Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1400,4,Mod(449,1400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1400.449");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1400.g (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: | \(82.6026740080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{73})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 37x^{2} + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 280) |
| 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
\( x^{4} + 37x^{2} + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 19\nu ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( 18\beta_{2} - 19\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).
| \(n\) | \(351\) | \(701\) | \(801\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 5.77200i | 0 | 0 | 0 | 7.00000i | 0 | −6.31601 | 0 | |||||||||||||||||||||||||||||
| 449.2 | 0 | − | 2.77200i | 0 | 0 | 0 | − | 7.00000i | 0 | 19.3160 | 0 | |||||||||||||||||||||||||||||
| 449.3 | 0 | 2.77200i | 0 | 0 | 0 | 7.00000i | 0 | 19.3160 | 0 | |||||||||||||||||||||||||||||||
| 449.4 | 0 | 5.77200i | 0 | 0 | 0 | − | 7.00000i | 0 | −6.31601 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1400.4.g.i | 4 | |
| 5.b | even | 2 | 1 | inner | 1400.4.g.i | 4 | |
| 5.c | odd | 4 | 1 | 280.4.a.e | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1400.4.a.j | 2 | ||
| 20.e | even | 4 | 1 | 560.4.a.s | 2 | ||
| 35.f | even | 4 | 1 | 1960.4.a.n | 2 | ||
| 40.i | odd | 4 | 1 | 2240.4.a.bp | 2 | ||
| 40.k | even | 4 | 1 | 2240.4.a.bm | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.a.e | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 560.4.a.s | 2 | 20.e | even | 4 | 1 | ||
| 1400.4.a.j | 2 | 5.c | odd | 4 | 1 | ||
| 1400.4.g.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 1400.4.g.i | 4 | 5.b | even | 2 | 1 | inner | |
| 1960.4.a.n | 2 | 35.f | even | 4 | 1 | ||
| 2240.4.a.bm | 2 | 40.k | even | 4 | 1 | ||
| 2240.4.a.bp | 2 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1400, [\chi])\):
|
\( T_{3}^{4} + 41T_{3}^{2} + 256 \)
|
|
\( T_{11}^{2} - 21T_{11} + 92 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 41T^{2} + 256 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{2} - 21 T + 92)^{2} \)
|
| $13$ |
\( T^{4} + 8477 T^{2} + 15792676 \)
|
| $17$ |
\( T^{4} + 16781 T^{2} + 59382436 \)
|
| $19$ |
\( (T^{2} + 102 T + 2528)^{2} \)
|
| $23$ |
\( T^{4} + 16452 T^{2} + 47775744 \)
|
| $29$ |
\( (T^{2} + 129 T - 5494)^{2} \)
|
| $31$ |
\( (T^{2} - 320 T + 20928)^{2} \)
|
| $37$ |
\( T^{4} + 59368 T^{2} + 824608656 \)
|
| $41$ |
\( (T^{2} + 94 T - 6624)^{2} \)
|
| $43$ |
\( T^{4} + 44276 T^{2} + 6431296 \)
|
| $47$ |
\( T^{4} + 21049 T^{2} + 109998144 \)
|
| $53$ |
\( T^{4} + 83188 T^{2} + 572549184 \)
|
| $59$ |
\( (T^{2} + 512 T - 163392)^{2} \)
|
| $61$ |
\( (T^{2} + 226 T - 25848)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 2884978944 \)
|
| $71$ |
\( (T^{2} - 56 T - 56448)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 55494167184 \)
|
| $79$ |
\( (T^{2} + 1299 T + 404312)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 2286047233024 \)
|
| $89$ |
\( (T^{2} + 546 T - 1066096)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 3289776123076 \)
|
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