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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 138x^{3} + 122x^{2} + 2304x + 2520 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 138x^{3} + 122x^{2} + 2304x + 2520 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{4} - 5\nu^{3} - 368\nu^{2} + 654\nu + 2892 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 7\nu^{3} + 128\nu^{2} - 794\nu - 1220 ) / 32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} + 5\nu^{3} + 432\nu^{2} - 718\nu - 6476 ) / 64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + \beta _1 + 56 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + 6\beta_{3} + 6\beta_{2} + 107\beta _1 - 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 121\beta_{4} + 10\beta_{3} + 298\beta_{2} + 83\beta _1 + 5892 \)
|
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 |
|
−10.9973 | −9.00000 | 88.9404 | −61.0146 | 98.9756 | −218.144 | −626.190 | 81.0000 | 670.996 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.93119 | −9.00000 | −23.4081 | −74.4476 | 26.3807 | 64.4251 | 162.412 | 81.0000 | 218.220 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.32128 | −9.00000 | −30.2542 | 78.2397 | 11.8915 | −203.626 | 82.2553 | 81.0000 | −103.377 | ||||||||||||||||||||||||||||||||||
| 1.4 | 5.50815 | −9.00000 | −1.66028 | 40.0248 | −49.5733 | 256.808 | −185.406 | 81.0000 | 220.463 | ||||||||||||||||||||||||||||||||||
| 1.5 | 10.7416 | −9.00000 | 83.3822 | 21.1978 | −96.6745 | 60.5370 | 551.928 | 81.0000 | 227.698 | ||||||||||||||||||||||||||||||||||
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.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 153.6.a.g | 5 | ||
| 4.b | odd | 2 | 1 | 816.6.a.u | 5 | ||
| 17.b | even | 2 | 1 | 867.6.a.i | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.6.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 153.6.a.g | 5 | 3.b | odd | 2 | 1 | ||
| 816.6.a.u | 5 | 4.b | odd | 2 | 1 | ||
| 867.6.a.i | 5 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - T_{2}^{4} - 138T_{2}^{3} + 122T_{2}^{2} + 2304T_{2} + 2520 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots + 2520 \)
|
| $3$ |
\( (T + 9)^{5} \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 301530168 \)
|
| $7$ |
\( T^{5} + \cdots - 44489834496 \)
|
| $11$ |
\( T^{5} + \cdots + 12261839113152 \)
|
| $13$ |
\( T^{5} + \cdots + 2121939133768 \)
|
| $17$ |
\( (T - 289)^{5} \)
|
| $19$ |
\( T^{5} + \cdots + 27911819012544 \)
|
| $23$ |
\( T^{5} + \cdots - 3036499623936 \)
|
| $29$ |
\( T^{5} + \cdots + 90\!\cdots\!28 \)
|
| $31$ |
\( T^{5} + \cdots - 27\!\cdots\!96 \)
|
| $37$ |
\( T^{5} + \cdots - 39\!\cdots\!44 \)
|
| $41$ |
\( T^{5} + \cdots - 35\!\cdots\!56 \)
|
| $43$ |
\( T^{5} + \cdots - 35\!\cdots\!36 \)
|
| $47$ |
\( T^{5} + \cdots - 12\!\cdots\!16 \)
|
| $53$ |
\( T^{5} + \cdots - 54\!\cdots\!52 \)
|
| $59$ |
\( T^{5} + \cdots + 11\!\cdots\!72 \)
|
| $61$ |
\( T^{5} + \cdots + 16\!\cdots\!16 \)
|
| $67$ |
\( T^{5} + \cdots + 27\!\cdots\!64 \)
|
| $71$ |
\( T^{5} + \cdots + 15\!\cdots\!96 \)
|
| $73$ |
\( T^{5} + \cdots - 22\!\cdots\!64 \)
|
| $79$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 37\!\cdots\!72 \)
|
| $89$ |
\( T^{5} + \cdots - 58\!\cdots\!60 \)
|
| $97$ |
\( T^{5} + \cdots - 24\!\cdots\!36 \)
|
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