Properties

Label 555.2.bd.a
Level $555$
Weight $2$
Character orbit 555.bd
Analytic conductor $4.432$
Analytic rank $0$
Dimension $152$
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(82,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bd (of order \(12\), degree \(4\), 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: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 152 q + 4 q^{2} - 76 q^{4} + 8 q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q + 4 q^{2} - 76 q^{4} + 8 q^{5} - 24 q^{8} - 4 q^{10} - 8 q^{13} + 16 q^{14} - 76 q^{16} - 36 q^{17} - 40 q^{20} - 40 q^{25} + 16 q^{26} + 16 q^{28} - 28 q^{29} - 8 q^{30} + 8 q^{31} + 28 q^{32} - 56 q^{35} + 4 q^{37} + 112 q^{38} - 24 q^{39} + 4 q^{40} + 60 q^{41} - 32 q^{43} + 72 q^{44} + 4 q^{45} + 96 q^{46} + 32 q^{48} + 48 q^{49} - 16 q^{50} + 8 q^{51} + 4 q^{52} - 48 q^{53} - 72 q^{56} - 128 q^{58} - 68 q^{60} - 32 q^{61} - 16 q^{62} + 200 q^{64} + 12 q^{65} - 32 q^{66} + 8 q^{67} + 8 q^{69} + 80 q^{70} - 16 q^{71} + 120 q^{73} + 40 q^{74} + 16 q^{75} - 120 q^{76} - 28 q^{77} - 24 q^{78} - 64 q^{79} - 72 q^{80} + 76 q^{81} - 12 q^{83} + 4 q^{89} - 72 q^{91} + 24 q^{93} + 48 q^{94} + 12 q^{95} - 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −1.29013 2.23457i −0.965926 + 0.258819i −2.32886 + 4.03370i 0.661101 2.13611i 1.82452 + 1.82452i −2.46133 + 0.659511i 6.85757 0.866025 0.500000i −5.62617 + 1.27857i
82.2 −1.28383 2.22366i 0.965926 0.258819i −2.29643 + 3.97754i 2.20849 0.350110i −1.81561 1.81561i −4.23915 + 1.13588i 6.65758 0.866025 0.500000i −3.61385 4.46144i
82.3 −1.27502 2.20839i −0.965926 + 0.258819i −2.25133 + 3.89941i −2.22698 0.201353i 1.80314 + 1.80314i 3.42337 0.917290i 6.38185 0.866025 0.500000i 2.39477 + 5.17478i
82.4 −1.23268 2.13507i 0.965926 0.258819i −2.03900 + 3.53166i −2.23494 0.0709518i −1.74327 1.74327i −0.0512180 + 0.0137238i 5.12305 0.866025 0.500000i 2.60348 + 4.85921i
82.5 −1.19267 2.06576i −0.965926 + 0.258819i −1.84490 + 3.19547i 0.0771478 + 2.23474i 1.68668 + 1.68668i −0.615378 + 0.164890i 4.03075 0.866025 0.500000i 4.52441 2.82466i
82.6 −1.15271 1.99655i 0.965926 0.258819i −1.65747 + 2.87082i 1.97430 + 1.04983i −1.63017 1.63017i 2.40277 0.643820i 3.03146 0.866025 0.500000i −0.179756 5.15193i
82.7 −0.936247 1.62163i 0.965926 0.258819i −0.753118 + 1.30444i −1.36154 + 1.77375i −1.32405 1.32405i 1.92977 0.517080i −0.924569 0.866025 0.500000i 4.15111 + 0.547248i
82.8 −0.774878 1.34213i −0.965926 + 0.258819i −0.200873 + 0.347922i 0.286465 2.21764i 1.09584 + 1.09584i 0.817142 0.218952i −2.47691 0.866025 0.500000i −3.19834 + 1.33393i
82.9 −0.771950 1.33706i −0.965926 + 0.258819i −0.191813 + 0.332230i 0.201731 + 2.22695i 1.09170 + 1.09170i 4.70754 1.26138i −2.49552 0.866025 0.500000i 2.82183 1.98882i
82.10 −0.723485 1.25311i 0.965926 0.258819i −0.0468616 + 0.0811666i 1.66683 1.49053i −1.02316 1.02316i 1.20508 0.322900i −2.75833 0.866025 0.500000i −3.07373 1.01035i
82.11 −0.711025 1.23153i 0.965926 0.258819i −0.0111120 + 0.0192466i 0.246144 + 2.22248i −1.00554 1.00554i −4.94627 + 1.32535i −2.81249 0.866025 0.500000i 2.56204 1.88337i
82.12 −0.702715 1.21714i −0.965926 + 0.258819i 0.0123832 0.0214483i 2.21949 + 0.271757i 0.993789 + 0.993789i −3.44735 + 0.923714i −2.84567 0.866025 0.500000i −1.22890 2.89240i
82.13 −0.567625 0.983155i −0.965926 + 0.258819i 0.355604 0.615925i −2.08976 + 0.795543i 0.802743 + 0.802743i −0.353460 + 0.0947093i −3.07790 0.866025 0.500000i 1.96834 + 1.60299i
82.14 −0.454649 0.787476i 0.965926 0.258819i 0.586588 1.01600i −1.51872 1.64118i −0.642971 0.642971i 4.76595 1.27703i −2.88537 0.866025 0.500000i −0.601903 + 1.94212i
82.15 −0.343761 0.595411i 0.965926 0.258819i 0.763657 1.32269i 2.00349 + 0.992985i −0.486151 0.486151i 0.629862 0.168771i −2.42510 0.866025 0.500000i −0.0974877 1.53425i
82.16 −0.261270 0.452533i 0.965926 0.258819i 0.863476 1.49558i −2.05205 0.888300i −0.369492 0.369492i −3.40688 + 0.912870i −1.94748 0.866025 0.500000i 0.134155 + 1.16071i
82.17 −0.224905 0.389547i −0.965926 + 0.258819i 0.898836 1.55683i −0.245397 + 2.22256i 0.318064 + 0.318064i −2.19780 + 0.588898i −1.70823 0.866025 0.500000i 0.920982 0.404272i
82.18 −0.0903978 0.156574i −0.965926 + 0.258819i 0.983656 1.70374i −1.59092 1.57130i 0.127842 + 0.127842i −2.75537 + 0.738298i −0.717272 0.866025 0.500000i −0.102209 + 0.391137i
82.19 0.0276845 + 0.0479509i −0.965926 + 0.258819i 0.998467 1.72940i 2.23456 0.0821691i −0.0391518 0.0391518i 2.39340 0.641311i 0.221306 0.866025 0.500000i 0.0658027 + 0.104874i
82.20 0.0430209 + 0.0745144i −0.965926 + 0.258819i 0.996298 1.72564i 0.208099 2.22636i −0.0608408 0.0608408i 2.96943 0.795658i 0.343530 0.866025 0.500000i 0.174849 0.0802738i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bd.a 152
5.c odd 4 1 555.2.bn.a yes 152
37.g odd 12 1 555.2.bn.a yes 152
185.p even 12 1 inner 555.2.bd.a 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bd.a 152 1.a even 1 1 trivial
555.2.bd.a 152 185.p even 12 1 inner
555.2.bn.a yes 152 5.c odd 4 1
555.2.bn.a yes 152 37.g odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).