Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [510,2,Mod(53,510)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(510, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([4, 6, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("510.53");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 510.z (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: | \(4.07237050309\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(307\) | \(341\) |
| \(\chi(n)\) | \(\zeta_{8}\) | \(\zeta_{8}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
1.00000i | 1.00000 | + | 1.41421i | −1.00000 | 2.12132 | − | 0.707107i | −1.41421 | + | 1.00000i | 1.70711 | + | 4.12132i | − | 1.00000i | −1.00000 | + | 2.82843i | 0.707107 | + | 2.12132i | |||||||||||||||||
| 77.1 | − | 1.00000i | 1.00000 | − | 1.41421i | −1.00000 | 2.12132 | + | 0.707107i | −1.41421 | − | 1.00000i | 1.70711 | − | 4.12132i | 1.00000i | −1.00000 | − | 2.82843i | 0.707107 | − | 2.12132i | ||||||||||||||||||
| 83.1 | 1.00000i | 1.00000 | − | 1.41421i | −1.00000 | −2.12132 | + | 0.707107i | 1.41421 | + | 1.00000i | 0.292893 | − | 0.121320i | − | 1.00000i | −1.00000 | − | 2.82843i | −0.707107 | − | 2.12132i | ||||||||||||||||||
| 467.1 | − | 1.00000i | 1.00000 | + | 1.41421i | −1.00000 | −2.12132 | − | 0.707107i | 1.41421 | − | 1.00000i | 0.292893 | + | 0.121320i | 1.00000i | −1.00000 | + | 2.82843i | −0.707107 | + | 2.12132i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 255.v | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 510.2.z.b | yes | 4 |
| 3.b | odd | 2 | 1 | 510.2.z.a | yes | 4 | |
| 5.c | odd | 4 | 1 | 510.2.w.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 510.2.w.b | yes | 4 | |
| 17.d | even | 8 | 1 | 510.2.w.b | yes | 4 | |
| 51.g | odd | 8 | 1 | 510.2.w.a | ✓ | 4 | |
| 85.n | odd | 8 | 1 | 510.2.z.a | yes | 4 | |
| 255.v | even | 8 | 1 | inner | 510.2.z.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.w.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 510.2.w.a | ✓ | 4 | 51.g | odd | 8 | 1 | |
| 510.2.w.b | yes | 4 | 15.e | even | 4 | 1 | |
| 510.2.w.b | yes | 4 | 17.d | even | 8 | 1 | |
| 510.2.z.a | yes | 4 | 3.b | odd | 2 | 1 | |
| 510.2.z.a | yes | 4 | 85.n | odd | 8 | 1 | |
| 510.2.z.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 510.2.z.b | yes | 4 | 255.v | even | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\):
|
\( T_{7}^{4} - 4T_{7}^{3} + 22T_{7}^{2} - 12T_{7} + 2 \)
|
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\( T_{11}^{4} - 12T_{11}^{3} + 54T_{11}^{2} - 108T_{11} + 162 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( (T^{2} - 2 T + 3)^{2} \)
|
| $5$ |
\( T^{4} - 8T^{2} + 25 \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
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| $11$ |
\( T^{4} - 12 T^{3} + \cdots + 162 \)
|
| $13$ |
\( T^{4} - 12 T^{3} + \cdots + 196 \)
|
| $17$ |
\( T^{4} + 2T^{2} + 289 \)
|
| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 1156 \)
|
| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 2 \)
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| $31$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $41$ |
\( T^{4} - 4 T^{3} + \cdots + 98 \)
|
| $43$ |
\( (T^{2} + 12 T + 4)^{2} \)
|
| $47$ |
\( T^{4} - 12 T^{3} + \cdots + 4 \)
|
| $53$ |
\( (T^{2} + 8 T + 8)^{2} \)
|
| $59$ |
\( T^{4} + 20 T^{3} + \cdots + 2116 \)
|
| $61$ |
\( T^{4} - 24 T^{3} + \cdots + 1058 \)
|
| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 196 \)
|
| $71$ |
\( T^{4} + 16 T^{3} + \cdots + 4802 \)
|
| $73$ |
\( T^{4} - 24 T^{3} + \cdots + 7938 \)
|
| $79$ |
\( T^{4} - 32 T^{3} + \cdots + 28322 \)
|
| $83$ |
\( (T^{2} - 12 T + 4)^{2} \)
|
| $89$ |
\( (T^{2} + 200)^{2} \)
|
| $97$ |
\( T^{4} + 50 T^{2} + \cdots + 1250 \)
|
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