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Properties

Label 555.2.bc.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 3 q^{4} - 6 q^{7} - 3 q^{8} - 9 q^{11} + 6 q^{12} + 9 q^{13} - 12 q^{14} + 21 q^{16} + 21 q^{17} + 3 q^{18} + 21 q^{19} + 3 q^{20} + 6 q^{21} + 36 q^{22} + 3 q^{23} + 3 q^{24} - 48 q^{26} + 18 q^{27} - 36 q^{28} + 12 q^{29} + 6 q^{30} + 36 q^{31} - 30 q^{32} - 3 q^{33} - 48 q^{34} + 3 q^{35} + 60 q^{36} - 57 q^{37} - 72 q^{38} + 18 q^{39} - 3 q^{40} + 45 q^{41} - 6 q^{42} + 90 q^{43} - 18 q^{44} - 18 q^{45} - 27 q^{46} + 18 q^{47} + 18 q^{48} + 18 q^{49} + 3 q^{50} + 9 q^{51} - 33 q^{52} + 6 q^{53} - 3 q^{54} + 12 q^{55} + 90 q^{56} - 3 q^{57} + 6 q^{58} + 18 q^{59} + 30 q^{60} - 42 q^{61} - 126 q^{62} + 9 q^{63} - 15 q^{64} + 9 q^{65} - 27 q^{67} + 12 q^{68} + 18 q^{69} - 3 q^{70} - 18 q^{71} - 3 q^{72} - 78 q^{73} + 57 q^{74} - 36 q^{75} + 12 q^{76} - 63 q^{77} + 33 q^{78} - 51 q^{79} + 36 q^{80} - 9 q^{82} - 39 q^{83} + 9 q^{84} - 9 q^{85} + 57 q^{86} - 9 q^{87} - 63 q^{88} - 18 q^{89} - 6 q^{90} + 84 q^{91} + 57 q^{92} + 6 q^{93} - 126 q^{94} - 24 q^{95} + 30 q^{96} - 27 q^{97} + 30 q^{98} + 12 q^{99}+O(q^{100}) \) 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 −2.45937 − 0.895139i 0.939693 − 0.342020i 3.71516 + 3.11739i 0.173648 − 0.984808i −2.61721 −0.569456 + 3.22955i −3.72926 − 6.45926i 0.766044 − 0.642788i −1.30861 + 2.26657i
16.2 −1.84769 − 0.672505i 0.939693 − 0.342020i 1.42961 + 1.19959i 0.173648 − 0.984808i −1.96627 0.671507 − 3.80831i 0.131517 + 0.227794i 0.766044 − 0.642788i −0.983136 + 1.70284i
16.3 −0.902500 − 0.328483i 0.939693 − 0.342020i −0.825484 − 0.692663i 0.173648 − 0.984808i −0.960420 −0.207651 + 1.17765i 1.47789 + 2.55978i 0.766044 − 0.642788i −0.480210 + 0.831748i
16.4 1.21754 + 0.443148i 0.939693 − 0.342020i −0.246066 − 0.206474i 0.173648 − 0.984808i 1.29568 −0.0187207 + 0.106170i −1.50378 − 2.60462i 0.766044 − 0.642788i 0.647839 − 1.12209i
16.5 2.12427 + 0.773169i 0.939693 − 0.342020i 2.38262 + 1.99926i 0.173648 − 0.984808i 2.26060 0.446624 − 2.53293i 1.25496 + 2.17366i 0.766044 − 0.642788i 1.13030 − 1.95773i
16.6 2.19411 + 0.798592i 0.939693 − 0.342020i 2.64429 + 2.21883i 0.173648 − 0.984808i 2.33493 −0.903555 + 5.12432i 1.69502 + 2.93585i 0.766044 − 0.642788i 1.16746 − 2.02211i
46.1 −0.456572 − 2.58935i −0.173648 + 0.984808i −4.61689 + 1.68041i 0.766044 + 0.642788i 2.62930 −0.424039 − 0.355811i 3.82982 + 6.63344i −0.939693 − 0.342020i 1.31465 − 2.27704i
46.2 −0.337824 − 1.91590i −0.173648 + 0.984808i −1.67715 + 0.610432i 0.766044 + 0.642788i 1.94545 1.11238 + 0.933394i −0.209347 − 0.362600i −0.939693 − 0.342020i 0.972726 − 1.68481i
