Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,3,Mod(74,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.74");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.i (of order \(6\), degree \(2\), not 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: | \(6.13080594811\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 45) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
74.1 | −1.86827 | + | 3.23594i | −1.45927 | + | 2.62117i | −4.98088 | − | 8.62715i | 0 | −5.75562 | − | 9.61919i | −6.28840 | − | 3.63061i | 22.2764 | −4.74103 | − | 7.65001i | 0 | ||||||
74.2 | −1.63532 | + | 2.83245i | 1.96733 | − | 2.26486i | −3.34853 | − | 5.79983i | 0 | 3.19790 | + | 9.27615i | 5.48940 | + | 3.16931i | 8.82112 | −1.25919 | − | 8.91148i | 0 | ||||||
74.3 | −1.54563 | + | 2.67710i | 1.01971 | + | 2.82138i | −2.77793 | − | 4.81151i | 0 | −9.12922 | − | 1.63094i | 1.91037 | + | 1.10296i | 4.80953 | −6.92040 | + | 5.75396i | 0 | ||||||
74.4 | −1.41438 | + | 2.44978i | −2.99795 | − | 0.110987i | −2.00096 | − | 3.46576i | 0 | 4.51214 | − | 7.18734i | −6.88441 | − | 3.97472i | 0.00541780 | 8.97536 | + | 0.665467i | 0 | ||||||
74.5 | −0.916957 | + | 1.58822i | 2.88508 | − | 0.822398i | 0.318381 | + | 0.551452i | 0 | −1.33934 | + | 5.33623i | 5.60142 | + | 3.23398i | −8.50342 | 7.64732 | − | 4.74536i | 0 | ||||||
74.6 | −0.696021 | + | 1.20554i | 1.88791 | + | 2.33148i | 1.03111 | + | 1.78593i | 0 | −4.12473 | + | 0.653207i | −7.63842 | − | 4.41004i | −8.43886 | −1.87156 | + | 8.80325i | 0 | ||||||
74.7 | −0.346451 | + | 0.600071i | 0.450312 | − | 2.96601i | 1.75994 | + | 3.04831i | 0 | 1.62381 | + | 1.29780i | −10.6367 | − | 6.14112i | −5.21055 | −8.59444 | − | 2.67126i | 0 | ||||||
74.8 | −0.0238054 | + | 0.0412321i | 2.42528 | − | 1.76580i | 1.99887 | + | 3.46214i | 0 | 0.0150729 | + | 0.142035i | −3.30399 | − | 1.90756i | −0.380778 | 2.76393 | − | 8.56508i | 0 | ||||||
74.9 | 0.0238054 | − | 0.0412321i | −2.42528 | + | 1.76580i | 1.99887 | + | 3.46214i | 0 | 0.0150729 | + | 0.142035i | 3.30399 | + | 1.90756i | 0.380778 | 2.76393 | − | 8.56508i | 0 | ||||||
74.10 | 0.346451 | − | 0.600071i | −0.450312 | + | 2.96601i | 1.75994 | + | 3.04831i | 0 | 1.62381 | + | 1.29780i | 10.6367 | + | 6.14112i | 5.21055 | −8.59444 | − | 2.67126i | 0 | ||||||
74.11 | 0.696021 | − | 1.20554i | −1.88791 | − | 2.33148i | 1.03111 | + | 1.78593i | 0 | −4.12473 | + | 0.653207i | 7.63842 | + | 4.41004i | 8.43886 | −1.87156 | + | 8.80325i | 0 | ||||||
74.12 | 0.916957 | − | 1.58822i | −2.88508 | + | 0.822398i | 0.318381 | + | 0.551452i | 0 | −1.33934 | + | 5.33623i | −5.60142 | − | 3.23398i | 8.50342 | 7.64732 | − | 4.74536i | 0 | ||||||
74.13 | 1.41438 | − | 2.44978i | 2.99795 | + | 0.110987i | −2.00096 | − | 3.46576i | 0 | 4.51214 | − | 7.18734i | 6.88441 | + | 3.97472i | −0.00541780 | 8.97536 | + | 0.665467i | 0 | ||||||
74.14 | 1.54563 | − | 2.67710i | −1.01971 | − | 2.82138i | −2.77793 | − | 4.81151i | 0 | −9.12922 | − | 1.63094i | −1.91037 | − | 1.10296i | −4.80953 | −6.92040 | + | 5.75396i | 0 | ||||||
74.15 | 1.63532 | − | 2.83245i | −1.96733 | + | 2.26486i | −3.34853 | − | 5.79983i | 0 | 3.19790 | + | 9.27615i | −5.48940 | − | 3.16931i | −8.82112 | −1.25919 | − | 8.91148i | 0 | ||||||
