Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3192,2,Mod(265,3192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3192.265");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3192.o (of order \(2\), degree \(1\), 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: | \(25.4882483252\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
265.1 | 0 | 1.00000 | 0 | − | 4.42440i | 0 | −2.30674 | − | 1.29575i | 0 | 1.00000 | 0 | |||||||||||||||
265.2 | 0 | 1.00000 | 0 | − | 3.81802i | 0 | −0.437937 | − | 2.60925i | 0 | 1.00000 | 0 | |||||||||||||||
265.3 | 0 | 1.00000 | 0 | − | 3.49542i | 0 | −2.01443 | + | 1.71525i | 0 | 1.00000 | 0 | |||||||||||||||
265.4 | 0 | 1.00000 | 0 | − | 3.24542i | 0 | 2.23477 | + | 1.41627i | 0 | 1.00000 | 0 | |||||||||||||||
265.5 | 0 | 1.00000 | 0 | − | 3.23567i | 0 | 2.43709 | − | 1.02986i | 0 | 1.00000 | 0 | |||||||||||||||
265.6 | 0 | 1.00000 | 0 | − | 3.12999i | 0 | 1.04022 | + | 2.43268i | 0 | 1.00000 | 0 | |||||||||||||||
265.7 | 0 | 1.00000 | 0 | − | 3.06066i | 0 | 2.33587 | − | 1.24247i | 0 | 1.00000 | 0 | |||||||||||||||
265.8 | 0 | 1.00000 | 0 | − | 2.93144i | 0 | −2.43014 | − | 1.04614i | 0 | 1.00000 | 0 | |||||||||||||||
265.9 | 0 | 1.00000 | 0 | − | 2.78430i | 0 | 0.562419 | + | 2.58528i | 0 | 1.00000 | 0 | |||||||||||||||
265.10 | 0 | 1.00000 | 0 | − | 2.21808i | 0 | 0.528301 | − | 2.59247i | 0 | 1.00000 | 0 | |||||||||||||||
265.11 | 0 | 1.00000 | 0 | − | 1.97676i | 0 | −2.49077 | + | 0.892215i | 0 | 1.00000 | 0 | |||||||||||||||
265.12 | 0 | 1.00000 | 0 | − | 1.91009i | 0 | −1.26524 | + | 2.32361i | 0 | 1.00000 | 0 | |||||||||||||||
265.13 | 0 | 1.00000 | 0 | − | 1.43577i | 0 | −2.08442 | − | 1.62948i | 0 | 1.00000 | 0 | |||||||||||||||
265.14 | 0 | 1.00000 | 0 | − | 1.33474i | 0 | −0.149381 | − | 2.64153i | 0 | 1.00000 | 0 | |||||||||||||||
265.15 | 0 | 1.00000 | 0 | − | 1.08803i | 0 | 2.24871 | + | 1.39402i | 0 | 1.00000 | 0 | |||||||||||||||
265.16 | 0 | 1.00000 | 0 | − | 1.01661i | 0 | 2.64290 | + | 0.122800i | 0 | 1.00000 | 0 | |||||||||||||||
265.17 | 0 | 1.00000 | 0 | − | 0.535557i | 0 | 0.357985 | − | 2.62142i | 0 | 1.00000 | 0 | |||||||||||||||
265.18 | 0 | 1.00000 | 0 | − | 0.442452i | 0 | −2.62394 | − | 0.339044i | 0 | 1.00000 | 0 | |||||||||||||||
265.19 | 0 | 1.00000 | 0 | − | 0.302259i | 0 | −1.47245 | + | 2.19816i | 0 | 1.00000 | 0 | |||||||||||||||
265.20 | 0 | 1.00000 | 0 | − | 0.0830565i | 0 | 1.88719 | + | 1.85432i | 0 | 1.00000 | 0 | |||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
133.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3192.2.o.b | yes | 40 |
7.b | odd | 2 | 1 | 3192.2.o.a | ✓ | 40 | |
19.b | odd | 2 | 1 | 3192.2.o.a | ✓ | 40 | |
133.c | even | 2 | 1 | inner | 3192.2.o.b | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3192.2.o.a | ✓ | 40 | 7.b | odd | 2 | 1 | |
3192.2.o.a | ✓ | 40 | 19.b | odd | 2 | 1 | |
3192.2.o.b | yes | 40 | 1.a | even | 1 | 1 | trivial |
3192.2.o.b | yes | 40 | 133.c | even | 2 | 1 | inner |