Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,5,Mod(134,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.134");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(32.5615383714\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
134.1 | −7.75434 | 0 | 44.1298 | 21.9687 | + | 11.9322i | 0 | − | 18.5203i | −218.128 | 0 | −170.353 | − | 92.5261i | |||||||||||||
134.2 | −7.75434 | 0 | 44.1298 | 21.9687 | − | 11.9322i | 0 | 18.5203i | −218.128 | 0 | −170.353 | + | 92.5261i | ||||||||||||||
134.3 | −7.26389 | 0 | 36.7640 | −21.6988 | + | 12.4163i | 0 | 18.5203i | −150.828 | 0 | 157.617 | − | 90.1902i | ||||||||||||||
134.4 | −7.26389 | 0 | 36.7640 | −21.6988 | − | 12.4163i | 0 | − | 18.5203i | −150.828 | 0 | 157.617 | + | 90.1902i | |||||||||||||
134.5 | −7.05397 | 0 | 33.7584 | −2.15368 | + | 24.9071i | 0 | 18.5203i | −125.267 | 0 | 15.1920 | − | 175.694i | ||||||||||||||
134.6 | −7.05397 | 0 | 33.7584 | −2.15368 | − | 24.9071i | 0 | − | 18.5203i | −125.267 | 0 | 15.1920 | + | 175.694i | |||||||||||||
134.7 | −6.05730 | 0 | 20.6909 | 21.8682 | + | 12.1154i | 0 | 18.5203i | −28.4144 | 0 | −132.462 | − | 73.3869i | ||||||||||||||
134.8 | −6.05730 | 0 | 20.6909 | 21.8682 | − | 12.1154i | 0 | − | 18.5203i | −28.4144 | 0 | −132.462 | + | 73.3869i | |||||||||||||
134.9 | −5.50164 | 0 | 14.2681 | 15.7292 | + | 19.4317i | 0 | − | 18.5203i | 9.52837 | 0 | −86.5364 | − | 106.907i | |||||||||||||
134.10 | −5.50164 | 0 | 14.2681 | 15.7292 | − | 19.4317i | 0 | 18.5203i | 9.52837 | 0 | −86.5364 | + | 106.907i | ||||||||||||||
134.11 | −5.17669 | 0 | 10.7981 | −5.01029 | − | 24.4928i | 0 | 18.5203i | 26.9285 | 0 | 25.9367 | + | 126.792i | ||||||||||||||
134.12 | −5.17669 | 0 | 10.7981 | −5.01029 | + | 24.4928i | 0 | − | 18.5203i | 26.9285 | 0 | 25.9367 | − | 126.792i | |||||||||||||
134.13 | −3.82553 | 0 | −1.36532 | 21.7643 | − | 12.3010i | 0 | − | 18.5203i | 66.4316 | 0 | −83.2601 | + | 47.0578i | |||||||||||||
134.14 | −3.82553 | 0 | −1.36532 | 21.7643 | + | 12.3010i | 0 | 18.5203i | 66.4316 | 0 | −83.2601 | − | 47.0578i | ||||||||||||||
134.15 | −3.57740 | 0 | −3.20217 | −24.6015 | + | 4.44567i | 0 | − | 18.5203i | 68.6940 | 0 | 88.0097 | − | 15.9040i | |||||||||||||
134.16 | −3.57740 | 0 | −3.20217 | −24.6015 | − | 4.44567i | 0 | 18.5203i | 68.6940 | 0 | 88.0097 | + | 15.9040i | ||||||||||||||
134.17 | −3.05766 | 0 | −6.65074 | −23.4191 | − | 8.74905i | 0 | − | 18.5203i | 69.2582 | 0 | 71.6075 | + | 26.7516i | |||||||||||||
134.18 | −3.05766 | 0 | −6.65074 | −23.4191 | + | 8.74905i | 0 | 18.5203i | 69.2582 | 0 | 71.6075 | − | 26.7516i | ||||||||||||||
134.19 | −2.67247 | 0 | −8.85789 | −7.43665 | + | 23.8683i | 0 | 18.5203i | 66.4320 | 0 | 19.8743 | − | 63.7874i | ||||||||||||||
134.20 | −2.67247 | 0 | −8.85789 | −7.43665 | − | 23.8683i | 0 | − | 18.5203i | 66.4320 | 0 | 19.8743 | + | 63.7874i | |||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.5.f.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 315.5.f.a | ✓ | 48 |
5.b | even | 2 | 1 | inner | 315.5.f.a | ✓ | 48 |
15.d | odd | 2 | 1 | inner | 315.5.f.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.5.f.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
315.5.f.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
315.5.f.a | ✓ | 48 | 5.b | even | 2 | 1 | inner |
315.5.f.a | ✓ | 48 | 15.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(315, [\chi])\).