Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [325,4,Mod(274,325)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("325.274");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 325.b (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: | \(19.1756207519\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 65) |
| 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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 274.1 |
|
− | 2.56155i | 8.24621i | 1.43845 | 0 | 21.1231 | − | 24.8769i | − | 24.1771i | −41.0000 | 0 | |||||||||||||||||||||||||||
| 274.2 | − | 1.56155i | 8.24621i | 5.56155 | 0 | 12.8769 | 33.1231i | − | 21.1771i | −41.0000 | 0 | |||||||||||||||||||||||||||||
| 274.3 | 1.56155i | − | 8.24621i | 5.56155 | 0 | 12.8769 | − | 33.1231i | 21.1771i | −41.0000 | 0 | |||||||||||||||||||||||||||||
| 274.4 | 2.56155i | − | 8.24621i | 1.43845 | 0 | 21.1231 | 24.8769i | 24.1771i | −41.0000 | 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 | 325.4.b.f | 4 | |
| 5.b | even | 2 | 1 | inner | 325.4.b.f | 4 | |
| 5.c | odd | 4 | 1 | 65.4.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 325.4.a.g | 2 | ||
| 15.e | even | 4 | 1 | 585.4.a.h | 2 | ||
| 20.e | even | 4 | 1 | 1040.4.a.k | 2 | ||
| 65.h | odd | 4 | 1 | 845.4.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.4.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 325.4.a.g | 2 | 5.c | odd | 4 | 1 | ||
| 325.4.b.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 325.4.b.f | 4 | 5.b | even | 2 | 1 | inner | |
| 585.4.a.h | 2 | 15.e | even | 4 | 1 | ||
| 845.4.a.d | 2 | 65.h | odd | 4 | 1 | ||
| 1040.4.a.k | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(325, [\chi])\):
|
\( T_{2}^{4} + 9T_{2}^{2} + 16 \)
|
|
\( T_{3}^{2} + 68 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 9T^{2} + 16 \)
|
| $3$ |
\( (T^{2} + 68)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 1716 T^{2} + 678976 \)
|
| $11$ |
\( (T^{2} + 86 T + 1832)^{2} \)
|
| $13$ |
\( (T^{2} + 169)^{2} \)
|
| $17$ |
\( T^{4} + 9096 T^{2} + 17272336 \)
|
| $19$ |
\( (T^{2} + 166 T + 4832)^{2} \)
|
| $23$ |
\( T^{4} + 36616 T^{2} + 272514064 \)
|
| $29$ |
\( (T^{2} + 120 T + 268)^{2} \)
|
| $31$ |
\( (T^{2} + 78 T - 65952)^{2} \)
|
| $37$ |
\( T^{4} + 75816 T^{2} + 723179664 \)
|
| $41$ |
\( (T^{2} + 72 T - 64052)^{2} \)
|
| $43$ |
\( T^{4} + 14568 T^{2} + 39891856 \)
|
| $47$ |
\( T^{4} + 98196 T^{2} + 269747776 \)
|
| $53$ |
\( T^{4} + 156744 T^{2} + 1296 \)
|
| $59$ |
\( (T^{2} + 18 T - 128592)^{2} \)
|
| $61$ |
\( (T - 442)^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 62248254016 \)
|
| $71$ |
\( (T^{2} - 634 T - 318544)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 229352872464 \)
|
| $79$ |
\( (T^{2} - 180 T + 7488)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 5554422784 \)
|
| $89$ |
\( (T^{2} - 852 T + 120276)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 8821208882704 \)
|
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