Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(125,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.125");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 462.s (of order \(10\), 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: | \(3.68908857338\) |
Analytic rank: | \(0\) |
Dimension: | \(128\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
125.1 | −0.587785 | + | 0.809017i | −1.72594 | + | 0.145412i | −0.309017 | − | 0.951057i | 0.826181 | − | 0.600255i | 0.896839 | − | 1.48178i | −1.44871 | + | 2.21388i | 0.951057 | + | 0.309017i | 2.95771 | − | 0.501942i | 1.02122i | ||
125.2 | −0.587785 | + | 0.809017i | −1.70765 | + | 0.289739i | −0.309017 | − | 0.951057i | 0.820027 | − | 0.595785i | 0.769325 | − | 1.55182i | −1.08488 | − | 2.41309i | 0.951057 | + | 0.309017i | 2.83210 | − | 0.989542i | 1.01361i | ||
125.3 | −0.587785 | + | 0.809017i | −1.63638 | − | 0.567690i | −0.309017 | − | 0.951057i | −3.26960 | + | 2.37551i | 1.42111 | − | 0.990177i | 1.78115 | + | 1.95640i | 0.951057 | + | 0.309017i | 2.35546 | + | 1.85791i | − | 4.04145i | |
125.4 | −0.587785 | + | 0.809017i | −1.24874 | − | 1.20027i | −0.309017 | − | 0.951057i | 2.87147 | − | 2.08625i | 1.70503 | − | 0.304749i | 1.88621 | − | 1.85532i | 0.951057 | + | 0.309017i | 0.118698 | + | 2.99765i | 3.54933i | ||
125.5 | −0.587785 | + | 0.809017i | −1.03869 | + | 1.38604i | −0.309017 | − | 0.951057i | −1.46380 | + | 1.06351i | −0.510803 | − | 1.65502i | 1.28572 | − | 2.31234i | 0.951057 | + | 0.309017i | −0.842230 | − | 2.87935i | − | 1.80935i | |
125.6 | −0.587785 | + | 0.809017i | −0.582733 | − | 1.63108i | −0.309017 | − | 0.951057i | 0.793236 | − | 0.576320i | 1.66209 | + | 0.487284i | 1.17577 | + | 2.37014i | 0.951057 | + | 0.309017i | −2.32084 | + | 1.90097i | 0.980494i | ||
125.7 | −0.587785 | + | 0.809017i | −0.389332 | − | 1.68773i | −0.309017 | − | 0.951057i | 0.0969774 | − | 0.0704582i | 1.59424 | + | 0.677045i | −2.45348 | + | 0.990165i | 0.951057 | + | 0.309017i | −2.69684 | + | 1.31417i | 0.119871i | ||
125.8 | −0.587785 | + | 0.809017i | −0.0377610 | + | 1.73164i | −0.309017 | − | 0.951057i | 2.99982 | − | 2.17950i | −1.37873 | − | 1.04838i | −2.21103 | − | 1.45305i | 0.951057 | + | 0.309017i | −2.99715 | − | 0.130777i | 3.70798i | ||
125.9 | −0.587785 | + | 0.809017i | 0.0377610 | − | 1.73164i | −0.309017 | − | 0.951057i | −2.99982 | + | 2.17950i | 1.37873 | + | 1.04838i | 0.934677 | − | 2.47515i | 0.951057 | + | 0.309017i | −2.99715 | − | 0.130777i | − | 3.70798i | |
125.10 | −0.587785 | + | 0.809017i | 0.389332 | + | 1.68773i | −0.309017 | − | 0.951057i | −0.0969774 | + | 0.0704582i | −1.59424 | − | 0.677045i | 2.56691 | − | 0.641060i | 0.951057 | + | 0.309017i | −2.69684 | + | 1.31417i | − | 0.119871i | |
125.11 | −0.587785 | + | 0.809017i | 0.582733 | + | 1.63108i | −0.309017 | − | 0.951057i | −0.793236 | + | 0.576320i | −1.66209 | − | 0.487284i | 0.441911 | + | 2.60858i | 0.951057 | + | 0.309017i | −2.32084 | + | 1.90097i | − | 0.980494i | |
125.12 | −0.587785 | + | 0.809017i | 1.03869 | − | 1.38604i | −0.309017 | − | 0.951057i | 1.46380 | − | 1.06351i | 0.510803 | + | 1.65502i | −2.39933 | − | 1.11499i | 0.951057 | + | 0.309017i | −0.842230 | − | 2.87935i | 1.80935i | ||
