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Properties

Label 927.2.ba.e
Level $927$
Weight $2$
Character orbit 927.ba
Analytic conductor $7.402$
Analytic rank $0$
Dimension $512$
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] = [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: \(512\)
Relative dimension: \(16\) over \(\Q(\zeta_{51})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 512 q + 18 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q + 18 q^{4} - 8 q^{7} - 36 q^{10} + 4 q^{13} + 18 q^{16} + 30 q^{19} + 40 q^{22} - 42 q^{25} - 110 q^{28} - 32 q^{31} - 110 q^{34} + 48 q^{37} - 22 q^{40} - 2 q^{43} + 152 q^{46} + 76 q^{49} + 68 q^{52} + 32 q^{55} - 44 q^{58} + 20 q^{61} - 170 q^{64} - 8 q^{67} - 38 q^{70} - 4 q^{73} + 188 q^{76} + 20 q^{79} + 212 q^{82} + 60 q^{85} + 164 q^{88} - 310 q^{91} + 228 q^{94} - 100 q^{97}+O(q^{100}) \) 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 −0.0826600 + 2.68292i 0 −5.19504 − 0.320419i −1.02242 + 0.822739i 0 −2.92879 − 1.34746i 0.793751 − 8.56593i 0 −2.12283 − 2.81109i
19.2 −0.0760154 + 2.46726i 0 −4.08537 − 0.251977i −1.39599 + 1.12335i 0 4.25142 + 1.95596i 0.476728 − 5.14471i 0 −2.66547 − 3.52965i
19.3 −0.0680488 + 2.20868i 0 −2.87744 − 0.177475i 0.999071 − 0.803949i 0 1.17408 + 0.540161i 0.180015 − 1.94267i 0 1.70768 + 2.26134i
19.4 −0.0484123 + 1.57133i 0 −0.470538 − 0.0290218i 0.160599 − 0.129233i 0 −1.39035 − 0.639663i −0.221724 + 2.39278i 0 0.195294 + 0.258610i
19.5 −0.0454754 + 1.47601i 0 −0.180334 − 0.0111226i −2.73378 + 2.19986i 0 0.720040 + 0.331271i −0.247890 + 2.67516i 0 −3.12270 − 4.13512i
19.6 −0.0396014 + 1.28536i 0 0.345635 + 0.0213180i 0.225933 − 0.181808i 0 −3.49557 − 1.60822i −0.278397 + 3.00438i 0 0.224740 + 0.297604i
19.7 −0.0218132 + 0.707997i 0 1.49542 + 0.0922345i 2.15225 − 1.73191i 0 1.83878 + 0.845971i −0.228635 + 2.46737i 0 1.17924 + 1.56156i
19.8 −0.00330016 + 0.107115i 0 1.98474 + 0.122415i 2.79409 − 2.24840i 0 −3.61100 − 1.66133i −0.0394383 + 0.425607i 0 0.231615 + 0.306708i
19.9 0.00330016 − 0.107115i 0 1.98474 + 0.122415i −2.79409 + 2.24840i 0 −3.61100 − 1.66133i 0.0394383 − 0.425607i 0 0.231615 + 0.306708i
19.10 0.0218132 − 0.707997i 0 1.49542 + 0.0922345i −2.15225 + 1.73191i 0 1.83878 + 0.845971i 0.228635 − 2.46737i 0 1.17924 + 1.56156i
19.11 0.0396014 − 1.28536i 0 0.345635 + 0.0213180i −0.225933 + 0.181808i 0 −3.49557 − 1.60822i 0.278397 − 3.00438i 0 0.224740 + 0.297604i
19.12 0.0454754 − 1.47601i 0 −0.180334 − 0.0111226i 2.73378 − 2.19986i 0 0.720040 + 0.331271i 0.247890 − 2.67516i 0 −3.12270 − 4.13512i
19.13 0.0484123 − 1.57133i 0 −0.470538 − 0.0290218i −0.160599 + 0.129233i 0 −1.39035 − 0.639663i 0.221724 − 2.39278i 0 0.195294 + 0.258610i
19.14 0.0680488 − 2.20868i 0 −2.87744 − 0.177475i −0.999071 + 0.803949i 0 1.17408 + 0.540161i −0.180015 + 1.94267i 0 1.70768 + 2.26134i
19.15 0.0760154 − 2.46726i 0 −4.08537 − 0.251977i 1.39599 − 1.12335i 0 4.25142 + 1.95596i −0.476728 + 5.14471i 0 −2.66547 − 3.52965i
19.16 0.0826600 − 2.68292i 0 −5.19504 − 0.320419i 1.02242 − 0.822739i 0 −2.92879 − 1.34746i −0.793751 + 8.56593i 0 −2.12283 − 2.81109i
28.1 −1.49514 + 2.25639i 0 −2.07629 − 4.90543i −3.86801 + 1.23050i 0 −0.664684 + 1.88624i 8.85150 + 1.65463i 0 3.00672 − 10.5675i
28.2 −1.27787 + 1.92850i 0 −1.30658 − 3.08692i −0.572956 + 0.182270i 0 1.23962 − 3.51778i 3.07465 + 0.574752i 0 0.380655 − 1.33786i
28.3 −1.18987 + 1.79570i 0 −1.02916 − 2.43149i 2.49701 − 0.794356i 0 −0.384886 + 1.09223i 1.35588 + 0.253457i 0 −1.54470 + 5.42906i
28.4 −1.14375 + 1.72609i 0 −0.891651 − 2.10661i 0.0369533 − 0.0117557i 0 −1.22063 + 3.46389i 0.585256 + 0.109403i 0 −0.0219739 + 0.0772301i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.g even 51 1 inner
309.n odd 102 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.ba.e ✓ 512
3.b odd 2 1 inner 927.2.ba.e ✓ 512
103.g even 51 1 inner 927.2.ba.e ✓ 512
309.n odd 102 1 inner 927.2.ba.e ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.ba.e ✓ 512 1.a even 1 1 trivial
927.2.ba.e ✓ 512 3.b odd 2 1 inner
927.2.ba.e ✓ 512 103.g even 51 1 inner
927.2.ba.e ✓ 512 309.n odd 102 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{512} - 25 T_{2}^{510} + 249 T_{2}^{508} - 499 T_{2}^{506} - 15692 T_{2}^{504} + \cdots + 16\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display