Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,4,Mod(1,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("637.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 637.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: | \(37.5842166737\) |
| 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: |
\( x^{6} - 2x^{5} - 27x^{4} + 42x^{3} + 154x^{2} - 156x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{6} - 2x^{5} - 27x^{4} + 42x^{3} + 154x^{2} - 156x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 27\nu^{3} + 38\nu^{2} + 158\nu - 92 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} - 55\nu^{3} + 56\nu^{2} + 328\nu - 148 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 3\nu^{4} + 57\nu^{3} - 58\nu^{2} - 360\nu + 164 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 16\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} - 4\beta_{3} + 21\beta_{2} + 4\beta _1 + 166 \)
|
| \(\nu^{5}\) | \(=\) |
\( 29\beta_{5} + 33\beta_{4} - 6\beta_{3} + 31\beta_{2} + 282\beta _1 + 98 \)
|
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 |
|
−4.20122 | −7.70488 | 9.65024 | −16.6197 | 32.3699 | 0 | −6.93300 | 32.3652 | 69.8228 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.68895 | 1.13312 | −0.769540 | 11.1663 | −3.04690 | 0 | 23.5809 | −25.7160 | −30.0256 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.412587 | −5.53235 | −7.82977 | −15.7299 | −2.28258 | 0 | −6.53115 | 3.60695 | −6.48995 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.498984 | 9.32638 | −7.75101 | −8.07429 | 4.65372 | 0 | −7.85951 | 59.9813 | −4.02894 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.32935 | −9.72173 | 3.08456 | 9.41246 | −32.3670 | 0 | −16.3652 | 67.5120 | 31.3374 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.64925 | −0.500533 | 13.6155 | −6.15490 | −2.32710 | 0 | 26.1080 | −26.7495 | −28.6157 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 637.4.a.f | 6 | |
| 7.b | odd | 2 | 1 | 91.4.a.d | ✓ | 6 | |
| 21.c | even | 2 | 1 | 819.4.a.l | 6 | ||
| 28.d | even | 2 | 1 | 1456.4.a.y | 6 | ||
| 35.c | odd | 2 | 1 | 2275.4.a.l | 6 | ||
| 91.b | odd | 2 | 1 | 1183.4.a.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.4.a.d | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 637.4.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 819.4.a.l | 6 | 21.c | even | 2 | 1 | ||
| 1183.4.a.g | 6 | 91.b | odd | 2 | 1 | ||
| 1456.4.a.y | 6 | 28.d | even | 2 | 1 | ||
| 2275.4.a.l | 6 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(637))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 27T_{2}^{4} + 42T_{2}^{3} + 154T_{2}^{2} - 156T_{2} + 36 \)
|
|
\( T_{3}^{6} + 13T_{3}^{5} - 52T_{3}^{4} - 1164T_{3}^{3} - 3092T_{3}^{2} + 3116T_{3} + 2192 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 36 \)
|
| $3$ |
\( T^{6} + 13 T^{5} + \cdots + 2192 \)
|
| $5$ |
\( T^{6} + 26 T^{5} + \cdots + 1365480 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 11 T^{5} + \cdots - 50674032 \)
|
| $13$ |
\( (T + 13)^{6} \)
|
| $17$ |
\( T^{6} + \cdots + 109733452128 \)
|
| $19$ |
\( T^{6} + \cdots + 2430493176 \)
|
| $23$ |
\( T^{6} + 75 T^{5} + \cdots + 50143752 \)
|
| $29$ |
\( T^{6} + \cdots + 30771381744 \)
|
| $31$ |
\( T^{6} + \cdots - 11538231146432 \)
|
| $37$ |
\( T^{6} + \cdots + 341114263632920 \)
|
| $41$ |
\( T^{6} + \cdots - 1208949930720 \)
|
| $43$ |
\( T^{6} + \cdots + 10083630522144 \)
|
| $47$ |
\( T^{6} + \cdots + 521676349537104 \)
|
| $53$ |
\( T^{6} + \cdots + 5376223832400 \)
|
| $59$ |
\( T^{6} + \cdots + 19\!\cdots\!32 \)
|
| $61$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 73\!\cdots\!80 \)
|
| $71$ |
\( T^{6} + \cdots + 39\!\cdots\!36 \)
|
| $73$ |
\( T^{6} + \cdots + 575424087273346 \)
|
| $79$ |
\( T^{6} + \cdots - 633670985363184 \)
|
| $83$ |
\( T^{6} + \cdots - 65\!\cdots\!36 \)
|
| $89$ |
\( T^{6} + \cdots + 40\!\cdots\!60 \)
|
| $97$ |
\( T^{6} + \cdots + 67\!\cdots\!54 \)
|
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