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: | no (minimal twist has level 138) |
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 | −5.01874 | + | 7.80931i | 0 | 5.11630 | − | 0.735612i | 2.71386 | + | 0.796860i | 0 | 12.9944 | + | 1.86832i | ||||||
19.2 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | −0.891807 | + | 1.38768i | 0 | 9.81972 | − | 1.41186i | 2.71386 | + | 0.796860i | 0 | 2.30905 | + | 0.331992i | ||||||
19.3 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | 0.858668 | − | 1.33611i | 0 | −8.36490 | + | 1.20269i | 2.71386 | + | 0.796860i | 0 | −2.22325 | − | 0.319655i | ||||||
19.4 | −0.587486 | − | 1.28641i | 0 | −1.30972 | + | 1.51150i | 4.44262 | − | 6.91285i | 0 | −10.3360 | + | 1.48610i | 2.71386 | + | 0.796860i | 0 | −11.5028 | − | 1.65385i | ||||||
19.5 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | −3.94925 | + | 6.14515i | 0 | −2.61314 | + | 0.375713i | −2.71386 | − | 0.796860i | 0 | −10.2253 | − | 1.47018i | ||||||
19.6 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | −0.791659 | + | 1.23185i | 0 | 8.65930 | − | 1.24502i | −2.71386 | − | 0.796860i | 0 | −2.04975 | − | 0.294710i | ||||||
19.7 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | 0.634924 | − | 0.987962i | 0 | 1.01096 | − | 0.145354i | −2.71386 | − | 0.796860i | 0 | 1.64394 | + | 0.236362i | ||||||
19.8 | 0.587486 | + | 1.28641i | 0 | −1.30972 | + | 1.51150i | 3.49672 | − | 5.44100i | 0 | 9.55110 | − | 1.37324i | −2.71386 | − | 0.796860i | 0 | 9.05366 | + | 1.30172i | ||||||
37.1 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | −8.41654 | + | 1.21012i | 0 | −8.45540 | − | 3.86145i | −1.85223 | + | 2.13758i | 0 | 10.9385 | − | 4.99544i | ||||||
37.2 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | −3.43203 | + | 0.493451i | 0 | 4.31074 | + | 1.96865i | −1.85223 | + | 2.13758i | 0 | 4.46041 | − | 2.03700i | ||||||
37.3 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | 1.17981 | − | 0.169632i | 0 | 7.87069 | + | 3.59442i | −1.85223 | + | 2.13758i | 0 | −1.53334 | + | 0.700252i | ||||||
37.4 | −1.35693 | + | 0.398430i | 0 | 1.68251 | − | 1.08128i | 7.67653 | − | 1.10372i | 0 | −11.6311 | − | 5.31175i | −1.85223 | + | 2.13758i | 0 | −9.97674 | + | 4.55623i | ||||||
37.5 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | −5.59508 | + | 0.804450i | 0 | 0.513410 | + | 0.234466i | 1.85223 | − | 2.13758i | 0 | −7.27160 | + | 3.32083i | ||||||
37.6 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | −4.69745 | + | 0.675391i | 0 | −5.04392 | − | 2.30348i | 1.85223 | − | 2.13758i | 0 | −6.10500 | + | 2.78806i | ||||||
37.7 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | 3.60161 | − | 0.517833i | 0 | −10.7332 | − | 4.90168i | 1.85223 | − | 2.13758i | 0 | 4.68080 | − | 2.13765i | ||||||
37.8 | 1.35693 | − | 0.398430i | 0 | 1.68251 | − | 1.08128i | 3.69869 | − | 0.531791i | 0 | 1.55981 | + | 0.712339i | 1.85223 | − | 2.13758i | 0 | 4.80697 | − | 2.19527i | ||||||
109.1 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | −5.01874 | − | 7.80931i | 0 | 5.11630 | + | 0.735612i | 2.71386 | − | 0.796860i | 0 | 12.9944 | − | 1.86832i | ||||||
109.2 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | −0.891807 | − | 1.38768i | 0 | 9.81972 | + | 1.41186i | 2.71386 | − | 0.796860i | 0 | 2.30905 | − | 0.331992i | ||||||
109.3 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | 0.858668 | + | 1.33611i | 0 | −8.36490 | − | 1.20269i | 2.71386 | − | 0.796860i | 0 | −2.22325 | + | 0.319655i | ||||||
109.4 | −0.587486 | + | 1.28641i | 0 | −1.30972 | − | 1.51150i | 4.44262 | + | 6.91285i | 0 | −10.3360 | − | 1.48610i | 2.71386 | − | 0.796860i | 0 | −11.5028 | + | 1.65385i | ||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 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.b | 80 | |
3.b | odd | 2 | 1 | 138.3.h.a | ✓ | 80 | |
23.d | odd | 22 | 1 | inner | 414.3.l.b | 80 | |
69.g | even | 22 | 1 | 138.3.h.a | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
138.3.h.a | ✓ | 80 | 3.b | odd | 2 | 1 | |
138.3.h.a | ✓ | 80 | 69.g | even | 22 | 1 | |
414.3.l.b | 80 | 1.a | even | 1 | 1 | trivial | |
414.3.l.b | 80 | 23.d | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{80} + 68 T_{5}^{78} - 484 T_{5}^{77} + 3194 T_{5}^{76} + 17160 T_{5}^{75} - 202264 T_{5}^{74} + 2039928 T_{5}^{73} - 24535009 T_{5}^{72} + 335882228 T_{5}^{71} - 956558316 T_{5}^{70} + \cdots + 11\!\cdots\!09 \)
acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\).