Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,10,Mod(1,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, 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("315.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 315.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: | \(162.236288392\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 307x^{2} - 270x + 8836 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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
\( x^{4} - 307x^{2} - 270x + 8836 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 246\nu + 258 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 6\nu - 614 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta _1 + 614 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} + 4\beta_{2} + 501\beta _1 + 810 ) / 4 \)
|
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 |
|
−33.9706 | 0 | 642.004 | 625.000 | 0 | −2401.00 | −4416.33 | 0 | −21231.6 | ||||||||||||||||||||||||||||||
| 1.2 | −14.5441 | 0 | −300.469 | 625.000 | 0 | −2401.00 | 11816.6 | 0 | −9090.08 | |||||||||||||||||||||||||||||||
| 1.3 | 8.30453 | 0 | −443.035 | 625.000 | 0 | −2401.00 | −7931.11 | 0 | 5190.33 | |||||||||||||||||||||||||||||||
| 1.4 | 32.2102 | 0 | 525.499 | 625.000 | 0 | −2401.00 | 434.806 | 0 | 20131.4 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 315.10.a.c | 4 | |
| 3.b | odd | 2 | 1 | 105.10.a.f | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.f | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.10.a.c | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 8T_{2}^{3} - 1204T_{2}^{2} - 7040T_{2} + 132160 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 8 T^{3} + \cdots + 132160 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T - 625)^{4} \)
|
| $7$ |
\( (T + 2401)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 32\!\cdots\!64 \)
|
| $13$ |
\( T^{4} + \cdots + 60\!\cdots\!60 \)
|
| $17$ |
\( T^{4} + \cdots - 18\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 33\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 59\!\cdots\!56 \)
|
| $31$ |
\( T^{4} + \cdots - 36\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 73\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots - 19\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots - 29\!\cdots\!40 \)
|
| $47$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 81\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots - 24\!\cdots\!60 \)
|
| $61$ |
\( T^{4} + \cdots - 75\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 23\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 83\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots - 53\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 11\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 14\!\cdots\!64 \)
|
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