Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(98.4012830275\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 584x^{3} + 2550x^{2} + 46220x - 155664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 584x^{3} + 2550x^{2} + 46220x - 155664 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 29\nu^{3} + 716\nu^{2} - 12922\nu - 32740 ) / 236 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 5\nu - 235 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{4} + 33\nu^{3} - 3596\nu^{2} - 878\nu + 194724 ) / 236 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 5\beta _1 + 235 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 6\beta_{3} + 7\beta_{2} + 417\beta _1 - 1264 \)
|
| \(\nu^{4}\) | \(=\) |
\( 29\beta_{4} + 542\beta_{3} - 33\beta_{2} - 4409\beta _1 + 98864 \)
|
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 |
|
−21.1208 | 0 | 318.088 | −125.000 | 0 | 343.000 | −4014.82 | 0 | 2640.10 | |||||||||||||||||||||||||||||||||
| 1.2 | −13.6859 | 0 | 59.3043 | −125.000 | 0 | 343.000 | 940.164 | 0 | 1710.74 | ||||||||||||||||||||||||||||||||||
| 1.3 | −5.21232 | 0 | −100.832 | −125.000 | 0 | 343.000 | 1192.74 | 0 | 651.540 | ||||||||||||||||||||||||||||||||||
| 1.4 | 7.04671 | 0 | −78.3439 | −125.000 | 0 | 343.000 | −1454.05 | 0 | −880.839 | ||||||||||||||||||||||||||||||||||
| 1.5 | 21.9723 | 0 | 354.783 | −125.000 | 0 | 343.000 | 4982.95 | 0 | −2746.54 | ||||||||||||||||||||||||||||||||||
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 | 315.8.a.m | 5 | |
| 3.b | odd | 2 | 1 | 35.8.a.d | ✓ | 5 | |
| 12.b | even | 2 | 1 | 560.8.a.s | 5 | ||
| 15.d | odd | 2 | 1 | 175.8.a.f | 5 | ||
| 15.e | even | 4 | 2 | 175.8.b.f | 10 | ||
| 21.c | even | 2 | 1 | 245.8.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.8.a.d | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 175.8.a.f | 5 | 15.d | odd | 2 | 1 | ||
| 175.8.b.f | 10 | 15.e | even | 4 | 2 | ||
| 245.8.a.f | 5 | 21.c | even | 2 | 1 | ||
| 315.8.a.m | 5 | 1.a | even | 1 | 1 | trivial | |
| 560.8.a.s | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 11T_{2}^{4} - 536T_{2}^{3} - 5950T_{2}^{2} + 29124T_{2} + 233280 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 11 T^{4} + \cdots + 233280 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T + 125)^{5} \)
|
| $7$ |
\( (T - 343)^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 85\!\cdots\!48 \)
|
| $13$ |
\( T^{5} + \cdots - 52\!\cdots\!40 \)
|
| $17$ |
\( T^{5} + \cdots + 16\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 14\!\cdots\!40 \)
|
| $23$ |
\( T^{5} + \cdots + 14\!\cdots\!52 \)
|
| $29$ |
\( T^{5} + \cdots - 59\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 24\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 35\!\cdots\!16 \)
|
| $41$ |
\( T^{5} + \cdots - 14\!\cdots\!12 \)
|
| $43$ |
\( T^{5} + \cdots - 26\!\cdots\!80 \)
|
| $47$ |
\( T^{5} + \cdots + 16\!\cdots\!64 \)
|
| $53$ |
\( T^{5} + \cdots + 20\!\cdots\!56 \)
|
| $59$ |
\( T^{5} + \cdots - 21\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 17\!\cdots\!48 \)
|
| $67$ |
\( T^{5} + \cdots + 63\!\cdots\!28 \)
|
| $71$ |
\( T^{5} + \cdots + 28\!\cdots\!40 \)
|
| $73$ |
\( T^{5} + \cdots - 99\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 51\!\cdots\!84 \)
|
| $89$ |
\( T^{5} + \cdots - 31\!\cdots\!60 \)
|
| $97$ |
\( T^{5} + \cdots + 44\!\cdots\!48 \)
|
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