Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,2,Mod(31,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 420.bi (of order \(6\), degree \(2\), 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: | \(3.35371688489\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.41270 | − | 0.0654937i | −0.500000 | + | 0.866025i | 1.99142 | + | 0.185045i | −0.866025 | + | 0.500000i | 0.763067 | − | 1.19068i | 2.61608 | − | 0.395104i | −2.80115 | − | 0.391838i | −0.500000 | − | 0.866025i | 1.25618 | − | 0.649629i |
31.2 | −1.41178 | + | 0.0829396i | −0.500000 | + | 0.866025i | 1.98624 | − | 0.234185i | 0.866025 | − | 0.500000i | 0.634062 | − | 1.26411i | −0.693280 | + | 2.55330i | −2.78471 | + | 0.495356i | −0.500000 | − | 0.866025i | −1.18117 | + | 0.777718i |
31.3 | −1.21645 | + | 0.721286i | −0.500000 | + | 0.866025i | 0.959493 | − | 1.75481i | −0.866025 | + | 0.500000i | −0.0164277 | − | 1.41412i | −2.64546 | + | 0.0390617i | 0.0985483 | + | 2.82671i | −0.500000 | − | 0.866025i | 0.692832 | − | 1.23288i |
31.4 | −1.10407 | + | 0.883758i | −0.500000 | + | 0.866025i | 0.437945 | − | 1.95146i | 0.866025 | − | 0.500000i | −0.213321 | − | 1.39803i | 2.27168 | − | 1.35627i | 1.24110 | + | 2.54159i | −0.500000 | − | 0.866025i | −0.514275 | + | 1.31739i |
31.5 | −1.08061 | − | 0.912296i | −0.500000 | + | 0.866025i | 0.335431 | + | 1.97167i | 0.866025 | − | 0.500000i | 1.33038 | − | 0.479687i | −2.63725 | − | 0.211889i | 1.43628 | − | 2.43662i | −0.500000 | − | 0.866025i | −1.39198 | − | 0.249767i |
31.6 | −0.668090 | + | 1.24646i | −0.500000 | + | 0.866025i | −1.10731 | − | 1.66549i | 0.866025 | − | 0.500000i | −0.745419 | − | 1.20181i | −1.71433 | − | 2.01521i | 2.81575 | − | 0.267520i | −0.500000 | − | 0.866025i | 0.0446459 | + | 1.41351i |
31.7 | −0.535729 | − | 1.30881i | −0.500000 | + | 0.866025i | −1.42599 | + | 1.40234i | −0.866025 | + | 0.500000i | 1.40133 | + | 0.190453i | 0.482420 | + | 2.60140i | 2.59934 | + | 1.11508i | −0.500000 | − | 0.866025i | 1.11836 | + | 0.865602i |
31.8 | 0.00862847 | + | 1.41419i | −0.500000 | + | 0.866025i | −1.99985 | + | 0.0244045i | −0.866025 | + | 0.500000i | −1.22904 | − | 0.699621i | 0.189415 | − | 2.63896i | −0.0517682 | − | 2.82795i | −0.500000 | − | 0.866025i | −0.714566 | − | 1.22041i |
31.9 | 0.252003 | − | 1.39158i | −0.500000 | + | 0.866025i | −1.87299 | − | 0.701364i | 0.866025 | − | 0.500000i | 1.07914 | + | 0.914031i | −2.29181 | − | 1.32197i | −1.44800 | + | 2.42967i | −0.500000 | − | 0.866025i | −0.477549 | − | 1.33114i |
31.10 | 0.483616 | − | 1.32895i | −0.500000 | + | 0.866025i | −1.53223 | − | 1.28541i | −0.866025 | + | 0.500000i | 0.909099 | + | 1.08330i | −1.79146 | + | 1.94696i | −2.44925 | + | 1.41462i | −0.500000 | − | 0.866025i | 0.245653 | + | 1.39271i |
31.11 | 0.905086 | − | 1.08666i | −0.500000 | + | 0.866025i | −0.361639 | − | 1.96703i | 0.866025 | − | 0.500000i | 0.488528 | + | 1.32715i | 2.52676 | + | 0.784530i | −2.46480 | − | 1.38736i | −0.500000 | − | 0.866025i | 0.240500 | − | 1.39361i |
31.12 | 1.14443 | + | 0.830825i | −0.500000 | + | 0.866025i | 0.619461 | + | 1.90165i | −0.866025 | + | 0.500000i | −1.29173 | + | 0.575697i | −2.50660 | + | 0.846737i | −0.871005 | + | 2.69098i | −0.500000 | − | 0.866025i | −1.40652 | − | 0.147298i |
