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Properties

Label 576.2.bd.c
Level $576$
Weight $2$
Character orbit 576.bd
Analytic conductor $4.599$
Analytic rank $0$
Dimension $128$
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] = [576,2,Mod(37,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.bd (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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 16 q^{22} + 80 q^{28} + 80 q^{34} + 80 q^{40} + 48 q^{52} - 64 q^{55} + 96 q^{64} - 32 q^{67} + 96 q^{70} + 16 q^{76} - 32 q^{79} - 80 q^{82} - 80 q^{88} - 96 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.41184 + 0.0818225i 0 1.98661 − 0.231041i −0.236808 + 1.19051i 0 0.813698 − 1.96444i −2.78588 + 0.488744i 0 0.236925 − 1.70019i
37.2 −1.36676 + 0.363273i 0 1.73607 − 0.993013i 0.466809 − 2.34681i 0 −0.714248 + 1.72435i −2.01205 + 1.98788i 0 0.214515 + 3.37710i
37.3 −1.22329 − 0.709617i 0 0.992888 + 1.73614i −0.111985 + 0.562985i 0 −1.81965 + 4.39303i 0.0174012 − 2.82837i 0 0.536493 − 0.609228i
37.4 −0.978490 + 1.02106i 0 −0.0851132 − 1.99819i −0.847770 + 4.26203i 0 −1.35580 + 3.27320i 2.12355 + 1.86830i 0 −3.52224 − 5.03598i
37.5 −0.947125 − 1.05022i 0 −0.205909 + 1.98937i −0.494578 + 2.48641i 0 1.33217 − 3.21615i 2.28429 − 1.66794i 0 3.07970 − 1.83553i
37.6 −0.838345 + 1.13894i 0 −0.594356 − 1.90964i 0.528008 − 2.65447i 0 0.715686 − 1.72782i 2.67324 + 0.924005i 0 2.58063 + 2.82673i
37.7 −0.365292 − 1.36622i 0 −1.73312 + 0.998139i 0.527257 − 2.65070i 0 0.735368 − 1.77534i 1.99677 + 2.00322i 0 −3.81405 + 0.247929i
37.8 −0.244654 + 1.39289i 0 −1.88029 − 0.681553i 0.142292 − 0.715349i 0 −0.472587 + 1.14093i 1.40935 − 2.45229i 0 0.961591 + 0.373210i
37.9 0.244654 − 1.39289i 0 −1.88029 − 0.681553i −0.142292 + 0.715349i 0 −0.472587 + 1.14093i −1.40935 + 2.45229i 0 0.961591 + 0.373210i
37.10 0.365292 + 1.36622i 0 −1.73312 + 0.998139i −0.527257 + 2.65070i 0 0.735368 − 1.77534i −1.99677 − 2.00322i 0 −3.81405 + 0.247929i
37.11 0.838345 − 1.13894i 0 −0.594356 − 1.90964i −0.528008 + 2.65447i 0 0.715686 − 1.72782i −2.67324 − 0.924005i 0 2.58063 + 2.82673i
37.12 0.947125 + 1.05022i 0 −0.205909 + 1.98937i 0.494578 − 2.48641i 0 1.33217 − 3.21615i −2.28429 + 1.66794i 0 3.07970 − 1.83553i
37.13 0.978490 − 1.02106i 0 −0.0851132 − 1.99819i 0.847770 − 4.26203i 0 −1.35580 + 3.27320i −2.12355 − 1.86830i 0 −3.52224 − 5.03598i
37.14 1.22329 + 0.709617i 0 0.992888 + 1.73614i 0.111985 − 0.562985i 0 −1.81965 + 4.39303i −0.0174012 + 2.82837i 0 0.536493 − 0.609228i
37.15 1.36676 − 0.363273i 0 1.73607 − 0.993013i −0.466809 + 2.34681i 0 −0.714248 + 1.72435i 2.01205 − 1.98788i 0 0.214515 + 3.37710i
37.16 1.41184 − 0.0818225i 0 1.98661 − 0.231041i 0.236808 − 1.19051i 0 0.813698 − 1.96444i 2.78588 − 0.488744i 0 0.236925 − 1.70019i
109.1 −1.41184 − 0.0818225i 0 1.98661 + 0.231041i −0.236808 − 1.19051i 0 0.813698 + 1.96444i −2.78588 − 0.488744i 0 0.236925 + 1.70019i
109.2 −1.36676 − 0.363273i 0 1.73607 + 0.993013i 0.466809 + 2.34681i 0 −0.714248 − 1.72435i −2.01205 − 1.98788i 0 0.214515 − 3.37710i
109.3 −1.22329 + 0.709617i 0 0.992888 − 1.73614i −0.111985 − 0.562985i 0 −1.81965 − 4.39303i 0.0174012 + 2.82837i 0 0.536493 + 0.609228i
109.4 −0.978490 − 1.02106i 0 −0.0851132 + 1.99819i −0.847770 − 4.26203i 0 −1.35580 − 3.27320i 2.12355 − 1.86830i 0 −3.52224 + 5.03598i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
64.i even 16 1 inner
192.q odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.bd.c ✓ 128
3.b odd 2 1 inner 576.2.bd.c ✓ 128
64.i even 16 1 inner 576.2.bd.c ✓ 128
192.q odd 16 1 inner 576.2.bd.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.bd.c ✓ 128 1.a even 1 1 trivial
576.2.bd.c ✓ 128 3.b odd 2 1 inner
576.2.bd.c ✓ 128 64.i even 16 1 inner
576.2.bd.c ✓ 128 192.q odd 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} + 1344 T_{5}^{122} - 420352 T_{5}^{118} + 6239040 T_{5}^{116} - 146681600 T_{5}^{114} + \cdots + 69\!\cdots\!36 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display