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Properties

Label 864.2.w.b
Level $864$
Weight $2$
Character orbit 864.w
Analytic conductor $6.899$
Analytic rank $0$
Dimension $128$
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] = [864,2,Mod(107,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.w (of order \(8\), 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 8 q^{10} + 32 q^{16} - 32 q^{22} + 64 q^{40} + 64 q^{46} + 40 q^{52} + 64 q^{55} + 64 q^{58} + 32 q^{61} + 96 q^{64} - 64 q^{67} - 48 q^{70} - 32 q^{76} - 32 q^{79} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 72 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.41338 − 0.0486157i 0 1.99527 + 0.137425i 0.817684 + 0.338696i 0 1.96438 − 1.96438i −2.81339 − 0.291234i 0 −1.13923 − 0.518457i
107.2 −1.40430 + 0.167138i 0 1.94413 − 0.469426i −3.49346 − 1.44704i 0 −2.54256 + 2.54256i −2.65169 + 0.984154i 0 5.14773 + 1.44819i
107.3 −1.35869 − 0.392392i 0 1.69206 + 1.06627i 2.69281 + 1.11540i 0 −1.39266 + 1.39266i −1.88058 − 2.11268i 0 −3.22102 − 2.57212i
107.4 −1.31895 + 0.510265i 0 1.47926 − 1.34603i 1.03442 + 0.428472i 0 −0.461188 + 0.461188i −1.26424 + 2.53016i 0 −1.58299 − 0.0373037i
107.5 −1.31642 − 0.516757i 0 1.46592 + 1.36054i −2.69312 − 1.11553i 0 0.0380585 − 0.0380585i −1.22670 − 2.54857i 0 2.96882 + 2.86019i
107.6 −1.25848 + 0.645153i 0 1.16756 − 1.62383i −2.13183 − 0.883034i 0 3.32511 − 3.32511i −0.421732 + 2.79681i 0 3.25256 − 0.264075i
107.7 −1.23132 + 0.695593i 0 1.03230 − 1.71300i 2.52121 + 1.04432i 0 −2.75999 + 2.75999i −0.0795431 + 2.82731i 0 −3.83084 + 0.467845i
107.8 −0.984674 − 1.01509i 0 −0.0608353 + 1.99907i 2.42976 + 1.00644i 0 3.62344 − 3.62344i 2.08915 − 1.90668i 0 −1.37089 − 3.45745i
107.9 −0.919792 − 1.07424i 0 −0.307965 + 1.97615i −1.92195 − 0.796096i 0 0.893559 − 0.893559i 2.40611 − 1.48682i 0 0.912596 + 2.79687i
107.10 −0.856765 + 1.12515i 0 −0.531906 − 1.92797i −2.84859 − 1.17993i 0 1.33794 − 1.33794i 2.62497 + 1.05335i 0 3.76817 − 2.19416i
107.11 −0.679670 − 1.24018i 0 −1.07610 + 1.68583i −0.301020 − 0.124687i 0 −2.88637 + 2.88637i 2.82212 + 0.188749i 0 0.0499604 + 0.458065i
107.12 −0.654624 + 1.25358i 0 −1.14293 − 1.64125i −0.0422264 − 0.0174907i 0 −1.54323 + 1.54323i 2.80563 − 0.358360i 0 0.0495684 − 0.0414843i
107.13 −0.535739 + 1.30881i 0 −1.42597 − 1.40236i −0.682524 − 0.282711i 0 −1.19295 + 1.19295i 2.59937 − 1.11502i 0 0.735670 − 0.741835i
107.14 −0.485421 − 1.32829i 0 −1.52873 + 1.28957i 3.81307 + 1.57942i 0 −1.13033 + 1.13033i 2.45500 + 1.40462i 0 0.246996 − 5.83156i
107.15 −0.0483362 + 1.41339i 0 −1.99533 − 0.136636i 3.11062 + 1.28846i 0 1.36902 − 1.36902i 0.289566 − 2.81357i 0 −1.97145 + 4.33423i
107.16 −0.0136252 − 1.41415i 0 −1.99963 + 0.0385360i −0.327253 − 0.135553i 0 1.35777 − 1.35777i 0.0817409 + 2.82725i 0 −0.187232 + 0.464631i
107.17 0.0136252 + 1.41415i 0 −1.99963 + 0.0385360i 0.327253 + 0.135553i 0 1.35777 − 1.35777i −0.0817409 − 2.82725i 0 −0.187232 + 0.464631i
107.18 0.0483362 − 1.41339i 0 −1.99533 − 0.136636i −3.11062 − 1.28846i 0 1.36902 − 1.36902i −0.289566 + 2.81357i 0 −1.97145 + 4.33423i
107.19 0.485421 + 1.32829i 0 −1.52873 + 1.28957i −3.81307 − 1.57942i 0 −1.13033 + 1.13033i −2.45500 − 1.40462i 0 0.246996 − 5.83156i
107.20 0.535739 − 1.30881i 0 −1.42597 − 1.40236i 0.682524 + 0.282711i 0 −1.19295 + 1.19295i −2.59937 + 1.11502i 0 0.735670 − 0.741835i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.w.b ✓ 128
3.b odd 2 1 inner 864.2.w.b ✓ 128
32.h odd 8 1 inner 864.2.w.b ✓ 128
96.o even 8 1 inner 864.2.w.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.w.b ✓ 128 1.a even 1 1 trivial
864.2.w.b ✓ 128 3.b odd 2 1 inner
864.2.w.b ✓ 128 32.h odd 8 1 inner
864.2.w.b ✓ 128 96.o even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} - 1536 T_{5}^{122} + 206080 T_{5}^{120} - 457088 T_{5}^{118} + 1179648 T_{5}^{116} + \cdots + 45\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display