Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,3,Mod(67,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 8]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 310.o (of order \(12\), 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: | \(8.44688819517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(251\) |
| \(\chi(n)\) | \(\zeta_{12}^{3}\) | \(-1 + \zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
1.00000 | + | 1.00000i | −1.09808 | − | 4.09808i | 2.00000i | −0.598076 | + | 4.96410i | 3.00000 | − | 5.19615i | 5.46410 | − | 1.46410i | −2.00000 | + | 2.00000i | −7.79423 | + | 4.50000i | −5.56218 | + | 4.36603i | ||||||||||||||
| 87.1 | 1.00000 | + | 1.00000i | 4.09808 | + | 1.09808i | 2.00000i | 4.59808 | − | 1.96410i | 3.00000 | + | 5.19615i | −1.46410 | + | 5.46410i | −2.00000 | + | 2.00000i | 7.79423 | + | 4.50000i | 6.56218 | + | 2.63397i | |||||||||||||||
| 253.1 | 1.00000 | − | 1.00000i | 4.09808 | − | 1.09808i | − | 2.00000i | 4.59808 | + | 1.96410i | 3.00000 | − | 5.19615i | −1.46410 | − | 5.46410i | −2.00000 | − | 2.00000i | 7.79423 | − | 4.50000i | 6.56218 | − | 2.63397i | ||||||||||||||
| 273.1 | 1.00000 | − | 1.00000i | −1.09808 | + | 4.09808i | − | 2.00000i | −0.598076 | − | 4.96410i | 3.00000 | + | 5.19615i | 5.46410 | + | 1.46410i | −2.00000 | − | 2.00000i | −7.79423 | − | 4.50000i | −5.56218 | − | 4.36603i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 31.c | even | 3 | 1 | inner |
| 155.o | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 310.3.o.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | inner | 310.3.o.b | ✓ | 4 |
| 31.c | even | 3 | 1 | inner | 310.3.o.b | ✓ | 4 |
| 155.o | odd | 12 | 1 | inner | 310.3.o.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.3.o.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 310.3.o.b | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 310.3.o.b | ✓ | 4 | 31.c | even | 3 | 1 | inner |
| 310.3.o.b | ✓ | 4 | 155.o | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 6T_{3}^{3} + 18T_{3}^{2} - 108T_{3} + 324 \)
acting on \(S_{3}^{\mathrm{new}}(310, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} - 6 T^{3} + \cdots + 324 \)
|
| $5$ |
\( T^{4} - 8 T^{3} + \cdots + 625 \)
|
| $7$ |
\( T^{4} - 8 T^{3} + \cdots + 1024 \)
|
| $11$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $13$ |
\( T^{4} - 10 T^{3} + \cdots + 2500 \)
|
| $17$ |
\( T^{4} - 16 T^{3} + \cdots + 16384 \)
|
| $19$ |
\( T^{4} - 576 T^{2} + 331776 \)
|
| $23$ |
\( (T^{2} - 30 T + 450)^{2} \)
|
| $29$ |
\( (T^{2} + 529)^{2} \)
|
| $31$ |
\( (T^{2} - 31 T + 961)^{2} \)
|
| $37$ |
\( T^{4} - 96 T^{3} + \cdots + 21233664 \)
|
| $41$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $43$ |
\( T^{4} + 120 T^{3} + \cdots + 51840000 \)
|
| $47$ |
\( (T^{2} + 16 T + 128)^{2} \)
|
| $53$ |
\( T^{4} + 66 T^{3} + \cdots + 4743684 \)
|
| $59$ |
\( T^{4} - 7921 T^{2} + 62742241 \)
|
| $61$ |
\( (T + 65)^{4} \)
|
| $67$ |
\( T^{4} - 54 T^{3} + \cdots + 2125764 \)
|
| $71$ |
\( (T^{2} + 7 T + 49)^{2} \)
|
| $73$ |
\( T^{4} + 48 T^{3} + \cdots + 1327104 \)
|
| $79$ |
\( T^{4} - 289 T^{2} + 83521 \)
|
| $83$ |
\( T^{4} - 16 T^{3} + \cdots + 16384 \)
|
| $89$ |
\( (T^{2} + 10609)^{2} \)
|
| $97$ |
\( (T^{2} + 24 T + 288)^{2} \)
|
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