↑

Properties

Label 896.2.z.a
Level $896$
Weight $2$
Character orbit 896.z
Analytic conductor $7.155$
Analytic rank $0$
Dimension $56$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [896,2,Mod(31,896)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.z (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{3} + 6 q^{5} + 8 q^{7} + 2 q^{11} - 12 q^{17} - 6 q^{19} + 10 q^{21} + 12 q^{23} + 24 q^{29} - 12 q^{33} - 2 q^{35} - 6 q^{37} + 4 q^{39} - 12 q^{45} - 8 q^{49} - 34 q^{51} - 6 q^{53} + 42 q^{59}+ \cdots - 16 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −0.825526 − 3.08091i 0 −1.86998 − 0.501060i 0 −2.17953 − 1.49988i 0 −6.21241 + 3.58674i 0
31.2 0 −0.700587 − 2.61463i 0 −0.450647 − 0.120751i 0 2.37326 + 1.16946i 0 −3.74737 + 2.16355i 0
31.3 0 −0.665801 − 2.48480i 0 3.12401 + 0.837076i 0 −1.56215 + 2.13534i 0 −3.13287 + 1.80877i 0
31.4 0 −0.388227 − 1.44888i 0 −2.81809 − 0.755106i 0 1.47726 + 2.19493i 0 0.649537 − 0.375010i 0
31.5 0 −0.282507 − 1.05433i 0 1.39922 + 0.374919i 0 0.298614 − 2.62885i 0 1.56627 − 0.904287i 0
31.6 0 −0.237312 − 0.885661i 0 −3.05579 − 0.818796i 0 0.640837 − 2.56697i 0 1.87000 − 1.07964i 0
31.7 0 −0.204601 − 0.763582i 0 3.81370 + 1.02188i 0 2.64575 − 0.00379639i 0 2.05688 − 1.18754i 0
31.8 0 −0.0661591 − 0.246909i 0 0.499339 + 0.133797i 0 −2.45011 + 0.998472i 0 2.54149 − 1.46733i 0
31.9 0 0.0530773 + 0.198087i 0 1.82029 + 0.487744i 0 −1.84933 − 1.89208i 0 2.56165 − 1.47897i 0
31.10 0 0.350301 + 1.30734i 0 −1.09872 − 0.294400i 0 1.56831 + 2.13082i 0 1.01165 − 0.584076i 0
31.11 0 0.449868 + 1.67893i 0 −0.731029 − 0.195879i 0 −2.52163 + 0.800849i 0 −0.0183525 + 0.0105958i 0
31.12 0 0.524583 + 1.95777i 0 −2.48890 − 0.666898i 0 2.38027 − 1.15512i 0 −0.959602 + 0.554026i 0
31.13 0 0.615336 + 2.29647i 0 0.835825 + 0.223959i 0 1.25761 − 2.32775i 0 −2.29704 + 1.32620i 0
31.14 0 0.743580 + 2.77508i 0 3.38680 + 0.907491i 0 −0.0791552 + 2.64457i 0 −4.55008 + 2.62699i 0
159.1 0 −3.08091 − 0.825526i 0 −0.501060 − 1.86998i 0 −2.17953 + 1.49988i 0 6.21241 + 3.58674i 0
159.2 0 −2.61463 − 0.700587i 0 −0.120751 − 0.450647i 0 2.37326 − 1.16946i 0 3.74737 + 2.16355i 0
159.3 0 −2.48480 − 0.665801i 0 0.837076 + 3.12401i 0 −1.56215 − 2.13534i 0 3.13287 + 1.80877i 0
159.4 0 −1.44888 − 0.388227i 0 −0.755106 − 2.81809i 0 1.47726 − 2.19493i 0 −0.649537 − 0.375010i 0
159.5 0 −1.05433 − 0.282507i 0 0.374919 + 1.39922i 0 0.298614 + 2.62885i 0 −1.56627 − 0.904287i 0
159.6 0 −0.885661 − 0.237312i 0 −0.818796 − 3.05579i 0 0.640837 + 2.56697i 0 −1.87000 − 1.07964i 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.a 56
4.b odd 2 1 896.2.z.b 56
7.d odd 6 1 inner 896.2.z.a 56
8.b even 2 1 448.2.z.a 56
8.d odd 2 1 112.2.v.a ✓ 56
16.e even 4 1 112.2.v.a ✓ 56
16.e even 4 1 896.2.z.b 56
16.f odd 4 1 448.2.z.a 56
16.f odd 4 1 inner 896.2.z.a 56
28.f even 6 1 896.2.z.b 56
56.e even 2 1 784.2.w.f 56
56.j odd 6 1 448.2.z.a 56
56.k odd 6 1 784.2.j.a 56
56.k odd 6 1 784.2.w.f 56
56.m even 6 1 112.2.v.a ✓ 56
56.m even 6 1 784.2.j.a 56
112.l odd 4 1 784.2.w.f 56
112.v even 12 1 448.2.z.a 56
112.v even 12 1 inner 896.2.z.a 56
112.w even 12 1 784.2.j.a 56
112.w even 12 1 784.2.w.f 56
112.x odd 12 1 112.2.v.a ✓ 56
112.x odd 12 1 784.2.j.a 56
112.x odd 12 1 896.2.z.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.v.a ✓ 56 8.d odd 2 1
112.2.v.a ✓ 56 16.e even 4 1
112.2.v.a ✓ 56 56.m even 6 1
112.2.v.a ✓ 56 112.x odd 12 1
448.2.z.a 56 8.b even 2 1
448.2.z.a 56 16.f odd 4 1
448.2.z.a 56 56.j odd 6 1
448.2.z.a 56 112.v even 12 1
784.2.j.a 56 56.k odd 6 1
784.2.j.a 56 56.m even 6 1
784.2.j.a 56 112.w even 12 1
784.2.j.a 56 112.x odd 12 1
784.2.w.f 56 56.e even 2 1
784.2.w.f 56 56.k odd 6 1
784.2.w.f 56 112.l odd 4 1
784.2.w.f 56 112.w even 12 1
896.2.z.a 56 1.a even 1 1 trivial
896.2.z.a 56 7.d odd 6 1 inner
896.2.z.a 56 16.f odd 4 1 inner
896.2.z.a 56 112.v even 12 1 inner
896.2.z.b 56 4.b odd 2 1
896.2.z.b 56 16.e even 4 1
896.2.z.b 56 28.f even 6 1
896.2.z.b 56 112.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} + 6 T_{3}^{55} + 18 T_{3}^{54} + 36 T_{3}^{53} - 115 T_{3}^{52} - 816 T_{3}^{51} + \cdots + 4100625 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display