Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [735,4,Mod(1,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, 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("735.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 735.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: | \(43.3664038542\) |
| 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: |
\( x^{5} - x^{4} - 30x^{3} + 2x^{2} + 164x + 84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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
\( x^{5} - x^{4} - 30x^{3} + 2x^{2} + 164x + 84 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 19\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 24\nu^{2} + 44\nu + 84 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 20\beta _1 + 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{3} + 27\beta_{2} + 40\beta _1 + 243 \)
|
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 |
|
−5.36047 | −3.00000 | 20.7346 | 5.00000 | 16.0814 | 0 | −68.2635 | 9.00000 | −26.8023 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.83495 | −3.00000 | 0.0369369 | 5.00000 | 8.50485 | 0 | 22.5749 | 9.00000 | −14.1747 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.544516 | −3.00000 | −7.70350 | 5.00000 | −1.63355 | 0 | −8.55081 | 9.00000 | 2.72258 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.37290 | −3.00000 | −2.36934 | 5.00000 | −7.11870 | 0 | −24.6054 | 9.00000 | 11.8645 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.27800 | −3.00000 | 10.3013 | 5.00000 | −12.8340 | 0 | 9.84488 | 9.00000 | 21.3900 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 735.4.a.x | 5 | |
| 3.b | odd | 2 | 1 | 2205.4.a.bv | 5 | ||
| 7.b | odd | 2 | 1 | 735.4.a.y | 5 | ||
| 7.d | odd | 6 | 2 | 105.4.i.e | ✓ | 10 | |
| 21.c | even | 2 | 1 | 2205.4.a.bw | 5 | ||
| 21.g | even | 6 | 2 | 315.4.j.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.i.e | ✓ | 10 | 7.d | odd | 6 | 2 | |
| 315.4.j.f | 10 | 21.g | even | 6 | 2 | ||
| 735.4.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
| 735.4.a.y | 5 | 7.b | odd | 2 | 1 | ||
| 2205.4.a.bv | 5 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.bw | 5 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{5} + T_{2}^{4} - 30T_{2}^{3} - 2T_{2}^{2} + 164T_{2} - 84 \)
|
|
\( T_{11}^{5} + 33T_{11}^{4} - 4896T_{11}^{3} - 131808T_{11}^{2} + 3902092T_{11} - 1409268 \)
|
|
\( T_{13}^{5} - 23T_{13}^{4} - 5870T_{13}^{3} + 134666T_{13}^{2} + 4614873T_{13} + 25478281 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} + \cdots - 84 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 33 T^{4} + \cdots - 1409268 \)
|
| $13$ |
\( T^{5} - 23 T^{4} + \cdots + 25478281 \)
|
| $17$ |
\( T^{5} + \cdots - 18600240000 \)
|
| $19$ |
\( T^{5} + \cdots - 1499415779 \)
|
| $23$ |
\( T^{5} + \cdots - 3734385060 \)
|
| $29$ |
\( T^{5} + \cdots - 12806983008 \)
|
| $31$ |
\( T^{5} + \cdots - 2020193658 \)
|
| $37$ |
\( T^{5} + \cdots + 12781138293 \)
|
| $41$ |
\( T^{5} + \cdots - 45007298820 \)
|
| $43$ |
\( T^{5} + \cdots + 1217258521160 \)
|
| $47$ |
\( T^{5} + \cdots + 126271211376 \)
|
| $53$ |
\( T^{5} + \cdots + 1034244891456 \)
|
| $59$ |
\( T^{5} + \cdots + 9210995795856 \)
|
| $61$ |
\( T^{5} + \cdots - 1704614400000 \)
|
| $67$ |
\( T^{5} + \cdots + 21938021087360 \)
|
| $71$ |
\( T^{5} + \cdots - 27239555642904 \)
|
| $73$ |
\( T^{5} + \cdots + 19578152376194 \)
|
| $79$ |
\( T^{5} + \cdots - 18750107484808 \)
|
| $83$ |
\( T^{5} + \cdots - 59475285699552 \)
|
| $89$ |
\( T^{5} + \cdots + 13240319676648 \)
|
| $97$ |
\( T^{5} + \cdots - 701198534208512 \)
|
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