Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(257,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 9, 17]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.257");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 912.cc (of order \(18\), degree \(6\), 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: | \(7.28235666434\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 456) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
257.1 | 0 | −1.73061 | − | 0.0706292i | 0 | −0.555150 | + | 1.52526i | 0 | 1.81489 | − | 3.14348i | 0 | 2.99002 | + | 0.244463i | 0 | ||||||||||
257.2 | 0 | −1.63447 | + | 0.573170i | 0 | 0.0679758 | − | 0.186762i | 0 | −2.39129 | + | 4.14183i | 0 | 2.34295 | − | 1.87365i | 0 | ||||||||||
257.3 | 0 | −1.35435 | − | 1.07969i | 0 | 1.42044 | − | 3.90263i | 0 | 0.0388591 | − | 0.0673059i | 0 | 0.668525 | + | 2.92456i | 0 | ||||||||||
257.4 | 0 | −0.537955 | − | 1.64639i | 0 | −0.621356 | + | 1.70716i | 0 | −0.144681 | + | 0.250595i | 0 | −2.42121 | + | 1.77137i | 0 | ||||||||||
257.5 | 0 | −0.183479 | + | 1.72231i | 0 | −1.21966 | + | 3.35098i | 0 | −0.702445 | + | 1.21667i | 0 | −2.93267 | − | 0.632013i | 0 | ||||||||||
257.6 | 0 | 0.858795 | + | 1.50415i | 0 | −0.0199730 | + | 0.0548754i | 0 | 1.38520 | − | 2.39923i | 0 | −1.52494 | + | 2.58352i | 0 | ||||||||||
257.7 | 0 | 0.898268 | + | 1.48092i | 0 | 0.978409 | − | 2.68816i | 0 | −1.73645 | + | 3.00762i | 0 | −1.38623 | + | 2.66052i | 0 | ||||||||||
257.8 | 0 | 1.01873 | − | 1.40078i | 0 | 0.293373 | − | 0.806036i | 0 | −0.627281 | + | 1.08648i | 0 | −0.924371 | − | 2.85404i | 0 | ||||||||||
257.9 | 0 | 1.70388 | − | 0.311134i | 0 | −1.11883 | + | 3.07396i | 0 | −0.429476 | + | 0.743873i | 0 | 2.80639 | − | 1.06027i | 0 | ||||||||||
257.10 | 0 | 1.72723 | − | 0.129127i | 0 | 0.774768 | − | 2.12866i | 0 | 2.02663 | − | 3.51023i | 0 | 2.96665 | − | 0.446064i | 0 | ||||||||||
401.1 | 0 | −1.73061 | + | 0.0706292i | 0 | −0.555150 | − | 1.52526i | 0 | 1.81489 | + | 3.14348i | 0 | 2.99002 | − | 0.244463i | 0 | ||||||||||
401.2 | 0 | −1.63447 | − | 0.573170i | 0 | 0.0679758 | + | 0.186762i | 0 | −2.39129 | − | 4.14183i | 0 | 2.34295 | + | 1.87365i | 0 | ||||||||||
401.3 | 0 | −1.35435 | + | 1.07969i | 0 | 1.42044 | + | 3.90263i | 0 | 0.0388591 | + | 0.0673059i | 0 | 0.668525 | − | 2.92456i | 0 | ||||||||||
401.4 | 0 | −0.537955 | + | 1.64639i | 0 | −0.621356 | − | 1.70716i | 0 | −0.144681 | − | 0.250595i | 0 | −2.42121 | − | 1.77137i | 0 | ||||||||||
401.5 | 0 | −0.183479 | − | 1.72231i | 0 | −1.21966 | − | 3.35098i | 0 | −0.702445 | − | 1.21667i | 0 | −2.93267 | + | 0.632013i | 0 | ||||||||||
401.6 | 0 | 0.858795 | − | 1.50415i | 0 | −0.0199730 | − | 0.0548754i | 0 | 1.38520 | + | 2.39923i | 0 | −1.52494 | − | 2.58352i | 0 | ||||||||||
401.7 | 0 | 0.898268 | − | 1.48092i | 0 | 0.978409 | + | 2.68816i | 0 | −1.73645 | − | 3.00762i | 0 | −1.38623 | − | 2.66052i | 0 | ||||||||||
401.8 | 0 | 1.01873 | + | 1.40078i | 0 | 0.293373 | + | 0.806036i | 0 | −0.627281 | − | 1.08648i | 0 | −0.924371 | + | 2.85404i | 0 | ||||||||||
401.9 | 0 | 1.70388 | + | 0.311134i | 0 | −1.11883 | − | 3.07396i | 0 | −0.429476 | − | 0.743873i | 0 | 2.80639 | + | 1.06027i | 0 | ||||||||||
401.10 | 0 | 1.72723 | + | 0.129127i | 0 | 0.774768 | + | 2.12866i | 0 | 2.02663 | + | 3.51023i | 0 | 2.96665 | + | 0.446064i | 0 | ||||||||||
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
57.j | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 912.2.cc.g | 60 | |
3.b | odd | 2 | 1 | 912.2.cc.h | 60 | ||
4.b | odd | 2 | 1 | 456.2.bm.b | yes | 60 | |
12.b | even | 2 | 1 | 456.2.bm.a | ✓ | 60 | |
19.f | odd | 18 | 1 | 912.2.cc.h | 60 | ||
57.j | even | 18 | 1 | inner | 912.2.cc.g | 60 | |
76.k | even | 18 | 1 | 456.2.bm.a | ✓ | 60 | |
228.u | odd | 18 | 1 | 456.2.bm.b | yes | 60 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
456.2.bm.a | ✓ | 60 | 12.b | even | 2 | 1 | |
456.2.bm.a | ✓ | 60 | 76.k | even | 18 | 1 | |
456.2.bm.b | yes | 60 | 4.b | odd | 2 | 1 | |
456.2.bm.b | yes | 60 | 228.u | odd | 18 | 1 | |
912.2.cc.g | 60 | 1.a | even | 1 | 1 | trivial | |
912.2.cc.g | 60 | 57.j | even | 18 | 1 | inner | |
912.2.cc.h | 60 | 3.b | odd | 2 | 1 | ||
912.2.cc.h | 60 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):
\( T_{5}^{60} - 3 T_{5}^{58} + 36 T_{5}^{57} + 3 T_{5}^{56} - 78 T_{5}^{55} - 3824 T_{5}^{54} + \cdots + 66324791296 \) |
\( T_{7}^{60} + 120 T_{7}^{58} + 14 T_{7}^{57} + 8106 T_{7}^{56} + 1581 T_{7}^{55} + 375711 T_{7}^{54} + \cdots + 1671828544 \) |