Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,3,Mod(2,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.2");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.g (of order \(8\), 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: | \(1.38964934824\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
1.70711 | − | 1.70711i | 2.12132 | − | 2.12132i | − | 1.82843i | −4.82843 | + | 2.00000i | − | 7.24264i | −3.82843 | + | 9.24264i | 3.70711 | + | 3.70711i | − | 9.00000i | −4.82843 | + | 11.6569i | |||||||||||||||
| 8.1 | 0.292893 | + | 0.292893i | −2.12132 | − | 2.12132i | − | 3.82843i | 0.828427 | − | 2.00000i | − | 1.24264i | 1.82843 | − | 0.757359i | 2.29289 | − | 2.29289i | 9.00000i | 0.828427 | − | 0.343146i | |||||||||||||||||
| 26.1 | 1.70711 | + | 1.70711i | 2.12132 | + | 2.12132i | 1.82843i | −4.82843 | − | 2.00000i | 7.24264i | −3.82843 | − | 9.24264i | 3.70711 | − | 3.70711i | 9.00000i | −4.82843 | − | 11.6569i | |||||||||||||||||||
| 32.1 | 0.292893 | − | 0.292893i | −2.12132 | + | 2.12132i | 3.82843i | 0.828427 | + | 2.00000i | 1.24264i | 1.82843 | + | 0.757359i | 2.29289 | + | 2.29289i | − | 9.00000i | 0.828427 | + | 0.343146i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 51.g | odd | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 51.3.g.b | yes | 4 |
| 3.b | odd | 2 | 1 | 51.3.g.a | ✓ | 4 | |
| 17.d | even | 8 | 1 | 51.3.g.a | ✓ | 4 | |
| 51.g | odd | 8 | 1 | inner | 51.3.g.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.3.g.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 51.3.g.a | ✓ | 4 | 17.d | even | 8 | 1 | |
| 51.3.g.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 51.3.g.b | yes | 4 | 51.g | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(51, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 81 \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 128 \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots + 392 \)
|
| $11$ |
\( T^{4} - 16 T^{3} + \cdots + 1922 \)
|
| $13$ |
\( T^{4} + 304 T^{2} + 18496 \)
|
| $17$ |
\( T^{4} - 24 T^{3} + \cdots + 83521 \)
|
| $19$ |
\( T^{4} + 40 T^{3} + \cdots + 33856 \)
|
| $23$ |
\( T^{4} - 28 T^{3} + \cdots + 102152 \)
|
| $29$ |
\( T^{4} + 1352 T^{2} + \cdots + 913952 \)
|
| $31$ |
\( T^{4} - 12 T^{3} + \cdots + 187272 \)
|
| $37$ |
\( T^{4} - 24 T^{3} + \cdots + 6272 \)
|
| $41$ |
\( T^{4} + 16 T^{3} + \cdots + 15842 \)
|
| $43$ |
\( T^{4} - 28 T^{3} + \cdots + 148996 \)
|
| $47$ |
\( (T^{2} - 28 T - 5212)^{2} \)
|
| $53$ |
\( T^{4} - 72 T^{3} + \cdots + 3429904 \)
|
| $59$ |
\( T^{4} + 300 T^{3} + \cdots + 125753796 \)
|
| $61$ |
\( T^{4} + 216 T^{3} + \cdots + 63528992 \)
|
| $67$ |
\( (T^{2} - 116 T + 3266)^{2} \)
|
| $71$ |
\( T^{4} - 132 T^{3} + \cdots + 14818568 \)
|
| $73$ |
\( T^{4} + 88 T^{3} + \cdots + 321602 \)
|
| $79$ |
\( T^{4} - 44 T^{3} + \cdots + 25948808 \)
|
| $83$ |
\( T^{4} - 36 T^{3} + \cdots + 142515844 \)
|
| $89$ |
\( (T^{2} - 88 T + 14)^{2} \)
|
| $97$ |
\( T^{4} - 204 T^{3} + \cdots + 49980002 \)
|
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