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Properties

Label 88.3.j.a
Level $88$
Weight $3$
Character orbit 88.j
Analytic conductor $2.398$
Analytic rank $0$
Dimension $24$
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] = [88,3,Mod(17,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 88.j (of order \(10\), 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: \(2.39782632637\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 6 q^{9} - 10 q^{11} + 60 q^{15} + 40 q^{17} + 30 q^{19} + 32 q^{23} - 102 q^{25} - 122 q^{27} - 80 q^{29} - 64 q^{31} - 86 q^{33} - 240 q^{35} + 48 q^{37} - 240 q^{39} - 60 q^{41} - 112 q^{45} + 180 q^{47} + 350 q^{49} + 550 q^{51} + 132 q^{53} + 388 q^{55} + 530 q^{57} - 26 q^{59} + 80 q^{61} + 80 q^{63} + 212 q^{67} - 252 q^{69} - 540 q^{71} - 580 q^{73} - 106 q^{75} - 540 q^{77} - 280 q^{79} - 228 q^{81} - 90 q^{83} - 120 q^{85} - 324 q^{89} + 228 q^{91} + 988 q^{93} + 280 q^{95} + 410 q^{97} + 318 q^{99}+O(q^{100}) \) 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 −4.52659 − 3.28876i 0 −1.81573 + 5.58823i 0 4.31792 + 5.94311i 0 6.89293 + 21.2143i 0
17.2 0 −2.12522 − 1.54406i 0 0.346461 − 1.06630i 0 −5.02508 − 6.91643i 0 −0.648721 − 1.99656i 0
17.3 0 −0.560350 − 0.407118i 0 2.08221 − 6.40839i 0 0.954558 + 1.31384i 0 −2.63291 − 8.10325i 0
17.4 0 0.819877 + 0.595676i 0 −1.97358 + 6.07406i 0 5.14561 + 7.08233i 0 −2.46378 − 7.58275i 0
17.5 0 3.41069 + 2.47801i 0 −1.29397 + 3.98244i 0 0.167326 + 0.230305i 0 2.71112 + 8.34397i 0
17.6 0 3.98159 + 2.89280i 0 2.65461 − 8.17003i 0 −1.08820 − 1.49778i 0 4.70366 + 14.4764i 0
41.1 0 −1.36332 − 4.19587i 0 −0.338630 − 0.246029i 0 −6.54605 − 2.12694i 0 −8.46554 + 6.15057i 0
41.2 0 −0.360016 − 1.10802i 0 −6.96735 − 5.06208i 0 −7.09062 − 2.30388i 0 6.18307 − 4.49226i 0
41.3 0 −0.310916 − 0.956901i 0 4.68625 + 3.40476i 0 1.71995 + 0.558844i 0 6.46216 − 4.69504i 0
41.4 0 0.0884657 + 0.272269i 0 −0.176098 − 0.127943i 0 12.2289 + 3.97342i 0 7.21485 − 5.24189i 0
41.5 0 1.26111 + 3.88130i 0 4.86723 + 3.53625i 0 −11.7577 − 3.82031i 0 −6.19297 + 4.49946i 0
41.6 0 1.68468 + 5.18490i 0 −2.07140 − 1.50496i 0 6.97337 + 2.26578i 0 −16.7639 + 12.1797i 0
57.1 0 −4.52659 + 3.28876i 0 −1.81573 − 5.58823i 0 4.31792 − 5.94311i 0 6.89293 − 21.2143i 0
57.2 0 −2.12522 + 1.54406i 0 0.346461 + 1.06630i 0 −5.02508 + 6.91643i 0 −0.648721 + 1.99656i 0
57.3 0 −0.560350 + 0.407118i 0 2.08221 + 6.40839i 0 0.954558 − 1.31384i 0 −2.63291 + 8.10325i 0
57.4 0 0.819877 − 0.595676i 0 −1.97358 − 6.07406i 0 5.14561 − 7.08233i 0 −2.46378 + 7.58275i 0
57.5 0 3.41069 − 2.47801i 0 −1.29397 − 3.98244i 0 0.167326 − 0.230305i 0 2.71112 − 8.34397i 0
57.6 0 3.98159 − 2.89280i 0 2.65461 + 8.17003i 0 −1.08820 + 1.49778i 0 4.70366 − 14.4764i 0
73.1 0 −1.36332 + 4.19587i 0 −0.338630 + 0.246029i 0 −6.54605 + 2.12694i 0 −8.46554 − 6.15057i 0
73.2 0 −0.360016 + 1.10802i 0 −6.96735 + 5.06208i 0 −7.09062 + 2.30388i 0 6.18307 + 4.49226i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.3.j.a ✓ 24
4.b odd 2 1 176.3.n.d 24
11.c even 5 1 968.3.h.e 24
11.d odd 10 1 inner 88.3.j.a ✓ 24
11.d odd 10 1 968.3.h.e 24
44.g even 10 1 176.3.n.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.3.j.a ✓ 24 1.a even 1 1 trivial
88.3.j.a ✓ 24 11.d odd 10 1 inner
176.3.n.d 24 4.b odd 2 1
176.3.n.d 24 44.g even 10 1
968.3.h.e 24 11.c even 5 1
968.3.h.e 24 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(88, [\chi])\).