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: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 652x + 4000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 25) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 652x + 4000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 63\nu - 454 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} - 6\nu + 872 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta _1 + 12 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -63\beta_{2} - 12\beta _1 + 17404 ) / 40 \)
|
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 | −268.664 | 0 | 0 | 0 | 637.237 | 0 | 52497.3 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −30.5073 | 0 | 0 | 0 | 4010.25 | 0 | −18752.3 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 210.171 | 0 | 0 | 0 | −9905.49 | 0 | 24489.0 | 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 | 400.10.a.u | 3 | |
| 4.b | odd | 2 | 1 | 25.10.a.d | yes | 3 | |
| 5.b | even | 2 | 1 | 400.10.a.y | 3 | ||
| 5.c | odd | 4 | 2 | 400.10.c.q | 6 | ||
| 12.b | even | 2 | 1 | 225.10.a.m | 3 | ||
| 20.d | odd | 2 | 1 | 25.10.a.c | ✓ | 3 | |
| 20.e | even | 4 | 2 | 25.10.b.c | 6 | ||
| 60.h | even | 2 | 1 | 225.10.a.p | 3 | ||
| 60.l | odd | 4 | 2 | 225.10.b.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.10.a.c | ✓ | 3 | 20.d | odd | 2 | 1 | |
| 25.10.a.d | yes | 3 | 4.b | odd | 2 | 1 | |
| 25.10.b.c | 6 | 20.e | even | 4 | 2 | ||
| 225.10.a.m | 3 | 12.b | even | 2 | 1 | ||
| 225.10.a.p | 3 | 60.h | even | 2 | 1 | ||
| 225.10.b.m | 6 | 60.l | odd | 4 | 2 | ||
| 400.10.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 400.10.a.y | 3 | 5.b | even | 2 | 1 | ||
| 400.10.c.q | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 89T_{3}^{2} - 54681T_{3} - 1722609 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 89 T^{2} + \cdots - 1722609 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 25313297688 \)
|
| $11$ |
\( T^{3} + \cdots + 75283351667163 \)
|
| $13$ |
\( T^{3} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( T^{3} + \cdots + 17\!\cdots\!07 \)
|
| $19$ |
\( T^{3} + \cdots - 22\!\cdots\!25 \)
|
| $23$ |
\( T^{3} + \cdots + 38\!\cdots\!16 \)
|
| $29$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 81\!\cdots\!48 \)
|
| $37$ |
\( T^{3} + \cdots - 17\!\cdots\!28 \)
|
| $41$ |
\( T^{3} + \cdots + 15\!\cdots\!47 \)
|
| $43$ |
\( T^{3} + \cdots - 33\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots + 14\!\cdots\!08 \)
|
| $53$ |
\( T^{3} + \cdots + 40\!\cdots\!84 \)
|
| $59$ |
\( T^{3} + \cdots + 49\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 62\!\cdots\!12 \)
|
| $67$ |
\( T^{3} + \cdots + 27\!\cdots\!93 \)
|
| $71$ |
\( T^{3} + \cdots - 11\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots + 13\!\cdots\!09 \)
|
| $79$ |
\( T^{3} + \cdots + 26\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 83\!\cdots\!41 \)
|
| $89$ |
\( T^{3} + \cdots - 19\!\cdots\!75 \)
|
| $97$ |
\( T^{3} + \cdots - 34\!\cdots\!08 \)
|
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