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Properties

Label 255.2.bg.a
Level $255$
Weight $2$
Character orbit 255.bg
Analytic conductor $2.036$
Analytic rank $0$
Dimension $192$
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] = [255,2,Mod(11,255)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("255.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 255 = 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 255.bg (of order \(16\), degree \(8\), 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: \(2.03618525154\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 48 q^{12} - 64 q^{18} - 48 q^{24} - 96 q^{28} - 32 q^{31} - 112 q^{34} - 64 q^{37} + 16 q^{39} + 32 q^{40} - 64 q^{43} + 32 q^{45} + 48 q^{46} + 160 q^{48} + 32 q^{49} + 64 q^{51} - 64 q^{54} + 32 q^{55} - 112 q^{57} + 64 q^{58} + 64 q^{60} + 32 q^{61} + 48 q^{63} + 32 q^{64} - 192 q^{66} - 192 q^{69} - 208 q^{72} - 64 q^{76} - 112 q^{81} + 96 q^{87} + 128 q^{88} - 96 q^{91} + 128 q^{93} + 192 q^{94} + 64 q^{97} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.04808 + 2.53028i 1.11113 − 1.32868i −3.88964 − 3.88964i −0.555570 + 0.831470i 2.19737 + 4.20404i 2.12576 − 1.42039i 8.85797 − 3.66909i −0.530762 − 2.95268i −1.52157 − 2.27719i
11.2 −0.928460 + 2.24150i −1.71090 + 0.269848i −2.74807 − 2.74807i 0.555570 − 0.831470i 0.983637 − 4.08553i 2.30703 − 1.54151i 4.22827 − 1.75141i 2.85436 − 0.923368i 1.34791 + 2.01730i
11.3 −0.878980 + 2.12205i −0.476712 − 1.66516i −2.31626 − 2.31626i 0.555570 − 0.831470i 3.95256 + 0.452035i −1.52928 + 1.02183i 2.70706 − 1.12130i −2.54549 + 1.58760i 1.27608 + 1.90979i
11.4 −0.831217 + 2.00674i −1.50735 − 0.853176i −1.92185 − 1.92185i −0.555570 + 0.831470i 2.96503 − 2.31567i −3.12757 + 2.08978i 1.44065 − 0.596737i 1.54418 + 2.57206i −1.20674 − 1.80601i
11.5 −0.693520 + 1.67431i −0.705307 + 1.58194i −0.908118 − 0.908118i −0.555570 + 0.831470i −2.15951 − 2.27801i −0.375953 + 0.251204i −1.19835 + 0.496371i −2.00508 − 2.23151i −1.00684 − 1.50684i
11.6 −0.667761 + 1.61212i 1.72180 + 0.188121i −0.738806 − 0.738806i 0.555570 − 0.831470i −1.45303 + 2.65013i 3.47691 − 2.32320i −1.53985 + 0.637826i 2.92922 + 0.647817i 0.969439 + 1.45087i
11.7 −0.521461 + 1.25892i 1.69661 − 0.348589i 0.101261 + 0.101261i −0.555570 + 0.831470i −0.445870 + 2.31767i −0.753030 + 0.503159i −2.69812 + 1.11760i 2.75697 − 1.18284i −0.757044 − 1.13300i
11.8 −0.380841 + 0.919431i −1.61022 + 0.638116i 0.713901 + 0.713901i 0.555570 − 0.831470i 0.0265334 − 1.72351i −2.56652 + 1.71489i −2.76713 + 1.14618i 2.18562 − 2.05502i 0.552895 + 0.827466i
11.9 −0.377068 + 0.910322i 1.03348 + 1.38993i 0.727707 + 0.727707i 0.555570 − 0.831470i −1.65498 + 0.416702i −3.62069 + 2.41927i −2.75749 + 1.14219i −0.863834 + 2.87294i 0.547418 + 0.819269i
