Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,3,Mod(12,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65.12");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 65.h (of order \(4\), degree \(2\), 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: | \(1.77112171834\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −2.66395 | + | 2.66395i | −2.12513 | + | 2.12513i | − | 10.1932i | −4.54019 | + | 2.09444i | − | 11.3225i | 2.95114 | − | 2.95114i | 16.4984 | + | 16.4984i | − | 0.0323960i | 6.51533 | − | 17.6743i | |||
12.2 | −2.14254 | + | 2.14254i | 1.85789 | − | 1.85789i | − | 5.18097i | 1.93662 | − | 4.60972i | 7.96121i | 5.84982 | − | 5.84982i | 2.53029 | + | 2.53029i | 2.09649i | 5.72722 | + | 14.0258i | |||||
12.3 | −1.78036 | + | 1.78036i | 0.433767 | − | 0.433767i | − | 2.33933i | 3.33534 | + | 3.72498i | 1.54452i | −5.63522 | + | 5.63522i | −2.95658 | − | 2.95658i | 8.62369i | −12.5699 | − | 0.693690i | |||||
12.4 | −1.30775 | + | 1.30775i | −3.70495 | + | 3.70495i | 0.579580i | 1.70825 | − | 4.69914i | − | 9.69031i | −2.51452 | + | 2.51452i | −5.98895 | − | 5.98895i | − | 18.4534i | 3.91133 | + | 8.37926i | ||||
12.5 | −0.474292 | + | 0.474292i | −0.839739 | + | 0.839739i | 3.55009i | −4.99852 | + | 0.121682i | − | 0.796563i | −1.35251 | + | 1.35251i | −3.58095 | − | 3.58095i | 7.58968i | 2.31304 | − | 2.42847i | |||||
12.6 | −0.456148 | + | 0.456148i | 3.37817 | − | 3.37817i | 3.58386i | −0.779969 | + | 4.93879i | 3.08189i | 7.82065 | − | 7.82065i | −3.45936 | − | 3.45936i | − | 13.8241i | −1.89704 | − | 2.60860i | |||||
12.7 | 0.456148 | − | 0.456148i | 3.37817 | − | 3.37817i | 3.58386i | 0.779969 | − | 4.93879i | − | 3.08189i | −7.82065 | + | 7.82065i | 3.45936 | + | 3.45936i | − | 13.8241i | −1.89704 | − | 2.60860i | ||||
12.8 | 0.474292 | − | 0.474292i | −0.839739 | + | 0.839739i | 3.55009i | 4.99852 | − | 0.121682i | 0.796563i | 1.35251 | − | 1.35251i | 3.58095 | + | 3.58095i | 7.58968i | 2.31304 | − | 2.42847i | ||||||
12.9 | 1.30775 | − | 1.30775i | −3.70495 | + | 3.70495i | 0.579580i | −1.70825 | + | 4.69914i | 9.69031i | 2.51452 | − | 2.51452i | 5.98895 | + | 5.98895i | − | 18.4534i | 3.91133 | + | 8.37926i | |||||
12.10 | 1.78036 | − | 1.78036i | 0.433767 | − | 0.433767i | − | 2.33933i | −3.33534 | − | 3.72498i | − | 1.54452i | 5.63522 | − | 5.63522i | 2.95658 | + | 2.95658i | 8.62369i | −12.5699 | − | 0.693690i | ||||
12.11 | 2.14254 | − | 2.14254i | 1.85789 | − | 1.85789i | − | 5.18097i | −1.93662 | + | 4.60972i | − | 7.96121i | −5.84982 | + | 5.84982i | −2.53029 | − | 2.53029i | 2.09649i | 5.72722 | + | 14.0258i | ||||
