↑

Properties

Label 555.2.bc.d
Level $555$
Weight $2$
Character orbit 555.bc
Analytic conductor $4.432$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [555,2,Mod(16,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bc (of order \(9\), degree \(6\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} + 12 q^{6} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} + 12 q^{6} - 21 q^{8} + 6 q^{10} + 9 q^{11} - 6 q^{12} - 15 q^{13} - 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{18} + 9 q^{19} - 3 q^{20} - 30 q^{22} - 9 q^{23} - 3 q^{24} - 36 q^{26} - 18 q^{27} - 48 q^{28} - 36 q^{29} + 6 q^{30} + 36 q^{31} - 30 q^{32} - 3 q^{33} + 60 q^{34} - 9 q^{35} + 36 q^{36} - 15 q^{37} + 12 q^{38} + 12 q^{39} + 3 q^{40} + 3 q^{41} - 18 q^{42} - 30 q^{43} - 90 q^{44} + 18 q^{45} - 3 q^{46} - 12 q^{47} - 18 q^{48} - 18 q^{49} + 3 q^{50} - 27 q^{51} + 75 q^{52} + 18 q^{53} + 3 q^{54} - 6 q^{55} + 126 q^{56} - 9 q^{57} - 36 q^{58} - 12 q^{59} + 18 q^{60} + 30 q^{61} + 12 q^{62} + 9 q^{63} - 3 q^{64} + 15 q^{65} + 15 q^{67} + 60 q^{68} + 12 q^{69} - 9 q^{70} - 30 q^{71} - 3 q^{72} - 42 q^{73} - 123 q^{74} + 36 q^{75} + 72 q^{76} + 27 q^{77} + 39 q^{78} + 45 q^{79} - 36 q^{80} - 15 q^{82} - 21 q^{83} + 21 q^{84} + 27 q^{85} + 9 q^{86} + 3 q^{87} + 21 q^{88} + 72 q^{89} + 6 q^{90} + 72 q^{91} - 45 q^{92} + 6 q^{93} - 18 q^{94} - 30 q^{96} - 45 q^{97} + 6 q^{98} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.57990 − 0.939008i −0.939693 + 0.342020i 4.24208 + 3.55952i −0.173648 + 0.984808i 2.74548 −0.803318 + 4.55584i −4.85625 − 8.41127i 0.766044 − 0.642788i 1.37274 − 2.37765i
16.2 −1.87276 − 0.681628i −0.939693 + 0.342020i 1.51051 + 1.26747i −0.173648 + 0.984808i 1.99295 0.112817 − 0.639817i 0.0280638 + 0.0486079i 0.766044 − 0.642788i 0.996473 − 1.72594i
16.3 −0.943825 − 0.343524i −0.939693 + 0.342020i −0.759293 − 0.637122i −0.173648 + 0.984808i 1.00440 0.633189 − 3.59099i 1.50217 + 2.60183i 0.766044 − 0.642788i 0.502199 − 0.869833i
16.4 0.584668 + 0.212802i −0.939693 + 0.342020i −1.23554 − 1.03674i −0.173648 + 0.984808i −0.622190 0.0487026 − 0.276206i −1.12395 − 1.94674i 0.766044 − 0.642788i −0.311095 + 0.538833i
16.5 0.978109 + 0.356002i −0.939693 + 0.342020i −0.702130 − 0.589157i −0.173648 + 0.984808i −1.04088 −0.822325 + 4.66363i −1.51790 − 2.62908i 0.766044 − 0.642788i −0.520441 + 0.901430i
16.6 2.28068 + 0.830098i −0.939693 + 0.342020i 2.98033 + 2.50079i −0.173648 + 0.984808i −2.42704 −0.282407 + 1.60161i 2.29422 + 3.97370i 0.766044 − 0.642788i −1.21352 + 2.10188i
46.1 −0.378392 − 2.14597i 0.173648 − 0.984808i −2.58261 + 0.939992i −0.766044 − 0.642788i −2.17907 1.36242 + 1.14321i 0.815359 + 1.41224i −0.939693 − 0.342020i −1.08954 + 1.88713i