46.3 −0.187109 − 1.06115i −0.173648 + 0.984808i 0.788357 − 0.286939i 0.766044 + 0.642788i 1.07752 −4.01790 − 3.37142i −1.52951 − 2.64919i −0.939693 − 0.342020i 0.538760 − 0.933159i
46.4 −0.0313002 − 0.177512i −0.173648 + 0.984808i 1.84885 − 0.672928i 0.766044 + 0.642788i 0.180251 −0.366857 − 0.307829i −0.357573 − 0.619335i −0.939693 − 0.342020i 0.0901253 − 0.156102i
46.5 0.285319 + 1.61813i −0.173648 + 0.984808i −0.657543 + 0.239326i 0.766044 + 0.642788i −1.64309 −0.290414 − 0.243686i 1.06822 + 1.85021i −0.939693 − 0.342020i −0.821545 + 1.42296i
46.6 0.461442 + 2.61697i −0.173648 + 0.984808i −4.75621 + 1.73112i 0.766044 + 0.642788i −2.65734 0.515051 + 0.432179i −4.06765 − 7.04538i −0.939693 − 0.342020i −1.32867 + 2.30132i
181.1 −0.456572 + 2.58935i −0.173648 − 0.984808i −4.61689 − 1.68041i 0.766044 − 0.642788i 2.62930 −0.424039 + 0.355811i 3.82982 − 6.63344i −0.939693 + 0.342020i 1.31465 + 2.27704i
181.2 −0.337824 + 1.91590i −0.173648 − 0.984808i −1.67715 − 0.610432i 0.766044 − 0.642788i 1.94545 1.11238 − 0.933394i −0.209347 + 0.362600i −0.939693 + 0.342020i 0.972726 + 1.68481i
181.3 −0.187109 + 1.06115i −0.173648 − 0.984808i 0.788357 + 0.286939i 0.766044 − 0.642788i 1.07752 −4.01790 + 3.37142i −1.52951 + 2.64919i −0.939693 + 0.342020i 0.538760 + 0.933159i
181.4 −0.0313002 + 0.177512i −0.173648 − 0.984808i 1.84885 + 0.672928i 0.766044 − 0.642788i 0.180251 −0.366857 + 0.307829i −0.357573 + 0.619335i −0.939693 + 0.342020i 0.0901253 + 0.156102i
181.5 0.285319 − 1.61813i −0.173648 − 0.984808i −0.657543 − 0.239326i 0.766044 − 0.642788i −1.64309 −0.290414 + 0.243686i 1.06822 − 1.85021i −0.939693 + 0.342020i −0.821545 − 1.42296i
181.6 0.461442 − 2.61697i −0.173648 − 0.984808i −4.75621 − 1.73112i 0.766044 − 0.642788i −2.65734 0.515051 − 0.432179i −4.06765 + 7.04538i −0.939693 + 0.342020i −1.32867 − 2.30132i
256.1 −1.81386 − 1.52201i −0.766044 + 0.642788i 0.626282 + 3.55182i −0.939693 − 0.342020i 2.36783 1.93274 + 0.703462i 1.90209 − 3.29452i 0.173648 − 0.984808i 1.18391 + 2.05060i
256.2 −1.35577 − 1.13763i −0.766044 + 0.642788i 0.196622 + 1.11510i −0.939693 − 0.342020i 1.76983 −3.47044 − 1.26314i −0.767841 + 1.32994i 0.173648 − 0.984808i 0.884915 + 1.53272i
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.c ✓ 36
37.f even 9 1 inner 555.2.bc.c ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bc.c ✓ 36 1.a even 1 1 trivial
555.2.bc.c ✓ 36 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{35} + 3 T_{2}^{34} + T_{2}^{33} - 15 T_{2}^{32} + 54 T_{2}^{31} + 243 T_{2}^{30} + \cdots + 1540081 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display