Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("276.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(44.2659342684\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 480x^{2} + 3169x + 6509 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 480x^{2} + 3169x + 6509 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{3} + 58\nu^{2} - 1156\nu - 5647 ) / 135 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 16\nu^{2} + 848\nu - 8089 ) / 135 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 16\nu^{2} + 1388\nu - 8224 ) / 135 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{3} + 9\beta_{2} + 3\beta _1 + 482 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 188\beta_{3} - 275\beta_{2} + 24\beta _1 - 4021 ) / 2 \)
|
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 | −9.00000 | 0 | −89.1864 | 0 | −18.1690 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −13.1070 | 0 | 65.1578 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | 19.9950 | 0 | −195.062 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.00000 | 0 | 104.299 | 0 | −5.92670 | 0 | 81.0000 | 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 | 276.6.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 828.6.a.a | 4 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.p | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 276.6.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 828.6.a.a | 4 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.p | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 22T_{5}^{3} - 9460T_{5}^{2} + 68032T_{5} + 2437824 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(276))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} - 22 T^{3} + \cdots + 2437824 \)
|
| $7$ |
\( T^{4} + 154 T^{3} + \cdots - 1368624 \)
|
| $11$ |
\( T^{4} + \cdots + 6963926400 \)
|
| $13$ |
\( T^{4} + \cdots + 38171312432 \)
|
| $17$ |
\( T^{4} + \cdots + 759348879840 \)
|
| $19$ |
\( T^{4} + \cdots + 127930639632 \)
|
| $23$ |
\( (T - 529)^{4} \)
|
| $29$ |
\( T^{4} + \cdots - 392814211889232 \)
|
| $31$ |
\( T^{4} + \cdots + 263943486949632 \)
|
| $37$ |
\( T^{4} + \cdots - 147587042708592 \)
|
| $41$ |
\( T^{4} + \cdots - 13\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots + 63\!\cdots\!28 \)
|
| $47$ |
\( T^{4} + \cdots - 10\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots - 14\!\cdots\!80 \)
|
| $59$ |
\( T^{4} + \cdots + 62\!\cdots\!28 \)
|
| $61$ |
\( T^{4} + \cdots - 61\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 32\!\cdots\!44 \)
|
| $71$ |
\( T^{4} + \cdots + 40\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots - 12\!\cdots\!72 \)
|
| $79$ |
\( T^{4} + \cdots + 39\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 33\!\cdots\!80 \)
|
| $97$ |
\( T^{4} + \cdots + 23\!\cdots\!64 \)
|
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