Properties

Label 3584.2.m.bn
Level $3584$
Weight $2$
Character orbit 3584.m
Analytic conductor $28.618$
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] = [3584,2,Mod(897,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3584.897");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.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: \(28.6183840844\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{5} - 8 q^{13} + 8 q^{29} + 32 q^{33} + 8 q^{35} - 40 q^{37} + 40 q^{45} - 24 q^{49} - 8 q^{53} - 40 q^{61} - 24 q^{63} - 16 q^{65} - 64 q^{67} - 56 q^{81} + 32 q^{85} + 8 q^{91} + 64 q^{95} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
897.1 0 −2.34057 + 2.34057i 0 1.01838 + 1.01838i 0 1.00000i 0 7.95654i 0
897.2 0 −1.63240 + 1.63240i 0 0.462322 + 0.462322i 0 1.00000i 0 2.32947i 0
897.3 0 −1.53441 + 1.53441i 0 −1.58168 1.58168i 0 1.00000i 0 1.70884i 0
897.4 0 −1.07038 + 1.07038i 0 2.97612 + 2.97612i 0 1.00000i 0 0.708572i 0
897.5 0 −0.482987 + 0.482987i 0 2.27930 + 2.27930i 0 1.00000i 0 2.53345i 0
897.6 0 −0.303325 + 0.303325i 0 −2.39032 2.39032i 0 1.00000i 0 2.81599i 0
897.7 0 0.125826 0.125826i 0 −1.32428 1.32428i 0 1.00000i 0 2.96834i 0
897.8 0 0.714770 0.714770i 0 0.214515 + 0.214515i 0 1.00000i 0 1.97821i 0
897.9 0 1.19058 1.19058i 0 0.370923 + 0.370923i 0 1.00000i 0 0.165048i 0
897.10 0 1.22968 1.22968i 0 1.60657 + 1.60657i 0 1.00000i 0 0.0242347i 0
897.11 0 1.77137 1.77137i 0 −1.80277 1.80277i 0 1.00000i 0 3.27553i 0
897.12 0 2.33184 2.33184i 0 2.17092 + 2.17092i 0 1.00000i 0 7.87499i 0
2689.1 0 −2.34057 2.34057i 0 1.01838 1.01838i 0 1.00000i 0 7.95654i 0
2689.2 0 −1.63240 1.63240i 0 0.462322 0.462322i 0 1.00000i 0 2.32947i 0
2689.3 0 −1.53441 1.53441i 0 −1.58168 + 1.58168i 0 1.00000i 0 1.70884i 0
2689.4 0 −1.07038 1.07038i 0 2.97612 2.97612i 0 1.00000i 0 0.708572i 0
2689.5 0 −0.482987 0.482987i 0 2.27930 2.27930i 0 1.00000i 0 2.53345i 0
2689.6 0 −0.303325 0.303325i 0 −2.39032 + 2.39032i 0 1.00000i 0 2.81599i 0
2689.7 0 0.125826 + 0.125826i 0 −1.32428 + 1.32428i 0 1.00000i 0 2.96834i 0
2689.8 0 0.714770 + 0.714770i 0 0.214515 0.214515i 0 1.00000i 0 1.97821i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 897.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.m.bn yes 24
4.b odd 2 1 3584.2.m.bm yes 24
8.b even 2 1 3584.2.m.bk 24
8.d odd 2 1 3584.2.m.bl yes 24
16.e even 4 1 3584.2.m.bk 24
16.e even 4 1 inner 3584.2.m.bn yes 24
16.f odd 4 1 3584.2.m.bl yes 24
16.f odd 4 1 3584.2.m.bm yes 24
32.g even 8 1 7168.2.a.bg 12
32.g even 8 1 7168.2.a.bk 12
32.h odd 8 1 7168.2.a.bh 12
32.h odd 8 1 7168.2.a.bl 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.bk 24 8.b even 2 1
3584.2.m.bk 24 16.e even 4 1
3584.2.m.bl yes 24 8.d odd 2 1
3584.2.m.bl yes 24 16.f odd 4 1
3584.2.m.bm yes 24 4.b odd 2 1
3584.2.m.bm yes 24 16.f odd 4 1
3584.2.m.bn yes 24 1.a even 1 1 trivial
3584.2.m.bn yes 24 16.e even 4 1 inner
7168.2.a.bg 12 32.g even 8 1
7168.2.a.bh 12 32.h odd 8 1
7168.2.a.bk 12 32.g even 8 1
7168.2.a.bl 12 32.h odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3584, [\chi])\):

\( T_{3}^{24} + 176 T_{3}^{20} + 7632 T_{3}^{16} + 64 T_{3}^{15} - 1408 T_{3}^{13} + 106048 T_{3}^{12} - 9728 T_{3}^{11} + 20480 T_{3}^{9} + 430656 T_{3}^{8} - 22016 T_{3}^{7} + 2048 T_{3}^{6} + 105472 T_{3}^{5} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{5}^{24} - 8 T_{5}^{23} + 32 T_{5}^{22} - 48 T_{5}^{21} + 248 T_{5}^{20} - 1696 T_{5}^{19} + 6784 T_{5}^{18} - 9984 T_{5}^{17} + 20224 T_{5}^{16} - 101952 T_{5}^{15} + 403456 T_{5}^{14} - 568448 T_{5}^{13} + 728512 T_{5}^{12} + \cdots + 262144 \) Copy content Toggle raw display
\( T_{11}^{24} + 1584 T_{11}^{20} + 128 T_{11}^{19} - 6144 T_{11}^{17} + 766016 T_{11}^{16} - 5120 T_{11}^{15} + 8192 T_{11}^{14} - 3514368 T_{11}^{13} + 107331584 T_{11}^{12} - 54820864 T_{11}^{11} + 5242880 T_{11}^{10} + \cdots + 67108864 \) Copy content Toggle raw display
\( T_{13}^{24} + 8 T_{13}^{23} + 32 T_{13}^{22} + 48 T_{13}^{21} + 1720 T_{13}^{20} + 13664 T_{13}^{19} + 55424 T_{13}^{18} + 78016 T_{13}^{17} + 834048 T_{13}^{16} + 6411968 T_{13}^{15} + 26374144 T_{13}^{14} + \cdots + 157856825344 \) Copy content Toggle raw display