Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,4,Mod(1,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("322.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.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: | \(18.9986150218\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{73}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). 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 |
|
−2.00000 | 0.227998 | 4.00000 | −21.0880 | −0.455996 | 7.00000 | −8.00000 | −26.9480 | 42.1760 | ||||||||||||||||||||||||
| 1.2 | −2.00000 | 8.77200 | 4.00000 | 13.0880 | −17.5440 | 7.00000 | −8.00000 | 49.9480 | −26.1760 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 322.4.a.b | ✓ | 2 |
| 7.b | odd | 2 | 1 | 2254.4.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.c | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 9T_{3} + 2 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} - 9T + 2 \)
|
| $5$ |
\( T^{2} + 8T - 276 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( T^{2} - 8T - 276 \)
|
| $13$ |
\( T^{2} + 57T - 1396 \)
|
| $17$ |
\( T^{2} - 78T + 864 \)
|
| $19$ |
\( T^{2} + 90T - 1552 \)
|
| $23$ |
\( (T - 23)^{2} \)
|
| $29$ |
\( T^{2} - 283T + 19566 \)
|
| $31$ |
\( T^{2} + 153T + 2768 \)
|
| $37$ |
\( T^{2} + 490T + 47688 \)
|
| $41$ |
\( T^{2} + 279T - 52974 \)
|
| $43$ |
\( T^{2} - 190T - 23168 \)
|
| $47$ |
\( T^{2} - 151T - 45564 \)
|
| $53$ |
\( T^{2} - 630T + 19728 \)
|
| $59$ |
\( T^{2} - 46T - 204528 \)
|
| $61$ |
\( T^{2} - 292T - 73292 \)
|
| $67$ |
\( T^{2} + 1080 T + 291308 \)
|
| $71$ |
\( T^{2} - 619T - 246744 \)
|
| $73$ |
\( T^{2} - 1283 T + 260394 \)
|
| $79$ |
\( T^{2} - 2210 T + 1219200 \)
|
| $83$ |
\( T^{2} - 1932 T + 804384 \)
|
| $89$ |
\( T^{2} + 1042 T - 415416 \)
|
| $97$ |
\( T^{2} + 1422 T + 370544 \)
|
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