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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 295x^{2} + 188x + 1420 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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
\( x^{4} - 2x^{3} - 295x^{2} + 188x + 1420 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 148 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 285\nu + 136 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 148 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 289\beta _1 + 160 \)
|
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 |
|
−14.7605 | 0 | 89.8711 | −125.000 | 0 | −343.000 | 562.800 | 0 | 1845.06 | ||||||||||||||||||||||||||||||
| 1.2 | 0.457550 | 0 | −127.791 | −125.000 | 0 | −343.000 | −117.037 | 0 | −57.1937 | |||||||||||||||||||||||||||||||
| 1.3 | 4.91945 | 0 | −103.799 | −125.000 | 0 | −343.000 | −1140.32 | 0 | −614.931 | |||||||||||||||||||||||||||||||
| 1.4 | 19.3835 | 0 | 247.719 | −125.000 | 0 | −343.000 | 2320.56 | 0 | −2422.93 | |||||||||||||||||||||||||||||||
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.l | 4 | |
| 3.b | odd | 2 | 1 | 105.8.a.d | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 525.8.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.8.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 315.8.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 525.8.a.m | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 10T_{2}^{3} - 259T_{2}^{2} + 1528T_{2} - 644 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 10 T^{3} + \cdots - 644 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 125)^{4} \)
|
| $7$ |
\( (T + 343)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 68367082577920 \)
|
| $13$ |
\( T^{4} + \cdots - 22578519713552 \)
|
| $17$ |
\( T^{4} + \cdots + 10\!\cdots\!28 \)
|
| $19$ |
\( T^{4} + \cdots - 22\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 62\!\cdots\!48 \)
|
| $29$ |
\( T^{4} + \cdots + 94\!\cdots\!40 \)
|
| $31$ |
\( T^{4} + \cdots - 95\!\cdots\!16 \)
|
| $37$ |
\( T^{4} + \cdots - 48\!\cdots\!32 \)
|
| $41$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 11\!\cdots\!32 \)
|
| $47$ |
\( T^{4} + \cdots - 12\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 10\!\cdots\!80 \)
|
| $59$ |
\( T^{4} + \cdots + 82\!\cdots\!20 \)
|
| $61$ |
\( T^{4} + \cdots - 64\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots - 77\!\cdots\!08 \)
|
| $71$ |
\( T^{4} + \cdots + 54\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 10\!\cdots\!72 \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots - 16\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 46\!\cdots\!08 \)
|
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