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: | \(1\) |
| 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: |
\( x^{6} - 3x^{5} - 570x^{4} - 276x^{3} + 67464x^{2} + 110400x - 1253696 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 3x^{5} - 570x^{4} - 276x^{3} + 67464x^{2} + 110400x - 1253696 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5\nu - 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 496\nu^{3} + 524\nu^{2} + 38320\nu + 14176 ) / 96 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 9\nu^{3} - 424\nu^{2} + 1668\nu + 22024 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 13\nu^{4} + 520\nu^{3} - 4588\nu^{2} - 55312\nu + 210112 ) / 96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 189 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 351\beta _1 + 822 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{5} + 3\beta_{4} + 9\beta_{3} + 487\beta_{2} + 3611\beta _1 + 65510 \)
|
| \(\nu^{5}\) | \(=\) |
\( 541\beta_{5} - 481\beta_{4} + 637\beta_{3} + 5383\beta_{2} + 151211\beta _1 + 622050 \)
|
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 |
|
−21.9671 | 0 | 354.553 | 125.000 | 0 | −343.000 | −4976.71 | 0 | −2745.89 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −14.9234 | 0 | 94.7076 | 125.000 | 0 | −343.000 | 496.836 | 0 | −1865.42 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −10.3287 | 0 | −21.3173 | 125.000 | 0 | −343.000 | 1542.26 | 0 | −1291.09 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.213623 | 0 | −127.954 | 125.000 | 0 | −343.000 | 54.6777 | 0 | −26.7029 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 7.75456 | 0 | −67.8668 | 125.000 | 0 | −343.000 | −1518.86 | 0 | 969.320 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 18.6783 | 0 | 220.878 | 125.000 | 0 | −343.000 | 1734.80 | 0 | 2334.78 | |||||||||||||||||||||||||||||||||||||
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.n | ✓ | 6 |
| 3.b | odd | 2 | 1 | 315.8.a.o | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.8.a.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 315.8.a.o | yes | 6 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 21T_{2}^{5} - 390T_{2}^{4} - 8596T_{2}^{3} + 11352T_{2}^{2} + 493248T_{2} + 104768 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 21 T^{5} + \cdots + 104768 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 125)^{6} \)
|
| $7$ |
\( (T + 343)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots + 19\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots - 36\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 94\!\cdots\!68 \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots + 38\!\cdots\!96 \)
|
| $43$ |
\( T^{6} + \cdots + 20\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots - 16\!\cdots\!72 \)
|
| $53$ |
\( T^{6} + \cdots + 24\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots - 27\!\cdots\!68 \)
|
| $61$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 37\!\cdots\!52 \)
|
| $71$ |
\( T^{6} + \cdots - 53\!\cdots\!04 \)
|
| $73$ |
\( T^{6} + \cdots - 83\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots + 15\!\cdots\!08 \)
|
| $83$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots - 13\!\cdots\!96 \)
|
| $97$ |
\( T^{6} + \cdots + 17\!\cdots\!32 \)
|
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