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Properties

Label 555.2.bc.b
Level $555$
Weight $2$
Character orbit 555.bc
Analytic conductor $4.432$
Analytic rank $0$
Dimension $36$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 12 q^{6} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 12 q^{6} + 12 q^{7} + 15 q^{8} + 6 q^{10} + 3 q^{11} - 6 q^{12} + 3 q^{13} + 21 q^{16} - 15 q^{17} + 3 q^{18} - 15 q^{19} + 3 q^{20} + 12 q^{21} - 12 q^{22} + 9 q^{23} + 15 q^{24} + 48 q^{26} - 18 q^{27} - 30 q^{28} + 18 q^{29} - 6 q^{30} - 12 q^{31} - 72 q^{32} - 9 q^{33} - 48 q^{34} + 3 q^{35} + 36 q^{36} - 21 q^{37} + 48 q^{38} - 24 q^{39} + 15 q^{40} - 3 q^{41} - 18 q^{42} - 42 q^{43} - 42 q^{44} - 18 q^{45} + 45 q^{46} + 12 q^{47} - 18 q^{48} + 18 q^{49} + 3 q^{50} + 21 q^{51} + 21 q^{52} - 12 q^{53} + 3 q^{54} + 18 q^{55} - 30 q^{56} + 3 q^{57} + 24 q^{58} + 6 q^{59} - 18 q^{60} - 6 q^{61} - 18 q^{62} + 3 q^{63} - 75 q^{64} + 3 q^{65} + 12 q^{66} - 9 q^{67} - 156 q^{68} + 6 q^{69} + 9 q^{70} + 6 q^{71} + 15 q^{72} + 30 q^{73} - 45 q^{74} + 36 q^{75} - 24 q^{76} - 3 q^{77} - 3 q^{78} - 51 q^{79} + 36 q^{80} - 39 q^{82} + 63 q^{83} + 3 q^{84} + 21 q^{85} + 51 q^{86} + 33 q^{87} - 75 q^{88} + 24 q^{89} - 6 q^{90} - 72 q^{91} + 87 q^{92} - 6 q^{93} - 6 q^{94} + 12 q^{95} - 72 q^{96} + 15 q^{97} + 96 q^{98} + 18 q^{99}+O(q^{100}) \) 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 −1.83739 − 0.668754i −0.939693 + 0.342020i 1.39667 + 1.17195i 0.173648 − 0.984808i 1.95531 0.398216 − 2.25840i 0.172824 + 0.299340i 0.766044 − 0.642788i −0.977654 + 1.69335i
16.2 −1.69492 − 0.616899i −0.939693 + 0.342020i 0.960090 + 0.805612i 0.173648 − 0.984808i 1.80369 −0.689225 + 3.90879i 0.673401 + 1.16636i 0.766044 − 0.642788i −0.901847 + 1.56204i
16.3 0.135196 + 0.0492072i −0.939693 + 0.342020i −1.51623 − 1.27227i 0.173648 − 0.984808i −0.143872 −0.0394454 + 0.223706i −0.286256 − 0.495809i 0.766044 − 0.642788i 0.0719361 − 0.124597i
16.4 1.44876 + 0.527305i −0.939693 + 0.342020i 0.288763 + 0.242301i 0.173648 − 0.984808i −1.54174 0.272364 − 1.54465i −1.25116 − 2.16707i 0.766044 − 0.642788i 0.770869 − 1.33518i
16.5 1.54799 + 0.563421i −0.939693 + 0.342020i 0.546732 + 0.458763i 0.173648 − 0.984808i −1.64733 −0.826864 + 4.68938i −1.05948 − 1.83507i 0.766044 − 0.642788i 0.823667 − 1.42663i
16.6 2.60610 + 0.948543i −0.939693 + 0.342020i 4.35993 + 3.65842i 0.173648 − 0.984808i −2.77335 0.239526 − 1.35842i 5.11890 + 8.86620i 0.766044 − 0.642788i 1.38668 − 2.40179i
46.1 −0.485786 − 2.75503i 0.173648 − 0.984808i −5.47483 + 1.99267i 0.766044 + 0.642788i −2.79753 3.13750 + 2.63268i 5.35195 + 9.26984i −0.939693 − 0.342020i 1.39877 − 2.42273i
