Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,4,Mod(2,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.2");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 45.l (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: | \(2.65508595026\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −1.41347 | − | 5.27515i | 4.81992 | − | 1.94123i | −18.9011 | + | 10.9125i | −3.32298 | − | 10.6751i | −17.0531 | − | 22.6819i | −4.22008 | + | 1.13077i | 53.3880 | + | 53.3880i | 19.4633 | − | 18.7131i | −51.6158 | + | 32.6182i |
2.2 | −1.27001 | − | 4.73973i | −2.23746 | + | 4.68975i | −13.9239 | + | 8.03899i | 1.75298 | + | 11.0421i | 25.0698 | + | 4.64896i | −21.2589 | + | 5.69629i | 28.0284 | + | 28.0284i | −16.9875 | − | 20.9863i | 50.1101 | − | 22.3322i |
2.3 | −1.05047 | − | 3.92041i | −3.51799 | − | 3.82410i | −7.33795 | + | 4.23656i | −9.79933 | + | 5.38267i | −11.2965 | + | 17.8091i | 22.8928 | − | 6.13411i | 1.35786 | + | 1.35786i | −2.24743 | + | 26.9063i | 31.3962 | + | 32.7631i |
2.4 | −0.771317 | − | 2.87860i | 0.414910 | + | 5.17956i | −0.763180 | + | 0.440622i | 1.89424 | − | 11.0187i | 14.5898 | − | 5.18944i | 31.3140 | − | 8.39055i | −15.0012 | − | 15.0012i | −26.6557 | + | 4.29810i | −33.1795 | + | 3.04617i |
2.5 | −0.748271 | − | 2.79258i | −0.352314 | − | 5.18419i | −0.310415 | + | 0.179218i | 11.0394 | − | 1.76941i | −14.2137 | + | 4.86305i | −18.7862 | + | 5.03374i | −15.6218 | − | 15.6218i | −26.7517 | + | 3.65293i | −13.2017 | − | 29.5046i |
2.6 | −0.689083 | − | 2.57169i | 5.10395 | + | 0.974542i | 0.789441 | − | 0.455784i | 6.87382 | + | 8.81763i | −1.01082 | − | 13.7973i | 1.49332 | − | 0.400135i | −16.7770 | − | 16.7770i | 25.1005 | + | 9.94802i | 17.9396 | − | 23.7534i |
2.7 | −0.429078 | − | 1.60134i | −4.91725 | + | 1.67949i | 4.54802 | − | 2.62580i | −7.92902 | − | 7.88230i | 4.79931 | + | 7.15356i | −30.4193 | + | 8.15082i | −15.5344 | − | 15.5344i | 21.3587 | − | 16.5169i | −9.22008 | + | 16.0792i |
2.8 | −0.126293 | − | 0.471331i | 3.65204 | − | 3.69630i | 6.72200 | − | 3.88095i | −11.1362 | + | 0.993015i | −2.20341 | − | 1.25450i | 5.71856 | − | 1.53228i | −5.43846 | − | 5.43846i | −0.325220 | − | 26.9980i | 1.87445 | + | 5.12340i |
2.9 | 0.0388957 | + | 0.145161i | −5.17295 | + | 0.490448i | 6.90864 | − | 3.98871i | 9.52728 | + | 5.85072i | −0.272400 | − | 0.731834i | 17.2584 | − | 4.62436i | 1.69784 | + | 1.69784i | 26.5189 | − | 5.07413i | −0.478726 | + | 1.61056i |
2.10 | 0.335877 | + | 1.25351i | 0.636882 | + | 5.15697i | 5.46973 | − | 3.15795i | −6.60187 | + | 9.02305i | −6.25041 | + | 2.53045i | 2.99997 | − | 0.803840i | 13.1367 | + | 13.1367i | −26.1888 | + | 6.56877i | −13.5279 | − | 5.24487i |
2.11 | 0.385641 | + | 1.43923i | 4.61491 | + | 2.38802i | 5.00553 | − | 2.88994i | 4.60989 | − | 10.1857i | −1.65722 | + | 7.56285i | −24.7344 | + | 6.62758i | 14.5184 | + | 14.5184i | 15.5947 | + | 22.0410i | 16.4374 | + | 2.70668i |
2.12 | 0.576664 | + | 2.15214i | −2.09209 | − | 4.75638i | 2.62904 | − | 1.51788i | 1.05576 | − | 11.1304i | 9.02996 | − | 7.24530i | 13.8105 | − | 3.70051i | 17.3866 | + | 17.3866i | −18.2463 | + | 19.9016i | 24.5629 | − | 4.14635i |
