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: | \(0\) |
| 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: |
\( x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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
\( x^{6} - 3x^{5} - 3018x^{4} + 3368x^{3} + 2066979x^{2} - 6329061x - 14714266 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 71\nu^{5} - 1106\nu^{4} - 175920\nu^{3} + 1251392\nu^{2} + 92091141\nu + 14392190 ) / 520728 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 71\nu^{5} - 1106\nu^{4} - 175920\nu^{3} + 1772120\nu^{2} + 90528957\nu - 509460178 ) / 520728 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 185\nu^{5} + 3230\nu^{4} - 628292\nu^{3} - 6470588\nu^{2} + 460995047\nu + 11708850 ) / 520728 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 37\nu^{5} + 646\nu^{4} - 229804\nu^{3} - 148516\nu^{2} + 262060483\nu - 940175910 ) / 520728 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 3\beta _1 + 1006 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{5} + \beta_{4} + 11\beta_{3} - 11\beta_{2} + 1664\beta _1 + 2016 \)
|
| \(\nu^{4}\) | \(=\) |
\( -139\beta_{5} + 113\beta_{4} + 1898\beta_{3} - 2120\beta_{2} + 14870\beta _1 + 1662018 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14554\beta_{5} + 4238\beta_{4} + 39196\beta_{3} - 35320\beta_{2} + 3004673\beta _1 + 12951466 \)
|
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 |
|
−42.7818 | 0 | 1318.28 | −625.000 | 0 | −2401.00 | −34494.1 | 0 | 26738.6 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −39.8299 | 0 | 1074.42 | −625.000 | 0 | −2401.00 | −22401.1 | 0 | 24893.7 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.54749 | 0 | −491.320 | −625.000 | 0 | −2401.00 | 4562.59 | 0 | 2842.18 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.69340 | 0 | −509.132 | −625.000 | 0 | −2401.00 | −1729.19 | 0 | −1058.38 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 26.3435 | 0 | 181.982 | −625.000 | 0 | −2401.00 | −8693.85 | 0 | −16464.7 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 44.1222 | 0 | 1434.77 | −625.000 | 0 | −2401.00 | 40714.7 | 0 | −27576.4 | |||||||||||||||||||||||||||||||||||||
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.l | 6 | |
| 3.b | odd | 2 | 1 | 35.10.a.e | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 175.10.a.g | 6 | ||
| 15.e | even | 4 | 2 | 175.10.b.g | 12 | ||
| 21.c | even | 2 | 1 | 245.10.a.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.10.a.e | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 175.10.a.g | 6 | 15.d | odd | 2 | 1 | ||
| 175.10.b.g | 12 | 15.e | even | 4 | 2 | ||
| 245.10.a.g | 6 | 21.c | even | 2 | 1 | ||
| 315.10.a.l | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 15T_{2}^{5} - 2928T_{2}^{4} - 32578T_{2}^{3} + 1934724T_{2}^{2} + 5838048T_{2} - 15252160 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 15 T^{5} + \cdots - 15252160 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 625)^{6} \)
|
| $7$ |
\( (T + 2401)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 78\!\cdots\!76 \)
|
| $13$ |
\( T^{6} + \cdots + 11\!\cdots\!60 \)
|
| $17$ |
\( T^{6} + \cdots - 10\!\cdots\!28 \)
|
| $19$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 52\!\cdots\!36 \)
|
| $29$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 42\!\cdots\!56 \)
|
| $41$ |
\( T^{6} + \cdots - 74\!\cdots\!64 \)
|
| $43$ |
\( T^{6} + \cdots - 56\!\cdots\!80 \)
|
| $47$ |
\( T^{6} + \cdots - 15\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots - 85\!\cdots\!88 \)
|
| $59$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 52\!\cdots\!16 \)
|
| $67$ |
\( T^{6} + \cdots - 68\!\cdots\!88 \)
|
| $71$ |
\( T^{6} + \cdots + 20\!\cdots\!40 \)
|
| $73$ |
\( T^{6} + \cdots - 49\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 74\!\cdots\!88 \)
|
| $89$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 27\!\cdots\!72 \)
|
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