Properties

Label 896.2.z.b
Level $896$
Weight $2$
Character orbit 896.z
Analytic conductor $7.155$
Analytic rank $0$
Dimension $56$
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] = [896,2,Mod(31,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.z (of order \(12\), degree \(4\), not 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} + 6 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} + 6 q^{5} - 8 q^{7} - 2 q^{11} - 12 q^{17} + 6 q^{19} + 10 q^{21} - 12 q^{23} + 24 q^{29} - 12 q^{33} + 2 q^{35} - 6 q^{37} - 4 q^{39} - 12 q^{45} - 8 q^{49} + 34 q^{51} - 6 q^{53} - 42 q^{59} + 6 q^{61} - 4 q^{65} - 6 q^{67} - 80 q^{71} - 24 q^{75} - 10 q^{77} - 8 q^{81} + 28 q^{85} - 12 q^{87} - 16 q^{91} - 10 q^{93} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −0.743580 2.77508i 0 3.38680 + 0.907491i 0 0.0791552 2.64457i 0 −4.55008 + 2.62699i 0
31.2 0 −0.615336 2.29647i 0 0.835825 + 0.223959i 0 −1.25761 + 2.32775i 0 −2.29704 + 1.32620i 0
31.3 0 −0.524583 1.95777i 0 −2.48890 0.666898i 0 −2.38027 + 1.15512i 0 −0.959602 + 0.554026i 0
31.4 0 −0.449868 1.67893i 0 −0.731029 0.195879i 0 2.52163 0.800849i 0 −0.0183525 + 0.0105958i 0
31.5 0 −0.350301 1.30734i 0 −1.09872 0.294400i 0 −1.56831 2.13082i 0 1.01165 0.584076i 0
31.6 0 −0.0530773 0.198087i 0 1.82029 + 0.487744i 0 1.84933 + 1.89208i 0 2.56165 1.47897i 0
31.7 0 0.0661591 + 0.246909i 0 0.499339 + 0.133797i 0 2.45011 0.998472i 0 2.54149 1.46733i 0
31.8 0 0.204601 + 0.763582i 0 3.81370 + 1.02188i 0 −2.64575 + 0.00379639i 0 2.05688 1.18754i 0
31.9 0 0.237312 + 0.885661i 0 −3.05579 0.818796i 0 −0.640837 + 2.56697i 0 1.87000 1.07964i 0
31.10 0 0.282507 + 1.05433i 0 1.39922 + 0.374919i 0 −0.298614 + 2.62885i 0 1.56627 0.904287i 0
31.11 0 0.388227 + 1.44888i 0 −2.81809 0.755106i 0 −1.47726 2.19493i 0 0.649537 0.375010i 0
31.12 0 0.665801 + 2.48480i 0 3.12401 + 0.837076i 0 1.56215 2.13534i 0 −3.13287 + 1.80877i 0
31.13 0 0.700587 + 2.61463i 0 −0.450647 0.120751i 0 −2.37326 1.16946i 0 −3.74737 + 2.16355i 0
31.14 0 0.825526 + 3.08091i 0 −1.86998 0.501060i 0 2.17953 + 1.49988i 0 −6.21241 + 3.58674i 0
159.1 0 −2.77508 0.743580i 0 0.907491 + 3.38680i 0 0.0791552 + 2.64457i 0 4.55008 + 2.62699i 0
159.2 0 −2.29647 0.615336i 0 0.223959 + 0.835825i 0 −1.25761 2.32775i 0 2.29704 + 1.32620i 0
159.3 0 −1.95777 0.524583i 0 −0.666898 2.48890i 0 −2.38027 1.15512i 0 0.959602 + 0.554026i 0
159.4 0 −1.67893 0.449868i 0 −0.195879 0.731029i 0 2.52163 + 0.800849i 0 0.0183525 + 0.0105958i 0
159.5 0 −1.30734 0.350301i 0 −0.294400 1.09872i 0 −1.56831 + 2.13082i 0 −1.01165 0.584076i 0
159.6 0 −0.198087 0.0530773i 0 0.487744 + 1.82029i 0 1.84933 1.89208i 0 −2.56165 1.47897i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.z.b 56
4.b odd 2 1 896.2.z.a 56
7.d odd 6 1 inner 896.2.z.b 56
8.b even 2 1 112.2.v.a 56
8.d odd 2 1 448.2.z.a 56
16.e even 4 1 448.2.z.a 56
16.e even 4 1 896.2.z.a 56
16.f odd 4 1 112.2.v.a 56
16.f odd 4 1 inner 896.2.z.b 56
28.f even 6 1 896.2.z.a 56
56.h odd 2 1 784.2.w.f 56
56.j odd 6 1 112.2.v.a 56
56.j odd 6 1 784.2.j.a 56
56.m even 6 1 448.2.z.a 56
56.p even 6 1 784.2.j.a 56
56.p even 6 1 784.2.w.f 56
112.j even 4 1 784.2.w.f 56
112.u odd 12 1 784.2.j.a 56
112.u odd 12 1 784.2.w.f 56
112.v even 12 1 112.2.v.a 56
112.v even 12 1 784.2.j.a 56
112.v even 12 1 inner 896.2.z.b 56
112.x odd 12 1 448.2.z.a 56
112.x odd 12 1 896.2.z.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.v.a 56 8.b even 2 1
112.2.v.a 56 16.f odd 4 1
112.2.v.a 56 56.j odd 6 1
112.2.v.a 56 112.v even 12 1
448.2.z.a 56 8.d odd 2 1
448.2.z.a 56 16.e even 4 1
448.2.z.a 56 56.m even 6 1
448.2.z.a 56 112.x odd 12 1
784.2.j.a 56 56.j odd 6 1
784.2.j.a 56 56.p even 6 1
784.2.j.a 56 112.u odd 12 1
784.2.j.a 56 112.v even 12 1
784.2.w.f 56 56.h odd 2 1
784.2.w.f 56 56.p even 6 1
784.2.w.f 56 112.j even 4 1
784.2.w.f 56 112.u odd 12 1
896.2.z.a 56 4.b odd 2 1
896.2.z.a 56 16.e even 4 1
896.2.z.a 56 28.f even 6 1
896.2.z.a 56 112.x odd 12 1
896.2.z.b 56 1.a even 1 1 trivial
896.2.z.b 56 7.d odd 6 1 inner
896.2.z.b 56 16.f odd 4 1 inner
896.2.z.b 56 112.v even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} - 6 T_{3}^{55} + 18 T_{3}^{54} - 36 T_{3}^{53} - 115 T_{3}^{52} + 816 T_{3}^{51} - 2178 T_{3}^{50} + 4662 T_{3}^{49} + 9200 T_{3}^{48} - 73254 T_{3}^{47} + 188550 T_{3}^{46} - 404976 T_{3}^{45} - 310773 T_{3}^{44} + \cdots + 4100625 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display