Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [315,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(18.5856016518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.22952.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 18x + 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 18x + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
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 |
|
−3.37989 | 0 | 3.42368 | 5.00000 | 0 | 7.00000 | 15.4675 | 0 | −16.8995 | |||||||||||||||||||||||||||
| 1.2 | 0.229795 | 0 | −7.94719 | 5.00000 | 0 | 7.00000 | −3.66459 | 0 | 1.14898 | ||||||||||||||||||||||||||||
| 1.3 | 5.15010 | 0 | 18.5235 | 5.00000 | 0 | 7.00000 | 54.1971 | 0 | 25.7505 | ||||||||||||||||||||||||||||
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 | 315.4.a.o | yes | 3 |
| 3.b | odd | 2 | 1 | 315.4.a.n | ✓ | 3 | |
| 5.b | even | 2 | 1 | 1575.4.a.bb | 3 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.bl | 3 | ||
| 15.d | odd | 2 | 1 | 1575.4.a.be | 3 | ||
| 21.c | even | 2 | 1 | 2205.4.a.bk | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 315.4.a.n | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 315.4.a.o | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 1575.4.a.bb | 3 | 5.b | even | 2 | 1 | ||
| 1575.4.a.be | 3 | 15.d | odd | 2 | 1 | ||
| 2205.4.a.bk | 3 | 21.c | even | 2 | 1 | ||
| 2205.4.a.bl | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2T_{2}^{2} - 17T_{2} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 2 T^{2} + \cdots + 4 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} + 20 T^{2} + \cdots - 32160 \)
|
| $13$ |
\( T^{3} - 1748T - 17328 \)
|
| $17$ |
\( T^{3} - 234 T^{2} + \cdots - 282328 \)
|
| $19$ |
\( T^{3} + 82 T^{2} + \cdots + 15280 \)
|
| $23$ |
\( T^{3} - 30 T^{2} + \cdots - 378280 \)
|
| $29$ |
\( T^{3} + 32 T^{2} + \cdots + 409600 \)
|
| $31$ |
\( T^{3} + 362 T^{2} + \cdots + 1543664 \)
|
| $37$ |
\( T^{3} - 302 T^{2} + \cdots + 1425496 \)
|
| $41$ |
\( T^{3} - 626 T^{2} + \cdots + 16079208 \)
|
| $43$ |
\( T^{3} + 68 T^{2} + \cdots - 10742464 \)
|
| $47$ |
\( T^{3} - 936 T^{2} + \cdots - 29520128 \)
|
| $53$ |
\( T^{3} - 298 T^{2} + \cdots + 19383056 \)
|
| $59$ |
\( T^{3} - 428 T^{2} + \cdots + 229584064 \)
|
| $61$ |
\( T^{3} - 1050 T^{2} + \cdots + 63221000 \)
|
| $67$ |
\( T^{3} + 52 T^{2} + \cdots - 93054016 \)
|
| $71$ |
\( T^{3} - 416 T^{2} + \cdots + 14345328 \)
|
| $73$ |
\( T^{3} + 312 T^{2} + \cdots - 43935536 \)
|
| $79$ |
\( T^{3} + 488 T^{2} + \cdots - 282894592 \)
|
| $83$ |
\( T^{3} - 948 T^{2} + \cdots + 431507200 \)
|
| $89$ |
\( T^{3} + 142 T^{2} + \cdots + 19655976 \)
|
| $97$ |
\( T^{3} + 3052 T^{2} + \cdots + 204161472 \)
|
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