Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,2,Mod(103,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 9, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.103");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.x (of order \(12\), degree \(4\), 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: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
103.1 | −0.965926 | + | 0.258819i | −0.258819 | − | 0.965926i | 0.866025 | − | 0.500000i | −2.23391 | − | 0.0982997i | 0.500000 | + | 0.866025i | −0.368637 | + | 0.368637i | −0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 2.18323 | − | 0.483227i |
103.2 | −0.965926 | + | 0.258819i | −0.258819 | − | 0.965926i | 0.866025 | − | 0.500000i | −1.02196 | − | 1.98887i | 0.500000 | + | 0.866025i | −0.613926 | + | 0.613926i | −0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | 1.50189 | + | 1.65660i |
103.3 | −0.965926 | + | 0.258819i | −0.258819 | − | 0.965926i | 0.866025 | − | 0.500000i | 0.0149760 | + | 2.23602i | 0.500000 | + | 0.866025i | 1.36100 | − | 1.36100i | −0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | −0.593190 | − | 2.15595i |
103.4 | −0.965926 | + | 0.258819i | −0.258819 | − | 0.965926i | 0.866025 | − | 0.500000i | 0.392079 | + | 2.20143i | 0.500000 | + | 0.866025i | −3.01170 | + | 3.01170i | −0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | −0.948490 | − | 2.02494i |
103.5 | −0.965926 | + | 0.258819i | −0.258819 | − | 0.965926i | 0.866025 | − | 0.500000i | 1.17578 | − | 1.90199i | 0.500000 | + | 0.866025i | −2.04097 | + | 2.04097i | −0.707107 | + | 0.707107i | −0.866025 | + | 0.500000i | −0.643442 | + | 2.14149i |
103.6 | 0.965926 | − | 0.258819i | 0.258819 | + | 0.965926i | 0.866025 | − | 0.500000i | −1.99201 | − | 1.01583i | 0.500000 | + | 0.866025i | −0.219288 | + | 0.219288i | 0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | −2.18705 | − | 0.465647i |
103.7 | 0.965926 | − | 0.258819i | 0.258819 | + | 0.965926i | 0.866025 | − | 0.500000i | −0.981839 | − | 2.00898i | 0.500000 | + | 0.866025i | 3.62953 | − | 3.62953i | 0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | −1.46835 | − | 1.68640i |
103.8 | 0.965926 | − | 0.258819i | 0.258819 | + | 0.965926i | 0.866025 | − | 0.500000i | 0.649262 | + | 2.13973i | 0.500000 | + | 0.866025i | −2.55332 | + | 2.55332i | 0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.18094 | + | 1.89878i |
103.9 | 0.965926 | − | 0.258819i | 0.258819 | + | 0.965926i | 0.866025 | − | 0.500000i | 1.89255 | + | 1.19091i | 0.500000 | + | 0.866025i | 2.32404 | − | 2.32404i | 0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 2.13629 | + | 0.660499i |
103.10 | 0.965926 | − | 0.258819i | 0.258819 | + | 0.965926i | 0.866025 | − | 0.500000i | 2.10507 | − | 0.754119i | 0.500000 | + | 0.866025i | −0.506739 | + | 0.506739i | 0.707107 | − | 0.707107i | −0.866025 | + | 0.500000i | 1.83816 | − | 1.27325i |
217.1 | −0.258819 | − | 0.965926i | −0.965926 | + | 0.258819i | −0.866025 | + | 0.500000i | −2.10253 | + | 0.761162i | 0.500000 | + | 0.866025i | −3.01170 | − | 3.01170i | 0.707107 | + | 0.707107i | 0.866025 | − | 0.500000i | 1.27940 | + | 1.83388i |
