Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(64.1535279252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1595208.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} + 33x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{4} - x^{3} - 20x^{2} + 33x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu^{3} + 40\nu - 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\nu^{3} + 40\nu^{2} - 240\nu - 141 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{3} + 40\nu^{2} + 120\nu - 516 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 5\beta _1 + 20 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + \beta_{2} - 3\beta _1 + 820 ) / 80 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 3\beta _1 - 38 ) / 4 \)
|
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 | −24.1383 | 0 | 0 | 0 | 179.876 | 0 | 339.657 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.49000 | 0 | 0 | 0 | −188.968 | 0 | −212.860 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.69449 | 0 | 0 | 0 | 10.2635 | 0 | −220.962 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 28.9338 | 0 | 0 | 0 | 146.828 | 0 | 594.165 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.6.a.ba | 4 | |
| 4.b | odd | 2 | 1 | 200.6.a.j | 4 | ||
| 5.b | even | 2 | 1 | 400.6.a.z | 4 | ||
| 5.c | odd | 4 | 2 | 80.6.c.d | 8 | ||
| 15.e | even | 4 | 2 | 720.6.f.n | 8 | ||
| 20.d | odd | 2 | 1 | 200.6.a.k | 4 | ||
| 20.e | even | 4 | 2 | 40.6.c.a | ✓ | 8 | |
| 40.i | odd | 4 | 2 | 320.6.c.i | 8 | ||
| 40.k | even | 4 | 2 | 320.6.c.j | 8 | ||
| 60.l | odd | 4 | 2 | 360.6.f.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.6.c.a | ✓ | 8 | 20.e | even | 4 | 2 | |
| 80.6.c.d | 8 | 5.c | odd | 4 | 2 | ||
| 200.6.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 200.6.a.k | 4 | 20.d | odd | 2 | 1 | ||
| 320.6.c.i | 8 | 40.i | odd | 4 | 2 | ||
| 320.6.c.j | 8 | 40.k | even | 4 | 2 | ||
| 360.6.f.b | 8 | 60.l | odd | 4 | 2 | ||
| 400.6.a.z | 4 | 5.b | even | 2 | 1 | ||
| 400.6.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 720.6.f.n | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} - 728T_{3}^{2} - 432T_{3} + 18000 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 18000 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 148 T^{3} + \cdots - 51222832 \)
|
| $11$ |
\( T^{4} + \cdots + 37397137664 \)
|
| $13$ |
\( T^{4} + \cdots + 10683053312 \)
|
| $17$ |
\( T^{4} + \cdots + 22118400000 \)
|
| $19$ |
\( T^{4} + \cdots + 1042985883904 \)
|
| $23$ |
\( T^{4} + \cdots - 5643370924592 \)
|
| $29$ |
\( T^{4} + \cdots - 97147517576176 \)
|
| $31$ |
\( T^{4} + \cdots + 244229603328000 \)
|
| $37$ |
\( T^{4} + \cdots - 16\!\cdots\!36 \)
|
| $41$ |
\( T^{4} + \cdots - 296811236945008 \)
|
| $43$ |
\( T^{4} + \cdots - 74\!\cdots\!72 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots - 11\!\cdots\!48 \)
|
| $61$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 71\!\cdots\!32 \)
|
| $71$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 15\!\cdots\!32 \)
|
| $79$ |
\( T^{4} + \cdots - 96\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 25\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 93\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots - 14\!\cdots\!84 \)
|
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