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Properties

Label 456.2.bp.a
Level $456$
Weight $2$
Character orbit 456.bp
Analytic conductor $3.641$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [456,2,Mod(67,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bp (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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{4} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{4} - 6 q^{6} - 12 q^{10} + 18 q^{14} - 6 q^{16} - 60 q^{20} + 36 q^{28} + 90 q^{32} - 66 q^{34} - 6 q^{36} - 96 q^{38} + 114 q^{40} + 60 q^{44} - 90 q^{46} - 24 q^{48} + 120 q^{49} - 54 q^{50} - 24 q^{51} + 54 q^{52} + 6 q^{54} + 72 q^{58} - 12 q^{60} - 132 q^{62} - 42 q^{64} - 48 q^{66} - 6 q^{68} - 108 q^{70} - 36 q^{72} + 24 q^{73} - 96 q^{74} - 72 q^{76} - 96 q^{78} - 114 q^{80} + 54 q^{82} - 108 q^{84} - 144 q^{86} - 72 q^{88} - 30 q^{90} + 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.40744 − 0.138283i −0.642788 − 0.766044i 1.96176 + 0.389248i −1.04100 + 2.86013i 0.798752 + 1.16705i −1.32147 − 0.762952i −2.70722 − 0.819119i −0.173648 + 0.984808i 1.86065 − 3.88150i
67.2 −1.40658 + 0.146775i 0.642788 + 0.766044i 1.95691 − 0.412902i 0.159496 − 0.438211i −1.01657 − 0.983155i −3.79549 − 2.19132i −2.69195 + 0.868004i −0.173648 + 0.984808i −0.160024 + 0.639787i
67.3 −1.38055 − 0.306710i −0.642788 − 0.766044i 1.81186 + 0.846859i 0.113716 − 0.312433i 0.652450 + 1.25471i 2.67726 + 1.54572i −2.24163 − 1.72485i −0.173648 + 0.984808i −0.252818 + 0.396452i
67.4 −1.37195 + 0.343154i 0.642788 + 0.766044i 1.76449 − 0.941579i −0.159496 + 0.438211i −1.14474 − 0.830399i 3.79549 + 2.19132i −2.09769 + 1.89729i −0.173648 + 0.984808i 0.0684464 − 0.655934i
67.5 −1.35218 − 0.414266i 0.642788 + 0.766044i 1.65677 + 1.12032i −1.47491 + 4.05228i −0.551816 − 1.30211i 1.38238 + 0.798120i −1.77613 − 2.20122i −0.173648 + 0.984808i 3.67306 − 4.86840i
67.6 −1.28634 − 0.587640i 0.642788 + 0.766044i 1.30936 + 1.51181i 0.113916 − 0.312981i −0.376687 − 1.36312i −2.26403 − 1.30714i −0.795881 − 2.71414i −0.173648 + 0.984808i −0.330455 + 0.335659i
67.7 −1.27526 + 0.611315i −0.642788 − 0.766044i 1.25259 − 1.55917i 1.04100 − 2.86013i 1.28802 + 0.583962i 1.32147 + 0.762952i −0.644232 + 2.75408i −0.173648 + 0.984808i 0.420889 + 4.28379i
67.8 −1.19240 + 0.760390i −0.642788 − 0.766044i 0.843614 − 1.81337i −0.113716 + 0.312433i 1.34895 + 0.424659i −2.67726 − 1.54572i 0.372949 + 2.80373i −0.173648 + 0.984808i −0.101976 − 0.459012i
67.9 −1.14198 − 0.834197i −0.642788 − 0.766044i 0.608230 + 1.90527i −0.568427 + 1.56174i 0.0950175 + 1.41102i −2.99105 − 1.72688i 0.894786 − 2.68316i −0.173648 + 0.984808i 1.95193 − 1.30929i
67.10 −1.12894 + 0.851755i 0.642788 + 0.766044i 0.549027 − 1.92317i 1.47491 − 4.05228i −1.37815 − 0.317323i −1.38238 − 0.798120i 1.01825 + 2.63878i −0.173648 + 0.984808i 1.78646 + 5.83106i
67.11 −1.00778 + 0.992156i 0.642788 + 0.766044i 0.0312511 − 1.99976i −0.113916 + 0.312981i −1.40783 − 0.134260i 2.26403 + 1.30714i 1.95258 + 2.04632i −0.173648 + 0.984808i −0.195724 − 0.428439i
67.12 −1.00089 − 0.999113i 0.642788 + 0.766044i 0.00354586 + 2.00000i 1.20942 − 3.32285i 0.122008 − 1.40894i 2.71241 + 1.56601i 1.99467 − 2.00531i −0.173648 + 0.984808i −4.53039 + 2.11745i
67.13 −0.787797 + 1.17447i −0.642788 − 0.766044i −0.758753 − 1.85048i 0.568427 − 1.56174i 1.40608 − 0.151447i 2.99105 + 1.72688i 2.77108 + 0.566674i −0.173648 + 0.984808i 1.38641 + 1.89793i
67.14 −0.767022 − 1.18814i 0.642788 + 0.766044i −0.823355 + 1.82266i −0.425460 + 1.16894i 0.417136 − 1.35129i −0.404754 − 0.233685i 2.79711 − 0.419758i −0.173648 + 0.984808i 1.71521 − 0.391098i
67.15 −0.598808 + 1.28118i 0.642788 + 0.766044i −1.28286 − 1.53437i −1.20942 + 3.32285i −1.36635 + 0.364814i −2.71241 − 1.56601i 2.73399 − 0.724783i −0.173648 + 0.984808i −3.53296 − 3.53923i
67.16 −0.473247 − 1.33268i −0.642788 − 0.766044i −1.55207 + 1.26137i −0.164021 + 0.450645i −0.716695 + 1.21916i −0.0379501 − 0.0219105i 2.41552 + 1.47148i −0.173648 + 0.984808i 0.678188 + 0.00532158i
67.17 −0.471286 − 1.33338i −0.642788 − 0.766044i −1.55578 + 1.25680i 0.721375 − 1.98196i −0.718488 + 1.21810i 1.20255 + 0.694291i 2.40901 + 1.48212i −0.173648 + 0.984808i −2.98267 − 0.0277931i
67.18 −0.314397 + 1.37882i 0.642788 + 0.766044i −1.80231 − 0.866996i 0.425460 − 1.16894i −1.25833 + 0.645449i 0.404754 + 0.233685i 1.76207 − 2.21249i −0.173648 + 0.984808i 1.47800 + 0.954147i
67.19 0.0110966 + 1.41417i −0.642788 − 0.766044i −1.99975 + 0.0313851i 0.164021 − 0.450645i 1.07618 − 0.917511i 0.0379501 + 0.0219105i −0.0665744 − 2.82764i −0.173648 + 0.984808i 0.639109 + 0.226953i
67.20 0.0131773 + 1.41415i −0.642788 − 0.766044i −1.99965 + 0.0372695i −0.721375 + 1.98196i 1.07483 − 0.919094i −1.20255 − 0.694291i −0.0790549 − 2.82732i −0.173648 + 0.984808i −2.81230 − 0.994018i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.f odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bp.a ✓ 240
8.d odd 2 1 inner 456.2.bp.a ✓ 240
19.f odd 18 1 inner 456.2.bp.a ✓ 240
152.v even 18 1 inner 456.2.bp.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bp.a ✓ 240 1.a even 1 1 trivial
456.2.bp.a ✓ 240 8.d odd 2 1 inner
456.2.bp.a ✓ 240 19.f odd 18 1 inner
456.2.bp.a ✓ 240 152.v even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(456, [\chi])\).