Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,3,Mod(61,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.61");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 310.c (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: | \(8.44688819517\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
61.1 | −1.41421 | − | 5.88545i | 2.00000 | −2.23607 | 8.32328i | 1.81095 | −2.82843 | −25.6385 | 3.16228 | |||||||||||||||||
61.2 | −1.41421 | − | 4.63999i | 2.00000 | 2.23607 | 6.56194i | −0.345649 | −2.82843 | −12.5295 | −3.16228 | |||||||||||||||||
61.3 | −1.41421 | − | 2.59054i | 2.00000 | 2.23607 | 3.66358i | 11.2758 | −2.82843 | 2.28909 | −3.16228 | |||||||||||||||||
61.4 | −1.41421 | − | 2.35136i | 2.00000 | −2.23607 | 3.32533i | 7.27991 | −2.82843 | 3.47110 | 3.16228 | |||||||||||||||||
61.5 | −1.41421 | − | 1.96026i | 2.00000 | 2.23607 | 2.77223i | −10.7520 | −2.82843 | 5.15737 | −3.16228 | |||||||||||||||||
61.6 | −1.41421 | − | 0.304421i | 2.00000 | −2.23607 | 0.430516i | −4.44058 | −2.82843 | 8.90733 | 3.16228 | |||||||||||||||||
61.7 | −1.41421 | 0.304421i | 2.00000 | −2.23607 | − | 0.430516i | −4.44058 | −2.82843 | 8.90733 | 3.16228 | |||||||||||||||||
61.8 | −1.41421 | 1.96026i | 2.00000 | 2.23607 | − | 2.77223i | −10.7520 | −2.82843 | 5.15737 | −3.16228 | |||||||||||||||||
61.9 | −1.41421 | 2.35136i | 2.00000 | −2.23607 | − | 3.32533i | 7.27991 | −2.82843 | 3.47110 | 3.16228 | |||||||||||||||||
61.10 | −1.41421 | 2.59054i | 2.00000 | 2.23607 | − | 3.66358i | 11.2758 | −2.82843 | 2.28909 | −3.16228 | |||||||||||||||||
61.11 | −1.41421 | 4.63999i | 2.00000 | 2.23607 | − | 6.56194i | −0.345649 | −2.82843 | −12.5295 | −3.16228 | |||||||||||||||||
61.12 | −1.41421 | 5.88545i | 2.00000 | −2.23607 | − | 8.32328i | 1.81095 | −2.82843 | −25.6385 | 3.16228 | |||||||||||||||||
61.13 | 1.41421 | − | 5.82697i | 2.00000 | 2.23607 | − | 8.24058i | −12.9826 | 2.82843 | −24.9536 | 3.16228 | ||||||||||||||||
61.14 | 1.41421 | − | 5.42696i | 2.00000 | −2.23607 | − | 7.67489i | 7.66369 | 2.82843 | −20.4519 | −3.16228 | ||||||||||||||||
61.15 | 1.41421 | − | 3.58942i | 2.00000 | 2.23607 | − | 5.07621i | 11.6269 | 2.82843 | −3.88397 | 3.16228 | ||||||||||||||||
61.16 | 1.41421 | − | 1.88454i | 2.00000 | 2.23607 | − | 2.66514i | −1.29462 | 2.82843 | 5.44852 | 3.16228 | ||||||||||||||||
61.17 | 1.41421 | − | 1.46742i | 2.00000 | −2.23607 | − | 2.07524i | −11.9745 | 2.82843 | 6.84669 | −3.16228 | ||||||||||||||||
61.18 | 1.41421 | − | 1.28940i | 2.00000 | −2.23607 | − | 1.82349i | 6.13262 | 2.82843 | 7.33744 | −3.16228 | ||||||||||||||||
61.19 | 1.41421 | 1.28940i | 2.00000 | −2.23607 | 1.82349i | 6.13262 | 2.82843 | 7.33744 | −3.16228 | ||||||||||||||||||
61.20 | 1.41421 | 1.46742i | 2.00000 | −2.23607 | 2.07524i | −11.9745 | 2.82843 | 6.84669 | −3.16228 | ||||||||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 310.3.c.a | ✓ | 24 |
5.b | even | 2 | 1 | 1550.3.c.d | 24 | ||
5.c | odd | 4 | 2 | 1550.3.d.c | 48 | ||
31.b | odd | 2 | 1 | inner | 310.3.c.a | ✓ | 24 |
155.c | odd | 2 | 1 | 1550.3.c.d | 24 | ||
155.f | even | 4 | 2 | 1550.3.d.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
310.3.c.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
310.3.c.a | ✓ | 24 | 31.b | odd | 2 | 1 | inner |
1550.3.c.d | 24 | 5.b | even | 2 | 1 | ||
1550.3.c.d | 24 | 155.c | odd | 2 | 1 | ||
1550.3.d.c | 48 | 5.c | odd | 4 | 2 | ||
1550.3.d.c | 48 | 155.f | even | 4 | 2 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(310, [\chi])\).