↑

Properties

Label 536.2.y.a
Level $536$
Weight $2$
Character orbit 536.y
Analytic conductor $4.280$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [536,2,Mod(17,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 0, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.y (of order \(33\), degree \(20\), 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: \(4.27998154834\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{3} + 2 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 2 q^{3} + 2 q^{5} - 14 q^{9} + q^{11} - 2 q^{13} + 12 q^{15} + q^{17} + 5 q^{19} + 49 q^{21} + 21 q^{23} - 10 q^{25} + 10 q^{27} + 2 q^{29} + 16 q^{31} + 33 q^{33} + 61 q^{35} + 52 q^{37} - 12 q^{39} + 13 q^{41} - 25 q^{43} - 49 q^{45} + 29 q^{47} + 14 q^{49} + 11 q^{51} + 12 q^{53} - 88 q^{55} - 23 q^{57} + 57 q^{59} + 107 q^{61} - 36 q^{63} - 11 q^{65} - 30 q^{67} - 85 q^{69} + q^{71} - 10 q^{73} + 70 q^{75} - 12 q^{77} - 59 q^{79} - 244 q^{81} + 42 q^{83} - 44 q^{85} + 143 q^{87} + 32 q^{89} + 44 q^{91} - 117 q^{93} - 115 q^{95} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −2.39266 + 1.53767i 0 −0.0712804 + 0.495766i 0 −0.572717 − 0.804269i 0 2.11416 − 4.62936i 0
17.2 0 −0.836023 + 0.537279i 0 0.449187 − 3.12416i 0 −0.634987 − 0.891715i 0 −0.835980 + 1.83054i 0
17.3 0 −0.699524 + 0.449556i 0 0.114263 − 0.794713i 0 −0.906588 − 1.27313i 0 −0.959013 + 2.09995i 0
17.4 0 −0.283093 + 0.181933i 0 −0.129416 + 0.900108i 0 2.46292 + 3.45868i 0 −1.19920 + 2.62589i 0
17.5 0 0.424570 − 0.272854i 0 −0.610178 + 4.24388i 0 −2.57882 − 3.62145i 0 −1.14044 + 2.49720i 0
17.6 0 1.43422 − 0.921716i 0 −0.160396 + 1.11558i 0 1.54103 + 2.16407i 0 −0.0388230 + 0.0850105i 0
17.7 0 1.87917 − 1.20767i 0 0.632855 − 4.40160i 0 0.981623 + 1.37850i 0 0.826563 − 1.80992i 0
17.8 0 2.57004 − 1.65166i 0 −0.0827182 + 0.575318i 0 −1.24371 − 1.74654i 0 2.63085 − 5.76076i 0
33.1 0 −2.48667 + 0.730151i 0 1.97321 + 2.27721i 0 1.10953 + 0.572002i 0 3.12663 − 2.00936i 0
33.2 0 −2.06437 + 0.606153i 0 −1.04261 − 1.20323i 0 2.18051 + 1.12413i 0 1.37043 − 0.880724i 0
33.3 0 −1.88359 + 0.553071i 0 −2.12177 − 2.44865i 0 −3.61041 − 1.86130i 0 0.718255 − 0.461594i 0
33.4 0 −0.0583028 + 0.0171193i 0 1.66522 + 1.92177i 0 −2.66847 − 1.37569i 0 −2.52065 + 1.61993i 0
33.5 0 0.129990 − 0.0381684i 0 0.180675 + 0.208510i 0 −3.00999 − 1.55176i 0 −2.50832 + 1.61200i 0
33.6 0 0.844311 − 0.247912i 0 0.511898 + 0.590762i 0 3.25666 + 1.67892i 0 −1.87236 + 1.20329i 0
33.7 0 2.22207 − 0.652458i 0 1.00580 + 1.16075i 0 1.10426 + 0.569284i 0 1.98812 − 1.27769i 0
33.8 0 2.73894 − 0.804224i 0 −1.51757 − 1.75136i 0 −1.17000 − 0.603176i 0 4.33124 − 2.78352i 0
49.1 0 −2.73496 − 1.75765i 0 0.208769 + 1.45202i 0 −2.99928 − 0.286396i 0 3.14442 + 6.88532i 0
49.2 0 −1.65312 − 1.06240i 0 0.0994144 + 0.691442i 0 2.99588 + 0.286072i 0 0.357876 + 0.783640i 0
49.3 0 −1.42282 − 0.914393i 0 −0.533520 − 3.71071i 0 −0.857588 − 0.0818897i 0 −0.0579333 − 0.126856i 0
49.4 0 −0.383058 − 0.246177i 0 0.570034 + 3.96467i 0 1.14676 + 0.109503i 0 −1.16011 − 2.54030i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.y.a ✓ 160
67.g even 33 1 inner 536.2.y.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.y.a ✓ 160 1.a even 1 1 trivial
536.2.y.a ✓ 160 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{160} + 2 T_{3}^{159} + 33 T_{3}^{158} + 64 T_{3}^{157} + 709 T_{3}^{156} + 962 T_{3}^{155} + \cdots + 77\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(536, [\chi])\). Copy content Toggle raw display