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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{7}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 7 \)
|
| 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 \(\beta = 2\sqrt{7}\). 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 | −7.29150 | 0 | 5.87451 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | 3.29150 | 0 | −25.8745 | 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.e | ✓ | 2 |
| 3.b | odd | 2 | 1 | 936.4.a.f | 2 | ||
| 4.b | odd | 2 | 1 | 624.4.a.k | 2 | ||
| 8.b | even | 2 | 1 | 2496.4.a.y | 2 | ||
| 8.d | odd | 2 | 1 | 2496.4.a.bh | 2 | ||
| 12.b | even | 2 | 1 | 1872.4.a.bf | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.4.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 624.4.a.k | 2 | 4.b | odd | 2 | 1 | ||
| 936.4.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 1872.4.a.bf | 2 | 12.b | even | 2 | 1 | ||
| 2496.4.a.y | 2 | 8.b | even | 2 | 1 | ||
| 2496.4.a.bh | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 4T_{5} - 24 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(312))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} + 4T - 24 \)
|
| $7$ |
\( T^{2} + 20T - 152 \)
|
| $11$ |
\( T^{2} + 60T + 452 \)
|
| $13$ |
\( (T + 13)^{2} \)
|
| $17$ |
\( T^{2} + 84T + 756 \)
|
| $19$ |
\( T^{2} + 60T - 7192 \)
|
| $23$ |
\( T^{2} + 72T - 9904 \)
|
| $29$ |
\( T^{2} + 124T - 50364 \)
|
| $31$ |
\( T^{2} + 108T - 75736 \)
|
| $37$ |
\( T^{2} - 36T - 58924 \)
|
| $41$ |
\( T^{2} + 52T - 61176 \)
|
| $43$ |
\( T^{2} + 32T - 8816 \)
|
| $47$ |
\( T^{2} + 428T - 91404 \)
|
| $53$ |
\( T^{2} - 380T - 39612 \)
|
| $59$ |
\( T^{2} + 1420 T + 471732 \)
|
| $61$ |
\( T^{2} - 1012 T + 254244 \)
|
| $67$ |
\( T^{2} + 844T + 176712 \)
|
| $71$ |
\( T^{2} + 868T + 172228 \)
|
| $73$ |
\( T^{2} + 60T - 663148 \)
|
| $79$ |
\( T^{2} - 272T + 7296 \)
|
| $83$ |
\( T^{2} + 1252T - 24876 \)
|
| $89$ |
\( T^{2} - 572T - 556632 \)
|
| $97$ |
\( T^{2} - 708T - 62956 \)
|
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