Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,90,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, 90, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 90);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 90 \) |
| 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: | \(50.1624291928\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3 x^{6} + \cdots + 56\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{83}\cdot 3^{29}\cdot 5^{9}\cdot 7^{5}\cdot 11^{2}\cdot 13^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3 x^{6} + \cdots + 56\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 48\nu - 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13\!\cdots\!73 \nu^{6} + \cdots + 12\!\cdots\!60 ) / 25\!\cdots\!88 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 15\!\cdots\!61 \nu^{6} + \cdots - 10\!\cdots\!64 ) / 95\!\cdots\!44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18\!\cdots\!99 \nu^{6} + \cdots - 44\!\cdots\!40 ) / 12\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 33\!\cdots\!43 \nu^{6} + \cdots - 52\!\cdots\!08 ) / 32\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 37\!\cdots\!79 \nu^{6} + \cdots - 44\!\cdots\!80 ) / 32\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 21 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 31239\beta_{2} - 4751692611940\beta _1 + 921949945033243971463488761 ) / 2304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 42 \beta_{6} + 8322 \beta_{5} - 10260742 \beta_{4} - 535021437249 \beta_{3} + \cdots - 27\!\cdots\!73 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 42626265665746 \beta_{6} + \cdots + 58\!\cdots\!79 ) / 20736 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 14\!\cdots\!94 \beta_{6} + \cdots - 13\!\cdots\!39 ) / 20736 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 21\!\cdots\!42 \beta_{6} + \cdots + 16\!\cdots\!17 ) / 6912 \)
|
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 |
|
−4.81096e13 | −1.58991e21 | 1.69556e27 | 1.07790e31 | 7.64900e34 | 5.70296e37 | −5.17945e40 | −3.81503e41 | −5.18572e44 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.19029e13 | 3.23465e21 | 3.98828e26 | 1.00222e30 | −1.03195e35 | 1.60489e37 | 7.02319e39 | 7.55364e42 | −3.19739e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.24646e13 | −6.44834e20 | −1.14310e26 | −1.81181e31 | 1.44860e34 | −2.76134e37 | 1.64729e40 | −2.49351e42 | 4.07017e44 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.03857e13 | −5.04839e20 | −5.11108e26 | 2.06848e31 | 5.24310e33 | −1.95248e37 | 1.17366e40 | −2.65446e42 | −2.14826e44 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.79872e13 | −3.11193e21 | −2.95431e26 | −4.75717e30 | −5.59749e34 | 3.90338e37 | −1.64475e40 | 6.77478e42 | −8.55680e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.92430e13 | 1.59613e21 | −2.48679e26 | −3.02831e30 | 3.07142e34 | 1.70526e37 | −1.66961e40 | −3.61697e41 | −5.82737e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.42254e13 | −3.38901e20 | 1.33691e27 | 3.67006e30 | −1.49880e34 | −4.35255e37 | 3.17513e40 | −2.79447e42 | 1.62310e44 | ||||||||||||||||||||||||||||||||||||||||
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 | 1.90.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9.90.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.90.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 9.90.a.b | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{90}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 54\!\cdots\!12 \)
|
| $3$ |
\( T^{7} + \cdots + 28\!\cdots\!36 \)
|
| $5$ |
\( T^{7} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 14\!\cdots\!48 \)
|
| $11$ |
\( T^{7} + \cdots - 39\!\cdots\!48 \)
|
| $13$ |
\( T^{7} + \cdots + 39\!\cdots\!36 \)
|
| $17$ |
\( T^{7} + \cdots - 15\!\cdots\!32 \)
|
| $19$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 11\!\cdots\!64 \)
|
| $29$ |
\( T^{7} + \cdots - 50\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!88 \)
|
| $37$ |
\( T^{7} + \cdots + 13\!\cdots\!08 \)
|
| $41$ |
\( T^{7} + \cdots + 32\!\cdots\!92 \)
|
| $43$ |
\( T^{7} + \cdots + 45\!\cdots\!36 \)
|
| $47$ |
\( T^{7} + \cdots + 12\!\cdots\!28 \)
|
| $53$ |
\( T^{7} + \cdots - 35\!\cdots\!64 \)
|
| $59$ |
\( T^{7} + \cdots + 20\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 77\!\cdots\!52 \)
|
| $67$ |
\( T^{7} + \cdots - 40\!\cdots\!32 \)
|
| $71$ |
\( T^{7} + \cdots - 57\!\cdots\!68 \)
|
| $73$ |
\( T^{7} + \cdots - 21\!\cdots\!64 \)
|
| $79$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 58\!\cdots\!36 \)
|
| $89$ |
\( T^{7} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 61\!\cdots\!72 \)
|
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