31.13 | 1.24143 | + | 0.677377i | −0.500000 | + | 0.866025i | 1.08232 | + | 1.68184i | 0.866025 | − | 0.500000i | −1.20734 | + | 0.736426i | 1.03822 | + | 2.43353i | 0.204394 | + | 2.82103i | −0.500000 | − | 0.866025i | 1.41380 | − | 0.0340920i |
31.14 | 1.39422 | + | 0.236963i | −0.500000 | + | 0.866025i | 1.88770 | + | 0.660757i | −0.866025 | + | 0.500000i | −0.902326 | + | 1.08895i | 2.15561 | − | 1.53406i | 2.47529 | + | 1.36855i | −0.500000 | − | 0.866025i | −1.32591 | + | 0.491894i |
271.1 | −1.41270 | + | 0.0654937i | −0.500000 | − | 0.866025i | 1.99142 | − | 0.185045i | −0.866025 | − | 0.500000i | 0.763067 | + | 1.19068i | 2.61608 | + | 0.395104i | −2.80115 | + | 0.391838i | −0.500000 | + | 0.866025i | 1.25618 | + | 0.649629i |
271.2 | −1.41178 | − | 0.0829396i | −0.500000 | − | 0.866025i | 1.98624 | + | 0.234185i | 0.866025 | + | 0.500000i | 0.634062 | + | 1.26411i | −0.693280 | − | 2.55330i | −2.78471 | − | 0.495356i | −0.500000 | + | 0.866025i | −1.18117 | − | 0.777718i |
271.3 | −1.21645 | − | 0.721286i | −0.500000 | − | 0.866025i | 0.959493 | + | 1.75481i | −0.866025 | − | 0.500000i | −0.0164277 | + | 1.41412i | −2.64546 | − | 0.0390617i | 0.0985483 | − | 2.82671i | −0.500000 | + | 0.866025i | 0.692832 | + | 1.23288i |
271.4 | −1.10407 | − | 0.883758i | −0.500000 | − | 0.866025i | 0.437945 | + | 1.95146i | 0.866025 | + | 0.500000i | −0.213321 | + | 1.39803i | 2.27168 | + | 1.35627i | 1.24110 | − | 2.54159i | −0.500000 | + | 0.866025i | −0.514275 | − | 1.31739i |
271.5 | −1.08061 | + | 0.912296i | −0.500000 | − | 0.866025i | 0.335431 | − | 1.97167i | 0.866025 | + | 0.500000i | 1.33038 | + | 0.479687i | −2.63725 | + | 0.211889i | 1.43628 | + | 2.43662i | −0.500000 | + | 0.866025i | −1.39198 | + | 0.249767i |
271.6 | −0.668090 | − | 1.24646i | −0.500000 | − | 0.866025i | −1.10731 | + | 1.66549i | 0.866025 | + | 0.500000i | −0.745419 | + | 1.20181i | −1.71433 | + | 2.01521i | 2.81575 | + | 0.267520i | −0.500000 | + | 0.866025i | 0.0446459 | − | 1.41351i |
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
28.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 420.2.bi.c | ✓ | 28 |
4.b | odd | 2 | 1 | 420.2.bi.d | yes | 28 | |
7.d | odd | 6 | 1 | 420.2.bi.d | yes | 28 | |
28.f | even | 6 | 1 | inner | 420.2.bi.c | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
420.2.bi.c | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
420.2.bi.c | ✓ | 28 | 28.f | even | 6 | 1 | inner |
420.2.bi.d | yes | 28 | 4.b | odd | 2 | 1 | |
420.2.bi.d | yes | 28 | 7.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):
\( T_{11}^{28} - 110 T_{11}^{26} + 7351 T_{11}^{24} - 720 T_{11}^{23} - 320470 T_{11}^{22} + 68952 T_{11}^{21} + 10366725 T_{11}^{20} - 4165440 T_{11}^{19} - 246715120 T_{11}^{18} + 156801984 T_{11}^{17} + \cdots + 12068340736 \) |
\( T_{19}^{28} - 14 T_{19}^{27} + 253 T_{19}^{26} - 2506 T_{19}^{25} + 30520 T_{19}^{24} - 261002 T_{19}^{23} + 2365083 T_{19}^{22} - 16637418 T_{19}^{21} + 119531269 T_{19}^{20} - 719735832 T_{19}^{19} + \cdots + 96477596934601 \) |