11.10 −0.245146 + 0.591834i −0.992918 − 1.41919i 1.12404 + 1.12404i 0.555570 − 0.831470i 1.08334 − 0.239733i 1.91020 − 1.27635i −2.12447 + 0.879984i −1.02823 + 2.81829i 0.355896 + 0.532637i
11.11 −0.160942 + 0.388548i 0.972393 + 1.43334i 1.28915 + 1.28915i −0.555570 + 0.831470i −0.713419 + 0.147137i 2.32037 − 1.55042i −1.48547 + 0.615301i −1.10891 + 2.78753i −0.233651 − 0.349684i
11.12 −0.0737228 + 0.177983i −1.62355 − 0.603383i 1.38797 + 1.38797i −0.555570 + 0.831470i 0.227084 − 0.244481i −0.167227 + 0.111738i −0.705325 + 0.292155i 2.27186 + 1.95925i −0.107029 − 0.160180i
11.13 0.0737228 − 0.177983i 1.26906 − 1.17876i 1.38797 + 1.38797i 0.555570 − 0.831470i −0.116240 − 0.312773i −0.167227 + 0.111738i 0.705325 − 0.292155i 0.221049 − 2.99185i −0.107029 − 0.160180i
11.14 0.160942 − 0.388548i −0.349860 + 1.69635i 1.28915 + 1.28915i 0.555570 − 0.831470i 0.602806 + 0.408951i 2.32037 − 1.55042i 1.48547 − 0.615301i −2.75520 − 1.18697i −0.233651 − 0.349684i
11.15 0.245146 − 0.591834i 0.374234 − 1.69114i 1.12404 + 1.12404i −0.555570 + 0.831470i −0.909131 − 0.636060i 1.91020 − 1.27635i 2.12447 − 0.879984i −2.71990 − 1.26576i 0.355896 + 0.532637i
11.16 0.377068 − 0.910322i −0.422907 + 1.67963i 0.727707 + 0.727707i −0.555570 + 0.831470i 1.36954 + 1.01832i −3.62069 + 2.41927i 2.75749 − 1.14219i −2.64230 − 1.42065i 0.547418 + 0.819269i
11.17 0.380841 − 0.919431i 1.73185 − 0.0266618i 0.713901 + 0.713901i −0.555570 + 0.831470i 0.635043 − 1.60247i −2.56652 + 1.71489i 2.76713 − 1.14618i 2.99858 − 0.0923482i 0.552895 + 0.827466i
11.18 0.521461 − 1.25892i −1.70086 + 0.327210i 0.101261 + 0.101261i 0.555570 − 0.831470i −0.475003 + 2.31187i −0.753030 + 0.503159i 2.69812 − 1.11760i 2.78587 − 1.11308i −0.757044 − 1.13300i
11.19 0.667761 − 1.61212i −1.51875 + 0.832708i −0.738806 − 0.738806i −0.555570 + 0.831470i 0.328261 + 3.00445i 3.47691 − 2.32320i 1.53985 − 0.637826i 1.61320 − 2.52935i 0.969439 + 1.45087i
11.20 0.693520 − 1.67431i 1.25700 + 1.19161i −0.908118 − 0.908118i 0.555570 − 0.831470i 2.86688 − 1.27820i −0.375953 + 0.251204i 1.19835 − 0.496371i 0.160109 + 2.99572i −1.00684 − 1.50684i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.e odd 16 1 inner
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 255.2.bg.a ✓ 192
3.b odd 2 1 inner 255.2.bg.a ✓ 192
17.e odd 16 1 inner 255.2.bg.a ✓ 192
51.i even 16 1 inner 255.2.bg.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.bg.a ✓ 192 1.a even 1 1 trivial
255.2.bg.a ✓ 192 3.b odd 2 1 inner
255.2.bg.a ✓ 192 17.e odd 16 1 inner
255.2.bg.a ✓ 192 51.i even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(255, [\chi])\).