12.12 | 2.66395 | − | 2.66395i | −2.12513 | + | 2.12513i | − | 10.1932i | 4.54019 | − | 2.09444i | 11.3225i | −2.95114 | + | 2.95114i | −16.4984 | − | 16.4984i | − | 0.0323960i | 6.51533 | − | 17.6743i | ||||
38.1 | −2.66395 | − | 2.66395i | −2.12513 | − | 2.12513i | 10.1932i | −4.54019 | − | 2.09444i | 11.3225i | 2.95114 | + | 2.95114i | 16.4984 | − | 16.4984i | 0.0323960i | 6.51533 | + | 17.6743i | ||||||
38.2 | −2.14254 | − | 2.14254i | 1.85789 | + | 1.85789i | 5.18097i | 1.93662 | + | 4.60972i | − | 7.96121i | 5.84982 | + | 5.84982i | 2.53029 | − | 2.53029i | − | 2.09649i | 5.72722 | − | 14.0258i | ||||
38.3 | −1.78036 | − | 1.78036i | 0.433767 | + | 0.433767i | 2.33933i | 3.33534 | − | 3.72498i | − | 1.54452i | −5.63522 | − | 5.63522i | −2.95658 | + | 2.95658i | − | 8.62369i | −12.5699 | + | 0.693690i | ||||
38.4 | −1.30775 | − | 1.30775i | −3.70495 | − | 3.70495i | − | 0.579580i | 1.70825 | + | 4.69914i | 9.69031i | −2.51452 | − | 2.51452i | −5.98895 | + | 5.98895i | 18.4534i | 3.91133 | − | 8.37926i | |||||
38.5 | −0.474292 | − | 0.474292i | −0.839739 | − | 0.839739i | − | 3.55009i | −4.99852 | − | 0.121682i | 0.796563i | −1.35251 | − | 1.35251i | −3.58095 | + | 3.58095i | − | 7.58968i | 2.31304 | + | 2.42847i | ||||
38.6 | −0.456148 | − | 0.456148i | 3.37817 | + | 3.37817i | − | 3.58386i | −0.779969 | − | 4.93879i | − | 3.08189i | 7.82065 | + | 7.82065i | −3.45936 | + | 3.45936i | 13.8241i | −1.89704 | + | 2.60860i | ||||
38.7 | 0.456148 | + | 0.456148i | 3.37817 | + | 3.37817i | − | 3.58386i | 0.779969 | + | 4.93879i | 3.08189i | −7.82065 | − | 7.82065i | 3.45936 | − | 3.45936i | 13.8241i | −1.89704 | + | 2.60860i | |||||
38.8 | 0.474292 | + | 0.474292i | −0.839739 | − | 0.839739i | − | 3.55009i | 4.99852 | + | 0.121682i | − | 0.796563i | 1.35251 | + | 1.35251i | 3.58095 | − | 3.58095i | − | 7.58968i | 2.31304 | + | 2.42847i | |||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
13.b | even | 2 | 1 | inner |
65.h | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 65.3.h.a | ✓ | 24 |
5.b | even | 2 | 1 | 325.3.h.b | 24 | ||
5.c | odd | 4 | 1 | inner | 65.3.h.a | ✓ | 24 |
5.c | odd | 4 | 1 | 325.3.h.b | 24 | ||
13.b | even | 2 | 1 | inner | 65.3.h.a | ✓ | 24 |
65.d | even | 2 | 1 | 325.3.h.b | 24 | ||
65.h | odd | 4 | 1 | inner | 65.3.h.a | ✓ | 24 |
65.h | odd | 4 | 1 | 325.3.h.b | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.3.h.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
65.3.h.a | ✓ | 24 | 5.c | odd | 4 | 1 | inner |
65.3.h.a | ✓ | 24 | 13.b | even | 2 | 1 | inner |
65.3.h.a | ✓ | 24 | 65.h | odd | 4 | 1 | inner |
325.3.h.b | 24 | 5.b | even | 2 | 1 | ||
325.3.h.b | 24 | 5.c | odd | 4 | 1 | ||
325.3.h.b | 24 | 65.d | even | 2 | 1 | ||
325.3.h.b | 24 | 65.h | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(65, [\chi])\).