Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(8.17957481046\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{145}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 36 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{145})\). 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 |
|
−9.52080 | 9.00000 | 58.6456 | −86.6040 | −85.6872 | 192.666 | −253.687 | 81.0000 | 824.539 | ||||||||||||||||||||||||
| 1.2 | 2.52080 | 9.00000 | −25.6456 | −26.3960 | 22.6872 | −192.666 | −145.313 | 81.0000 | −66.5390 | |||||||||||||||||||||||||
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 | 51.6.a.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 153.6.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 816.6.a.f | 2 | ||
| 17.b | even | 2 | 1 | 867.6.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.6.a.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 153.6.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 816.6.a.f | 2 | 4.b | odd | 2 | 1 | ||
| 867.6.a.d | 2 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 7T_{2} - 24 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 7T - 24 \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} + 113T + 2286 \)
|
| $7$ |
\( T^{2} - 37120 \)
|
| $11$ |
\( T^{2} - 297T + 19116 \)
|
| $13$ |
\( T^{2} + 1479 T + 479834 \)
|
| $17$ |
\( (T - 289)^{2} \)
|
| $19$ |
\( T^{2} + 2147 T + 369076 \)
|
| $23$ |
\( T^{2} + 6395 T + 8356080 \)
|
| $29$ |
\( T^{2} + 4584 T + 4997484 \)
|
| $31$ |
\( T^{2} + 1110 T - 12655120 \)
|
| $37$ |
\( T^{2} - 14790 T + 23063120 \)
|
| $41$ |
\( T^{2} + 15435 T + 57312270 \)
|
| $43$ |
\( T^{2} - 3723 T - 47782204 \)
|
| $47$ |
\( T^{2} - 5286T + 31104 \)
|
| $53$ |
\( T^{2} + 21120 T - 20453220 \)
|
| $59$ |
\( T^{2} + \cdots - 1405542744 \)
|
| $61$ |
\( T^{2} + 56654 T + 600862984 \)
|
| $67$ |
\( T^{2} + 25776 T - 538253776 \)
|
| $71$ |
\( T^{2} + \cdots + 5270424384 \)
|
| $73$ |
\( T^{2} - 28760 T + 166666380 \)
|
| $79$ |
\( T^{2} + 28990 T + 199456080 \)
|
| $83$ |
\( T^{2} + 82606 T - 38986536 \)
|
| $89$ |
\( T^{2} + \cdots - 4631690664 \)
|
| $97$ |
\( T^{2} - 85714 T - 264798656 \)
|
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