Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1205,2,Mod(481,1205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1205.481");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1205 = 5 \cdot 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1205.d (of order \(2\), degree \(1\), 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: | \(9.62197344356\) |
Analytic rank: | \(0\) |
Dimension: | \(42\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
481.1 | −2.65266 | −2.55371 | 5.03660 | −1.00000 | 6.77412 | − | 3.09402i | −8.05508 | 3.52143 | 2.65266 | |||||||||||||||||
481.2 | −2.65266 | −2.55371 | 5.03660 | −1.00000 | 6.77412 | 3.09402i | −8.05508 | 3.52143 | 2.65266 | ||||||||||||||||||
481.3 | −2.59746 | 0.184723 | 4.74680 | −1.00000 | −0.479810 | − | 1.63057i | −7.13471 | −2.96588 | 2.59746 | |||||||||||||||||
481.4 | −2.59746 | 0.184723 | 4.74680 | −1.00000 | −0.479810 | 1.63057i | −7.13471 | −2.96588 | 2.59746 | ||||||||||||||||||
481.5 | −2.43203 | 2.71572 | 3.91476 | −1.00000 | −6.60471 | 0.796450i | −4.65676 | 4.37513 | 2.43203 | ||||||||||||||||||
481.6 | −2.43203 | 2.71572 | 3.91476 | −1.00000 | −6.60471 | − | 0.796450i | −4.65676 | 4.37513 | 2.43203 | |||||||||||||||||
481.7 | −1.96118 | 1.54545 | 1.84621 | −1.00000 | −3.03090 | − | 3.56501i | 0.301608 | −0.611577 | 1.96118 | |||||||||||||||||
481.8 | −1.96118 | 1.54545 | 1.84621 | −1.00000 | −3.03090 | 3.56501i | 0.301608 | −0.611577 | 1.96118 | ||||||||||||||||||
481.9 | −1.77159 | −1.65886 | 1.13852 | −1.00000 | 2.93882 | 1.22826i | 1.52619 | −0.248169 | 1.77159 | ||||||||||||||||||
481.10 | −1.77159 | −1.65886 | 1.13852 | −1.00000 | 2.93882 | − | 1.22826i | 1.52619 | −0.248169 | 1.77159 | |||||||||||||||||
481.11 | −1.43869 | −3.24335 | 0.0698400 | −1.00000 | 4.66619 | − | 2.49639i | 2.77691 | 7.51934 | 1.43869 | |||||||||||||||||
481.12 | −1.43869 | −3.24335 | 0.0698400 | −1.00000 | 4.66619 | 2.49639i | 2.77691 | 7.51934 | 1.43869 | ||||||||||||||||||
481.13 | −1.42616 | −0.123830 | 0.0339350 | −1.00000 | 0.176601 | 2.22466i | 2.80393 | −2.98467 | 1.42616 | ||||||||||||||||||
481.14 | −1.42616 | −0.123830 | 0.0339350 | −1.00000 | 0.176601 | − | 2.22466i | 2.80393 | −2.98467 | 1.42616 | |||||||||||||||||
481.15 | −1.14954 | 3.06063 | −0.678558 | −1.00000 | −3.51831 | − | 4.38979i | 3.07911 | 6.36744 | 1.14954 | |||||||||||||||||
481.16 | −1.14954 | 3.06063 | −0.678558 | −1.00000 | −3.51831 | 4.38979i | 3.07911 | 6.36744 | 1.14954 | ||||||||||||||||||
481.17 | −0.563416 | −1.90443 | −1.68256 | −1.00000 | 1.07299 | − | 5.09769i | 2.07481 | 0.626872 | 0.563416 | |||||||||||||||||
481.18 | −0.563416 | −1.90443 | −1.68256 | −1.00000 | 1.07299 | 5.09769i | 2.07481 | 0.626872 | 0.563416 | ||||||||||||||||||
481.19 | −0.410869 | 0.874016 | −1.83119 | −1.00000 | −0.359106 | − | 2.07132i | 1.57411 | −2.23610 | 0.410869 | |||||||||||||||||
481.20 | −0.410869 | 0.874016 | −1.83119 | −1.00000 | −0.359106 | 2.07132i | 1.57411 | −2.23610 | 0.410869 | ||||||||||||||||||
See all 42 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
241.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1205.2.d.c | ✓ | 42 |
241.b | even | 2 | 1 | inner | 1205.2.d.c | ✓ | 42 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1205.2.d.c | ✓ | 42 | 1.a | even | 1 | 1 | trivial |
1205.2.d.c | ✓ | 42 | 241.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 33 T_{2}^{19} - T_{2}^{18} + 460 T_{2}^{17} + 29 T_{2}^{16} - 3532 T_{2}^{15} - 346 T_{2}^{14} + \cdots - 27 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\).