Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,4,Mod(1,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, 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("51.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.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: | \(3.00909741029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.5912.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 14x - 10 \)
|
| 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 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
\( x^{3} - x^{2} - 14x - 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 9 \)
|
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 |
|
−2.55528 | 3.00000 | −1.47057 | 8.47057 | −7.66583 | 3.66117 | 24.1999 | 9.00000 | −21.6446 | |||||||||||||||||||||||||||
| 1.2 | 2.79543 | 3.00000 | −0.185590 | 7.18559 | 8.38628 | 19.9241 | −22.8822 | 9.00000 | 20.0868 | ||||||||||||||||||||||||||||
| 1.3 | 4.75985 | 3.00000 | 14.6562 | −7.65616 | 14.2795 | −31.5852 | 31.6823 | 9.00000 | −36.4422 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 51.4.a.e | ✓ | 3 |
| 3.b | odd | 2 | 1 | 153.4.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 816.4.a.s | 3 | ||
| 5.b | even | 2 | 1 | 1275.4.a.q | 3 | ||
| 7.b | odd | 2 | 1 | 2499.4.a.n | 3 | ||
| 12.b | even | 2 | 1 | 2448.4.a.bd | 3 | ||
| 17.b | even | 2 | 1 | 867.4.a.k | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.4.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 153.4.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 816.4.a.s | 3 | 4.b | odd | 2 | 1 | ||
| 867.4.a.k | 3 | 17.b | even | 2 | 1 | ||
| 1275.4.a.q | 3 | 5.b | even | 2 | 1 | ||
| 2448.4.a.bd | 3 | 12.b | even | 2 | 1 | ||
| 2499.4.a.n | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(51))\):
|
\( T_{2}^{3} - 5T_{2}^{2} - 6T_{2} + 34 \)
|
|
\( T_{5}^{3} - 8T_{5}^{2} - 59T_{5} + 466 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 5 T^{2} + \cdots + 34 \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots + 466 \)
|
| $7$ |
\( T^{3} + 8 T^{2} + \cdots + 2304 \)
|
| $11$ |
\( T^{3} - 34 T^{2} + \cdots - 8964 \)
|
| $13$ |
\( T^{3} - 36 T^{2} + \cdots + 122698 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} + 142 T^{2} + \cdots + 8244 \)
|
| $23$ |
\( T^{3} - 110 T^{2} + \cdots - 53240 \)
|
| $29$ |
\( T^{3} - 90 T^{2} + \cdots - 415320 \)
|
| $31$ |
\( T^{3} + 148 T^{2} + \cdots - 640448 \)
|
| $37$ |
\( T^{3} - 110 T^{2} + \cdots - 5969792 \)
|
| $41$ |
\( T^{3} - 720 T^{2} + \cdots - 10440042 \)
|
| $43$ |
\( T^{3} + 146 T^{2} + \cdots - 62624916 \)
|
| $47$ |
\( T^{3} - 500 T^{2} + \cdots + 30472896 \)
|
| $53$ |
\( T^{3} - 610 T^{2} + \cdots + 80447688 \)
|
| $59$ |
\( T^{3} + 216 T^{2} + \cdots + 1302384 \)
|
| $61$ |
\( T^{3} + 18 T^{2} + \cdots + 8127200 \)
|
| $67$ |
\( T^{3} + 1404 T^{2} + \cdots + 62069312 \)
|
| $71$ |
\( T^{3} + 960 T^{2} + \cdots - 227624576 \)
|
| $73$ |
\( T^{3} + 794 T^{2} + \cdots - 227482344 \)
|
| $79$ |
\( T^{3} + 276 T^{2} + \cdots - 220814208 \)
|
| $83$ |
\( T^{3} + 1552 T^{2} + \cdots + 11261392 \)
|
| $89$ |
\( T^{3} - 1394 T^{2} + \cdots + 278458912 \)
|
| $97$ |
\( T^{3} + \cdots + 2026068032 \)
|
| show more | |
| show less |