Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,70,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, 70, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 70);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 70 \) |
| 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: | \(30.1514953292\) |
| 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: |
\( x^{5} - x^{4} + \cdots - 94\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{43}\cdot 3^{17}\cdot 5^{5}\cdot 7^{2}\cdot 17\cdot 23 \) |
| 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
\( x^{5} - x^{4} + \cdots - 94\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 288\nu - 58 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 13089 \nu^{4} + 125836814991 \nu^{3} + \cdots - 55\!\cdots\!80 ) / 12\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 7840311 \nu^{4} + 75376252179609 \nu^{3} + \cdots - 10\!\cdots\!84 ) / 12\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 133380259041 \nu^{4} + \cdots - 15\!\cdots\!12 ) / 12\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 58 ) / 288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 599\beta_{2} + 5343840437\beta _1 + 828972018021666404459 ) / 82944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 279232 \beta_{4} + 447604165 \beta_{3} + 2577330631373 \beta_{2} + \cdots + 13\!\cdots\!31 ) / 746496 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2684518719808 \beta_{4} + \cdots + 13\!\cdots\!31 ) / 746496 \)
|
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 |
|
−4.75433e10 | −9.70523e15 | 1.67007e21 | −1.40726e24 | 4.61419e26 | −1.18429e27 | −5.13361e31 | −7.40194e32 | 6.69057e34 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.83774e10 | 2.28315e16 | −2.52566e20 | 1.12358e24 | −4.19583e26 | −8.82897e28 | 1.54896e31 | −3.13110e32 | −2.06484e34 | ||||||||||||||||||||||||||||||||||
| 1.3 | −6.72544e9 | −4.13926e16 | −5.45064e20 | −6.57446e23 | 2.78383e26 | 1.09182e29 | 7.63579e30 | 8.78961e32 | 4.42161e33 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.12545e10 | 3.67245e16 | −1.38540e20 | −2.01597e24 | 7.80563e26 | 2.05778e29 | −1.54911e31 | 5.14307e32 | −4.28486e34 | ||||||||||||||||||||||||||||||||||
| 1.5 | 3.33859e10 | −1.33163e16 | 5.24322e20 | 1.09328e24 | −4.44575e26 | −1.48686e29 | −2.20259e30 | −6.57062e32 | 3.65000e34 | ||||||||||||||||||||||||||||||||||
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 | 1.70.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 9.70.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.70.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 9.70.a.b | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 41\!\cdots\!68 \)
|
| $3$ |
\( T^{5} + \cdots + 44\!\cdots\!24 \)
|
| $5$ |
\( T^{5} + \cdots + 22\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 34\!\cdots\!68 \)
|
| $11$ |
\( T^{5} + \cdots - 77\!\cdots\!32 \)
|
| $13$ |
\( T^{5} + \cdots - 72\!\cdots\!76 \)
|
| $17$ |
\( T^{5} + \cdots - 31\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 35\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 72\!\cdots\!24 \)
|
| $29$ |
\( T^{5} + \cdots + 11\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 40\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots - 13\!\cdots\!32 \)
|
| $41$ |
\( T^{5} + \cdots + 12\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots + 61\!\cdots\!24 \)
|
| $47$ |
\( T^{5} + \cdots + 32\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots - 48\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 16\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 50\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots + 11\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 37\!\cdots\!76 \)
|
| $79$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 16\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 93\!\cdots\!32 \)
|
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