Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [456,2,Mod(379,456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(456, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 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("456.379");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 456 = 2^{3} \cdot 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 456.e (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: | \(3.64117833217\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
379.1 | −1.41195 | − | 0.0800606i | 1.00000i | 1.98718 | + | 0.226082i | 1.12499i | 0.0800606 | − | 1.41195i | 4.22432i | −2.78769 | − | 0.478311i | −1.00000 | 0.0900673 | − | 1.58842i | ||||||||
379.2 | −1.41195 | + | 0.0800606i | − | 1.00000i | 1.98718 | − | 0.226082i | − | 1.12499i | 0.0800606 | + | 1.41195i | − | 4.22432i | −2.78769 | + | 0.478311i | −1.00000 | 0.0900673 | + | 1.58842i | |||||
379.3 | −1.38840 | − | 0.268989i | 1.00000i | 1.85529 | + | 0.746926i | − | 3.66131i | 0.268989 | − | 1.38840i | − | 0.477575i | −2.37496 | − | 1.53608i | −1.00000 | −0.984850 | + | 5.08335i | ||||||
379.4 | −1.38840 | + | 0.268989i | − | 1.00000i | 1.85529 | − | 0.746926i | 3.66131i | 0.268989 | + | 1.38840i | 0.477575i | −2.37496 | + | 1.53608i | −1.00000 | −0.984850 | − | 5.08335i | |||||||
379.5 | −1.33983 | − | 0.452615i | − | 1.00000i | 1.59028 | + | 1.21285i | 0.393257i | −0.452615 | + | 1.33983i | 1.35370i | −1.58175 | − | 2.34480i | −1.00000 | 0.177994 | − | 0.526897i | |||||||
379.6 | −1.33983 | + | 0.452615i | 1.00000i | 1.59028 | − | 1.21285i | − | 0.393257i | −0.452615 | − | 1.33983i | − | 1.35370i | −1.58175 | + | 2.34480i | −1.00000 | 0.177994 | + | 0.526897i | ||||||
379.7 | −1.10595 | − | 0.881408i | 1.00000i | 0.446239 | + | 1.94958i | − | 0.946513i | 0.881408 | − | 1.10595i | − | 1.89799i | 1.22486 | − | 2.54945i | −1.00000 | −0.834264 | + | 1.04679i | ||||||
379.8 | −1.10595 | + | 0.881408i | − | 1.00000i | 0.446239 | − | 1.94958i | 0.946513i | 0.881408 | + | 1.10595i | 1.89799i | 1.22486 | + | 2.54945i | −1.00000 | −0.834264 | − | 1.04679i | |||||||
379.9 | −0.939794 | − | 1.05678i | − | 1.00000i | −0.233575 | + | 1.98631i | − | 1.70737i | −1.05678 | + | 0.939794i | 4.15645i | 2.31861 | − | 1.61989i | −1.00000 | −1.80431 | + | 1.60457i | ||||||
379.10 | −0.939794 | + | 1.05678i | 1.00000i | −0.233575 | − | 1.98631i | 1.70737i | −1.05678 | − | 0.939794i | − | 4.15645i | 2.31861 | + | 1.61989i | −1.00000 | −1.80431 | − | 1.60457i | |||||||
379.11 | −0.906554 | − | 1.08543i | 1.00000i | −0.356321 | + | 1.96800i | 3.19828i | 1.08543 | − | 0.906554i | 0.607439i | 2.45916 | − | 1.39734i | −1.00000 | 3.47151 | − | 2.89941i | ||||||||
379.12 | −0.906554 | + | 1.08543i | − | 1.00000i | −0.356321 | − | 1.96800i | − | 3.19828i | 1.08543 | + | 0.906554i | − | 0.607439i | 2.45916 | + | 1.39734i | −1.00000 | 3.47151 | + | 2.89941i | |||||
379.13 | −0.612280 | − | 1.27480i | − | 1.00000i | −1.25023 | + | 1.56107i | − | 4.23483i | −1.27480 | + | 0.612280i | − | 3.84958i | 2.75554 | + | 0.637979i | −1.00000 | −5.39856 | + | 2.59290i | |||||
