q^{67} - 135 q^{68} - 54 q^{70} - 54 q^{71} - 54 q^{73} - 162 q^{74} - 54 q^{76} - 162 q^{77} - 54 q^{79} - 351 q^{80} - 27 q^{82} - 54 q^{83} - 54 q^{85} - 162 q^{86} - 54 q^{88} - 81 q^{89} - 54 q^{91} - 270 q^{92} - 54 q^{94} - 54 q^{95} - 54 q^{97} - 54 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.66910 − 0.417440i 0 5.04536 + 1.61773i 2.30401 + 1.04730i 0 0.333383 + 2.44079i −7.96289 − 3.99911i 0 −5.71246 − 3.75714i
10.2 −2.39830 − 0.375087i 0 3.70666 + 1.18849i −2.06902 − 0.940484i 0 −0.267953 − 1.96176i −4.10538 − 2.06180i 0 4.60936 + 3.03163i
10.3 −2.24863 − 0.351679i 0 3.02815 + 0.970937i −0.139024 − 0.0631943i 0 0.0427965 + 0.313326i −2.39997 − 1.20531i 0 0.290389 + 0.190992i
10.4 −1.82656 − 0.285668i 0 1.35020 + 0.432926i 0.984150 + 0.447351i 0 0.0949117 + 0.694877i 0.961673 + 0.482970i 0 −1.66981 − 1.09825i
10.5 −1.71636 − 0.268434i 0 0.969337 + 0.310805i 2.65183 + 1.20540i 0 −0.504558 − 3.69402i 1.52458 + 0.765673i 0 −4.22791 − 2.78074i
10.6 −1.66709 − 0.260728i 0 0.806720 + 0.258664i −3.14934 − 1.43155i 0 0.253071 + 1.85280i 1.73832 + 0.873017i 0 4.87699 + 3.20765i
10.7 −1.44758 − 0.226398i 0 0.139748 + 0.0448082i −3.00496 − 1.36592i 0 −0.603573 − 4.41893i 2.42652 + 1.21864i 0 4.04068 + 2.65760i
10.8 −1.35949 − 0.212621i 0 −0.101477 − 0.0325373i 3.22186 + 1.46451i 0 0.584445 + 4.27889i 2.59035 + 1.30092i 0 −4.06871 − 2.67603i
10.9 −1.11070 − 0.173710i 0 −0.701016 − 0.224772i −0.235615 − 0.107100i 0 0.578210 + 4.23325i 2.74882 + 1.38051i 0 0.243093 + 0.159885i
10.10 −0.593472 − 0.0928174i 0 −1.56090 − 0.500483i 0.621532 + 0.282521i 0 −0.302306 − 2.21327i 1.95348 + 0.981077i 0 −0.342639 − 0.225357i
10.11 −0.506689 − 0.0792447i 0 −1.65404 − 0.530347i −2.15085 − 0.977679i 0 0.325167 + 2.38064i 1.71265 + 0.860127i 0 1.01233 + 0.665822i
10.12 −0.269146 − 0.0420937i 0 −1.83383 − 0.587993i −0.567739 − 0.258069i 0 −0.486494 − 3.56177i 0.955700 + 0.479971i 0 0.141942 + 0.0933565i
10.13 0.172340 + 0.0269534i 0 −1.87552 − 0.601362i 3.61353 + 1.64255i 0 −0.239963 − 1.75684i −0.618779 − 0.310762i 0 0.578482 + 0.380474i
10.14 0.378569 + 0.0592071i 0 −1.76469 − 0.565824i −1.07712 − 0.489610i 0 0.336279 + 2.46200i −1.31938 − 0.662619i 0 −0.378775 − 0.249124i
10.15 0.504222 + 0.0788589i 0 −1.65647 − 0.531127i −3.35970 − 1.52717i 0 0.174210 + 1.27544i −1.70548 − 0.856524i 0 −1.57360 − 1.03498i
10.16 0.545687 + 0.0853439i 0 −1.61401 − 0.517510i 2.31192 + 1.05090i 0 0.332789 + 2.43645i −1.82372 − 0.915906i 0 1.17190 + 0.770770i
10.17 0.940500 + 0.147092i 0 −1.04159 − 0.333973i −0.357057 − 0.162302i 0 −0.150392 − 1.10107i −2.63185 − 1.32176i 0 −0.311939 − 0.205165i
10.18 1.31239 + 0.205254i 0 −0.224260 − 0.0719060i −1.56403 − 0.710937i 0 0.108987 + 0.797926i −2.65366 − 1.33272i 0 −1.90669 − 1.25405i
10.19 1.42734 + 0.223232i 0 0.0829775 + 0.0266057i 2.36005 + 1.07277i 0 −0.546388 − 4.00027i −2.46955 − 1.24026i 0 3.12912 + 2.05805i
10.20 1.58827 + 0.248401i 0 0.556399 + 0.178402i −0.870960 − 0.395900i 0 −0.334115 − 2.44616i −2.03377 − 1.02140i 0 −1.28498 − 0.845142i
See next 80 embeddings (of 1404 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
243.i even 81 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.i.a 1404
3.b odd 2 1 243.2.i.a ✓ 1404
243.i even 81 1 inner 729.2.i.a 1404
243.j odd 162 1 243.2.i.a ✓ 1404
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.i.a ✓ 1404 3.b odd 2 1
243.2.i.a ✓ 1404 243.j odd 162 1
729.2.i.a 1404 1.a even 1 1 trivial
729.2.i.a 1404 243.i even 81 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(729, [\chi])\).