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: |
\( x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{5} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 19\nu + 22 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 23\nu^{2} + 29\nu + 54 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 25\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{3} + 32\beta_{2} + 92\beta _1 + 264 \)
|
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 |
|
−4.80647 | 3.00000 | 15.1022 | 5.00000 | −14.4194 | 0 | −34.1365 | 9.00000 | −24.0324 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.44043 | 3.00000 | −5.92517 | 5.00000 | −4.32128 | 0 | 20.0582 | 9.00000 | −7.20214 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.461250 | 3.00000 | −7.78725 | 5.00000 | 1.38375 | 0 | −7.28187 | 9.00000 | 2.30625 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.67994 | 3.00000 | 5.54199 | 5.00000 | 11.0398 | 0 | −9.04535 | 9.00000 | 18.3997 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.10571 | 3.00000 | 18.0682 | 5.00000 | 15.3171 | 0 | 51.4055 | 9.00000 | 25.5285 | ||||||||||||||||||||||||||||||||||
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.ba | 5 | |
| 3.b | odd | 2 | 1 | 2205.4.a.br | 5 | ||
| 7.b | odd | 2 | 1 | 735.4.a.z | 5 | ||
| 7.c | even | 3 | 2 | 105.4.i.d | ✓ | 10 | |
| 21.c | even | 2 | 1 | 2205.4.a.bs | 5 | ||
| 21.h | odd | 6 | 2 | 315.4.j.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.i.d | ✓ | 10 | 7.c | even | 3 | 2 | |
| 315.4.j.h | 10 | 21.h | odd | 6 | 2 | ||
| 735.4.a.z | 5 | 7.b | odd | 2 | 1 | ||
| 735.4.a.ba | 5 | 1.a | even | 1 | 1 | trivial | |
| 2205.4.a.br | 5 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.bs | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 28T_{2}^{3} + 70T_{2}^{2} + 104T_{2} - 60 \)
|
|
\( T_{11}^{5} - 43T_{11}^{4} - 3788T_{11}^{3} + 74940T_{11}^{2} + 4542452T_{11} + 35201580 \)
|
|
\( T_{13}^{5} - 123T_{13}^{4} - 1074T_{13}^{3} + 566590T_{13}^{2} - 14427235T_{13} - 125988247 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 60 \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( (T - 5)^{5} \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 43 T^{4} + \cdots + 35201580 \)
|
| $13$ |
\( T^{5} - 123 T^{4} + \cdots - 125988247 \)
|
| $17$ |
\( T^{5} - 124 T^{4} + \cdots + 20514480 \)
|
| $19$ |
\( T^{5} - 37 T^{4} + \cdots - 235295225 \)
|
| $23$ |
\( T^{5} - 77 T^{4} + \cdots + 152403300 \)
|
| $29$ |
\( T^{5} + \cdots - 15483733056 \)
|
| $31$ |
\( T^{5} + \cdots - 69079322070 \)
|
| $37$ |
\( T^{5} + \cdots - 212593667013 \)
|
| $41$ |
\( T^{5} + \cdots - 914820763500 \)
|
| $43$ |
\( T^{5} + \cdots + 45414054020 \)
|
| $47$ |
\( T^{5} + \cdots + 439521150192 \)
|
| $53$ |
\( T^{5} + \cdots + 12693539474880 \)
|
| $59$ |
\( T^{5} + \cdots - 1309411612752 \)
|
| $61$ |
\( T^{5} + \cdots - 16475195520000 \)
|
| $67$ |
\( T^{5} + \cdots - 74697164857140 \)
|
| $71$ |
\( T^{5} + \cdots + 12244368636072 \)
|
| $73$ |
\( T^{5} + \cdots + 65528941041926 \)
|
| $79$ |
\( T^{5} + \cdots + 1149495333516 \)
|
| $83$ |
\( T^{5} + \cdots + 217219935694608 \)
|
| $89$ |
\( T^{5} + \cdots - 8415535021416 \)
|
| $97$ |
\( T^{5} + \cdots + 207081920604160 \)
|
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