Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,2,Mod(626,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1045.626");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1045 = 5 \cdot 11 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1045.f (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: | \(8.34436701122\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
626.1 | −2.71965 | − | 2.28311i | 5.39648 | −1.00000 | 6.20925i | 2.59337i | −9.23723 | −2.21259 | 2.71965 | |||||||||||||||||
626.2 | −2.71965 | 2.28311i | 5.39648 | −1.00000 | − | 6.20925i | − | 2.59337i | −9.23723 | −2.21259 | 2.71965 | ||||||||||||||||
626.3 | −2.52972 | − | 0.384887i | 4.39948 | −1.00000 | 0.973657i | 1.35297i | −6.07000 | 2.85186 | 2.52972 | |||||||||||||||||
626.4 | −2.52972 | 0.384887i | 4.39948 | −1.00000 | − | 0.973657i | − | 1.35297i | −6.07000 | 2.85186 | 2.52972 | ||||||||||||||||
626.5 | −2.08652 | 2.80799i | 2.35358 | −1.00000 | − | 5.85893i | − | 0.950248i | −0.737755 | −4.88478 | 2.08652 | ||||||||||||||||
626.6 | −2.08652 | − | 2.80799i | 2.35358 | −1.00000 | 5.85893i | 0.950248i | −0.737755 | −4.88478 | 2.08652 | |||||||||||||||||
626.7 | −2.04337 | − | 0.838315i | 2.17535 | −1.00000 | 1.71299i | − | 4.78057i | −0.358306 | 2.29723 | 2.04337 | ||||||||||||||||
626.8 | −2.04337 | 0.838315i | 2.17535 | −1.00000 | − | 1.71299i | 4.78057i | −0.358306 | 2.29723 | 2.04337 | |||||||||||||||||
626.9 | −1.55769 | 2.37450i | 0.426394 | −1.00000 | − | 3.69874i | − | 4.79166i | 2.45119 | −2.63826 | 1.55769 | ||||||||||||||||
626.10 | −1.55769 | − | 2.37450i | 0.426394 | −1.00000 | 3.69874i | 4.79166i | 2.45119 | −2.63826 | 1.55769 | |||||||||||||||||
626.11 | −1.53521 | − | 1.36588i | 0.356884 | −1.00000 | 2.09692i | − | 1.36858i | 2.52254 | 1.13438 | 1.53521 | ||||||||||||||||
626.12 | −1.53521 | 1.36588i | 0.356884 | −1.00000 | − | 2.09692i | 1.36858i | 2.52254 | 1.13438 | 1.53521 | |||||||||||||||||
626.13 | −1.42582 | 1.05301i | 0.0329672 | −1.00000 | − | 1.50140i | 0.901840i | 2.80464 | 1.89118 | 1.42582 | |||||||||||||||||
626.14 | −1.42582 | − | 1.05301i | 0.0329672 | −1.00000 | 1.50140i | − | 0.901840i | 2.80464 | 1.89118 | 1.42582 | ||||||||||||||||
626.15 | −0.743263 | − | 1.99663i | −1.44756 | −1.00000 | 1.48402i | − | 2.79593i | 2.56244 | −0.986525 | 0.743263 | ||||||||||||||||
626.16 | −0.743263 | 1.99663i | −1.44756 | −1.00000 | − | 1.48402i | 2.79593i | 2.56244 | −0.986525 | 0.743263 | |||||||||||||||||
626.17 | −0.512926 | − | 3.19882i | −1.73691 | −1.00000 | 1.64076i | − | 1.77223i | 1.91676 | −7.23242 | 0.512926 | ||||||||||||||||
626.18 | −0.512926 | 3.19882i | −1.73691 | −1.00000 | − | 1.64076i | 1.77223i | 1.91676 | −7.23242 | 0.512926 | |||||||||||||||||
626.19 | −0.208163 | 0.469109i | −1.95667 | −1.00000 | − | 0.0976512i | 2.46634i | 0.823632 | 2.77994 | 0.208163 | |||||||||||||||||
626.20 | −0.208163 | − | 0.469109i | −1.95667 | −1.00000 | 0.0976512i | − | 2.46634i | 0.823632 | 2.77994 | 0.208163 | ||||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
209.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1045.2.f.a | ✓ | 40 |
11.b | odd | 2 | 1 | inner | 1045.2.f.a | ✓ | 40 |
19.b | odd | 2 | 1 | inner | 1045.2.f.a | ✓ | 40 |
209.d | even | 2 | 1 | inner | 1045.2.f.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.2.f.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1045.2.f.a | ✓ | 40 | 11.b | odd | 2 | 1 | inner |
1045.2.f.a | ✓ | 40 | 19.b | odd | 2 | 1 | inner |
1045.2.f.a | ✓ | 40 | 209.d | even | 2 | 1 | inner |