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Properties

Label 927.2.ba.c
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 103)
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.

\(\operatorname{Tr}(f)(q) = \) \( 256 q + 33 q^{2} - 25 q^{4} + 37 q^{5} - 29 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q + 33 q^{2} - 25 q^{4} + 37 q^{5} - 29 q^{7} + 40 q^{8} + 50 q^{10} + 32 q^{11} - 26 q^{13} - 6 q^{14} - 31 q^{16} + 34 q^{17} - 25 q^{19} + 43 q^{20} - 62 q^{22} - q^{23} + 29 q^{25} + 54 q^{26} + 52 q^{28} + 25 q^{29} - 20 q^{31} + 37 q^{32} + 46 q^{34} + 41 q^{35} - q^{37} - 15 q^{38} - 39 q^{40} + 32 q^{41} - 26 q^{43} + 11 q^{44} + 92 q^{46} - 130 q^{47} + 31 q^{49} + 9 q^{50} - 45 q^{52} + 21 q^{53} - 15 q^{55} - 88 q^{56} + 40 q^{58} + 9 q^{59} - 6 q^{61} + 17 q^{62} + 58 q^{64} + 35 q^{65} + 4 q^{67} - 168 q^{68} - 44 q^{70} - 28 q^{71} + 51 q^{73} + 73 q^{74} + 12 q^{76} - 13 q^{77} - 42 q^{79} - 248 q^{80} + 207 q^{82} + 28 q^{83} - 100 q^{85} - 39 q^{86} - 147 q^{88} + 9 q^{89} - 47 q^{91} + 187 q^{92} - 72 q^{94} - 50 q^{95} - 26 q^{97} + 304 q^{98}+O(q^{100}) \) 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 −0.0729778 + 2.36867i 0 −3.60904 − 0.222598i 0.858639 − 0.690944i 0 0.474734 + 0.218412i 0.353327 − 3.81301i 0 1.57395 + 2.08425i
19.2 −0.0629228 + 2.04231i 0 −2.17085 − 0.133893i −3.21717 + 2.58885i 0 −2.18014 − 1.00302i 0.0329872 − 0.355988i 0 −5.08479 − 6.73335i
19.3 −0.0456089 + 1.48034i 0 −0.193126 − 0.0119116i 2.76833 − 2.22767i 0 3.94007 + 1.81272i −0.246866 + 2.66410i 0 3.17145 + 4.19968i
19.4 −0.0149386 + 0.484867i 0 1.76133 + 0.108635i 0.110357 − 0.0888040i 0 −1.07053 − 0.492520i −0.168504 + 1.81845i 0 0.0414096 + 0.0548352i
19.5 0.00404914 − 0.131424i 0 1.97895 + 0.122058i 2.52656 − 2.03312i 0 0.398452 + 0.183317i 0.0483185 − 0.521439i 0 −0.256971 − 0.340284i
19.6 0.0525209 − 1.70469i 0 −0.907001 − 0.0559419i −2.37758 + 1.91323i 0 −2.69832 − 1.24143i 0.171727 − 1.85323i 0 3.13659 + 4.15352i
19.7 0.0599351 − 1.94533i 0 −1.78452 − 0.110066i 2.26370 − 1.82159i 0 −0.510466 − 0.234852i 0.0380859 − 0.411013i 0 −3.40793 − 4.51282i
19.8 0.0818395 − 2.65629i 0 −5.05299 − 0.311658i −2.33806 + 1.88143i 0 2.85707 + 1.31446i −0.750974 + 8.10430i 0 4.80629 + 6.36455i
28.1 −1.39654 + 2.10759i 0 −1.71203 − 4.04484i 3.03692 − 0.966113i 0 −1.01748 + 2.88741i 5.94530 + 1.11137i 0 −2.20500 + 7.74978i
28.2 −1.15768 + 1.74711i 0 −0.932615 − 2.20339i −0.925657 + 0.294473i 0 −0.421691 + 1.19667i 0.808894 + 0.151208i 0 0.557137 − 1.95813i
28.3 −0.661407 + 0.998163i 0 0.220701 + 0.521428i 0.267709 − 0.0851643i 0 0.196917 − 0.558810i −3.02048 − 0.564626i 0 −0.0920566 + 0.323545i
28.4 0.217051 − 0.327562i 0 0.719386 + 1.69962i −2.66574 + 0.848034i 0 1.59191 − 4.51751i 1.48539 + 0.277667i 0 −0.300817 + 1.05726i
28.5 0.458214 − 0.691514i 0 0.511340 + 1.20809i 2.17340 − 0.691407i 0 0.779382 − 2.21173i 2.70056 + 0.504823i 0 0.517763 − 1.81975i
28.6 0.698872 − 1.05470i 0 0.155594 + 0.367605i 1.28791 − 0.409713i 0 −1.73442 + 4.92194i 2.98384 + 0.557776i 0 0.467957 − 1.64470i
28.7 1.02398 − 1.54534i 0 −0.559965 − 1.32297i −3.65578 + 1.16299i 0 −0.0906297 + 0.257188i 1.02666 + 0.191915i 0 −1.94623 + 6.84028i
28.8 1.42772 − 2.15465i 0 −1.82456 − 4.31069i 0.913131 − 0.290488i 0 0.945678 − 2.68364i −6.81154 − 1.27330i 0 0.677799 − 2.38222i
55.1 −1.86905 + 1.92752i 0 −0.160402 − 5.20622i 1.21909 − 3.45952i 0 0.0557294 + 0.254468i 6.36657 + 5.80389i 0 4.38976 + 8.81583i
55.2 −0.883344 + 0.910979i 0 0.0120045 + 0.389635i 0.470815 − 1.33608i 0 −0.262056 − 1.19658i −2.24105 − 2.04299i 0 0.801245 + 1.60912i
55.3 −0.859619 + 0.886512i 0 0.0146323 + 0.474926i −0.125028 + 0.354805i 0 −0.774228 − 3.53523i −2.25873 − 2.05911i 0 −0.207062 − 0.415836i
55.4 0.0787398 − 0.0812031i 0 0.0611961 + 1.98626i −0.841052 + 2.38673i 0 0.875389 + 3.99715i 0.333288 + 0.303832i 0 0.127586 + 0.256227i
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.c 256
3.b odd 2 1 103.2.g.a ✓ 256
103.g even 51 1 inner 927.2.ba.c 256
309.n odd 102 1 103.2.g.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.2.g.a ✓ 256 3.b odd 2 1
103.2.g.a ✓ 256 309.n odd 102 1
927.2.ba.c 256 1.a even 1 1 trivial
927.2.ba.c 256 103.g even 51 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{256} - 33 T_{2}^{255} + 549 T_{2}^{254} - 6154 T_{2}^{253} + 52380 T_{2}^{252} + \cdots + 48\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display