Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(58.6902423003\) |
| 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: |
\( x^{5} - x^{4} - 1624322x^{3} - 425897108x^{2} + 341963527256x + 109228040475424 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4}\cdot 5\cdot 7^{2}\cdot 19 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 1624322x^{3} - 425897108x^{2} + 341963527256x + 109228040475424 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 1578\nu - 2598600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 755\nu^{3} - 1296972\nu^{2} + 520664260\nu + 254103417136 ) / 3360 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\nu^{4} - 7625\nu^{3} - 27565668\nu^{2} + 2957191180\nu + 5004873397744 ) / 33600 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 789\beta _1 + 2598600 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{4} - 38\beta_{3} + 435\beta_{2} + 2407331\beta _1 + 1025088338 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 15100\beta_{4} - 15250\beta_{3} + 1625397\beta_{2} + 1799517293\beta _1 + 3127839465846 ) / 4 \)
|
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 |
|
−1967.31 | −59049.0 | 1.77317e6 | −1.05945e7 | 1.16168e8 | 2.82475e8 | 6.37379e8 | 3.48678e9 | 2.08427e10 | |||||||||||||||||||||||||||||||||
| 1.2 | −966.130 | −59049.0 | −1.16375e6 | −1.73436e7 | 5.70490e7 | 2.82475e8 | 3.15045e9 | 3.48678e9 | 1.67562e10 | ||||||||||||||||||||||||||||||||||
| 1.3 | −761.540 | −59049.0 | −1.51721e6 | 3.46280e7 | 4.49682e7 | 2.82475e8 | 2.75248e9 | 3.48678e9 | −2.63706e10 | ||||||||||||||||||||||||||||||||||
| 1.4 | 975.791 | −59049.0 | −1.14498e6 | 9.00667e6 | −5.76195e7 | 2.82475e8 | −3.16365e9 | 3.48678e9 | 8.78863e9 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2581.19 | −59049.0 | 4.56540e6 | 8.67068e6 | −1.52417e8 | 2.82475e8 | 6.37102e9 | 3.48678e9 | 2.23807e10 | ||||||||||||||||||||||||||||||||||
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 | 21.22.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 63.22.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.22.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 63.22.a.c | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 138T_{2}^{4} - 6489672T_{2}^{3} - 3952738944T_{2}^{2} + 5265335808000T_{2} + 3645683116277760 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(21))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 36\!\cdots\!60 \)
|
| $3$ |
\( (T + 59049)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 49\!\cdots\!00 \)
|
| $7$ |
\( (T - 282475249)^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 15\!\cdots\!96 \)
|
| $13$ |
\( T^{5} + \cdots + 13\!\cdots\!20 \)
|
| $17$ |
\( T^{5} + \cdots + 28\!\cdots\!04 \)
|
| $19$ |
\( T^{5} + \cdots + 67\!\cdots\!44 \)
|
| $23$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots - 11\!\cdots\!28 \)
|
| $31$ |
\( T^{5} + \cdots - 38\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 19\!\cdots\!04 \)
|
| $41$ |
\( T^{5} + \cdots - 85\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots + 15\!\cdots\!40 \)
|
| $47$ |
\( T^{5} + \cdots - 31\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 43\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots + 36\!\cdots\!20 \)
|
| $61$ |
\( T^{5} + \cdots - 13\!\cdots\!88 \)
|
| $67$ |
\( T^{5} + \cdots + 71\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 16\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots - 95\!\cdots\!84 \)
|
| $83$ |
\( T^{5} + \cdots - 27\!\cdots\!48 \)
|
| $89$ |
\( T^{5} + \cdots + 32\!\cdots\!36 \)
|
| $97$ |
\( T^{5} + \cdots - 50\!\cdots\!12 \)
|
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