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Properties

Label 896.2.x.b
Level $896$
Weight $2$
Character orbit 896.x
Analytic conductor $7.155$
Analytic rank $0$
Dimension $112$
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(111,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 7, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.x (of order \(8\), 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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 224)
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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 4 q^{7} - 8 q^{9} + 8 q^{11} + 16 q^{15} - 4 q^{21} + 48 q^{23} - 8 q^{25} - 8 q^{29} - 20 q^{35} - 8 q^{37} + 8 q^{39} - 32 q^{43} + 32 q^{51} - 32 q^{53} - 8 q^{57} - 16 q^{65} + 64 q^{67} - 56 q^{71} + 52 q^{77} + 16 q^{79} - 48 q^{85} + 52 q^{91} - 32 q^{93} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0 −1.25862 + 3.03857i 0 0.824079 + 1.98950i 0 −2.64443 + 0.0837494i 0 −5.52747 − 5.52747i 0
111.2 0 −1.15961 + 2.79954i 0 −0.0324155 − 0.0782579i 0 1.05883 − 2.42464i 0 −4.37141 − 4.37141i 0
111.3 0 −1.08865 + 2.62824i 0 1.22393 + 2.95482i 0 1.92921 + 1.81057i 0 −3.60116 − 3.60116i 0
111.4 0 −1.01638 + 2.45376i 0 −1.11188 − 2.68430i 0 −2.64539 − 0.0435134i 0 −2.86660 − 2.86660i 0
111.5 0 −0.935381 + 2.25821i 0 −0.395947 − 0.955900i 0 2.47479 + 0.935627i 0 −2.10326 − 2.10326i 0
111.6 0 −0.884481 + 2.13533i 0 −1.60624 − 3.87780i 0 2.56119 − 0.663558i 0 −1.65599 − 1.65599i 0
111.7 0 −0.656051 + 1.58385i 0 0.145433 + 0.351105i 0 −0.525039 + 2.59313i 0 0.0431539 + 0.0431539i 0
111.8 0 −0.613776 + 1.48179i 0 1.25513 + 3.03015i 0 −1.85488 − 1.88664i 0 0.302349 + 0.302349i 0
111.9 0 −0.592616 + 1.43070i 0 0.188106 + 0.454128i 0 0.152070 − 2.64138i 0 0.425604 + 0.425604i 0
111.10 0 −0.581678 + 1.40429i 0 −0.535546 − 1.29292i 0 0.124247 − 2.64283i 0 0.487628 + 0.487628i 0
111.11 0 −0.517660 + 1.24974i 0 −0.818546 − 1.97615i 0 −2.17823 + 1.50177i 0 0.827440 + 0.827440i 0
111.12 0 −0.311327 + 0.751610i 0 1.37157 + 3.31126i 0 −1.33176 + 2.28614i 0 1.65333 + 1.65333i 0
111.13 0 −0.167131 + 0.403490i 0 0.883342 + 2.13258i 0 2.48955 − 0.895629i 0 1.98645 + 1.98645i 0
111.14 0 −0.0852474 + 0.205805i 0 1.18326 + 2.85664i 0 2.49406 + 0.882988i 0 2.08623 + 2.08623i 0
111.15 0 0.0852474 − 0.205805i 0 −1.18326 − 2.85664i 0 0.882988 + 2.49406i 0 2.08623 + 2.08623i 0
111.16 0 0.167131 − 0.403490i 0 −0.883342 − 2.13258i 0 −0.895629 + 2.48955i 0 1.98645 + 1.98645i 0
111.17 0 0.311327 − 0.751610i 0 −1.37157 − 3.31126i 0 2.28614 − 1.33176i 0 1.65333 + 1.65333i 0
111.18 0 0.517660 − 1.24974i 0 0.818546 + 1.97615i 0 1.50177 − 2.17823i 0 0.827440 + 0.827440i 0
111.19 0 0.581678 − 1.40429i 0 0.535546 + 1.29292i 0 −2.64283 + 0.124247i 0 0.487628 + 0.487628i 0
111.20 0 0.592616 − 1.43070i 0 −0.188106 − 0.454128i 0 −2.64138 + 0.152070i 0 0.425604 + 0.425604i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
32.h odd 8 1 inner
224.x even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 896.2.x.b 112
4.b odd 2 1 224.2.x.b ✓ 112
7.b odd 2 1 inner 896.2.x.b 112
28.d even 2 1 224.2.x.b ✓ 112
32.g even 8 1 224.2.x.b ✓ 112
32.h odd 8 1 inner 896.2.x.b 112
224.v odd 8 1 224.2.x.b ✓ 112
224.x even 8 1 inner 896.2.x.b 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.b ✓ 112 4.b odd 2 1
224.2.x.b ✓ 112 28.d even 2 1
224.2.x.b ✓ 112 32.g even 8 1
224.2.x.b ✓ 112 224.v odd 8 1
896.2.x.b 112 1.a even 1 1 trivial
896.2.x.b 112 7.b odd 2 1 inner
896.2.x.b 112 32.h odd 8 1 inner
896.2.x.b 112 224.x even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} + 4 T_{3}^{110} + 8 T_{3}^{108} - 168 T_{3}^{106} + 27036 T_{3}^{104} + \cdots + 50\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\). Copy content Toggle raw display