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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 2759x^{4} + 14856x^{3} + 1722956x^{2} - 8110848x - 126461952 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 2759x^{4} + 14856x^{3} + 1722956x^{2} - 8110848x - 126461952 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 5\nu - 922 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 12\nu^{2} - 1432\nu - 5899 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} - 50\nu^{4} + 6725\nu^{3} + 61632\nu^{2} - 2957236\nu - 8372464 ) / 464 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 22\nu^{4} + 2783\nu^{3} - 50796\nu^{2} - 1422524\nu + 12322200 ) / 232 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 5\beta _1 + 922 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 12\beta_{2} + 1492\beta _1 - 5165 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} - 4\beta_{4} - 14\beta_{3} + 2013\beta_{2} - 18817\beta _1 + 1382546 \)
|
| \(\nu^{5}\) | \(=\) |
\( -100\beta_{5} - 88\beta_{4} + 2475\beta_{3} - 39906\beta_{2} + 2569718\beta _1 - 18469895 \)
|
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 |
|
−42.7325 | 0 | 1314.07 | −625.000 | 0 | 2401.00 | −34274.4 | 0 | 26707.8 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −35.1091 | 0 | 720.650 | −625.000 | 0 | 2401.00 | −7325.52 | 0 | 21943.2 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −16.2163 | 0 | −249.031 | −625.000 | 0 | 2401.00 | 12341.1 | 0 | 10135.2 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.87951 | 0 | −503.708 | −625.000 | 0 | 2401.00 | −2924.74 | 0 | −1799.69 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 23.1049 | 0 | 21.8344 | −625.000 | 0 | 2401.00 | −11325.2 | 0 | −14440.5 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 42.0736 | 0 | 1258.19 | −625.000 | 0 | 2401.00 | 31394.8 | 0 | −26296.0 | |||||||||||||||||||||||||||||||||||||
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.k | 6 | |
| 3.b | odd | 2 | 1 | 105.10.a.h | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.h | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 315.10.a.k | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 26T_{2}^{5} - 2479T_{2}^{4} - 57400T_{2}^{3} + 1284940T_{2}^{2} + 20483808T_{2} - 68102208 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 26 T^{5} + \cdots - 68102208 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 625)^{6} \)
|
| $7$ |
\( (T - 2401)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 50\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots - 24\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots - 55\!\cdots\!36 \)
|
| $19$ |
\( T^{6} + \cdots + 90\!\cdots\!40 \)
|
| $23$ |
\( T^{6} + \cdots + 60\!\cdots\!72 \)
|
| $29$ |
\( T^{6} + \cdots + 16\!\cdots\!60 \)
|
| $31$ |
\( T^{6} + \cdots - 21\!\cdots\!88 \)
|
| $37$ |
\( T^{6} + \cdots - 24\!\cdots\!64 \)
|
| $41$ |
\( T^{6} + \cdots + 41\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 20\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots + 67\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 82\!\cdots\!40 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $61$ |
\( T^{6} + \cdots - 15\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 21\!\cdots\!48 \)
|
| $71$ |
\( T^{6} + \cdots - 65\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots - 42\!\cdots\!48 \)
|
| $79$ |
\( T^{6} + \cdots + 86\!\cdots\!40 \)
|
| $83$ |
\( T^{6} + \cdots + 55\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
|
| $97$ |
\( T^{6} + \cdots + 28\!\cdots\!44 \)
|
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