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Properties

Label 816.2.br.b
Level $816$
Weight $2$
Character orbit 816.br
Analytic conductor $6.516$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(287,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.br (of order \(8\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{9} + 56 q^{45} + 96 q^{49} + 40 q^{57} + 16 q^{61} + 64 q^{69} + 32 q^{73} - 16 q^{85} + 56 q^{93} + 176 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1 0 −1.71029 + 0.273700i 0 0.841614 + 2.03184i 0 −0.473897 + 1.14409i 0 2.85018 − 0.936213i 0
287.2 0 −1.62423 − 0.601571i 0 −0.153275 − 0.370038i 0 0.0612364 − 0.147838i 0 2.27622 + 1.95418i 0
287.3 0 −1.61426 + 0.627832i 0 1.42471 + 3.43956i 0 1.17763 − 2.84305i 0 2.21165 − 2.02696i 0
287.4 0 −1.58540 + 0.697508i 0 −1.42471 − 3.43956i 0 1.17763 − 2.84305i 0 2.02696 − 2.21165i 0
287.5 0 −1.56223 − 0.747946i 0 −0.458628 − 1.10723i 0 −1.27004 + 3.06615i 0 1.88115 + 2.33693i 0
287.6 0 −1.40289 + 1.01582i 0 −0.841614 − 2.03184i 0 −0.473897 + 1.14409i 0 0.936213 − 2.85018i 0
287.7 0 −1.26023 − 1.18820i 0 0.378157 + 0.912953i 0 0.959554 − 2.31657i 0 0.176370 + 2.99481i 0
287.8 0 −1.20960 − 1.23971i 0 −1.51775 − 3.66417i 0 1.27981 − 3.08973i 0 −0.0737392 + 2.99909i 0
287.9 0 −0.723126 + 1.57388i 0 0.153275 + 0.370038i 0 0.0612364 − 0.147838i 0 −1.95418 − 2.27622i 0
287.10 0 −0.575789 + 1.63354i 0 0.458628 + 1.10723i 0 −1.27004 + 3.06615i 0 −2.33693 − 1.88115i 0
287.11 0 −0.0509356 + 1.73130i 0 −0.378157 − 0.912953i 0 0.959554 − 2.31657i 0 −2.99481 − 0.176370i 0
287.12 0 −0.0212883 − 1.73192i 0 1.51775 + 3.66417i 0 −1.27981 + 3.08973i 0 −2.99909 + 0.0737392i 0
287.13 0 0.0212883 + 1.73192i 0 1.51775 + 3.66417i 0 1.27981 − 3.08973i 0 −2.99909 + 0.0737392i 0
287.14 0 0.0509356 − 1.73130i 0 −0.378157 − 0.912953i 0 −0.959554 + 2.31657i 0 −2.99481 − 0.176370i 0
287.15 0 0.575789 − 1.63354i 0 0.458628 + 1.10723i 0 1.27004 − 3.06615i 0 −2.33693 − 1.88115i 0
287.16 0 0.723126 − 1.57388i 0 0.153275 + 0.370038i 0 −0.0612364 + 0.147838i 0 −1.95418 − 2.27622i 0
287.17 0 1.20960 + 1.23971i 0 −1.51775 − 3.66417i 0 −1.27981 + 3.08973i 0 −0.0737392 + 2.99909i 0
287.18 0 1.26023 + 1.18820i 0 0.378157 + 0.912953i 0 −0.959554 + 2.31657i 0 0.176370 + 2.99481i 0
287.19 0 1.40289 − 1.01582i 0 −0.841614 − 2.03184i 0 0.473897 − 1.14409i 0 0.936213 − 2.85018i 0
287.20 0 1.56223 + 0.747946i 0 −0.458628 − 1.10723i 0 1.27004 − 3.06615i 0 1.88115 + 2.33693i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
17.d even 8 1 inner
51.g odd 8 1 inner
68.g odd 8 1 inner
204.p even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.br.b ✓ 96
3.b odd 2 1 inner 816.2.br.b ✓ 96
4.b odd 2 1 inner 816.2.br.b ✓ 96
12.b even 2 1 inner 816.2.br.b ✓ 96
17.d even 8 1 inner 816.2.br.b ✓ 96
51.g odd 8 1 inner 816.2.br.b ✓ 96
68.g odd 8 1 inner 816.2.br.b ✓ 96
204.p even 8 1 inner 816.2.br.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.br.b ✓ 96 1.a even 1 1 trivial
816.2.br.b ✓ 96 3.b odd 2 1 inner
816.2.br.b ✓ 96 4.b odd 2 1 inner
816.2.br.b ✓ 96 12.b even 2 1 inner
816.2.br.b ✓ 96 17.d even 8 1 inner
816.2.br.b ✓ 96 51.g odd 8 1 inner
816.2.br.b ✓ 96 68.g odd 8 1 inner
816.2.br.b ✓ 96 204.p even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{48} - 440 T_{5}^{42} + 89457 T_{5}^{40} - 91160 T_{5}^{38} + 96800 T_{5}^{36} - 22209728 T_{5}^{34} + \cdots + 1212153856 \) acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\). Copy content Toggle raw display