46.2 −0.383979 − 2.17765i 0.173648 − 0.984808i −2.71535 + 0.988307i 0.766044 + 0.642788i −2.21125 −1.49189 − 1.25184i 0.983581 + 1.70361i −0.939693 − 0.342020i 1.10562 − 1.91500i
46.3 −0.141502 − 0.802498i 0.173648 − 0.984808i 1.25541 − 0.456930i 0.766044 + 0.642788i −0.814878 −1.82670 − 1.53278i −1.35921 − 2.35421i −0.939693 − 0.342020i 0.407439 − 0.705705i
46.4 −0.0760254 − 0.431162i 0.173648 − 0.984808i 1.69926 − 0.618482i 0.766044 + 0.642788i −0.437813 2.26854 + 1.90353i −0.833666 − 1.44395i −0.939693 − 0.342020i 0.218906 − 0.379157i
46.5 0.163670 + 0.928217i 0.173648 − 0.984808i 1.04459 − 0.380198i 0.766044 + 0.642788i 0.942537 0.806987 + 0.677142i 1.46641 + 2.53990i −0.939693 − 0.342020i −0.471268 + 0.816261i
46.6 0.310282 + 1.75970i 0.173648 − 0.984808i −1.12088 + 0.407967i 0.766044 + 0.642788i 1.78685 0.392550 + 0.329389i 0.721156 + 1.24908i −0.939693 − 0.342020i −0.893423 + 1.54745i
181.1 −0.485786 + 2.75503i 0.173648 + 0.984808i −5.47483 − 1.99267i 0.766044 − 0.642788i −2.79753 3.13750 − 2.63268i 5.35195 − 9.26984i −0.939693 + 0.342020i 1.39877 + 2.42273i
181.2 −0.383979 + 2.17765i 0.173648 + 0.984808i −2.71535 − 0.988307i 0.766044 − 0.642788i −2.21125 −1.49189 + 1.25184i 0.983581 − 1.70361i −0.939693 + 0.342020i 1.10562 + 1.91500i
181.3 −0.141502 + 0.802498i 0.173648 + 0.984808i 1.25541 + 0.456930i 0.766044 − 0.642788i −0.814878 −1.82670 + 1.53278i −1.35921 + 2.35421i −0.939693 + 0.342020i 0.407439 + 0.705705i
181.4 −0.0760254 + 0.431162i 0.173648 + 0.984808i 1.69926 + 0.618482i 0.766044 − 0.642788i −0.437813 2.26854 − 1.90353i −0.833666 + 1.44395i −0.939693 + 0.342020i 0.218906 + 0.379157i
181.5 0.163670 − 0.928217i 0.173648 + 0.984808i 1.04459 + 0.380198i 0.766044 − 0.642788i 0.942537 0.806987 − 0.677142i 1.46641 − 2.53990i −0.939693 + 0.342020i −0.471268 − 0.816261i
181.6 0.310282 − 1.75970i 0.173648 + 0.984808i −1.12088 − 0.407967i 0.766044 − 0.642788i 1.78685 0.392550 − 0.329389i 0.721156 − 1.24908i −0.939693 + 0.342020i −0.893423 − 1.54745i
256.1 −1.76720 − 1.48286i 0.766044 − 0.642788i 0.576837 + 3.27141i −0.939693 − 0.342020i −2.30692 2.45772 + 0.894538i 1.52473 − 2.64090i 0.173648 − 0.984808i 1.15346 + 1.99785i
256.2 −0.965174 − 0.809877i 0.766044 − 0.642788i −0.0716366 − 0.406272i −0.939693 − 0.342020i −1.25994 3.26300 + 1.18763i −1.51983 + 2.63243i 0.173648 − 0.984808i 0.629972 + 1.09114i
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.b ✓ 36
37.f even 9 1 inner 555.2.bc.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bc.b ✓ 36 1.a even 1 1 trivial
555.2.bc.b ✓ 36 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{35} + 3 T_{2}^{34} - 13 T_{2}^{33} + 27 T_{2}^{32} + 78 T_{2}^{31} + 67 T_{2}^{30} + \cdots + 3249 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display