Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [450,6,Mod(107,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("450.107");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 450.f (of order \(4\), degree \(2\), 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: | \(72.1727189158\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| 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: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 90) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 107.1 |
|
−2.82843 | + | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 52.0000 | + | 52.0000i | 45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||
| 107.2 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | 0 | 0 | 52.0000 | + | 52.0000i | −45.2548 | − | 45.2548i | 0 | 0 | ||||||||||||||||||||||||
| 143.1 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 0 | 0 | 52.0000 | − | 52.0000i | 45.2548 | − | 45.2548i | 0 | 0 | |||||||||||||||||||||||||
| 143.2 | 2.82843 | + | 2.82843i | 0 | 16.0000i | 0 | 0 | 52.0000 | − | 52.0000i | −45.2548 | + | 45.2548i | 0 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 450.6.f.d | 4 | |
| 3.b | odd | 2 | 1 | inner | 450.6.f.d | 4 | |
| 5.b | even | 2 | 1 | 90.6.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 90.6.f.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 450.6.f.d | 4 | |
| 15.d | odd | 2 | 1 | 90.6.f.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 90.6.f.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | inner | 450.6.f.d | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.6.f.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 90.6.f.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 90.6.f.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 90.6.f.a | ✓ | 4 | 15.e | even | 4 | 1 | |
| 450.6.f.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 450.6.f.d | 4 | 3.b | odd | 2 | 1 | inner | |
| 450.6.f.d | 4 | 5.c | odd | 4 | 1 | inner | |
| 450.6.f.d | 4 | 15.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 104T_{7} + 5408 \)
acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 256 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 104 T + 5408)^{2} \)
|
| $11$ |
\( (T^{2} + 15488)^{2} \)
|
| $13$ |
\( (T^{2} + 366 T + 66978)^{2} \)
|
| $17$ |
\( T^{4} + 60523872256 \)
|
| $19$ |
\( (T^{2} + 4443664)^{2} \)
|
| $23$ |
\( T^{4} + 43218738217216 \)
|
| $29$ |
\( (T^{2} - 48985202)^{2} \)
|
| $31$ |
\( (T - 6784)^{4} \)
|
| $37$ |
\( (T^{2} + 7446 T + 27721458)^{2} \)
|
| $41$ |
\( (T^{2} + 149057378)^{2} \)
|
| $43$ |
\( (T^{2} + 6360 T + 20224800)^{2} \)
|
| $47$ |
\( T^{4} + 14\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + 10\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} - 1285042208)^{2} \)
|
| $61$ |
\( (T - 13740)^{4} \)
|
| $67$ |
\( (T^{2} + 68128 T + 2320712192)^{2} \)
|
| $71$ |
\( (T^{2} + 2307106592)^{2} \)
|
| $73$ |
\( (T^{2} - 59782 T + 1786943762)^{2} \)
|
| $79$ |
\( (T^{2} + 3519811584)^{2} \)
|
| $83$ |
\( T^{4} + 84\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} - 290549618)^{2} \)
|
| $97$ |
\( (T^{2} + 192186 T + 18467729298)^{2} \)
|
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