2.13 | 0.956489 | + | 3.56967i | 3.31660 | − | 4.00002i | −4.89944 | + | 2.82870i | 6.37460 | + | 9.18501i | 17.4510 | + | 8.01320i | −4.17492 | + | 1.11867i | 6.12165 | + | 6.12165i | −5.00028 | − | 26.5329i | −26.6902 | + | 31.5406i |
2.14 | 1.11026 | + | 4.14355i | −4.87317 | − | 1.80338i | −9.00816 | + | 5.20086i | −9.88574 | + | 5.22228i | 2.06189 | − | 22.1945i | −14.5845 | + | 3.90791i | −7.28515 | − | 7.28515i | 20.4957 | + | 17.5763i | −32.6146 | − | 35.1640i |
2.15 | 1.15194 | + | 4.29910i | −2.52446 | + | 4.54171i | −10.2271 | + | 5.90461i | 10.7022 | − | 3.23468i | −22.4333 | − | 5.62113i | −7.13532 | + | 1.91190i | −11.9883 | − | 11.9883i | −14.2542 | − | 22.9307i | 26.2345 | + | 42.2836i |
2.16 | 1.30825 | + | 4.88245i | 4.86054 | + | 1.83715i | −15.1986 | + | 8.77490i | −10.1192 | − | 4.75411i | −2.61101 | + | 26.1348i | 28.4601 | − | 7.62587i | −34.1329 | − | 34.1329i | 20.2497 | + | 17.8591i | 9.97324 | − | 55.6261i |
23.1 | −1.41347 | + | 5.27515i | 4.81992 | + | 1.94123i | −18.9011 | − | 10.9125i | −3.32298 | + | 10.6751i | −17.0531 | + | 22.6819i | −4.22008 | − | 1.13077i | 53.3880 | − | 53.3880i | 19.4633 | + | 18.7131i | −51.6158 | − | 32.6182i |
23.2 | −1.27001 | + | 4.73973i | −2.23746 | − | 4.68975i | −13.9239 | − | 8.03899i | 1.75298 | − | 11.0421i | 25.0698 | − | 4.64896i | −21.2589 | − | 5.69629i | 28.0284 | − | 28.0284i | −16.9875 | + | 20.9863i | 50.1101 | + | 22.3322i |
23.3 | −1.05047 | + | 3.92041i | −3.51799 | + | 3.82410i | −7.33795 | − | 4.23656i | −9.79933 | − | 5.38267i | −11.2965 | − | 17.8091i | 22.8928 | + | 6.13411i | 1.35786 | − | 1.35786i | −2.24743 | − | 26.9063i | 31.3962 | − | 32.7631i |
23.4 | −0.771317 | + | 2.87860i | 0.414910 | − | 5.17956i | −0.763180 | − | 0.440622i | 1.89424 | + | 11.0187i | 14.5898 | + | 5.18944i | 31.3140 | + | 8.39055i | −15.0012 | + | 15.0012i | −26.6557 | − | 4.29810i | −33.1795 | − | 3.04617i |
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 45.4.l.a | ✓ | 64 |
3.b | odd | 2 | 1 | 135.4.m.a | 64 | ||
5.b | even | 2 | 1 | 225.4.p.b | 64 | ||
5.c | odd | 4 | 1 | inner | 45.4.l.a | ✓ | 64 |
5.c | odd | 4 | 1 | 225.4.p.b | 64 | ||
9.c | even | 3 | 1 | 135.4.m.a | 64 | ||
9.d | odd | 6 | 1 | inner | 45.4.l.a | ✓ | 64 |
15.e | even | 4 | 1 | 135.4.m.a | 64 | ||
45.h | odd | 6 | 1 | 225.4.p.b | 64 | ||
45.k | odd | 12 | 1 | 135.4.m.a | 64 | ||
45.l | even | 12 | 1 | inner | 45.4.l.a | ✓ | 64 |
45.l | even | 12 | 1 | 225.4.p.b | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.4.l.a | ✓ | 64 | 1.a | even | 1 | 1 | trivial |
45.4.l.a | ✓ | 64 | 5.c | odd | 4 | 1 | inner |
45.4.l.a | ✓ | 64 | 9.d | odd | 6 | 1 | inner |
45.4.l.a | ✓ | 64 | 45.l | even | 12 | 1 | inner |
135.4.m.a | 64 | 3.b | odd | 2 | 1 | ||
135.4.m.a | 64 | 9.c | even | 3 | 1 | ||
135.4.m.a | 64 | 15.e | even | 4 | 1 | ||
135.4.m.a | 64 | 45.k | odd | 12 | 1 | ||
225.4.p.b | 64 | 5.b | even | 2 | 1 | ||
225.4.p.b | 64 | 5.c | odd | 4 | 1 | ||
225.4.p.b | 64 | 45.h | odd | 6 | 1 | ||
225.4.p.b | 64 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(45, [\chi])\).