Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,66,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 66, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 66);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 66 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(26.7572356472\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - x^{4} + \cdots - 10\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{12}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13^{2} \) |
| 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
\( x^{5} - x^{4} + \cdots - 10\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 48\nu - 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 539 \nu^{4} - 69605886173 \nu^{3} + \cdots - 78\!\cdots\!96 ) / 10\!\cdots\!52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 81389 \nu^{4} + 10510488812123 \nu^{3} + \cdots - 36\!\cdots\!08 ) / 63\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2432657389 \nu^{4} + \cdots - 36\!\cdots\!24 ) / 51\!\cdots\!76 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 10 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2416\beta_{2} + 93896648\beta _1 + 59648754527074037862 ) / 2304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 120736 \beta_{4} + 137629895 \beta_{3} + 1422344336592 \beta_{2} + \cdots + 35\!\cdots\!82 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 15591718502752 \beta_{4} + \cdots + 62\!\cdots\!94 ) / 6912 \)
|
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 |
|
−1.09073e10 | 4.46080e15 | 8.20747e19 | 5.92338e22 | −4.86551e25 | −3.00901e27 | −4.92803e29 | 9.59772e30 | −6.46079e32 | |||||||||||||||||||||||||||||||||
| 1.2 | −7.49786e9 | −4.87294e15 | 1.93244e19 | −6.28165e22 | 3.65366e25 | −4.41462e27 | 1.31731e29 | 1.34445e31 | 4.70989e32 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.32566e9 | 9.15766e14 | −3.51361e19 | 9.89719e21 | −1.21399e24 | 2.98681e27 | 9.54865e28 | −9.46242e30 | −1.31203e31 | ||||||||||||||||||||||||||||||||||
| 1.4 | 7.69444e9 | −5.56240e15 | 2.23108e19 | 6.36641e22 | −4.27995e25 | 1.82854e27 | −1.12205e29 | 2.06393e31 | 4.89860e32 | ||||||||||||||||||||||||||||||||||
| 1.5 | 8.07662e9 | 2.82746e15 | 2.83384e19 | −4.30990e22 | 2.28363e25 | −4.31488e27 | −6.90965e28 | −2.30651e30 | −3.48095e32 | ||||||||||||||||||||||||||||||||||
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 | 1.66.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9.66.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.66.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9.66.a.b | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 67\!\cdots\!68 \)
|
| $3$ |
\( T^{5} + \cdots - 31\!\cdots\!76 \)
|
| $5$ |
\( T^{5} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 31\!\cdots\!68 \)
|
| $11$ |
\( T^{5} + \cdots + 49\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots + 24\!\cdots\!24 \)
|
| $17$ |
\( T^{5} + \cdots - 20\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 67\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 45\!\cdots\!24 \)
|
| $29$ |
\( T^{5} + \cdots + 37\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 77\!\cdots\!32 \)
|
| $37$ |
\( T^{5} + \cdots + 80\!\cdots\!68 \)
|
| $41$ |
\( T^{5} + \cdots - 59\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots - 26\!\cdots\!76 \)
|
| $47$ |
\( T^{5} + \cdots - 15\!\cdots\!32 \)
|
| $53$ |
\( T^{5} + \cdots + 75\!\cdots\!24 \)
|
| $59$ |
\( T^{5} + \cdots + 78\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 25\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots + 13\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots - 13\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 15\!\cdots\!24 \)
|
| $79$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 17\!\cdots\!76 \)
|
| $89$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 37\!\cdots\!68 \)
|
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