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Properties

Label 555.2.bg.a
Level $555$
Weight $2$
Character orbit 555.bg
Analytic conductor $4.432$
Analytic rank $0$
Dimension $208$
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] = [555,2,Mod(236,555)]
 
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([6, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.236");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bg (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: \(208\)
Relative dimension: \(52\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 208 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 24 q^{13} + 96 q^{16} - 20 q^{18} + 16 q^{19} + 36 q^{21} + 32 q^{22} + 56 q^{24} - 168 q^{28} - 32 q^{31} - 8 q^{34} - 80 q^{37} + 28 q^{39} - 44 q^{42} + 32 q^{43} + 16 q^{45} - 160 q^{49} + 84 q^{51} + 72 q^{52} - 112 q^{54} + 16 q^{55} + 36 q^{57} + 28 q^{60} - 56 q^{61} - 40 q^{63} - 84 q^{66} - 24 q^{69} - 48 q^{72} - 40 q^{76} - 72 q^{78} - 72 q^{79} - 16 q^{81} - 64 q^{82} - 112 q^{84} - 16 q^{87} - 56 q^{88} + 160 q^{91} - 24 q^{93} - 40 q^{94} - 12 q^{96} - 88 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1 −0.716589 − 2.67435i −1.38941 + 1.03419i −4.90658 + 2.83282i −0.258819 + 0.965926i 3.76142 + 2.97466i −0.150243 − 0.260229i 7.17643 + 7.17643i 0.860897 − 2.87382i 2.76869
236.2 −0.714181 − 2.66536i −1.22970 − 1.21977i −4.86203 + 2.80710i 0.258819 − 0.965926i −2.37289 + 4.14873i 1.20409 + 2.08554i 7.05193 + 7.05193i 0.0243269 + 2.99990i −2.75938
236.3 −0.666142 − 2.48607i 1.24145 + 1.20781i −4.00477 + 2.31216i 0.258819 − 0.965926i 2.17571 − 3.89091i 0.197086 + 0.341362i 4.77607 + 4.77607i 0.0824104 + 2.99887i −2.57377
236.4 −0.645965 − 2.41077i 1.65004 − 0.526669i −3.66251 + 2.11455i −0.258819 + 0.965926i −2.33554 − 3.63765i 2.43251 + 4.21323i 3.93393 + 3.93393i 2.44524 − 1.73805i 2.49582
236.5 −0.637866 − 2.38055i 0.984164 + 1.42528i −3.52808 + 2.03694i −0.258819 + 0.965926i 2.76518 − 3.25199i −0.557119 − 0.964959i 3.61411 + 3.61411i −1.06284 + 2.80542i 2.46452
236.6 −0.576989 − 2.15335i −0.656442 + 1.60284i −2.57195 + 1.48492i 0.258819 − 0.965926i 3.83023 + 0.488732i 2.61105 + 4.52248i 1.52881 + 1.52881i −2.13817 − 2.10434i −2.22931
236.7 −0.568331 − 2.12104i −1.72935 + 0.0966046i −2.44376 + 1.41091i 0.258819 − 0.965926i 1.18775 + 3.61313i −1.76066 − 3.04956i 1.27604 + 1.27604i 2.98134 − 0.334127i −2.19586
236.8 −0.556991 − 2.07872i 0.200230 − 1.72044i −2.27878 + 1.31565i 0.258819 − 0.965926i −3.68783 + 0.542046i −0.0340667 − 0.0590052i 0.960675 + 0.960675i −2.91982 − 0.688967i −2.15205
236.9 −0.549620 − 2.05121i −1.42399 − 0.986027i −2.17333 + 1.25477i −0.258819 + 0.965926i −1.23989 + 3.46284i 0.905744 + 1.56879i 0.765127 + 0.765127i 1.05550 + 2.80819i 2.12357
236.10 −0.539900 − 2.01493i 0.0740940 − 1.73047i −2.03642 + 1.17573i −0.258819 + 0.965926i −3.52678 + 0.784984i 0.451152 + 0.781418i 0.518402 + 0.518402i −2.98902 − 0.256434i 2.08601
236.11 −0.511921 − 1.91052i 1.72896 − 0.103362i −1.65596 + 0.956069i 0.258819 − 0.965926i −1.08257 − 3.25030i −1.90576 − 3.30087i −0.122884 − 0.122884i 2.97863 − 0.357418i −1.97791
236.12 −0.471868 − 1.76103i −1.27761 + 1.16950i −1.14653 + 0.661951i −0.258819 + 0.965926i 2.66239 + 1.69806i 0.380851 + 0.659654i −0.871603 − 0.871603i 0.264554 − 2.98831i 1.82316
236.13 −0.441379 − 1.64725i 0.173644 + 1.72332i −0.786558 + 0.454119i −0.258819 + 0.965926i 2.76210 − 1.04667i −1.80985 − 3.13475i −1.31652 − 1.31652i −2.93970 + 0.598491i 1.70536
236.14 −0.429920 − 1.60448i 1.60668 − 0.646974i −0.657486 + 0.379600i 0.258819 − 0.965926i −1.72880 − 2.29975i 1.22968 + 2.12987i −1.45740 − 1.45740i 2.16285 − 2.07896i −1.66108
236.15 −0.351270 − 1.31096i −0.811137 − 1.53038i 0.136832 − 0.0790000i 0.258819 − 0.965926i −1.72133 + 1.60094i −0.779091 − 1.34943i −2.07101 − 2.07101i −1.68411 + 2.48269i −1.35720
236.16 −0.337417 − 1.25926i 1.35480 − 1.07913i 0.260171 − 0.150210i −0.258819 + 0.965926i −1.81604 − 1.34192i −1.16307 − 2.01450i −2.12062 − 2.12062i 0.670947 − 2.92401i 1.30368
236.17 −0.313766 − 1.17099i −1.73194 − 0.0197021i 0.459283 − 0.265167i 0.258819 − 0.965926i 0.520352 + 2.03426i 1.67324 + 2.89814i −2.16906 − 2.16906i 2.99922 + 0.0682456i −1.21230
236.18 −0.307912 − 1.14914i 0.833473 + 1.51833i 0.506327 − 0.292328i −0.258819 + 0.965926i 1.48814 − 1.42529i 1.85947 + 3.22070i −2.17430 − 2.17430i −1.61065 + 2.53097i 1.18968
236.19 −0.281959 − 1.05229i −1.72601 + 0.144508i 0.704245 − 0.406596i −0.258819 + 0.965926i 0.638730 + 1.77551i −2.27350 − 3.93782i −2.16708 − 2.16708i 2.95823 − 0.498846i 1.08941
236.20 −0.188590 − 0.703829i −1.46415 − 0.925342i 1.27224 − 0.734530i −0.258819 + 0.965926i −0.375158 + 1.20502i 1.22562 + 2.12284i −1.78739 − 1.78739i 1.28748 + 2.70968i 0.728657
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
37.g odd 12 1 inner
111.m 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.bg.a ✓ 208
3.b odd 2 1 inner 555.2.bg.a ✓ 208
37.g odd 12 1 inner 555.2.bg.a ✓ 208
111.m even 12 1 inner 555.2.bg.a ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bg.a ✓ 208 1.a even 1 1 trivial
555.2.bg.a ✓ 208 3.b odd 2 1 inner
555.2.bg.a ✓ 208 37.g odd 12 1 inner
555.2.bg.a ✓ 208 111.m even 12 1 inner

Hecke kernels

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