Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,8,Mod(1,276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(276, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("276.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 276.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: | \(86.2182670336\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - x^{4} - 82397x^{3} + 7346801x^{2} + 425424304x - 21076942492 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{5} - x^{4} - 82397x^{3} + 7346801x^{2} + 425424304x - 21076942492 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -19\nu^{4} - 192\nu^{3} + 1769105\nu^{2} - 79028394\nu - 14102318872 ) / 25580880 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -10\nu^{4} - 1971\nu^{3} + 501020\nu^{2} + 31717395\nu - 1413769426 ) / 2558088 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -802\nu^{4} - 129651\nu^{3} + 43521530\nu^{2} + 1010516613\nu - 99352314166 ) / 12790440 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -8\beta_{4} + 104\beta_{3} + 128\beta_{2} - 131\beta _1 + 65900 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1840\beta_{4} - 26656\beta_{3} - 15040\beta_{2} + 69335\beta _1 - 8730612 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -763480\beta_{4} + 9952888\beta_{3} + 9377440\beta_{2} - 17057551\beta _1 + 4739771540 ) / 2 \)
|
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 |
|
0 | −27.0000 | 0 | −386.322 | 0 | −1140.12 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | −156.453 | 0 | 660.254 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 155.923 | 0 | −137.396 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −27.0000 | 0 | 292.009 | 0 | 1498.76 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −27.0000 | 0 | 346.844 | 0 | −467.495 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(23\) | \(1\) |
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 | 276.8.a.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 276.8.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 252T_{5}^{4} - 170050T_{5}^{3} + 45210400T_{5}^{2} + 3570710000T_{5} - 954494730000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(276))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 954494730000 \)
|
| $7$ |
\( T^{5} + \cdots + 72467814851536 \)
|
| $11$ |
\( T^{5} + \cdots + 55\!\cdots\!84 \)
|
| $13$ |
\( T^{5} + \cdots - 19\!\cdots\!80 \)
|
| $17$ |
\( T^{5} + \cdots + 37\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots - 87\!\cdots\!60 \)
|
| $23$ |
\( (T + 12167)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 35\!\cdots\!92 \)
|
| $31$ |
\( T^{5} + \cdots - 12\!\cdots\!48 \)
|
| $37$ |
\( T^{5} + \cdots - 85\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 87\!\cdots\!80 \)
|
| $43$ |
\( T^{5} + \cdots - 12\!\cdots\!28 \)
|
| $47$ |
\( T^{5} + \cdots - 18\!\cdots\!88 \)
|
| $53$ |
\( T^{5} + \cdots - 39\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots - 59\!\cdots\!48 \)
|
| $61$ |
\( T^{5} + \cdots + 64\!\cdots\!12 \)
|
| $67$ |
\( T^{5} + \cdots - 21\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots - 39\!\cdots\!60 \)
|
| $73$ |
\( T^{5} + \cdots - 71\!\cdots\!52 \)
|
| $79$ |
\( T^{5} + \cdots - 84\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 70\!\cdots\!96 \)
|
| $89$ |
\( T^{5} + \cdots - 46\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots + 64\!\cdots\!28 \)
|
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