Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,2,Mod(109,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.109");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 432.k (of order \(4\), degree \(2\), 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: | \(3.44953736732\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 | −1.40946 | + | 0.115854i | 0 | 1.97316 | − | 0.326583i | −1.78448 | + | 1.78448i | 0 | 4.77575i | −2.74325 | + | 0.688904i | 0 | 2.30841 | − | 2.72189i | ||||||||
109.2 | −1.37261 | − | 0.340511i | 0 | 1.76810 | + | 0.934777i | 2.75203 | − | 2.75203i | 0 | 0.113492i | −2.10861 | − | 1.88514i | 0 | −4.71455 | + | 2.84036i | ||||||||
109.3 | −1.20402 | + | 0.741841i | 0 | 0.899344 | − | 1.78639i | −0.431497 | + | 0.431497i | 0 | 0.145064i | 0.242383 | + | 2.81802i | 0 | 0.199430 | − | 0.839634i | ||||||||
109.4 | −1.15725 | − | 0.812885i | 0 | 0.678437 | + | 1.88142i | −2.39401 | + | 2.39401i | 0 | − | 2.42931i | 0.744255 | − | 2.72875i | 0 | 4.71651 | − | 0.824404i | |||||||
109.5 | −0.936391 | − | 1.05980i | 0 | −0.246343 | + | 1.98477i | 1.12951 | − | 1.12951i | 0 | 1.33035i | 2.33413 | − | 1.59745i | 0 | −2.25471 | − | 0.139389i | ||||||||
109.6 | −0.680919 | + | 1.23950i | 0 | −1.07270 | − | 1.68799i | −2.24693 | + | 2.24693i | 0 | − | 3.93534i | 2.82268 | − | 0.180219i | 0 | −1.25508 | − | 4.31505i | |||||||
109.7 | 0.680919 | − | 1.23950i | 0 | −1.07270 | − | 1.68799i | 2.24693 | − | 2.24693i | 0 | − | 3.93534i | −2.82268 | + | 0.180219i | 0 | −1.25508 | − | 4.31505i | |||||||
109.8 | 0.936391 | + | 1.05980i | 0 | −0.246343 | + | 1.98477i | −1.12951 | + | 1.12951i | 0 | 1.33035i | −2.33413 | + | 1.59745i | 0 | −2.25471 | − | 0.139389i | ||||||||
109.9 | 1.15725 | + | 0.812885i | 0 | 0.678437 | + | 1.88142i | 2.39401 | − | 2.39401i | 0 | − | 2.42931i | −0.744255 | + | 2.72875i | 0 | 4.71651 | − | 0.824404i | |||||||
109.10 | 1.20402 | − | 0.741841i | 0 | 0.899344 | − | 1.78639i | 0.431497 | − | 0.431497i | 0 | 0.145064i | −0.242383 | − | 2.81802i | 0 | 0.199430 | − | 0.839634i | ||||||||
109.11 | 1.37261 | + | 0.340511i | 0 | 1.76810 | + | 0.934777i | −2.75203 | + | 2.75203i | 0 | 0.113492i | 2.10861 | + | 1.88514i | 0 | −4.71455 | + | 2.84036i | ||||||||
109.12 | 1.40946 | − | 0.115854i | 0 | 1.97316 | − | 0.326583i | 1.78448 | − | 1.78448i | 0 | 4.77575i | 2.74325 | − | 0.688904i | 0 | 2.30841 | − | 2.72189i | ||||||||
325.1 | −1.40946 | − | 0.115854i | 0 | 1.97316 | + | 0.326583i | −1.78448 | − | 1.78448i | 0 | − | 4.77575i | −2.74325 | − | 0.688904i | 0 | 2.30841 | + | 2.72189i | |||||||
325.2 | −1.37261 | + | 0.340511i | 0 | 1.76810 | − | 0.934777i | 2.75203 | + | 2.75203i | 0 | − | 0.113492i | −2.10861 | + | 1.88514i | 0 | −4.71455 | − | 2.84036i | |||||||
325.3 | −1.20402 | − | 0.741841i | 0 | 0.899344 | + | 1.78639i | −0.431497 | − | 0.431497i | 0 | − | 0.145064i | 0.242383 | − | 2.81802i | 0 | 0.199430 | + | 0.839634i | |||||||
325.4 | −1.15725 | + | 0.812885i | 0 | 0.678437 | − | 1.88142i | −2.39401 | − | 2.39401i | 0 | 2.42931i | 0.744255 | + | 2.72875i | 0 | 4.71651 | + | 0.824404i | ||||||||
325.5 | −0.936391 | + | 1.05980i | 0 | −0.246343 | − | 1.98477i | 1.12951 | + | 1.12951i | 0 | − | 1.33035i | 2.33413 | + | 1.59745i | 0 | −2.25471 | + | 0.139389i | |||||||
325.6 | −0.680919 | − | 1.23950i | 0 | −1.07270 | + | 1.68799i | −2.24693 | − | 2.24693i | 0 | 3.93534i | 2.82268 | + | 0.180219i | 0 | −1.25508 | + | 4.31505i | ||||||||
325.7 | 0.680919 | + | 1.23950i | 0 | −1.07270 | + | 1.68799i | 2.24693 | + | 2.24693i | 0 | 3.93534i | −2.82268 | − | 0.180219i | 0 | −1.25508 | + | 4.31505i | ||||||||
325.8 | 0.936391 | − | 1.05980i | 0 | −0.246343 | − | 1.98477i | −1.12951 | − | 1.12951i | 0 | − | 1.33035i | −2.33413 | − | 1.59745i | 0 | −2.25471 | + | 0.139389i | |||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 432.2.k.c | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 432.2.k.c | ✓ | 24 |
4.b | odd | 2 | 1 | 1728.2.k.c | 24 | ||
12.b | even | 2 | 1 | 1728.2.k.c | 24 | ||
16.e | even | 4 | 1 | inner | 432.2.k.c | ✓ | 24 |
16.f | odd | 4 | 1 | 1728.2.k.c | 24 | ||
48.i | odd | 4 | 1 | inner | 432.2.k.c | ✓ | 24 |
48.k | even | 4 | 1 | 1728.2.k.c | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
432.2.k.c | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
432.2.k.c | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
432.2.k.c | ✓ | 24 | 16.e | even | 4 | 1 | inner |
432.2.k.c | ✓ | 24 | 48.i | odd | 4 | 1 | inner |
1728.2.k.c | 24 | 4.b | odd | 2 | 1 | ||
1728.2.k.c | 24 | 12.b | even | 2 | 1 | ||
1728.2.k.c | 24 | 16.f | odd | 4 | 1 | ||
1728.2.k.c | 24 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(432, [\chi])\):
\( T_{5}^{24} + 510T_{5}^{20} + 89055T_{5}^{16} + 6358980T_{5}^{12} + 163237055T_{5}^{8} + 834181822T_{5}^{4} + 112550881 \) |
\( T_{11}^{24} + 1494 T_{11}^{20} + 659871 T_{11}^{16} + 78092340 T_{11}^{12} + 1777212719 T_{11}^{8} + \cdots + 244140625 \) |