Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(277,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 5, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.277");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 930.bj (of order \(20\), degree \(8\), 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: | \(7.42608738798\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
277.1 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | −1.90277 | + | 1.17451i | 1.00000i | 1.72531 | − | 3.38612i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | 1.45771 | + | 1.69561i | ||
277.2 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | −1.82843 | + | 1.28719i | 1.00000i | −2.04975 | + | 4.02286i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | 1.55737 | + | 1.60455i | ||
277.3 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | −1.74178 | − | 1.40221i | 1.00000i | −0.335285 | + | 0.658034i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | −1.11247 | + | 1.93969i | ||
277.4 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | −1.67223 | − | 1.48447i | 1.00000i | 1.10706 | − | 2.17272i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | −1.20460 | + | 1.88386i | ||
277.5 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | 0.244623 | + | 2.22265i | 1.00000i | 0.795365 | − | 1.56099i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | 2.15702 | − | 0.589310i | ||
277.6 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | 1.48798 | − | 1.66911i | 1.00000i | −0.506200 | + | 0.993473i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | −1.88133 | − | 1.20856i | ||
277.7 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | 1.61098 | − | 1.55073i | 1.00000i | −1.36477 | + | 2.67852i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | −1.78365 | − | 1.34855i | ||
277.8 | −0.156434 | − | 0.987688i | −0.987688 | − | 0.156434i | −0.951057 | + | 0.309017i | 2.17207 | + | 0.531157i | 1.00000i | 0.369895 | − | 0.725959i | 0.453990 | + | 0.891007i | 0.951057 | + | 0.309017i | 0.184831 | − | 2.22842i | ||
277.9 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | −2.23524 | − | 0.0607951i | 1.00000i | 0.773249 | − | 1.51759i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | −0.289622 | − | 2.21723i | ||
277.10 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | −2.20784 | + | 0.354206i | 1.00000i | 0.622042 | − | 1.22083i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | −0.695226 | − | 2.12524i | ||
277.11 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | −1.01777 | − | 1.99102i | 1.00000i | −0.681739 | + | 1.33799i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | 1.80729 | − | 1.31670i | ||
277.12 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | −0.869928 | + | 2.05991i | 1.00000i | −2.25496 | + | 4.42561i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | −2.17063 | − | 0.536977i | ||
277.13 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | 0.145083 | + | 2.23136i | 1.00000i | 0.996252 | − | 1.95526i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | −2.18119 | + | 0.492358i | ||
277.14 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | 1.00799 | − | 1.99598i | 1.00000i | 1.85184 | − | 3.63444i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | 2.12909 | + | 0.683342i | ||
277.15 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | 2.22022 | + | 0.265769i | 1.00000i | 1.65375 | − | 3.24566i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | 0.0848219 | + | 2.23446i | ||
277.16 | 0.156434 | + | 0.987688i | 0.987688 | + | 0.156434i | −0.951057 | + | 0.309017i | 2.23590 | + | 0.0275636i | 1.00000i | −1.08401 | + | 2.12750i | −0.453990 | − | 0.891007i | 0.951057 | + | 0.309017i | 0.322547 | + | 2.21268i | ||
337.1 | −0.453990 | + | 0.891007i | 0.891007 | − | 0.453990i | −0.587785 | − | 0.809017i | −2.20246 | + | 0.386226i | 1.00000i | 3.54459 | + | 0.561408i | 0.987688 | − | 0.156434i | 0.587785 | − | 0.809017i | 0.655766 | − | 2.13775i | ||
337.2 | −0.453990 | + | 0.891007i | 0.891007 | − | 0.453990i | −0.587785 | − | 0.809017i | −1.81236 | − | 1.30972i | 1.00000i | −2.09145 | − | 0.331253i | 0.987688 | − | 0.156434i | 0.587785 | − | 0.809017i | 1.98976 | − | 1.02022i | ||
337.3 | −0.453990 | + | 0.891007i | 0.891007 | − | 0.453990i | −0.587785 | − | 0.809017i | −1.39059 | + | 1.75108i | 1.00000i | −2.23420 | − | 0.353863i | 0.987688 | − | 0.156434i | 0.587785 | − | 0.809017i | −0.928908 | − | 2.03399i | ||
337.4 | −0.453990 | + | 0.891007i | 0.891007 | − | 0.453990i | −0.587785 | − | 0.809017i | −0.354402 | − | 2.20780i | 1.00000i | 5.01200 | + | 0.793822i | 0.987688 | − | 0.156434i | 0.587785 | − | 0.809017i | 2.12806 | + | 0.686548i | ||
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
155.r | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 930.2.bj.b | yes | 128 |
5.c | odd | 4 | 1 | 930.2.bj.a | ✓ | 128 | |
31.f | odd | 10 | 1 | 930.2.bj.a | ✓ | 128 | |
155.r | even | 20 | 1 | inner | 930.2.bj.b | yes | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
930.2.bj.a | ✓ | 128 | 5.c | odd | 4 | 1 | |
930.2.bj.a | ✓ | 128 | 31.f | odd | 10 | 1 | |
930.2.bj.b | yes | 128 | 1.a | even | 1 | 1 | trivial |
930.2.bj.b | yes | 128 | 155.r | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{128} - 4 T_{7}^{127} - 42 T_{7}^{126} + 84 T_{7}^{125} + 475 T_{7}^{124} + 3256 T_{7}^{123} + \cdots + 34\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).