Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,30,Mod(1,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 30);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.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: | \(111.883889004\) |
| 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: |
\( x^{6} - 110200839 x^{4} - 84515300136 x^{3} + \cdots - 12\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{8}\cdot 5^{2}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 110200839 x^{4} - 84515300136 x^{3} + \cdots - 12\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{2} - 18408\nu - 587737808 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 404935 \nu^{5} - 579698838 \nu^{4} - 41735794063989 \nu^{3} + \cdots - 10\!\cdots\!24 ) / 75744294807552 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 312163 \nu^{5} - 1124954862 \nu^{4} - 24630414484425 \nu^{3} + \cdots - 56\!\cdots\!64 ) / 113616442211328 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1212725 \nu^{5} + 8282972802 \nu^{4} + 114843506785023 \nu^{3} + \cdots + 60\!\cdots\!64 ) / 227232884422656 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 4602\beta _1 + 587737808 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 29\beta_{5} + 140\beta_{4} - 43\beta_{3} + 802\beta_{2} + 114783130\beta _1 + 338061200544 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 645256 \beta_{5} + 434080 \beta_{4} + 421064 \beta_{3} + 82193003 \beta_{2} + 559830704094 \beta _1 + 33\!\cdots\!68 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3450838047 \beta_{5} + 14740215108 \beta_{4} - 2634100857 \beta_{3} + 107725010466 \beta_{2} + \cdots + 41\!\cdots\!80 ) / 8 \)
|
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 |
|
−30888.8 | 4.78297e6 | 4.17246e8 | −8.65686e9 | −1.47740e11 | 6.78223e11 | 3.69508e12 | 2.28768e13 | 2.67400e14 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −20705.9 | 4.78297e6 | −1.08136e8 | 1.18612e10 | −9.90357e10 | 6.78223e11 | 1.33555e13 | 2.28768e13 | −2.45596e14 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −12401.0 | 4.78297e6 | −3.83087e8 | −2.55800e10 | −5.93134e10 | 6.78223e11 | 1.14084e13 | 2.28768e13 | 3.17217e14 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 7564.59 | 4.78297e6 | −4.79648e8 | −7.91331e9 | 3.61812e10 | 6.78223e11 | −7.68955e12 | 2.28768e13 | −5.98610e13 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 24052.8 | 4.78297e6 | 4.16645e7 | 1.31167e10 | 1.15044e11 | 6.78223e11 | −1.19111e13 | 2.28768e13 | 3.15492e14 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 36842.3 | 4.78297e6 | 8.20484e8 | −1.68846e10 | 1.76216e11 | 6.78223e11 | 1.04489e13 | 2.28768e13 | −6.22068e14 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 21.30.a.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.30.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 4464 T_{2}^{5} - 1754910384 T_{2}^{4} - 169892674560 T_{2}^{3} + \cdots - 53\!\cdots\!68 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 53\!\cdots\!68 \)
|
| $3$ |
\( (T - 4782969)^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $7$ |
\( (T - 678223072849)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 20\!\cdots\!40 \)
|
| $13$ |
\( T^{6} + \cdots + 37\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots + 34\!\cdots\!44 \)
|
| $19$ |
\( T^{6} + \cdots - 55\!\cdots\!40 \)
|
| $23$ |
\( T^{6} + \cdots + 16\!\cdots\!72 \)
|
| $29$ |
\( T^{6} + \cdots + 11\!\cdots\!80 \)
|
| $31$ |
\( T^{6} + \cdots - 77\!\cdots\!28 \)
|
| $37$ |
\( T^{6} + \cdots + 32\!\cdots\!76 \)
|
| $41$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 53\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots - 32\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots - 66\!\cdots\!80 \)
|
| $59$ |
\( T^{6} + \cdots - 90\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 41\!\cdots\!36 \)
|
| $67$ |
\( T^{6} + \cdots - 14\!\cdots\!48 \)
|
| $71$ |
\( T^{6} + \cdots - 95\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!12 \)
|
| $79$ |
\( T^{6} + \cdots - 16\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots + 85\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots - 27\!\cdots\!40 \)
|
| $97$ |
\( T^{6} + \cdots - 16\!\cdots\!36 \)
|
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