Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(6.56008553517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2910828.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 303x - 490 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 303x - 490 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 202 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 202 \)
|
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 |
|
−17.5337 | 27.0000 | 179.431 | −511.401 | −473.410 | 343.000 | −901.776 | 729.000 | 8966.76 | |||||||||||||||||||||||||||
| 1.2 | −2.63149 | 27.0000 | −121.075 | 328.047 | −71.0503 | 343.000 | 655.440 | 729.000 | −863.253 | ||||||||||||||||||||||||||||
| 1.3 | 17.1652 | 27.0000 | 166.644 | 69.3548 | 463.461 | 343.000 | 663.336 | 729.000 | 1190.49 | ||||||||||||||||||||||||||||
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 | 21.8.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 63.8.a.g | 3 | ||
| 4.b | odd | 2 | 1 | 336.8.a.r | 3 | ||
| 5.b | even | 2 | 1 | 525.8.a.g | 3 | ||
| 7.b | odd | 2 | 1 | 147.8.a.e | 3 | ||
| 7.c | even | 3 | 2 | 147.8.e.i | 6 | ||
| 7.d | odd | 6 | 2 | 147.8.e.j | 6 | ||
| 21.c | even | 2 | 1 | 441.8.a.n | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 63.8.a.g | 3 | 3.b | odd | 2 | 1 | ||
| 147.8.a.e | 3 | 7.b | odd | 2 | 1 | ||
| 147.8.e.i | 6 | 7.c | even | 3 | 2 | ||
| 147.8.e.j | 6 | 7.d | odd | 6 | 2 | ||
| 336.8.a.r | 3 | 4.b | odd | 2 | 1 | ||
| 441.8.a.n | 3 | 21.c | even | 2 | 1 | ||
| 525.8.a.g | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 3T_{2}^{2} - 300T_{2} - 792 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3 T^{2} + \cdots - 792 \)
|
| $3$ |
\( (T - 27)^{3} \)
|
| $5$ |
\( T^{3} + 114 T^{2} + \cdots + 11635200 \)
|
| $7$ |
\( (T - 343)^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 21104274672 \)
|
| $13$ |
\( T^{3} + \cdots + 755769903784 \)
|
| $17$ |
\( T^{3} + \cdots + 12467114860032 \)
|
| $19$ |
\( T^{3} + \cdots - 5437419408320 \)
|
| $23$ |
\( T^{3} + \cdots - 102906745004448 \)
|
| $29$ |
\( T^{3} + \cdots + 208394349111720 \)
|
| $31$ |
\( T^{3} + \cdots + 24\!\cdots\!64 \)
|
| $37$ |
\( T^{3} + \cdots + 14\!\cdots\!08 \)
|
| $41$ |
\( T^{3} + \cdots + 24\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 30\!\cdots\!36 \)
|
| $47$ |
\( T^{3} + \cdots + 22\!\cdots\!84 \)
|
| $53$ |
\( T^{3} + \cdots + 56\!\cdots\!32 \)
|
| $59$ |
\( T^{3} + \cdots - 31\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 37\!\cdots\!52 \)
|
| $67$ |
\( T^{3} + \cdots - 42\!\cdots\!36 \)
|
| $71$ |
\( T^{3} + \cdots - 56\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 28\!\cdots\!52 \)
|
| $79$ |
\( T^{3} + \cdots + 35\!\cdots\!60 \)
|
| $83$ |
\( T^{3} + \cdots + 79\!\cdots\!52 \)
|
| $89$ |
\( T^{3} + \cdots + 26\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots + 70\!\cdots\!88 \)
|
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