Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,3,Mod(19,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 15]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 414.l (of order \(22\), degree \(10\), 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: | \(11.2806829445\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | −5.16660 | + | 8.03938i | 0 | −10.8325 | + | 1.55748i | 2.71386 | + | 0.796860i | 0 | 13.3773 | + | 1.92336i | ||||||
19.2 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | −1.30307 | + | 2.02762i | 0 | 4.20587 | − | 0.604713i | 2.71386 | + | 0.796860i | 0 | 3.37389 | + | 0.485092i | ||||||
19.3 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | 2.90981 | − | 4.52775i | 0 | 0.515290 | − | 0.0740875i | 2.71386 | + | 0.796860i | 0 | −7.53403 | − | 1.08323i | ||||||
19.4 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | 3.55986 | − | 5.53925i | 0 | −0.310301 | + | 0.0446146i | 2.71386 | + | 0.796860i | 0 | −9.21713 | − | 1.32522i | ||||||
19.5 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | −3.55986 | + | 5.53925i | 0 | −0.310301 | + | 0.0446146i | −2.71386 | − | 0.796860i | 0 | −9.21713 | − | 1.32522i | ||||||
19.6 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | −2.90981 | + | 4.52775i | 0 | 0.515290 | − | 0.0740875i | −2.71386 | − | 0.796860i | 0 | −7.53403 | − | 1.08323i | ||||||
19.7 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | 1.30307 | − | 2.02762i | 0 | 4.20587 | − | 0.604713i | −2.71386 | − | 0.796860i | 0 | 3.37389 | + | 0.485092i | ||||||
19.8 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | 5.16660 | − | 8.03938i | 0 | −10.8325 | + | 1.55748i | −2.71386 | − | 0.796860i | 0 | 13.3773 | + | 1.92336i | ||||||
37.1 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | −8.16225 | + | 1.17355i | 0 | −2.09488 | − | 0.956699i | −1.85223 | + | 2.13758i | 0 | 10.6080 | − | 4.84451i | ||||||
37.2 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | −2.79643 | + | 0.402066i | 0 | 9.94408 | + | 4.54131i | −1.85223 | + | 2.13758i | 0 | 3.63436 | − | 1.65976i | ||||||
37.3 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | 3.02216 | − | 0.434521i | 0 | −4.90669 | − | 2.24081i | −1.85223 | + | 2.13758i | 0 | −3.92772 | + | 1.79373i | ||||||
37.4 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | 7.93652 | − | 1.14110i | 0 | 7.86197 | + | 3.59044i | −1.85223 | + | 2.13758i | 0 | −10.3146 | + | 4.71054i | ||||||
37.5 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | −7.93652 | + | 1.14110i | 0 | 7.86197 | + | 3.59044i | 1.85223 | − | 2.13758i | 0 | −10.3146 | + | 4.71054i | ||||||
37.6 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | −3.02216 | + | 0.434521i | 0 | −4.90669 | − | 2.24081i | 1.85223 | − | 2.13758i | 0 | −3.92772 | + | 1.79373i | ||||||
37.7 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | 2.79643 | − | 0.402066i | 0 | 9.94408 | + | 4.54131i | 1.85223 | − | 2.13758i | 0 | 3.63436 | − | 1.65976i | ||||||
37.8 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | 8.16225 | − | 1.17355i | 0 | −2.09488 | − | 0.956699i | 1.85223 | − | 2.13758i | 0 | 10.6080 | − | 4.84451i | ||||||
109.1 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | −5.16660 | − | 8.03938i | 0 | −10.8325 | − | 1.55748i | 2.71386 | − | 0.796860i | 0 | 13.3773 | − | 1.92336i | ||||||
109.2 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | −1.30307 | − | 2.02762i | 0 | 4.20587 | + | 0.604713i | 2.71386 | − | 0.796860i | 0 | 3.37389 | − | 0.485092i | ||||||
109.3 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | 2.90981 | + | 4.52775i | 0 | 0.515290 | + | 0.0740875i | 2.71386 | − | 0.796860i | 0 | −7.53403 | + | 1.08323i | ||||||
109.4 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | 3.55986 | + | 5.53925i | 0 | −0.310301 | − | 0.0446146i | 2.71386 | − | 0.796860i | 0 | −9.21713 | + | 1.32522i | ||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 414.3.l.c | ✓ | 80 |
3.b | odd | 2 | 1 | inner | 414.3.l.c | ✓ | 80 |
23.d | odd | 22 | 1 | inner | 414.3.l.c | ✓ | 80 |
69.g | even | 22 | 1 | inner | 414.3.l.c | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
414.3.l.c | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
414.3.l.c | ✓ | 80 | 3.b | odd | 2 | 1 | inner |
414.3.l.c | ✓ | 80 | 23.d | odd | 22 | 1 | inner |
414.3.l.c | ✓ | 80 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} - 220 T_{5}^{78} + 26666 T_{5}^{76} - 2658876 T_{5}^{74} + 266330727 T_{5}^{72} + \cdots + 11\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).