Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,3,Mod(343,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.343");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 684.g (of order \(2\), degree \(1\), 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: | \(18.6376500822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 76) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{3} - 2\beta _1 + 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(343\) | \(533\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 343.1 |
|
1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −6.54983 | 0 | 1.31342i | −8.00000 | 0 | −6.54983 | + | 11.3446i | ||||||||||||||||||||||||
| 343.2 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | 8.54983 | 0 | − | 3.04547i | −8.00000 | 0 | 8.54983 | − | 14.8087i | ||||||||||||||||||||||||
| 343.3 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | −6.54983 | 0 | − | 1.31342i | −8.00000 | 0 | −6.54983 | − | 11.3446i | ||||||||||||||||||||||||
| 343.4 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | 8.54983 | 0 | 3.04547i | −8.00000 | 0 | 8.54983 | + | 14.8087i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 684.3.g.a | 4 | |
| 3.b | odd | 2 | 1 | 76.3.b.a | ✓ | 4 | |
| 4.b | odd | 2 | 1 | inner | 684.3.g.a | 4 | |
| 12.b | even | 2 | 1 | 76.3.b.a | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1216.3.d.a | 4 | ||
| 24.h | odd | 2 | 1 | 1216.3.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.3.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 76.3.b.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 684.3.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 684.3.g.a | 4 | 4.b | odd | 2 | 1 | inner | |
| 1216.3.d.a | 4 | 24.f | even | 2 | 1 | ||
| 1216.3.d.a | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} - 56 \)
acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 2 T - 56)^{2} \)
|
| $7$ |
\( T^{4} + 11T^{2} + 16 \)
|
| $11$ |
\( T^{4} + 188T^{2} + 3136 \)
|
| $13$ |
\( (T^{2} + 21 T - 18)^{2} \)
|
| $17$ |
\( (T^{2} - 7 T - 2)^{2} \)
|
| $19$ |
\( (T^{2} + 19)^{2} \)
|
| $23$ |
\( T^{4} + 2139 T^{2} + 1140624 \)
|
| $29$ |
\( (T^{2} - 29 T - 146)^{2} \)
|
| $31$ |
\( T^{4} + 2816 T^{2} + 1048576 \)
|
| $37$ |
\( (T^{2} + 12 T - 876)^{2} \)
|
| $41$ |
\( (T^{2} - 62 T - 464)^{2} \)
|
| $43$ |
\( T^{4} + 3296 T^{2} + 614656 \)
|
| $47$ |
\( T^{4} + 4064 T^{2} + 2027776 \)
|
| $53$ |
\( (T^{2} - T - 698)^{2} \)
|
| $59$ |
\( T^{4} + 8027 T^{2} + 12588304 \)
|
| $61$ |
\( (T^{2} + 64 T + 796)^{2} \)
|
| $67$ |
\( T^{4} + 13659 T^{2} + 12362256 \)
|
| $71$ |
\( T^{4} + 5612 T^{2} + 3211264 \)
|
| $73$ |
\( (T^{2} + 169 T + 6442)^{2} \)
|
| $79$ |
\( T^{4} + 23852 T^{2} + 141324544 \)
|
| $83$ |
\( T^{4} + 22076 T^{2} + 3136 \)
|
| $89$ |
\( (T^{2} + 10 T - 2768)^{2} \)
|
| $97$ |
\( (T^{2} + 24 T - 11028)^{2} \)
|
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