Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [896,2,Mod(113,896)]
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([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("896.113");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 896 = 2^{7} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 896.u (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: | \(52\) |
Relative dimension: | \(13\) 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.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
113.1 | 0 | −1.23966 | − | 2.99281i | 0 | 2.25399 | + | 0.933632i | 0 | 0.707107 | + | 0.707107i | 0 | −5.29885 | + | 5.29885i | 0 | ||||||||||
113.2 | 0 | −1.19114 | − | 2.87567i | 0 | −3.07070 | − | 1.27193i | 0 | 0.707107 | + | 0.707107i | 0 | −4.72935 | + | 4.72935i | 0 | ||||||||||
113.3 | 0 | −0.738703 | − | 1.78339i | 0 | −0.509656 | − | 0.211106i | 0 | 0.707107 | + | 0.707107i | 0 | −0.513466 | + | 0.513466i | 0 | ||||||||||
113.4 | 0 | −0.694865 | − | 1.67755i | 0 | −0.286541 | − | 0.118689i | 0 | 0.707107 | + | 0.707107i | 0 | −0.210023 | + | 0.210023i | 0 | ||||||||||
113.5 | 0 | −0.424336 | − | 1.02444i | 0 | 3.70330 | + | 1.53396i | 0 | 0.707107 | + | 0.707107i | 0 | 1.25191 | − | 1.25191i | 0 | ||||||||||
113.6 | 0 | −0.264188 | − | 0.637807i | 0 | −1.89692 | − | 0.785729i | 0 | 0.707107 | + | 0.707107i | 0 | 1.78432 | − | 1.78432i | 0 | ||||||||||
113.7 | 0 | −0.0427122 | − | 0.103116i | 0 | 2.38883 | + | 0.989485i | 0 | 0.707107 | + | 0.707107i | 0 | 2.11251 | − | 2.11251i | 0 | ||||||||||
113.8 | 0 | 0.439285 | + | 1.06053i | 0 | −3.40891 | − | 1.41202i | 0 | 0.707107 | + | 0.707107i | 0 | 1.18957 | − | 1.18957i | 0 | ||||||||||
113.9 | 0 | 0.587552 | + | 1.41848i | 0 | 2.93480 | + | 1.21564i | 0 | 0.707107 | + | 0.707107i | 0 | 0.454461 | − | 0.454461i | 0 | ||||||||||
113.10 | 0 | 0.622222 | + | 1.50218i | 0 | −0.781648 | − | 0.323769i | 0 | 0.707107 | + | 0.707107i | 0 | 0.251948 | − | 0.251948i | 0 | ||||||||||
113.11 | 0 | 0.728858 | + | 1.75962i | 0 | −1.90564 | − | 0.789342i | 0 | 0.707107 | + | 0.707107i | 0 | −0.443707 | + | 0.443707i | 0 | ||||||||||
113.12 | 0 | 0.951433 | + | 2.29696i | 0 | 2.43243 | + | 1.00754i | 0 | 0.707107 | + | 0.707107i | 0 | −2.24949 | + | 2.24949i | 0 | ||||||||||
113.13 | 0 | 1.26626 | + | 3.05702i | 0 | −1.85332 | − | 0.767672i | 0 | 0.707107 | + | 0.707107i | 0 | −5.62065 | + | 5.62065i | 0 | ||||||||||
337.1 | 0 | −2.91760 | − | 1.20851i | 0 | −0.629460 | − | 1.51965i | 0 | −0.707107 | + | 0.707107i | 0 | 4.93055 | + | 4.93055i | 0 | ||||||||||
337.2 | 0 | −2.47594 | − | 1.02557i | 0 | −1.23251 | − | 2.97553i | 0 | −0.707107 | + | 0.707107i | 0 | 2.95715 | + | 2.95715i | 0 | ||||||||||
337.3 | 0 | −2.18713 | − | 0.905938i | 0 | 0.797931 | + | 1.92638i | 0 | −0.707107 | + | 0.707107i | 0 | 1.84149 | + | 1.84149i | 0 | ||||||||||
337.4 | 0 | −1.74965 | − | 0.724727i | 0 | 1.44353 | + | 3.48498i | 0 | −0.707107 | + | 0.707107i | 0 | 0.414709 | + | 0.414709i | 0 | ||||||||||
337.5 | 0 | −0.999166 | − | 0.413868i | 0 | 0.523077 | + | 1.26282i | 0 | −0.707107 | + | 0.707107i | 0 | −1.29427 | − | 1.29427i | 0 | ||||||||||
337.6 | 0 | −0.496926 | − | 0.205834i | 0 | −0.334218 | − | 0.806875i | 0 | −0.707107 | + | 0.707107i | 0 | −1.91675 | − | 1.91675i | 0 | ||||||||||
337.7 | 0 | 0.301186 | + | 0.124755i | 0 | −0.107860 | − | 0.260396i | 0 | −0.707107 | + | 0.707107i | 0 | −2.04617 | − | 2.04617i | 0 | ||||||||||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
32.g | 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.u.c | 52 | |
4.b | odd | 2 | 1 | 224.2.u.c | ✓ | 52 | |
32.g | even | 8 | 1 | inner | 896.2.u.c | 52 | |
32.h | odd | 8 | 1 | 224.2.u.c | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.2.u.c | ✓ | 52 | 4.b | odd | 2 | 1 | |
224.2.u.c | ✓ | 52 | 32.h | odd | 8 | 1 | |
896.2.u.c | 52 | 1.a | even | 1 | 1 | trivial | |
896.2.u.c | 52 | 32.g | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{52} - 8 T_{3}^{49} + 72 T_{3}^{47} - 40 T_{3}^{46} + 88 T_{3}^{45} + 16440 T_{3}^{44} + \cdots + 151519232 \) acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\).