Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,2,Mod(53,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.53");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.ce (of order \(12\), degree \(4\), 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: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 | −0.695956 | + | 2.59734i | 0 | −4.52979 | − | 2.61528i | 1.27254 | − | 1.83865i | 0 | −1.50727 | − | 2.17442i | 6.14253 | − | 6.14253i | 0 | 3.88997 | + | 4.58485i | ||||||
53.2 | −0.604483 | + | 2.25596i | 0 | −2.99191 | − | 1.72738i | −0.254820 | + | 2.22150i | 0 | −2.19240 | + | 1.48100i | 2.40251 | − | 2.40251i | 0 | −4.85759 | − | 1.91772i | ||||||
53.3 | −0.543545 | + | 2.02854i | 0 | −2.08747 | − | 1.20520i | −1.61611 | − | 1.54538i | 0 | 1.07923 | − | 2.41563i | 0.609443 | − | 0.609443i | 0 | 4.01329 | − | 2.43835i | ||||||
53.4 | −0.494677 | + | 1.84616i | 0 | −1.43155 | − | 0.826508i | 0.541988 | + | 2.16939i | 0 | 2.64533 | + | 0.0471386i | −0.468944 | + | 0.468944i | 0 | −4.27315 | − | 0.0725500i | ||||||
53.5 | −0.364480 | + | 1.36026i | 0 | 0.0145936 | + | 0.00842564i | −2.23601 | − | 0.0163814i | 0 | −2.58285 | − | 0.573503i | −2.00834 | + | 2.00834i | 0 | 0.837263 | − | 3.03558i | ||||||
53.6 | −0.271660 | + | 1.01385i | 0 | 0.777962 | + | 0.449157i | 0.310035 | − | 2.21447i | 0 | −1.26800 | + | 2.32211i | −2.15109 | + | 2.15109i | 0 | 2.16091 | + | 0.915911i | ||||||
53.7 | −0.0991680 | + | 0.370100i | 0 | 1.60491 | + | 0.926596i | 1.78935 | + | 1.34098i | 0 | 2.39134 | + | 1.13202i | −1.04395 | + | 1.04395i | 0 | −0.673742 | + | 0.529258i | ||||||
53.8 | −0.0362581 | + | 0.135317i | 0 | 1.71505 | + | 0.990187i | −1.87468 | + | 1.21884i | 0 | 0.702570 | − | 2.55076i | −0.394292 | + | 0.394292i | 0 | −0.0969571 | − | 0.297870i | ||||||
53.9 | 0.0362581 | − | 0.135317i | 0 | 1.71505 | + | 0.990187i | 1.87468 | − | 1.21884i | 0 | 0.702570 | − | 2.55076i | 0.394292 | − | 0.394292i | 0 | −0.0969571 | − | 0.297870i | ||||||
53.10 | 0.0991680 | − | 0.370100i | 0 | 1.60491 | + | 0.926596i | −1.78935 | − | 1.34098i | 0 | 2.39134 | + | 1.13202i | 1.04395 | − | 1.04395i | 0 | −0.673742 | + | 0.529258i | ||||||
53.11 | 0.271660 | − | 1.01385i | 0 | 0.777962 | + | 0.449157i | −0.310035 | + | 2.21447i | 0 | −1.26800 | + | 2.32211i | 2.15109 | − | 2.15109i | 0 | 2.16091 | + | 0.915911i | ||||||
53.12 | 0.364480 | − | 1.36026i | 0 | 0.0145936 | + | 0.00842564i | 2.23601 | + | 0.0163814i | 0 | −2.58285 | − | 0.573503i | 2.00834 | − | 2.00834i | 0 | 0.837263 | − | 3.03558i | ||||||
53.13 | 0.494677 | − | 1.84616i | 0 | −1.43155 | − | 0.826508i | −0.541988 | − | 2.16939i | 0 | 2.64533 | + | 0.0471386i | 0.468944 | − | 0.468944i | 0 | −4.27315 | − | 0.0725500i | ||||||
53.14 | 0.543545 | − | 2.02854i | 0 | −2.08747 | − | 1.20520i | 1.61611 | + | 1.54538i | 0 | 1.07923 | − | 2.41563i | −0.609443 | + | 0.609443i | 0 | 4.01329 | − | 2.43835i | ||||||
53.15 | 0.604483 | − | 2.25596i | 0 | −2.99191 | − | 1.72738i | 0.254820 | − | 2.22150i | 0 | −2.19240 | + | 1.48100i | −2.40251 | + | 2.40251i | 0 | −4.85759 | − | 1.91772i | ||||||
53.16 | 0.695956 | − | 2.59734i | 0 | −4.52979 | − | 2.61528i | −1.27254 | + | 1.83865i | 0 | −1.50727 | − | 2.17442i | −6.14253 | + | 6.14253i | 0 | 3.88997 | + | 4.58485i | ||||||
107.1 | −0.695956 | − | 2.59734i | 0 | −4.52979 | + | 2.61528i | 1.27254 | + | 1.83865i | 0 | −1.50727 | + | 2.17442i | 6.14253 | + | 6.14253i | 0 | 3.88997 | − | 4.58485i | ||||||
107.2 | −0.604483 | − | 2.25596i | 0 | −2.99191 | + | 1.72738i | −0.254820 | − | 2.22150i | 0 | −2.19240 | − | 1.48100i | 2.40251 | + | 2.40251i | 0 | −4.85759 | + | 1.91772i | ||||||
107.3 | −0.543545 | − | 2.02854i | 0 | −2.08747 | + | 1.20520i | −1.61611 | + | 1.54538i | 0 | 1.07923 | + | 2.41563i | 0.609443 | + | 0.609443i | 0 | 4.01329 | + | 2.43835i | ||||||
107.4 | −0.494677 | − | 1.84616i | 0 | −1.43155 | + | 0.826508i | 0.541988 | − | 2.16939i | 0 | 2.64533 | − | 0.0471386i | −0.468944 | − | 0.468944i | 0 | −4.27315 | + | 0.0725500i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
7.c | even | 3 | 1 | inner |
15.e | even | 4 | 1 | inner |
21.h | odd | 6 | 1 | inner |
35.l | odd | 12 | 1 | inner |
105.x | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.ce.a | ✓ | 64 |
3.b | odd | 2 | 1 | inner | 315.2.ce.a | ✓ | 64 |
5.c | odd | 4 | 1 | inner | 315.2.ce.a | ✓ | 64 |
7.c | even | 3 | 1 | inner | 315.2.ce.a | ✓ | 64 |
15.e | even | 4 | 1 | inner | 315.2.ce.a | ✓ | 64 |
21.h | odd | 6 | 1 | inner | 315.2.ce.a | ✓ | 64 |
35.l | odd | 12 | 1 | inner | 315.2.ce.a | ✓ | 64 |
105.x | even | 12 | 1 | inner | 315.2.ce.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.ce.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
315.2.ce.a | ✓ | 64 | 3.b | odd | 2 | 1 | inner |
315.2.ce.a | ✓ | 64 | 5.c | odd | 4 | 1 | inner |
315.2.ce.a | ✓ | 64 | 7.c | even | 3 | 1 | inner |
315.2.ce.a | ✓ | 64 | 15.e | even | 4 | 1 | inner |
315.2.ce.a | ✓ | 64 | 21.h | odd | 6 | 1 | inner |
315.2.ce.a | ✓ | 64 | 35.l | odd | 12 | 1 | inner |
315.2.ce.a | ✓ | 64 | 105.x | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).