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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(\beta = 4\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−5.31371 | 0 | −99.7645 | −125.000 | 0 | 343.000 | 1210.27 | 0 | 664.214 | ||||||||||||||||||||||||
| 1.2 | 17.3137 | 0 | 171.765 | −125.000 | 0 | 343.000 | 757.726 | 0 | −2164.21 | |||||||||||||||||||||||||
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.d | 2 | |
| 3.b | odd | 2 | 1 | 105.8.a.c | ✓ | 2 | |
| 15.d | odd | 2 | 1 | 525.8.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.8.a.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 315.8.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 525.8.a.f | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 12T_{2} - 92 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 12T - 92 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 125)^{2} \)
|
| $7$ |
\( (T - 343)^{2} \)
|
| $11$ |
\( T^{2} - 10976 T + 29829344 \)
|
| $13$ |
\( T^{2} + 2796 T + 1362532 \)
|
| $17$ |
\( T^{2} - 8284 T - 11058908 \)
|
| $19$ |
\( T^{2} + 8096 T + 15731936 \)
|
| $23$ |
\( T^{2} + \cdots + 1167902176 \)
|
| $29$ |
\( T^{2} + \cdots - 2533764092 \)
|
| $31$ |
\( T^{2} + \cdots - 45934979312 \)
|
| $37$ |
\( T^{2} + \cdots - 83604374812 \)
|
| $41$ |
\( T^{2} + \cdots - 31383664700 \)
|
| $43$ |
\( T^{2} + \cdots - 41087241488 \)
|
| $47$ |
\( T^{2} + \cdots + 701642111264 \)
|
| $53$ |
\( T^{2} + \cdots - 3142721699644 \)
|
| $59$ |
\( T^{2} + \cdots + 1671072643216 \)
|
| $61$ |
\( T^{2} + \cdots - 4537899892316 \)
|
| $67$ |
\( T^{2} + \cdots + 2164616944 \)
|
| $71$ |
\( T^{2} + \cdots - 33999492976 \)
|
| $73$ |
\( T^{2} + \cdots + 6268678876004 \)
|
| $79$ |
\( T^{2} + \cdots - 28634836096 \)
|
| $83$ |
\( T^{2} + \cdots - 23626557722224 \)
|
| $89$ |
\( T^{2} + \cdots + 4683494838532 \)
|
| $97$ |
\( T^{2} + \cdots - 157167991209052 \)
|
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