379.14 | −0.612280 | + | 1.27480i | 1.00000i | −1.25023 | − | 1.56107i | 4.23483i | −1.27480 | − | 0.612280i | 3.84958i | 2.75554 | − | 0.637979i | −1.00000 | −5.39856 | − | 2.59290i | ||||||||
379.15 | −0.558288 | − | 1.29935i | − | 1.00000i | −1.37663 | + | 1.45082i | 1.71991i | −1.29935 | + | 0.558288i | 0.342554i | 2.65369 | + | 0.978748i | −1.00000 | 2.23477 | − | 0.960208i | |||||||
379.16 | −0.558288 | + | 1.29935i | 1.00000i | −1.37663 | − | 1.45082i | − | 1.71991i | −1.29935 | − | 0.558288i | − | 0.342554i | 2.65369 | − | 0.978748i | −1.00000 | 2.23477 | + | 0.960208i | ||||||
379.17 | −0.388306 | − | 1.35986i | 1.00000i | −1.69844 | + | 1.05608i | − | 1.84435i | 1.35986 | − | 0.388306i | − | 2.17959i | 2.09564 | + | 1.89955i | −1.00000 | −2.50805 | + | 0.716171i | ||||||
379.18 | −0.388306 | + | 1.35986i | − | 1.00000i | −1.69844 | − | 1.05608i | 1.84435i | 1.35986 | + | 0.388306i | 2.17959i | 2.09564 | − | 1.89955i | −1.00000 | −2.50805 | − | 0.716171i | |||||||
379.19 | −0.134537 | − | 1.40780i | 1.00000i | −1.96380 | + | 0.378802i | − | 2.61555i | 1.40780 | − | 0.134537i | 4.81248i | 0.797481 | + | 2.71367i | −1.00000 | −3.68217 | + | 0.351888i | |||||||
379.20 | −0.134537 | + | 1.40780i | − | 1.00000i | −1.96380 | − | 0.378802i | 2.61555i | 1.40780 | + | 0.134537i | − | 4.81248i | 0.797481 | − | 2.71367i | −1.00000 | −3.68217 | − | 0.351888i | ||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
152.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 456.2.e.a | ✓ | 40 |
3.b | odd | 2 | 1 | 1368.2.e.g | 40 | ||
4.b | odd | 2 | 1 | 1824.2.e.a | 40 | ||
8.b | even | 2 | 1 | 1824.2.e.a | 40 | ||
8.d | odd | 2 | 1 | inner | 456.2.e.a | ✓ | 40 |
12.b | even | 2 | 1 | 5472.2.e.g | 40 | ||
19.b | odd | 2 | 1 | inner | 456.2.e.a | ✓ | 40 |
24.f | even | 2 | 1 | 1368.2.e.g | 40 | ||
24.h | odd | 2 | 1 | 5472.2.e.g | 40 | ||
57.d | even | 2 | 1 | 1368.2.e.g | 40 | ||
76.d | even | 2 | 1 | 1824.2.e.a | 40 | ||
152.b | even | 2 | 1 | inner | 456.2.e.a | ✓ | 40 |
152.g | odd | 2 | 1 | 1824.2.e.a | 40 | ||
228.b | odd | 2 | 1 | 5472.2.e.g | 40 | ||
456.l | odd | 2 | 1 | 1368.2.e.g | 40 | ||
456.p | even | 2 | 1 | 5472.2.e.g | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
456.2.e.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
456.2.e.a | ✓ | 40 | 8.d | odd | 2 | 1 | inner |
456.2.e.a | ✓ | 40 | 19.b | odd | 2 | 1 | inner |
456.2.e.a | ✓ | 40 | 152.b | even | 2 | 1 | inner |
1368.2.e.g | 40 | 3.b | odd | 2 | 1 | ||
1368.2.e.g | 40 | 24.f | even | 2 | 1 | ||
1368.2.e.g | 40 | 57.d | even | 2 | 1 | ||
1368.2.e.g | 40 | 456.l | odd | 2 | 1 | ||
1824.2.e.a | 40 | 4.b | odd | 2 | 1 | ||
1824.2.e.a | 40 | 8.b | even | 2 | 1 | ||
1824.2.e.a | 40 | 76.d | even | 2 | 1 | ||
1824.2.e.a | 40 | 152.g | odd | 2 | 1 | ||
5472.2.e.g | 40 | 12.b | even | 2 | 1 | ||
5472.2.e.g | 40 | 24.h | odd | 2 | 1 | ||
5472.2.e.g | 40 | 228.b | odd | 2 | 1 | ||
5472.2.e.g | 40 | 456.p | even | 2 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(456, [\chi])\).