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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{43}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 43 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{43}\). 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 |
|
0 | 3.00000 | 0 | −7.11488 | 0 | 8.88512 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 19.1149 | 0 | 35.1149 | 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.f | ✓ | 2 |
| 3.b | odd | 2 | 1 | 936.4.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 624.4.a.l | 2 | ||
| 8.b | even | 2 | 1 | 2496.4.a.u | 2 | ||
| 8.d | odd | 2 | 1 | 2496.4.a.bd | 2 | ||
| 12.b | even | 2 | 1 | 1872.4.a.v | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.4.a.f | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 624.4.a.l | 2 | 4.b | odd | 2 | 1 | ||
| 936.4.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 1872.4.a.v | 2 | 12.b | even | 2 | 1 | ||
| 2496.4.a.u | 2 | 8.b | even | 2 | 1 | ||
| 2496.4.a.bd | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 12T_{5} - 136 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} - 12T - 136 \)
|
| $7$ |
\( T^{2} - 44T + 312 \)
|
| $11$ |
\( (T - 26)^{2} \)
|
| $13$ |
\( (T + 13)^{2} \)
|
| $17$ |
\( T^{2} + 20T - 588 \)
|
| $19$ |
\( T^{2} + 60T - 3400 \)
|
| $23$ |
\( T^{2} + 8T - 24752 \)
|
| $29$ |
\( T^{2} - 132T - 20412 \)
|
| $31$ |
\( T^{2} + 140T - 33800 \)
|
| $37$ |
\( T^{2} - 68T - 32556 \)
|
| $41$ |
\( T^{2} - 28T - 8232 \)
|
| $43$ |
\( T^{2} - 33712 \)
|
| $47$ |
\( T^{2} - 36T - 5868 \)
|
| $53$ |
\( T^{2} - 668T + 108804 \)
|
| $59$ |
\( T^{2} + 508T - 18732 \)
|
| $61$ |
\( T^{2} - 340T + 17892 \)
|
| $67$ |
\( T^{2} + 940T - 97128 \)
|
| $71$ |
\( T^{2} - 300T - 772828 \)
|
| $73$ |
\( T^{2} - 1124 T + 309652 \)
|
| $79$ |
\( T^{2} + 1520 T + 244608 \)
|
| $83$ |
\( T^{2} - 524T - 234764 \)
|
| $89$ |
\( T^{2} - 1900 T + 902328 \)
|
| $97$ |
\( T^{2} + 1436 T + 459796 \)
|
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