Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,4,Mod(1,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, 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("39.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.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: | \(2.30107449022\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{14}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 14 \)
|
| 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 \(\beta = \sqrt{14}\). 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 |
|
−2.74166 | −3.00000 | −0.483315 | 19.4833 | 8.22497 | 7.48331 | 23.2583 | 9.00000 | −53.4166 | ||||||||||||||||||||||||
| 1.2 | 4.74166 | −3.00000 | 14.4833 | 4.51669 | −14.2250 | −7.48331 | 30.7417 | 9.00000 | 21.4166 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 39.4.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 117.4.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 624.4.a.r | 2 | ||
| 5.b | even | 2 | 1 | 975.4.a.j | 2 | ||
| 7.b | odd | 2 | 1 | 1911.4.a.h | 2 | ||
| 8.b | even | 2 | 1 | 2496.4.a.bc | 2 | ||
| 8.d | odd | 2 | 1 | 2496.4.a.s | 2 | ||
| 12.b | even | 2 | 1 | 1872.4.a.t | 2 | ||
| 13.b | even | 2 | 1 | 507.4.a.f | 2 | ||
| 13.d | odd | 4 | 2 | 507.4.b.f | 4 | ||
| 39.d | odd | 2 | 1 | 1521.4.a.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 117.4.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 507.4.a.f | 2 | 13.b | even | 2 | 1 | ||
| 507.4.b.f | 4 | 13.d | odd | 4 | 2 | ||
| 624.4.a.r | 2 | 4.b | odd | 2 | 1 | ||
| 975.4.a.j | 2 | 5.b | even | 2 | 1 | ||
| 1521.4.a.s | 2 | 39.d | odd | 2 | 1 | ||
| 1872.4.a.t | 2 | 12.b | even | 2 | 1 | ||
| 1911.4.a.h | 2 | 7.b | odd | 2 | 1 | ||
| 2496.4.a.s | 2 | 8.d | odd | 2 | 1 | ||
| 2496.4.a.bc | 2 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} - 13 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(39))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T - 13 \)
|
| $3$ |
\( (T + 3)^{2} \)
|
| $5$ |
\( T^{2} - 24T + 88 \)
|
| $7$ |
\( T^{2} - 56 \)
|
| $11$ |
\( T^{2} + 44T - 1532 \)
|
| $13$ |
\( (T + 13)^{2} \)
|
| $17$ |
\( T^{2} - 164T + 6500 \)
|
| $19$ |
\( T^{2} - 48T + 520 \)
|
| $23$ |
\( T^{2} - 8T - 32240 \)
|
| $29$ |
\( T^{2} - 404T + 32740 \)
|
| $31$ |
\( T^{2} - 40T - 9064 \)
|
| $37$ |
\( T^{2} + 100T - 8476 \)
|
| $41$ |
\( T^{2} - 200T - 113704 \)
|
| $43$ |
\( T^{2} + 616T + 57008 \)
|
| $47$ |
\( T^{2} + 324T + 11908 \)
|
| $53$ |
\( T^{2} + 164T - 194876 \)
|
| $59$ |
\( T^{2} - 140T - 17500 \)
|
| $61$ |
\( T^{2} - 628T - 160348 \)
|
| $67$ |
\( T^{2} + 472T - 348904 \)
|
| $71$ |
\( T^{2} - 428T - 52988 \)
|
| $73$ |
\( T^{2} + 900T + 121636 \)
|
| $79$ |
\( T^{2} + 432T - 61760 \)
|
| $83$ |
\( T^{2} + 1388 T + 424292 \)
|
| $89$ |
\( T^{2} - 960T - 275000 \)
|
| $97$ |
\( T^{2} + 532T - 606844 \)
|
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