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Properties

Label 555.2.bt.a
Level $555$
Weight $2$
Character orbit 555.bt
Analytic conductor $4.432$
Analytic rank $0$
Dimension $72$
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(136,555)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bt (of order \(18\), 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: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} + 6 q^{4} - 12 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 6 q^{2} + 6 q^{4} - 12 q^{7} + 18 q^{8} - 12 q^{11} - 12 q^{12} + 24 q^{13} + 18 q^{14} - 6 q^{16} - 18 q^{17} + 6 q^{18} + 18 q^{19} - 12 q^{21} - 30 q^{22} - 12 q^{24} + 12 q^{26} - 36 q^{27} - 18 q^{28} + 18 q^{29} + 6 q^{32} - 36 q^{33} - 48 q^{34} + 18 q^{35} - 72 q^{36} + 60 q^{37} + 60 q^{38} + 12 q^{39} + 18 q^{40} - 12 q^{42} - 24 q^{44} - 18 q^{46} - 12 q^{47} - 36 q^{48} + 24 q^{49} - 6 q^{50} + 18 q^{51} - 42 q^{53} - 6 q^{54} + 24 q^{56} + 18 q^{57} - 6 q^{59} + 12 q^{61} - 18 q^{62} - 12 q^{63} - 6 q^{64} + 18 q^{65} - 18 q^{67} - 30 q^{69} + 12 q^{72} - 60 q^{73} - 72 q^{75} + 48 q^{76} - 60 q^{77} + 36 q^{78} + 6 q^{79} + 18 q^{82} - 6 q^{83} + 24 q^{84} - 24 q^{85} + 108 q^{86} + 24 q^{87} + 54 q^{88} - 60 q^{89} - 24 q^{91} - 162 q^{92} + 12 q^{93} + 132 q^{94} - 12 q^{95} - 6 q^{96} + 18 q^{97} + 150 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1 −1.76882 − 2.10800i 0.766044 + 0.642788i −0.967639 + 5.48775i 0.342020 + 0.939693i − 2.75180i −0.431966 + 0.157223i 8.51352 − 4.91528i 0.173648 + 0.984808i 1.37590 − 2.38313i
136.2 −1.45730 − 1.73675i 0.766044 + 0.642788i −0.545260 + 3.09232i −0.342020 − 0.939693i − 2.26716i 1.94023 − 0.706187i 2.23835 − 1.29231i 0.173648 + 0.984808i −1.13358 + 1.96342i
136.3 −1.10266 − 1.31410i 0.766044 + 0.642788i −0.163701 + 0.928394i −0.342020 − 0.939693i − 1.71543i −2.40627 + 0.875812i −1.57071 + 0.906851i 0.173648 + 0.984808i −0.857717 + 1.48561i
136.4 −0.946249 − 1.12770i 0.766044 + 0.642788i −0.0290136 + 0.164545i 0.342020 + 0.939693i − 1.47210i −2.22291 + 0.809072i −2.33674 + 1.34912i 0.173648 + 0.984808i 0.736051 − 1.27488i
136.5 −0.409461 − 0.487977i 0.766044 + 0.642788i 0.276833 − 1.57000i −0.342020 − 0.939693i − 0.637009i −2.61740 + 0.952654i −1.98281 + 1.14477i 0.173648 + 0.984808i −0.318504 + 0.551666i
136.6 −0.121402 − 0.144682i 0.766044 + 0.642788i 0.341102 − 1.93449i 0.342020 + 0.939693i − 0.188868i 2.31793 − 0.843658i −0.648425 + 0.374368i 0.173648 + 0.984808i 0.0944342 − 0.163565i
136.7 0.498592 + 0.594199i 0.766044 + 0.642788i 0.242818 − 1.37709i −0.342020 − 0.939693i 0.775672i −0.517437 + 0.188332i 2.28284 − 1.31800i 0.173648 + 0.984808i 0.387836 − 0.671751i