74.16 | 1.86827 | − | 3.23594i | 1.45927 | − | 2.62117i | −4.98088 | − | 8.62715i | 0 | −5.75562 | − | 9.61919i | 6.28840 | + | 3.63061i | −22.2764 | −4.74103 | − | 7.65001i | 0 | ||||||
149.1 | −1.86827 | − | 3.23594i | −1.45927 | − | 2.62117i | −4.98088 | + | 8.62715i | 0 | −5.75562 | + | 9.61919i | −6.28840 | + | 3.63061i | 22.2764 | −4.74103 | + | 7.65001i | 0 | ||||||
149.2 | −1.63532 | − | 2.83245i | 1.96733 | + | 2.26486i | −3.34853 | + | 5.79983i | 0 | 3.19790 | − | 9.27615i | 5.48940 | − | 3.16931i | 8.82112 | −1.25919 | + | 8.91148i | 0 | ||||||
149.3 | −1.54563 | − | 2.67710i | 1.01971 | − | 2.82138i | −2.77793 | + | 4.81151i | 0 | −9.12922 | + | 1.63094i | 1.91037 | − | 1.10296i | 4.80953 | −6.92040 | − | 5.75396i | 0 | ||||||
149.4 | −1.41438 | − | 2.44978i | −2.99795 | + | 0.110987i | −2.00096 | + | 3.46576i | 0 | 4.51214 | + | 7.18734i | −6.88441 | + | 3.97472i | 0.00541780 | 8.97536 | − | 0.665467i | 0 | ||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.3.i.b | 32 | |
3.b | odd | 2 | 1 | 675.3.i.c | 32 | ||
5.b | even | 2 | 1 | inner | 225.3.i.b | 32 | |
5.c | odd | 4 | 1 | 45.3.i.a | ✓ | 16 | |
5.c | odd | 4 | 1 | 225.3.j.b | 16 | ||
9.c | even | 3 | 1 | 675.3.i.c | 32 | ||
9.d | odd | 6 | 1 | inner | 225.3.i.b | 32 | |
15.d | odd | 2 | 1 | 675.3.i.c | 32 | ||
15.e | even | 4 | 1 | 135.3.i.a | 16 | ||
15.e | even | 4 | 1 | 675.3.j.b | 16 | ||
20.e | even | 4 | 1 | 720.3.bs.c | 16 | ||
45.h | odd | 6 | 1 | inner | 225.3.i.b | 32 | |
45.j | even | 6 | 1 | 675.3.i.c | 32 | ||
45.k | odd | 12 | 1 | 135.3.i.a | 16 | ||
45.k | odd | 12 | 1 | 405.3.c.a | 16 | ||
45.k | odd | 12 | 1 | 675.3.j.b | 16 | ||
45.l | even | 12 | 1 | 45.3.i.a | ✓ | 16 | |
45.l | even | 12 | 1 | 225.3.j.b | 16 | ||
45.l | even | 12 | 1 | 405.3.c.a | 16 | ||
60.l | odd | 4 | 1 | 2160.3.bs.c | 16 | ||
180.v | odd | 12 | 1 | 720.3.bs.c | 16 | ||
180.x | even | 12 | 1 | 2160.3.bs.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.3.i.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
45.3.i.a | ✓ | 16 | 45.l | even | 12 | 1 | |
135.3.i.a | 16 | 15.e | even | 4 | 1 | ||
135.3.i.a | 16 | 45.k | odd | 12 | 1 | ||
225.3.i.b | 32 | 1.a | even | 1 | 1 | trivial | |
225.3.i.b | 32 | 5.b | even | 2 | 1 | inner | |
225.3.i.b | 32 | 9.d | odd | 6 | 1 | inner | |
225.3.i.b | 32 | 45.h | odd | 6 | 1 | inner | |
225.3.j.b | 16 | 5.c | odd | 4 | 1 | ||
225.3.j.b | 16 | 45.l | even | 12 | 1 | ||
405.3.c.a | 16 | 45.k | odd | 12 | 1 | ||
405.3.c.a | 16 | 45.l | even | 12 | 1 | ||
675.3.i.c | 32 | 3.b | odd | 2 | 1 | ||
675.3.i.c | 32 | 9.c | even | 3 | 1 | ||
675.3.i.c | 32 | 15.d | odd | 2 | 1 | ||
675.3.i.c | 32 | 45.j | even | 6 | 1 | ||
675.3.j.b | 16 | 15.e | even | 4 | 1 | ||
675.3.j.b | 16 | 45.k | odd | 12 | 1 | ||
720.3.bs.c | 16 | 20.e | even | 4 | 1 | ||
720.3.bs.c | 16 | 180.v | odd | 12 | 1 | ||
2160.3.bs.c | 16 | 60.l | odd | 4 | 1 | ||
2160.3.bs.c | 16 | 180.x | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} + 48 T_{2}^{30} + 1392 T_{2}^{28} + 26368 T_{2}^{26} + 370350 T_{2}^{24} + 3861288 T_{2}^{22} + 30996744 T_{2}^{20} + 186329952 T_{2}^{18} + 848102067 T_{2}^{16} + 2735420800 T_{2}^{14} + \cdots + 6561 \)
acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\).