217.2 | −0.258819 | − | 0.965926i | −0.965926 | + | 0.258819i | −0.866025 | + | 0.500000i | −1.94394 | + | 1.10504i | 0.500000 | + | 0.866025i | 1.36100 | + | 1.36100i | 0.707107 | + | 0.707107i | 0.866025 | − | 0.500000i | 1.57051 | + | 1.59169i |
217.3 | −0.258819 | − | 0.965926i | −0.965926 | + | 0.258819i | −0.866025 | + | 0.500000i | 1.05928 | − | 1.96924i | 0.500000 | + | 0.866025i | −2.04097 | − | 2.04097i | 0.707107 | + | 0.707107i | 0.866025 | − | 0.500000i | −2.17631 | − | 0.513508i |
217.4 | −0.258819 | − | 0.965926i | −0.965926 | + | 0.258819i | −0.866025 | + | 0.500000i | 1.20208 | + | 1.88547i | 0.500000 | + | 0.866025i | −0.368637 | − | 0.368637i | 0.707107 | + | 0.707107i | 0.866025 | − | 0.500000i | 1.51010 | − | 1.64912i |
217.5 | −0.258819 | − | 0.965926i | −0.965926 | + | 0.258819i | −0.866025 | + | 0.500000i | 2.23339 | − | 0.109394i | 0.500000 | + | 0.866025i | −0.613926 | − | 0.613926i | 0.707107 | + | 0.707107i | 0.866025 | − | 0.500000i | −0.683710 | − | 2.12898i |
217.6 | 0.258819 | + | 0.965926i | 0.965926 | − | 0.258819i | −0.866025 | + | 0.500000i | −2.17769 | + | 0.507589i | 0.500000 | + | 0.866025i | −2.55332 | − | 2.55332i | −0.707107 | − | 0.707107i | 0.866025 | − | 0.500000i | −1.05392 | − | 1.97212i |
217.7 | 0.258819 | + | 0.965926i | 0.965926 | − | 0.258819i | −0.866025 | + | 0.500000i | −1.97763 | − | 1.04354i | 0.500000 | + | 0.866025i | 2.32404 | + | 2.32404i | −0.707107 | − | 0.707107i | 0.866025 | − | 0.500000i | 0.496137 | − | 2.18033i |
217.8 | 0.258819 | + | 0.965926i | 0.965926 | − | 0.258819i | −0.866025 | + | 0.500000i | −0.399447 | − | 2.20010i | 0.500000 | + | 0.866025i | −0.506739 | − | 0.506739i | −0.707107 | − | 0.707107i | 0.866025 | − | 0.500000i | 2.02175 | − | 0.955264i |
217.9 | 0.258819 | + | 0.965926i | 0.965926 | − | 0.258819i | −0.866025 | + | 0.500000i | 1.87574 | + | 1.21721i | 0.500000 | + | 0.866025i | −0.219288 | − | 0.219288i | −0.707107 | − | 0.707107i | 0.866025 | − | 0.500000i | −0.690261 | + | 2.12686i |
217.10 | 0.258819 | + | 0.965926i | 0.965926 | − | 0.258819i | −0.866025 | + | 0.500000i | 2.23075 | − | 0.154191i | 0.500000 | + | 0.866025i | 3.62953 | + | 3.62953i | −0.707107 | − | 0.707107i | 0.866025 | − | 0.500000i | 0.726296 | + | 2.11483i |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.d | odd | 6 | 1 | inner |
95.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.x.b | ✓ | 40 |
5.c | odd | 4 | 1 | inner | 570.2.x.b | ✓ | 40 |
19.d | odd | 6 | 1 | inner | 570.2.x.b | ✓ | 40 |
95.l | even | 12 | 1 | inner | 570.2.x.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.x.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
570.2.x.b | ✓ | 40 | 5.c | odd | 4 | 1 | inner |
570.2.x.b | ✓ | 40 | 19.d | odd | 6 | 1 | inner |
570.2.x.b | ✓ | 40 | 95.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 4 T_{7}^{19} + 8 T_{7}^{18} + 24 T_{7}^{17} + 761 T_{7}^{16} + 3400 T_{7}^{15} + \cdots + 21025 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).