Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,10,Mod(1,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.a (trivial) |
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: | yes |
| Analytic conductor: | \(206.014334466\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.49740556.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 45x^{2} + 304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 45x^{2} + 304 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{3} - 51\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} - 74\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 120\nu^{2} - 2700 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{2} + 4\beta_1 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2700 ) / 120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -51\beta_{2} + 148\beta_1 ) / 120 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −179.263 | 0 | 0 | 0 | −8712.99 | 0 | 12452.2 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −37.6407 | 0 | 0 | 0 | −5315.22 | 0 | −18266.2 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 37.6407 | 0 | 0 | 0 | 5315.22 | 0 | −18266.2 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 179.263 | 0 | 0 | 0 | 8712.99 | 0 | 12452.2 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 400.10.a.ba | 4 | |
| 4.b | odd | 2 | 1 | 25.10.a.e | 4 | ||
| 5.b | even | 2 | 1 | inner | 400.10.a.ba | 4 | |
| 5.c | odd | 4 | 2 | 80.10.c.c | 4 | ||
| 12.b | even | 2 | 1 | 225.10.a.s | 4 | ||
| 20.d | odd | 2 | 1 | 25.10.a.e | 4 | ||
| 20.e | even | 4 | 2 | 5.10.b.a | ✓ | 4 | |
| 60.h | even | 2 | 1 | 225.10.a.s | 4 | ||
| 60.l | odd | 4 | 2 | 45.10.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.10.b.a | ✓ | 4 | 20.e | even | 4 | 2 | |
| 25.10.a.e | 4 | 4.b | odd | 2 | 1 | ||
| 25.10.a.e | 4 | 20.d | odd | 2 | 1 | ||
| 45.10.b.b | 4 | 60.l | odd | 4 | 2 | ||
| 80.10.c.c | 4 | 5.c | odd | 4 | 2 | ||
| 225.10.a.s | 4 | 12.b | even | 2 | 1 | ||
| 225.10.a.s | 4 | 60.h | even | 2 | 1 | ||
| 400.10.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.10.a.ba | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 33552T_{3}^{2} + 45529776 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 33552 T^{2} + 45529776 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 21\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} + 54984 T + 464570064)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 29\!\cdots\!56 \)
|
| $17$ |
\( T^{4} + \cdots + 88\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} + 318440 T - 139618977200)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 47\!\cdots\!36 \)
|
| $29$ |
\( (T^{2} + \cdots - 2063356400700)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 6585677277184)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 17\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + \cdots - 433450792618956)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 46\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 73\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 67\!\cdots\!64)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 56\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} + \cdots - 59\!\cdots\!76)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
| $79$ |
\( (T^{2} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $89$ |
\( (T^{2} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 42\!\cdots\!36 \)
|
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