136.8 0.789563 + 0.940964i 0.766044 + 0.642788i 0.0852921 − 0.483716i 0.342020 + 0.939693i 1.22834i 2.92724 − 1.06543i 2.65005 − 1.53001i 0.173648 + 0.984808i −0.614171 + 1.06377i
136.9 0.845612 + 1.00776i 0.766044 + 0.642788i 0.0467736 − 0.265266i 0.342020 + 0.939693i 1.31554i −4.53221 + 1.64959i 2.58546 − 1.49271i 0.173648 + 0.984808i −0.657769 + 1.13929i
136.10 1.25682 + 1.49782i 0.766044 + 0.642788i −0.316570 + 1.79536i −0.342020 − 0.939693i 1.95526i 3.07167 − 1.11799i 0.299621 − 0.172986i 0.173648 + 0.984808i 0.977632 − 1.69331i
136.11 1.64099 + 1.95566i 0.766044 + 0.642788i −0.784449 + 4.44883i 0.342020 + 0.939693i 2.55293i 0.841107 − 0.306138i −5.56586 + 3.21345i 0.173648 + 0.984808i −1.27647 + 2.21090i
136.12 1.65371 + 1.97081i 0.766044 + 0.642788i −0.802054 + 4.54867i −0.342020 − 0.939693i 2.57271i −3.43416 + 1.24993i −5.83487 + 3.36876i 0.173648 + 0.984808i 1.28636 − 2.22803i
151.1 −1.76882 + 2.10800i 0.766044 − 0.642788i −0.967639 − 5.48775i 0.342020 − 0.939693i 2.75180i −0.431966 − 0.157223i 8.51352 + 4.91528i 0.173648 − 0.984808i 1.37590 + 2.38313i
151.2 −1.45730 + 1.73675i 0.766044 − 0.642788i −0.545260 − 3.09232i −0.342020 + 0.939693i 2.26716i 1.94023 + 0.706187i 2.23835 + 1.29231i 0.173648 − 0.984808i −1.13358 − 1.96342i
151.3 −1.10266 + 1.31410i 0.766044 − 0.642788i −0.163701 − 0.928394i −0.342020 + 0.939693i 1.71543i −2.40627 − 0.875812i −1.57071 − 0.906851i 0.173648 − 0.984808i −0.857717 − 1.48561i
151.4 −0.946249 + 1.12770i 0.766044 − 0.642788i −0.0290136 − 0.164545i 0.342020 − 0.939693i 1.47210i −2.22291 − 0.809072i −2.33674 − 1.34912i 0.173648 − 0.984808i 0.736051 + 1.27488i
151.5 −0.409461 + 0.487977i 0.766044 − 0.642788i 0.276833 + 1.57000i −0.342020 + 0.939693i 0.637009i −2.61740 − 0.952654i −1.98281 − 1.14477i 0.173648 − 0.984808i −0.318504 − 0.551666i
151.6 −0.121402 + 0.144682i 0.766044 − 0.642788i 0.341102 + 1.93449i 0.342020 − 0.939693i 0.188868i 2.31793 + 0.843658i −0.648425 − 0.374368i 0.173648 − 0.984808i 0.0944342 + 0.163565i
151.7 0.498592 − 0.594199i 0.766044 − 0.642788i 0.242818 + 1.37709i −0.342020 + 0.939693i − 0.775672i −0.517437 − 0.188332i 2.28284 + 1.31800i 0.173648 − 0.984808i 0.387836 + 0.671751i
151.8 0.789563 − 0.940964i 0.766044 − 0.642788i 0.0852921 + 0.483716i 0.342020 − 0.939693i − 1.22834i 2.92724 + 1.06543i 2.65005 + 1.53001i 0.173648 − 0.984808i −0.614171 − 1.06377i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bt.a ✓ 72
37.h even 18 1 inner 555.2.bt.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bt.a ✓ 72 1.a even 1 1 trivial
555.2.bt.a ✓ 72 37.h even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 6 T_{2}^{71} + 15 T_{2}^{70} + 12 T_{2}^{69} - 21 T_{2}^{68} + 42 T_{2}^{67} + 43 T_{2}^{66} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display