Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,4,Mod(1,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, 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("435.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 435.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: | \(25.6658308525\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 17\nu^{2} + 18\nu + 16 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 15\nu + 12 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 25\nu^{2} - 26\nu - 96 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 25\nu^{3} + 26\nu^{2} + 128\nu - 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 17\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21\beta_{4} + 4\beta_{3} + 29\beta_{2} + 33\beta _1 + 170 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{5} + 66\beta_{4} + 58\beta_{3} + 82\beta_{2} + 337\beta _1 + 304 \)
|
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 |
|
−4.98965 | −3.00000 | 16.8966 | 5.00000 | 14.9690 | 22.5076 | −44.3911 | 9.00000 | −24.9483 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.33404 | −3.00000 | 3.11579 | 5.00000 | 10.0021 | 1.26363 | 16.2841 | 9.00000 | −16.6702 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.691666 | −3.00000 | −7.52160 | 5.00000 | 2.07500 | −14.5720 | 10.7358 | 9.00000 | −3.45833 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.886494 | −3.00000 | −7.21413 | 5.00000 | −2.65948 | 17.9759 | −13.4872 | 9.00000 | 4.43247 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.16884 | −3.00000 | 2.04152 | 5.00000 | −9.50651 | −6.36645 | −18.8814 | 9.00000 | 15.8442 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 3.96002 | −3.00000 | 7.68179 | 5.00000 | −11.8801 | 2.19133 | −1.26012 | 9.00000 | 19.8001 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 435.4.a.g | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1305.4.a.i | 6 | ||
| 5.b | even | 2 | 1 | 2175.4.a.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1305.4.a.i | 6 | 3.b | odd | 2 | 1 | ||
| 2175.4.a.l | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 31T_{2}^{4} - 9T_{2}^{3} + 230T_{2}^{2} - 32T_{2} - 128 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(435))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots - 128 \)
|
| $3$ |
\( (T + 3)^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 23 T^{5} + \cdots + 103936 \)
|
| $11$ |
\( T^{6} + \cdots + 4447644416 \)
|
| $13$ |
\( T^{6} + 83 T^{5} + \cdots - 755576576 \)
|
| $17$ |
\( T^{6} + \cdots + 35167804928 \)
|
| $19$ |
\( T^{6} + \cdots + 70167660032 \)
|
| $23$ |
\( T^{6} + \cdots - 17557826816 \)
|
| $29$ |
\( (T + 29)^{6} \)
|
| $31$ |
\( T^{6} + \cdots + 679337811968 \)
|
| $37$ |
\( T^{6} + \cdots - 11015322431488 \)
|
| $41$ |
\( T^{6} + \cdots - 2844920728576 \)
|
| $43$ |
\( T^{6} + \cdots + 1852550152192 \)
|
| $47$ |
\( T^{6} + \cdots - 36309886959616 \)
|
| $53$ |
\( T^{6} + \cdots + 32\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots - 31\!\cdots\!44 \)
|
| $61$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 82\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots - 99\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + \cdots + 136347629123072 \)
|
| $79$ |
\( T^{6} + \cdots - 859390658797568 \)
|
| $83$ |
\( T^{6} + \cdots - 17\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots - 34\!\cdots\!76 \)
|
| $97$ |
\( T^{6} + \cdots - 45\!\cdots\!08 \)
|
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