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Properties

Label 512.2.i.a
Level $512$
Weight $2$
Character orbit 512.i
Analytic conductor $4.088$
Analytic rank $0$
Dimension $56$
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] = [512,2,Mod(33,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.i (of order \(16\), degree \(8\), not 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: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{3} + 8 q^{5} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{3} + 8 q^{5} + 8 q^{7} - 8 q^{9} - 8 q^{11} + 8 q^{13} + 8 q^{15} - 8 q^{17} - 8 q^{19} + 8 q^{21} + 8 q^{23} - 8 q^{25} - 8 q^{27} + 8 q^{29} - 8 q^{35} + 8 q^{37} + 8 q^{39} - 8 q^{41} - 8 q^{43} + 8 q^{45} + 8 q^{47} - 8 q^{49} + 24 q^{51} + 8 q^{53} - 56 q^{55} - 8 q^{57} + 56 q^{59} + 8 q^{61} - 64 q^{63} - 16 q^{65} + 72 q^{67} + 8 q^{69} - 56 q^{71} - 8 q^{73} + 56 q^{75} + 8 q^{77} - 24 q^{79} - 8 q^{81} - 8 q^{83} + 8 q^{85} + 8 q^{87} - 8 q^{89} - 8 q^{91} - 16 q^{93} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1 0 −2.03902 + 1.36243i 0 1.53851 − 0.306028i 0 −1.01301 − 2.44562i 0 1.15334 − 2.78440i 0
33.2 0 −2.00147 + 1.33734i 0 −0.756852 + 0.150547i 0 1.69148 + 4.08359i 0 1.06935 − 2.58163i 0
33.3 0 −1.31138 + 0.876237i 0 −3.52249 + 0.700667i 0 −1.02503 − 2.47464i 0 −0.196120 + 0.473476i 0
33.4 0 0.0799701 − 0.0534343i 0 3.47403 − 0.691028i 0 −1.22800 − 2.96465i 0 −1.14451 + 2.76309i 0
33.5 0 0.306211 − 0.204603i 0 −1.42470 + 0.283390i 0 0.666723 + 1.60961i 0 −1.09615 + 2.64634i 0
33.6 0 1.06920 − 0.714416i 0 0.330507 − 0.0657419i 0 0.739314 + 1.78486i 0 −0.515254 + 1.24393i 0
33.7 0 2.51381 − 1.67967i 0 2.28487 − 0.454489i 0 −0.303950 − 0.733799i 0 2.34987 − 5.67309i 0
97.1 0 −0.599600 − 3.01439i 0 1.78465 − 1.19247i 0 −1.99271 − 0.825409i 0 −5.95540 + 2.46681i 0
97.2 0 −0.435353 − 2.18867i 0 −0.649649 + 0.434082i 0 3.64486 + 1.50975i 0 −1.82909 + 0.757635i 0
97.3 0 −0.216111 − 1.08646i 0 −1.50133 + 1.00316i 0 −1.15320 − 0.477669i 0 1.63794 − 0.678457i 0
97.4 0 0.123576 + 0.621259i 0 −0.660623 + 0.441414i 0 0.860072 + 0.356253i 0 2.40095 − 0.994505i 0
97.5 0 0.152968 + 0.769021i 0 2.78737 − 1.86246i 0 3.13672 + 1.29927i 0 2.20364 − 0.912779i 0
97.6 0 0.344545 + 1.73215i 0 2.21982 − 1.48324i 0 −2.90595 − 1.20368i 0 −0.109979 + 0.0455548i 0
97.7 0 0.553854 + 2.78441i 0 −2.59756 + 1.73564i 0 1.96508 + 0.813965i 0 −4.67456 + 1.93627i 0
161.1 0 −2.98137 − 0.593031i 0 −0.914126 + 1.36809i 0 −2.65574 + 1.10004i 0 5.76522 + 2.38803i 0
161.2 0 −2.23702 − 0.444970i 0 2.33237 − 3.49064i 0 1.63661 − 0.677907i 0 2.03460 + 0.842759i 0
161.3 0 −1.22190 − 0.243052i 0 −0.884671 + 1.32400i 0 2.40727 − 0.997123i 0 −1.33766 − 0.554078i 0
161.4 0 0.191980 + 0.0381873i 0 0.967135 − 1.44742i 0 −4.53283 + 1.87756i 0 −2.73624 − 1.13339i 0
161.5 0 0.416408 + 0.0828287i 0 −1.82421 + 2.73012i 0 −0.00395016 + 0.00163621i 0 −2.60510 − 1.07907i 0
161.6 0 1.93660 + 0.385213i 0 0.787711 − 1.17889i 0 2.16489 − 0.896725i 0 0.830380 + 0.343954i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.i.a 56
4.b odd 2 1 512.2.i.b 56
8.b even 2 1 256.2.i.a 56
8.d odd 2 1 64.2.i.a ✓ 56
24.f even 2 1 576.2.bd.a 56
64.i even 16 1 256.2.i.a 56
64.i even 16 1 inner 512.2.i.a 56
64.j odd 16 1 64.2.i.a ✓ 56
64.j odd 16 1 512.2.i.b 56
192.s even 16 1 576.2.bd.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.i.a ✓ 56 8.d odd 2 1
64.2.i.a ✓ 56 64.j odd 16 1
256.2.i.a 56 8.b even 2 1
256.2.i.a 56 64.i even 16 1
512.2.i.a 56 1.a even 1 1 trivial
512.2.i.a 56 64.i even 16 1 inner
512.2.i.b 56 4.b odd 2 1
512.2.i.b 56 64.j odd 16 1
576.2.bd.a 56 24.f even 2 1
576.2.bd.a 56 192.s even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} + 8 T_{3}^{55} + 36 T_{3}^{54} + 120 T_{3}^{53} + 330 T_{3}^{52} + 936 T_{3}^{51} + \cdots + 2064512 \) acting on \(S_{2}^{\mathrm{new}}(512, [\chi])\). Copy content Toggle raw display