125.13 | −0.587785 | + | 0.809017i | 1.24874 | + | 1.20027i | −0.309017 | − | 0.951057i | −2.87147 | + | 2.08625i | −1.70503 | + | 0.304749i | −2.61650 | − | 0.392306i | 0.951057 | + | 0.309017i | 0.118698 | + | 2.99765i | − | 3.54933i | |
125.14 | −0.587785 | + | 0.809017i | 1.63638 | + | 0.567690i | −0.309017 | − | 0.951057i | 3.26960 | − | 2.37551i | −1.42111 | + | 0.990177i | −0.291040 | + | 2.62970i | 0.951057 | + | 0.309017i | 2.35546 | + | 1.85791i | 4.04145i | ||
125.15 | −0.587785 | + | 0.809017i | 1.70765 | − | 0.289739i | −0.309017 | − | 0.951057i | −0.820027 | + | 0.595785i | −0.769325 | + | 1.55182i | −0.540692 | − | 2.58991i | 0.951057 | + | 0.309017i | 2.83210 | − | 0.989542i | − | 1.01361i | |
125.16 | −0.587785 | + | 0.809017i | 1.72594 | − | 0.145412i | −0.309017 | − | 0.951057i | −0.826181 | + | 0.600255i | −0.896839 | + | 1.48178i | 2.47331 | + | 0.939534i | 0.951057 | + | 0.309017i | 2.95771 | − | 0.501942i | − | 1.02122i | |
125.17 | 0.587785 | − | 0.809017i | −1.71575 | + | 0.237049i | −0.309017 | − | 0.951057i | −2.87147 | + | 2.08625i | −0.816717 | + | 1.52741i | 1.88621 | − | 1.85532i | −0.951057 | − | 0.309017i | 2.88762 | − | 0.813436i | 3.54933i | ||
125.18 | 0.587785 | − | 0.809017i | −1.65754 | − | 0.502567i | −0.309017 | − | 0.951057i | 3.26960 | − | 2.37551i | −1.38086 | + | 1.04557i | 1.78115 | + | 1.95640i | −0.951057 | − | 0.309017i | 2.49485 | + | 1.66605i | − | 4.04145i | |
125.19 | 0.587785 | − | 0.809017i | −1.43017 | + | 0.977050i | −0.309017 | − | 0.951057i | −0.793236 | + | 0.576320i | −0.0501805 | + | 1.73132i | 1.17577 | + | 2.37014i | −0.951057 | − | 0.309017i | 1.09075 | − | 2.79469i | 0.980494i | ||
125.20 | 0.587785 | − | 0.809017i | −1.31084 | − | 1.13212i | −0.309017 | − | 0.951057i | −0.826181 | + | 0.600255i | −1.68640 | + | 0.395049i | −1.44871 | + | 2.21388i | −0.951057 | − | 0.309017i | 0.436607 | + | 2.96806i | 1.02122i | ||
See next 80 embeddings (of 128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
11.c | even | 5 | 1 | inner |
21.c | even | 2 | 1 | inner |
33.h | odd | 10 | 1 | inner |
77.j | odd | 10 | 1 | inner |
231.u | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 462.2.s.a | ✓ | 128 |
3.b | odd | 2 | 1 | inner | 462.2.s.a | ✓ | 128 |
7.b | odd | 2 | 1 | inner | 462.2.s.a | ✓ | 128 |
11.c | even | 5 | 1 | inner | 462.2.s.a | ✓ | 128 |
21.c | even | 2 | 1 | inner | 462.2.s.a | ✓ | 128 |
33.h | odd | 10 | 1 | inner | 462.2.s.a | ✓ | 128 |
77.j | odd | 10 | 1 | inner | 462.2.s.a | ✓ | 128 |
231.u | even | 10 | 1 | inner | 462.2.s.a | ✓ | 128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
462.2.s.a | ✓ | 128 | 1.a | even | 1 | 1 | trivial |
462.2.s.a | ✓ | 128 | 3.b | odd | 2 | 1 | inner |
462.2.s.a | ✓ | 128 | 7.b | odd | 2 | 1 | inner |
462.2.s.a | ✓ | 128 | 11.c | even | 5 | 1 | inner |
462.2.s.a | ✓ | 128 | 21.c | even | 2 | 1 | inner |
462.2.s.a | ✓ | 128 | 33.h | odd | 10 | 1 | inner |
462.2.s.a | ✓ | 128 | 77.j | odd | 10 | 1 | inner |
462.2.s.a | ✓ | 128 | 231.u | even | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(462, [\chi])\).