Properties

Label 1088.2.m.i
Level $1088$
Weight $2$
Character orbit 1088.m
Analytic conductor $8.688$
Analytic rank $0$
Dimension $24$
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] = [1088,2,Mod(225,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1088.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.m (of order \(4\), degree \(2\), 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: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{17} - 24 q^{19} + 16 q^{33} - 16 q^{41} + 32 q^{43} + 40 q^{51} - 48 q^{57} - 64 q^{59} - 16 q^{65} - 8 q^{73} + 32 q^{75} + 88 q^{81} - 144 q^{83} + 24 q^{89} + 48 q^{91} + 32 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1 0 −1.74370 1.74370i 0 −2.07081 2.07081i 0 0.691235 + 0.691235i 0 3.08101i 0
225.2 0 −1.74370 1.74370i 0 2.07081 + 2.07081i 0 −0.691235 0.691235i 0 3.08101i 0
225.3 0 −1.36000 1.36000i 0 −1.15824 1.15824i 0 2.45561 + 2.45561i 0 0.699218i 0
225.4 0 −1.36000 1.36000i 0 1.15824 + 1.15824i 0 −2.45561 2.45561i 0 0.699218i 0
225.5 0 −0.391124 0.391124i 0 2.42619 + 2.42619i 0 3.52212 + 3.52212i 0 2.69404i 0
225.6 0 −0.391124 0.391124i 0 −2.42619 2.42619i 0 −3.52212 3.52212i 0 2.69404i 0
225.7 0 0.514752 + 0.514752i 0 −1.38937 1.38937i 0 0.251879 + 0.251879i 0 2.47006i 0
225.8 0 0.514752 + 0.514752i 0 1.38937 + 1.38937i 0 −0.251879 0.251879i 0 2.47006i 0
225.9 0 1.13543 + 1.13543i 0 −2.26352 2.26352i 0 1.15552 + 1.15552i 0 0.421600i 0
225.10 0 1.13543 + 1.13543i 0 2.26352 + 2.26352i 0 −1.15552 1.15552i 0 0.421600i 0
225.11 0 1.84465 + 1.84465i 0 0.655716 + 0.655716i 0 2.58615 + 2.58615i 0 3.80548i 0
225.12 0 1.84465 + 1.84465i 0 −0.655716 0.655716i 0 −2.58615 2.58615i 0 3.80548i 0
353.1 0 −1.74370 + 1.74370i 0 −2.07081 + 2.07081i 0 0.691235 0.691235i 0 3.08101i 0
353.2 0 −1.74370 + 1.74370i 0 2.07081 2.07081i 0 −0.691235 + 0.691235i 0 3.08101i 0
353.3 0 −1.36000 + 1.36000i 0 −1.15824 + 1.15824i 0 2.45561 2.45561i 0 0.699218i 0
353.4 0 −1.36000 + 1.36000i 0 1.15824 1.15824i 0 −2.45561 + 2.45561i 0 0.699218i 0
353.5 0 −0.391124 + 0.391124i 0 2.42619 2.42619i 0 3.52212 3.52212i 0 2.69404i 0
353.6 0 −0.391124 + 0.391124i 0 −2.42619 + 2.42619i 0 −3.52212 + 3.52212i 0 2.69404i 0
353.7 0 0.514752 0.514752i 0 −1.38937 + 1.38937i 0 0.251879 0.251879i 0 2.47006i 0
353.8 0 0.514752 0.514752i 0 1.38937 1.38937i 0 −0.251879 + 0.251879i 0 2.47006i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
68.f odd 4 1 inner
136.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.m.i 24
4.b odd 2 1 1088.2.m.j yes 24
8.b even 2 1 1088.2.m.j yes 24
8.d odd 2 1 inner 1088.2.m.i 24
17.c even 4 1 1088.2.m.j yes 24
68.f odd 4 1 inner 1088.2.m.i 24
136.i even 4 1 inner 1088.2.m.i 24
136.j odd 4 1 1088.2.m.j yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1088.2.m.i 24 1.a even 1 1 trivial
1088.2.m.i 24 8.d odd 2 1 inner
1088.2.m.i 24 68.f odd 4 1 inner
1088.2.m.i 24 136.i even 4 1 inner
1088.2.m.j yes 24 4.b odd 2 1
1088.2.m.j yes 24 8.b even 2 1
1088.2.m.j yes 24 17.c even 4 1
1088.2.m.j yes 24 136.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1088, [\chi])\):

\( T_{3}^{12} + 52T_{3}^{8} + 4T_{3}^{7} - 40T_{3}^{5} + 416T_{3}^{4} - 96T_{3}^{3} + 8T_{3}^{2} + 32T_{3} + 64 \) Copy content Toggle raw display
\( T_{5}^{24} + 340T_{5}^{20} + 39840T_{5}^{16} + 1851520T_{5}^{12} + 28493056T_{5}^{8} + 134931456T_{5}^{4} + 84934656 \) Copy content Toggle raw display