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: |
\( x^{4} - x^{3} - 988x^{2} - 844x + 192256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| 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
\( x^{4} - x^{3} - 988x^{2} - 844x + 192256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 11\nu^{2} + 542\nu - 4240 ) / 168 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 11\nu^{2} + 1382\nu - 4408 ) / 84 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 431\nu^{2} - 718\nu - 211468 ) / 84 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 2 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 3\beta_{2} - 10\beta _1 + 4940 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 22\beta_{3} + 575\beta_{2} - 2874\beta _1 + 13024 ) / 10 \)
|
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 |
|
−16.5158 | 0 | −239.228 | −625.000 | 0 | 2401.00 | 12407.1 | 0 | 10322.4 | ||||||||||||||||||||||||||||||
| 1.2 | −8.08732 | 0 | −446.595 | −625.000 | 0 | 2401.00 | 7752.47 | 0 | 5054.57 | |||||||||||||||||||||||||||||||
| 1.3 | 27.3730 | 0 | 237.282 | −625.000 | 0 | 2401.00 | −7519.86 | 0 | −17108.1 | |||||||||||||||||||||||||||||||
| 1.4 | 38.2301 | 0 | 949.541 | −625.000 | 0 | 2401.00 | 16727.3 | 0 | −23893.8 | |||||||||||||||||||||||||||||||
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.i | 4 | |
| 3.b | odd | 2 | 1 | 105.10.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.10.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 41T_{2}^{3} - 434T_{2}^{2} + 16984T_{2} + 139776 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 41 T^{3} + \cdots + 139776 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 625)^{4} \)
|
| $7$ |
\( (T - 2401)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 14\!\cdots\!96 \)
|
| $13$ |
\( T^{4} + \cdots - 55\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots - 96\!\cdots\!64 \)
|
| $19$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $23$ |
\( T^{4} + \cdots + 91\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 79\!\cdots\!64 \)
|
| $31$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 19\!\cdots\!76 \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!16 \)
|
| $43$ |
\( T^{4} + \cdots - 33\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 62\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots - 84\!\cdots\!84 \)
|
| $61$ |
\( T^{4} + \cdots - 45\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $71$ |
\( T^{4} + \cdots - 42\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 16\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 31\!\cdots\!36 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $97$ |
\( T^{4} + \cdots - 43\!\cdots\!24 \)
|
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