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Properties

Label 927.2.ba.b
Level $927$
Weight $2$
Character orbit 927.ba
Analytic conductor $7.402$
Analytic rank $0$
Dimension $256$
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] = [927,2,Mod(19,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([0, 80]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.ba (of order \(51\), degree \(32\), 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: \(7.40213226737\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(8\) over \(\Q(\zeta_{51})\)
Twist minimal: no (minimal twist has level 309)
Sato-Tate group: $\mathrm{SU}(2)[C_{51}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q + q^{2} + 3 q^{4} + q^{5} - 4 q^{7} - 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q + q^{2} + 3 q^{4} + q^{5} - 4 q^{7} - 57 q^{8} - 58 q^{10} + 32 q^{11} + 10 q^{13} - 57 q^{14} - 7 q^{16} - 15 q^{19} - 46 q^{20} + 54 q^{22} + 21 q^{23} - 27 q^{25} - 14 q^{26} - 49 q^{28} - 2 q^{29} - 92 q^{31} - 8 q^{32} - 35 q^{34} - 58 q^{35} + 4 q^{37} + 51 q^{38} + 41 q^{40} + 216 q^{41} + 3 q^{43} + 133 q^{44} - 36 q^{46} - 2 q^{47} + 26 q^{49} + 60 q^{50} + 62 q^{52} + 7 q^{53} + 16 q^{55} - 19 q^{56} - 32 q^{58} - 73 q^{59} - 58 q^{61} + 20 q^{62} - 269 q^{64} - 41 q^{65} - 10 q^{67} + 8 q^{68} + 89 q^{70} + 178 q^{71} - 84 q^{73} + 239 q^{74} + 58 q^{76} - 33 q^{77} - 42 q^{79} + 24 q^{80} - 198 q^{82} - 31 q^{83} + 130 q^{85} + 86 q^{86} + 66 q^{88} - 73 q^{89} - 27 q^{91} - 146 q^{92} - 446 q^{94} - 48 q^{95} + 100 q^{97} - 363 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.0842717 + 2.73524i 0 −5.47821 − 0.337884i −2.18470 + 1.75802i 0 −0.394746 − 0.181612i 0.880861 − 9.50600i 0 −4.62449 − 6.12382i
19.2 −0.0613793 + 1.99221i 0 −1.96893 − 0.121439i 2.92758 − 2.35582i 0 −2.58881 − 1.19104i −0.00502627 + 0.0542421i 0 4.51359 + 5.97696i
19.3 −0.0516101 + 1.67513i 0 −0.807183 − 0.0497854i −0.867291 + 0.697906i 0 2.94095 + 1.35305i −0.184214 + 1.98798i 0 −1.12432 − 1.48884i
19.4 −0.0239795 + 0.778310i 0 1.39101 + 0.0857948i −1.02955 + 0.828474i 0 −3.38571 − 1.55767i −0.243826 + 2.63130i 0 −0.620122 − 0.821174i
19.5 0.0194847 − 0.632423i 0 1.59663 + 0.0984766i 0.661288 − 0.532137i 0 3.88646 + 1.78805i 0.210149 − 2.26787i 0 −0.323650 − 0.428582i
19.6 0.0276830 − 0.898517i 0 1.18964 + 0.0733745i −1.64821 + 1.32631i 0 −0.499473 − 0.229794i 0.264749 − 2.85710i 0 1.14609 + 1.51766i
19.7 0.0682910 − 2.21654i 0 −2.91219 − 0.179618i −0.760660 + 0.612101i 0 2.29828 + 1.05738i −0.187779 + 2.02646i 0 1.30480 + 1.72784i
19.8 0.0749870 − 2.43388i 0 −3.92193 − 0.241896i 1.89710 − 1.52659i 0 −0.357317 − 0.164392i −0.433486 + 4.67806i 0 −3.57327 − 4.73178i
28.1 −1.06654 + 1.60956i 0 −0.673627 − 1.59151i −2.28349 + 0.726431i 0 −0.823820 + 2.33783i −0.515877 − 0.0964341i 0 1.26619 − 4.45019i
28.2 −0.871046 + 1.31454i 0 −0.189723 − 0.448237i 1.90386 − 0.605660i 0 1.37839 − 3.91158i −2.34569 − 0.438486i 0 −0.862181 + 3.03025i
28.3 −0.569905 + 0.860073i 0 0.364638 + 0.861491i 3.33355 − 1.06048i 0 −1.29144 + 3.66484i −2.97713 − 0.556522i 0 −0.987718 + 3.47147i
28.4 −0.119403 + 0.180198i 0 0.761358 + 1.79878i −2.20954 + 0.702907i 0 0.943216 − 2.67665i −0.840018 − 0.157026i 0 0.137165 − 0.482084i
28.5 0.0261402 − 0.0394496i 0 0.778699 + 1.83975i 0.375918 − 0.119588i 0 −0.251099 + 0.712566i 0.185969 + 0.0347637i 0 0.00510888 − 0.0179558i
28.6 0.620441 − 0.936339i 0 0.287788 + 0.679926i −2.30166 + 0.732211i 0 −0.679991 + 1.92967i 3.02343 + 0.565178i 0 −0.742447 + 2.60943i
28.7 1.10614 − 1.66933i 0 −0.783541 − 1.85119i 2.51174 − 0.799043i 0 −0.479401 + 1.36044i −0.0200429 − 0.00374667i 0 1.44446 − 5.07677i
28.8 1.42654 − 2.15286i 0 −1.82024 − 4.30048i −1.34837 + 0.428948i 0 0.607192 − 1.72309i −6.77772 − 1.26698i 0 −1.00004 + 3.51478i
55.1 −1.62060 + 1.67130i 0 −0.105306 − 3.41795i −0.404469 + 1.14780i 0 0.683648 + 3.12163i 2.44224 + 2.22640i 0 −1.26283 − 2.53611i
55.2 −1.61901 + 1.66966i 0 −0.104979 − 3.40733i 0.444264 − 1.26073i 0 −0.689965 − 3.15048i 2.42160 + 2.20758i 0 1.38572 + 2.78290i
55.3 −0.786787 + 0.811401i 0 0.0222524 + 0.722253i 1.19217 − 3.38314i 0 0.227918 + 1.04071i −2.27403 − 2.07306i 0 1.80710 + 3.62914i
55.4 −0.632379 + 0.652163i 0 0.0361774 + 1.17422i −1.11268 + 3.15755i 0 −0.103672 − 0.473380i −2.13132 − 1.94295i 0 −1.35560 − 2.72242i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.g even 51 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.ba.b 256
3.b odd 2 1 309.2.m.a ✓ 256
103.g even 51 1 inner 927.2.ba.b 256
309.n odd 102 1 309.2.m.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.m.a ✓ 256 3.b odd 2 1
309.2.m.a ✓ 256 309.n odd 102 1
927.2.ba.b 256 1.a even 1 1 trivial
927.2.ba.b 256 103.g even 51 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{256} - T_{2}^{255} - 9 T_{2}^{254} + 31 T_{2}^{253} - 227 T_{2}^{251} + 1291 T_{2}^{250} + \cdots + 22005481 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display