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: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{6} - 7\nu^{5} + 115\nu^{4} + 125\nu^{3} - 1654\nu^{2} + 897\nu + 3640 ) / 159 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{6} - 55\nu^{5} - 164\nu^{4} + 1709\nu^{3} + 65\nu^{2} - 9102\nu - 974 ) / 318 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{6} - 41\nu^{5} - 394\nu^{4} + 1141\nu^{3} + 3373\nu^{2} - 4536\nu - 6982 ) / 318 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 17\nu^{6} - 20\nu^{5} - 580\nu^{4} + 448\nu^{3} + 4042\nu^{2} - 867\nu - 4069 ) / 159 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + 20\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - 4\beta_{5} + 2\beta_{4} + \beta_{3} + 27\beta_{2} + 27\beta _1 + 218 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{6} - 39\beta_{5} + 29\beta_{4} - 31\beta_{3} + 9\beta_{2} + 474\beta _1 + 118 \)
|
| \(\nu^{6}\) | \(=\) |
\( 47\beta_{6} - 156\beta_{5} + 76\beta_{4} + 24\beta_{3} + 694\beta_{2} + 765\beta _1 + 5095 \)
|
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 |
|
−5.24052 | −3.00000 | 19.4631 | 5.00000 | 15.7216 | −18.4475 | −60.0724 | 9.00000 | −26.2026 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.57720 | −3.00000 | 4.79637 | 5.00000 | 10.7316 | 15.0281 | 11.4600 | 9.00000 | −17.8860 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.27711 | −3.00000 | −2.81476 | 5.00000 | 6.83134 | −26.8439 | 24.6264 | 9.00000 | −11.3856 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.218728 | −3.00000 | −7.95216 | 5.00000 | −0.656184 | −21.0000 | −3.48919 | 9.00000 | 1.09364 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.40909 | −3.00000 | −6.01448 | 5.00000 | −4.22726 | 22.4156 | −19.7476 | 9.00000 | 7.04543 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.58378 | −3.00000 | −1.32408 | 5.00000 | −7.75134 | −26.6398 | −24.0914 | 9.00000 | 12.9189 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.88324 | −3.00000 | 15.8460 | 5.00000 | −14.6497 | 5.48750 | 38.3141 | 9.00000 | 24.4162 | ||||||||||||||||||||||||||||||||||||||||
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.i | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1305.4.a.n | 7 | ||
| 5.b | even | 2 | 1 | 2175.4.a.n | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.i | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1305.4.a.n | 7 | 3.b | odd | 2 | 1 | ||
| 2175.4.a.n | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 2T_{2}^{6} - 37T_{2}^{5} - 55T_{2}^{4} + 336T_{2}^{3} + 227T_{2}^{2} - 824T_{2} + 166 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(435))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots + 166 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( (T - 5)^{7} \)
|
| $7$ |
\( T^{7} + 50 T^{6} + \cdots - 512109952 \)
|
| $11$ |
\( T^{7} - 76 T^{6} + \cdots + 340000992 \)
|
| $13$ |
\( T^{7} + \cdots - 330782597664 \)
|
| $17$ |
\( T^{7} + \cdots + 4628388110976 \)
|
| $19$ |
\( T^{7} + \cdots - 1327363884544 \)
|
| $23$ |
\( T^{7} + \cdots - 61779908096 \)
|
| $29$ |
\( (T - 29)^{7} \)
|
| $31$ |
\( T^{7} + \cdots + 206419680059392 \)
|
| $37$ |
\( T^{7} + \cdots - 20\!\cdots\!76 \)
|
| $41$ |
\( T^{7} + \cdots - 56\!\cdots\!48 \)
|
| $43$ |
\( T^{7} + \cdots - 52\!\cdots\!36 \)
|
| $47$ |
\( T^{7} + \cdots - 46\!\cdots\!96 \)
|
| $53$ |
\( T^{7} + \cdots - 84\!\cdots\!08 \)
|
| $59$ |
\( T^{7} + \cdots - 50\!\cdots\!08 \)
|
| $61$ |
\( T^{7} + \cdots - 44\!\cdots\!36 \)
|
| $67$ |
\( T^{7} + \cdots - 53\!\cdots\!76 \)
|
| $71$ |
\( T^{7} + \cdots - 21\!\cdots\!32 \)
|
| $73$ |
\( T^{7} + \cdots + 88\!\cdots\!68 \)
|
| $79$ |
\( T^{7} + \cdots + 49\!\cdots\!36 \)
|
| $83$ |
\( T^{7} + \cdots - 11\!\cdots\!68 \)
|
| $89$ |
\( T^{7} + \cdots - 50\!\cdots\!48 \)
|
| $97$ |
\( T^{7} + \cdots - 22\!\cdots\!96 \)
|
| show more | |
| show less |