46.2 −0.245089 − 1.38997i 0.173648 − 0.984808i 0.00743392 − 0.00270572i −0.766044 − 0.642788i −1.41141 −3.07891 − 2.58351i −1.41700 − 2.45431i −0.939693 − 0.342020i −0.705707 + 1.22232i
46.3 −0.118139 − 0.670001i 0.173648 − 0.984808i 1.44444 − 0.525734i −0.766044 − 0.642788i −0.680336 2.91940 + 2.44967i −1.20322 − 2.08404i −0.939693 − 0.342020i −0.340168 + 0.589189i
46.4 0.0313736 + 0.177928i 0.173648 − 0.984808i 1.84871 − 0.672876i −0.766044 − 0.642788i 0.180673 −0.972021 − 0.815622i 0.358397 + 0.620762i −0.939693 − 0.342020i 0.0903366 − 0.156468i
46.5 0.374910 + 2.12622i 0.173648 − 0.984808i −2.50087 + 0.910242i −0.766044 − 0.642788i 2.15902 −3.21334 − 2.69631i −0.713956 − 1.23661i −0.939693 − 0.342020i 1.07951 − 1.86977i
46.6 0.416589 + 2.36259i 0.173648 − 0.984808i −3.52891 + 1.28442i −0.766044 − 0.642788i 2.39904 2.39004 + 2.00548i −2.10562 − 3.64705i −0.939693 − 0.342020i 1.19952 − 2.07763i
181.1 −0.378392 + 2.14597i 0.173648 + 0.984808i −2.58261 − 0.939992i −0.766044 + 0.642788i −2.17907 1.36242 − 1.14321i 0.815359 − 1.41224i −0.939693 + 0.342020i −1.08954 − 1.88713i
181.2 −0.245089 + 1.38997i 0.173648 + 0.984808i 0.00743392 + 0.00270572i −0.766044 + 0.642788i −1.41141 −3.07891 + 2.58351i −1.41700 + 2.45431i −0.939693 + 0.342020i −0.705707 − 1.22232i
181.3 −0.118139 + 0.670001i 0.173648 + 0.984808i 1.44444 + 0.525734i −0.766044 + 0.642788i −0.680336 2.91940 − 2.44967i −1.20322 + 2.08404i −0.939693 + 0.342020i −0.340168 − 0.589189i
181.4 0.0313736 − 0.177928i 0.173648 + 0.984808i 1.84871 + 0.672876i −0.766044 + 0.642788i 0.180673 −0.972021 + 0.815622i 0.358397 − 0.620762i −0.939693 + 0.342020i 0.0903366 + 0.156468i
181.5 0.374910 − 2.12622i 0.173648 + 0.984808i −2.50087 − 0.910242i −0.766044 + 0.642788i 2.15902 −3.21334 + 2.69631i −0.713956 + 1.23661i −0.939693 + 0.342020i 1.07951 + 1.86977i
181.6 0.416589 − 2.36259i 0.173648 + 0.984808i −3.52891 − 1.28442i −0.766044 + 0.642788i 2.39904 2.39004 − 2.00548i −2.10562 + 3.64705i −0.939693 + 0.342020i 1.19952 + 2.07763i
256.1 −1.47461 − 1.23735i 0.766044 − 0.642788i 0.296157 + 1.67959i 0.939693 + 0.342020i −1.92497 1.12156 + 0.408213i −0.283451 + 0.490951i 0.173648 − 0.984808i −0.962484 − 1.66707i
256.2 −0.322408 − 0.270532i 0.766044 − 0.642788i −0.316537 − 1.79517i 0.939693 + 0.342020i −0.420874 4.49612 + 1.63645i −0.804472 + 1.39339i 0.173648 − 0.984808i −0.210437 − 0.364487i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bc.d ✓ 36
37.f even 9 1 inner 555.2.bc.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bc.d ✓ 36 1.a even 1 1 trivial
555.2.bc.d ✓ 36 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{35} + 3 T_{2}^{34} + 15 T_{2}^{33} - 51 T_{2}^{32} + 78 T_{2}^{31} + 299 T_{2}^{30} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display