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 | −2.69018 | − | 1.09158i | 5.23709 | 1.00000 | 2.93656i | − | 4.34570i | −8.70838 | 1.80845 | −2.69018 | ||||||||||||||||
626.2 | −2.69018 | 1.09158i | 5.23709 | 1.00000 | − | 2.93656i | 4.34570i | −8.70838 | 1.80845 | −2.69018 | |||||||||||||||||
626.3 | −2.36088 | 2.84862i | 3.57375 | 1.00000 | − | 6.72526i | − | 0.816397i | −3.71543 | −5.11466 | −2.36088 | ||||||||||||||||
626.4 | −2.36088 | − | 2.84862i | 3.57375 | 1.00000 | 6.72526i | 0.816397i | −3.71543 | −5.11466 | −2.36088 | |||||||||||||||||
626.5 | −2.34872 | 1.68631i | 3.51646 | 1.00000 | − | 3.96066i | − | 0.129031i | −3.56174 | 0.156361 | −2.34872 | ||||||||||||||||
626.6 | −2.34872 | − | 1.68631i | 3.51646 | 1.00000 | 3.96066i | 0.129031i | −3.56174 | 0.156361 | −2.34872 | |||||||||||||||||
626.7 | −2.17500 | 1.39639i | 2.73061 | 1.00000 | − | 3.03714i | − | 0.972912i | −1.58909 | 1.05010 | −2.17500 | ||||||||||||||||
626.8 | −2.17500 | − | 1.39639i | 2.73061 | 1.00000 | 3.03714i | 0.972912i | −1.58909 | 1.05010 | −2.17500 | |||||||||||||||||
626.9 | −1.83534 | 0.556027i | 1.36849 | 1.00000 | − | 1.02050i | − | 4.08954i | 1.15904 | 2.69083 | −1.83534 | ||||||||||||||||
626.10 | −1.83534 | − | 0.556027i | 1.36849 | 1.00000 | 1.02050i | 4.08954i | 1.15904 | 2.69083 | −1.83534 | |||||||||||||||||
626.11 | −1.23717 | 3.34487i | −0.469419 | 1.00000 | − | 4.13816i | − | 3.14849i | 3.05508 | −8.18815 | −1.23717 | ||||||||||||||||
626.12 | −1.23717 | − | 3.34487i | −0.469419 | 1.00000 | 4.13816i | 3.14849i | 3.05508 | −8.18815 | −1.23717 | |||||||||||||||||
626.13 | −1.10652 | − | 0.279240i | −0.775612 | 1.00000 | 0.308985i | 2.99424i | 3.07127 | 2.92202 | −1.10652 | |||||||||||||||||
626.14 | −1.10652 | 0.279240i | −0.775612 | 1.00000 | − | 0.308985i | − | 2.99424i | 3.07127 | 2.92202 | −1.10652 | ||||||||||||||||
626.15 | −0.863466 | − | 0.924425i | −1.25443 | 1.00000 | 0.798209i | − | 0.285490i | 2.81009 | 2.14544 | −0.863466 | ||||||||||||||||
626.16 | −0.863466 | 0.924425i | −1.25443 | 1.00000 | − | 0.798209i | 0.285490i | 2.81009 | 2.14544 | −0.863466 | |||||||||||||||||
626.17 | −0.233551 | − | 2.00145i | −1.94545 | 1.00000 | 0.467439i | − | 1.43314i | 0.921463 | −1.00579 | −0.233551 | ||||||||||||||||
626.18 | −0.233551 | 2.00145i | −1.94545 | 1.00000 | − | 0.467439i | 1.43314i | 0.921463 | −1.00579 | −0.233551 | |||||||||||||||||
626.19 | −0.136022 | 2.54256i | −1.98150 | 1.00000 | − | 0.345844i | − | 4.55489i | 0.541571 | −3.46461 | −0.136022 | ||||||||||||||||
626.20 | −0.136022 | − | 2.54256i | −1.98150 | 1.00000 | 0.345844i | 4.55489i | 0.541571 | −3.46461 | −0.136022 | |||||||||||||||||
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.b | ✓ | 40 |
11.b | odd | 2 | 1 | inner | 1045.2.f.b | ✓ | 40 |
19.b | odd | 2 | 1 | inner | 1045.2.f.b | ✓ | 40 |
209.d | even | 2 | 1 | inner | 1045.2.f.b | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1045.2.f.b | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1045.2.f.b | ✓ | 40 | 11.b | odd | 2 | 1 | inner |
1045.2.f.b | ✓ | 40 | 19.b | odd | 2 | 1 | inner |
1045.2.f.b | ✓ | 40 | 209.d | even | 2 | 1 | inner |