Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,3,Mod(721,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.721");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.o (of order \(2\), degree \(1\), not 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: | \(24.8502001097\) |
| 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: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 57) |
| 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 \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 721.1 |
|
0 | − | 1.73205i | 0 | −6.27492 | 0 | −12.2749 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 721.2 | 0 | − | 1.73205i | 0 | 1.27492 | 0 | −4.72508 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 721.3 | 0 | 1.73205i | 0 | −6.27492 | 0 | −12.2749 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 721.4 | 0 | 1.73205i | 0 | 1.27492 | 0 | −4.72508 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.3.o.b | 4 | |
| 3.b | odd | 2 | 1 | 2736.3.o.l | 4 | ||
| 4.b | odd | 2 | 1 | 57.3.c.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 171.3.c.f | 4 | ||
| 19.b | odd | 2 | 1 | inner | 912.3.o.b | 4 | |
| 57.d | even | 2 | 1 | 2736.3.o.l | 4 | ||
| 76.d | even | 2 | 1 | 57.3.c.b | ✓ | 4 | |
| 228.b | odd | 2 | 1 | 171.3.c.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.3.c.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 57.3.c.b | ✓ | 4 | 76.d | even | 2 | 1 | |
| 171.3.c.f | 4 | 12.b | even | 2 | 1 | ||
| 171.3.c.f | 4 | 228.b | odd | 2 | 1 | ||
| 912.3.o.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 912.3.o.b | 4 | 19.b | odd | 2 | 1 | inner | |
| 2736.3.o.l | 4 | 3.b | odd | 2 | 1 | ||
| 2736.3.o.l | 4 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 5T_{5} - 8 \)
acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( (T^{2} + 5 T - 8)^{2} \)
|
| $7$ |
\( (T^{2} + 17 T + 58)^{2} \)
|
| $11$ |
\( (T^{2} - 7 T - 2)^{2} \)
|
| $13$ |
\( T^{4} + 188T^{2} + 3136 \)
|
| $17$ |
\( (T^{2} + 3 T - 354)^{2} \)
|
| $19$ |
\( T^{4} + 494 T^{2} + 130321 \)
|
| $23$ |
\( (T^{2} + 26 T + 112)^{2} \)
|
| $29$ |
\( T^{4} + 752 T^{2} + 50176 \)
|
| $31$ |
\( T^{4} + 956 T^{2} + 222784 \)
|
| $37$ |
\( T^{4} + 6108 T^{2} + 7354944 \)
|
| $41$ |
\( T^{4} + 992 T^{2} + 12544 \)
|
| $43$ |
\( (T^{2} + 17 T - 3134)^{2} \)
|
| $47$ |
\( (T^{2} + 43 T - 692)^{2} \)
|
| $53$ |
\( T^{4} + 6768 T^{2} + 4064256 \)
|
| $59$ |
\( T^{4} + 4832T^{2} + 256 \)
|
| $61$ |
\( (T^{2} + 35 T + 178)^{2} \)
|
| $67$ |
\( T^{4} + 25904 T^{2} + 162205696 \)
|
| $71$ |
\( T^{4} + 11616 T^{2} + 112896 \)
|
| $73$ |
\( (T^{2} - 45 T + 378)^{2} \)
|
| $79$ |
\( T^{4} + 8396 T^{2} + 5456896 \)
|
| $83$ |
\( (T^{2} + 32 T - 10916)^{2} \)
|
| $89$ |
\( T^{4} + 24288 T^{2} + 94945536 \)
|
| $97$ |
\( T^{4} + 14048 T^{2} + 21086464 \)
|
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