Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,4,Mod(1,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("460.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 460.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: | \(27.1408786026\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 118x^{4} + 155x^{3} + 3095x^{2} - 6472x + 2800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 118x^{4} + 155x^{3} + 3095x^{2} - 6472x + 2800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} + 122\nu^{3} - 307\nu^{2} - 3307\nu + 1436 ) / 288 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{4} + 138\nu^{3} + 205\nu^{2} - 4451\nu + 2012 ) / 192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 110\nu^{3} - 53\nu^{2} + 2557\nu - 2036 ) / 72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - \nu^{4} - 658\nu^{3} + 575\nu^{2} + 19967\nu - 31372 ) / 576 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta_{2} + 39 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} + 5\beta_{3} + 2\beta_{2} + 67\beta _1 - 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( 62\beta_{5} + 6\beta_{4} + 50\beta_{3} + 104\beta_{2} - 9\beta _1 + 2504 \)
|
| \(\nu^{5}\) | \(=\) |
\( -119\beta_{5} + 396\beta_{4} + 553\beta_{3} + 169\beta_{2} + 4822\beta _1 - 1921 \)
|
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 | −8.89275 | 0 | 5.00000 | 0 | −3.33881 | 0 | 52.0810 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.61024 | 0 | 5.00000 | 0 | 31.5484 | 0 | 4.47474 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.50639 | 0 | 5.00000 | 0 | −34.1899 | 0 | −24.7308 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.618658 | 0 | 5.00000 | 0 | 7.66036 | 0 | −26.6173 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 6.89901 | 0 | 5.00000 | 0 | 27.4822 | 0 | 20.5963 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.72903 | 0 | 5.00000 | 0 | −5.16219 | 0 | 49.1960 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 460.4.a.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 1840.4.a.t | 6 | ||
| 5.b | even | 2 | 1 | 2300.4.a.f | 6 | ||
| 5.c | odd | 4 | 2 | 2300.4.c.f | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.4.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1840.4.a.t | 6 | 4.b | odd | 2 | 1 | ||
| 2300.4.a.f | 6 | 5.b | even | 2 | 1 | ||
| 2300.4.c.f | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + T_{3}^{5} - 118T_{3}^{4} - 155T_{3}^{3} + 3095T_{3}^{2} + 6472T_{3} + 2800 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(460))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots + 2800 \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 24 T^{5} + \cdots - 3913824 \)
|
| $11$ |
\( T^{6} - 117 T^{5} + \cdots - 1658400 \)
|
| $13$ |
\( T^{6} + \cdots - 5281482996 \)
|
| $17$ |
\( T^{6} + \cdots - 72263824416 \)
|
| $19$ |
\( T^{6} + \cdots - 4665066336 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 3737986034040 \)
|
| $31$ |
\( T^{6} + \cdots - 1551436429200 \)
|
| $37$ |
\( T^{6} + \cdots - 245607645467520 \)
|
| $41$ |
\( T^{6} + \cdots - 12067610347602 \)
|
| $43$ |
\( T^{6} + \cdots + 16885856051200 \)
|
| $47$ |
\( T^{6} + \cdots + 26318668967424 \)
|
| $53$ |
\( T^{6} + \cdots + 38672088621984 \)
|
| $59$ |
\( T^{6} + \cdots + 34\!\cdots\!72 \)
|
| $61$ |
\( T^{6} + \cdots + 40\!\cdots\!36 \)
|
| $67$ |
\( T^{6} + \cdots - 30\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots - 79\!\cdots\!80 \)
|
| $73$ |
\( T^{6} + \cdots + 31\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots - 22\!\cdots\!20 \)
|
| $83$ |
\( T^{6} + \cdots + 99\!\cdots\!60 \)
|
| $89$ |
\( T^{6} + \cdots - 37\!\cdots\!80 \)
|
| $97$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
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