Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [312,4,Mod(1,312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("312.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 312.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: | \(18.4085959218\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.13916.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 16x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{3} - x^{2} - 16x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 4\nu - 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 22 ) / 2 \)
|
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 |
|
0 | 3.00000 | 0 | −20.1466 | 0 | −19.6988 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 3.18652 | 0 | 23.9341 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 12.9600 | 0 | 1.76468 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 312.4.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 936.4.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 624.4.a.s | 3 | ||
| 8.b | even | 2 | 1 | 2496.4.a.bm | 3 | ||
| 8.d | odd | 2 | 1 | 2496.4.a.bq | 3 | ||
| 12.b | even | 2 | 1 | 1872.4.a.bl | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.4.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 624.4.a.s | 3 | 4.b | odd | 2 | 1 | ||
| 936.4.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 1872.4.a.bl | 3 | 12.b | even | 2 | 1 | ||
| 2496.4.a.bm | 3 | 8.b | even | 2 | 1 | ||
| 2496.4.a.bq | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 4T_{5}^{2} - 284T_{5} + 832 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} + 4 T^{2} + \cdots + 832 \)
|
| $7$ |
\( T^{3} - 6 T^{2} + \cdots + 832 \)
|
| $11$ |
\( T^{3} - 32 T^{2} + \cdots + 2304 \)
|
| $13$ |
\( (T - 13)^{3} \)
|
| $17$ |
\( T^{3} - 158 T^{2} + \cdots + 759704 \)
|
| $19$ |
\( T^{3} - 70 T^{2} + \cdots + 1313312 \)
|
| $23$ |
\( T^{3} - 176 T^{2} + \cdots + 763904 \)
|
| $29$ |
\( T^{3} - 222 T^{2} + \cdots + 887032 \)
|
| $31$ |
\( T^{3} - 54 T^{2} + \cdots - 4437952 \)
|
| $37$ |
\( T^{3} + 90 T^{2} + \cdots - 22952 \)
|
| $41$ |
\( T^{3} - 104 T^{2} + \cdots - 2239952 \)
|
| $43$ |
\( T^{3} - 140 T^{2} + \cdots + 1118656 \)
|
| $47$ |
\( T^{3} - 328 T^{2} + \cdots + 55193088 \)
|
| $53$ |
\( T^{3} + 358 T^{2} + \cdots - 22509432 \)
|
| $59$ |
\( T^{3} - 1060 T^{2} + \cdots - 33152064 \)
|
| $61$ |
\( T^{3} + 186 T^{2} + \cdots - 8664104 \)
|
| $67$ |
\( T^{3} - 354 T^{2} + \cdots + 106208096 \)
|
| $71$ |
\( T^{3} - 1692 T^{2} + \cdots + 354160512 \)
|
| $73$ |
\( T^{3} + 974 T^{2} + \cdots + 2944584 \)
|
| $79$ |
\( T^{3} - 776 T^{2} + \cdots + 401260544 \)
|
| $83$ |
\( T^{3} - 2340 T^{2} + \cdots - 431863872 \)
|
| $89$ |
\( T^{3} + 684 T^{2} + \cdots - 23701376 \)
|
| $97$ |
\( T^{3} + \cdots - 1708711256 \)
|
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