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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.51264.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 15x^{2} + 16x + 46 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 2x^{3} - 15x^{2} + 16x + 46 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 8 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 11\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + \beta _1 + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 3\beta_{2} + 12\beta _1 + 8 \)
|
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 |
|
−4.91605 | 3.00000 | 16.1676 | −5.00000 | −14.7482 | 0 | −40.1522 | 9.00000 | 24.5803 | ||||||||||||||||||||||||||||||
| 1.2 | −3.62028 | 3.00000 | 5.10642 | −5.00000 | −10.8608 | 0 | 10.4756 | 9.00000 | 18.1014 | |||||||||||||||||||||||||||||||
| 1.3 | −0.912375 | 3.00000 | −7.16757 | −5.00000 | −2.73712 | 0 | 13.8385 | 9.00000 | 4.56187 | |||||||||||||||||||||||||||||||
| 1.4 | 3.44871 | 3.00000 | 3.89358 | −5.00000 | 10.3461 | 0 | −14.1618 | 9.00000 | −17.2435 | |||||||||||||||||||||||||||||||
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.u | yes | 4 |
| 3.b | odd | 2 | 1 | 2205.4.a.bq | 4 | ||
| 7.b | odd | 2 | 1 | 735.4.a.t | ✓ | 4 | |
| 21.c | even | 2 | 1 | 2205.4.a.bp | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 735.4.a.t | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 735.4.a.u | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 2205.4.a.bp | 4 | 21.c | even | 2 | 1 | ||
| 2205.4.a.bq | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{4} + 6T_{2}^{3} - 7T_{2}^{2} - 72T_{2} - 56 \)
|
|
\( T_{11}^{4} + 24T_{11}^{3} - 58T_{11}^{2} - 2808T_{11} - 10808 \)
|
|
\( T_{13}^{4} - 1096T_{13}^{2} - 3072T_{13} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6 T^{3} + \cdots - 56 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 24 T^{3} + \cdots - 10808 \)
|
| $13$ |
\( T^{4} - 1096 T^{2} + \cdots + 1024 \)
|
| $17$ |
\( T^{4} + 88 T^{3} + \cdots - 1153424 \)
|
| $19$ |
\( T^{4} - 72 T^{3} + \cdots + 5456016 \)
|
| $23$ |
\( T^{4} - 11698 T^{2} + \cdots + 9614584 \)
|
| $29$ |
\( T^{4} + 636 T^{3} + \cdots + 349146808 \)
|
| $31$ |
\( T^{4} - 228 T^{3} + \cdots - 24242148 \)
|
| $37$ |
\( T^{4} + 68 T^{3} + \cdots + 170898952 \)
|
| $41$ |
\( T^{4} + 56 T^{3} + \cdots + 126356848 \)
|
| $43$ |
\( T^{4} - 20 T^{3} + \cdots + 454079200 \)
|
| $47$ |
\( T^{4} + 744 T^{3} + \cdots + 276948792 \)
|
| $53$ |
\( T^{4} + 360 T^{3} + \cdots + 111504456 \)
|
| $59$ |
\( T^{4} + \cdots - 32798657504 \)
|
| $61$ |
\( T^{4} + \cdots - 37509812744 \)
|
| $67$ |
\( T^{4} + \cdots - 149047690736 \)
|
| $71$ |
\( T^{4} + \cdots + 56245124128 \)
|
| $73$ |
\( T^{4} + \cdots - 4390308576 \)
|
| $79$ |
\( T^{4} + \cdots + 4846395904 \)
|
| $83$ |
\( T^{4} + \cdots - 201571856352 \)
|
| $89$ |
\( T^{4} + \cdots - 9397074032 \)
|
| $97$ |
\( T^{4} + \cdots + 72223884864 \)
|
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