Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,104,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, 104, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 104);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 104 \) |
| 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: | \(67.1843880807\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3 x^{7} + \cdots + 10\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | multiple of \( 2^{104}\cdot 3^{40}\cdot 5^{12}\cdot 7^{8}\cdot 11\cdot 13^{3}\cdot 17^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3 x^{7} + \cdots + 10\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 16\!\cdots\!41 \nu^{7} + \cdots - 21\!\cdots\!46 ) / 59\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 57\!\cdots\!09 \nu^{7} + \cdots - 86\!\cdots\!86 ) / 59\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 44\!\cdots\!29 \nu^{7} + \cdots - 27\!\cdots\!50 ) / 82\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17\!\cdots\!41 \nu^{7} + \cdots + 19\!\cdots\!50 ) / 14\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 55\!\cdots\!79 \nu^{7} + \cdots + 19\!\cdots\!50 ) / 94\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 36\!\cdots\!03 \nu^{7} + \cdots + 40\!\cdots\!50 ) / 48\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 9 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 34249\beta_{2} + 90668611228163\beta _1 + 15880774323327381508482134600472 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 134 \beta_{6} - 75477 \beta_{5} + 783459941 \beta_{4} + 731911083234300 \beta_{3} + \cdots + 14\!\cdots\!36 ) / 13824 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 127486023401493 \beta_{7} + \cdots + 51\!\cdots\!96 ) / 41472 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 15\!\cdots\!27 \beta_{7} + \cdots + 68\!\cdots\!64 ) / 31104 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 23\!\cdots\!71 \beta_{7} + \cdots + 43\!\cdots\!04 ) / 6912 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12\!\cdots\!93 \beta_{7} + \cdots + 98\!\cdots\!88 ) / 41472 \)
|
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 |
|
−5.46018e15 | −1.30099e24 | 1.96724e31 | −5.30727e35 | 7.10366e39 | 2.61912e43 | −5.20418e46 | −1.22226e49 | 2.89786e51 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.84396e15 | 3.76369e24 | 4.63479e30 | 1.31095e36 | −1.44675e40 | −5.54417e43 | 2.11664e46 | 2.50204e47 | −5.03922e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.58307e15 | −4.68333e24 | −7.63510e30 | 1.63346e35 | 7.41404e39 | −1.85939e42 | 2.81411e46 | 8.01842e48 | −2.58588e50 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.38660e15 | 4.94875e24 | −8.21853e30 | −1.47562e36 | −6.86195e39 | 3.27966e43 | 2.54577e46 | 1.05749e49 | 2.04610e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.03451e15 | 2.78778e24 | −6.00197e30 | 1.50905e36 | 5.67177e39 | 3.17629e43 | −3.28435e46 | −6.14348e48 | 3.07017e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.09018e15 | −9.30646e23 | −5.91967e29 | −7.61227e35 | −2.87587e39 | −3.39343e43 | −3.31675e46 | −1.30491e49 | −2.35233e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.46569e15 | −5.92176e24 | 1.97326e31 | 6.19706e35 | −3.23665e40 | 6.38506e43 | 5.24235e46 | 2.11521e49 | 3.38712e51 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 6.07235e15 | 6.42408e24 | 2.67322e31 | −2.83827e35 | 3.90093e40 | −2.16284e43 | 1.00747e47 | 2.73537e49 | −1.72350e51 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1.104.a.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 9.104.a.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.104.a.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 9.104.a.a | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{104}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 96\!\cdots\!36 \)
|
| $3$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $5$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $11$ |
\( T^{8} + \cdots + 90\!\cdots\!76 \)
|
| $13$ |
\( T^{8} + \cdots + 70\!\cdots\!96 \)
|
| $17$ |
\( T^{8} + \cdots - 30\!\cdots\!24 \)
|
| $19$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 17\!\cdots\!36 \)
|
| $29$ |
\( T^{8} + \cdots - 31\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 14\!\cdots\!84 \)
|
| $37$ |
\( T^{8} + \cdots + 32\!\cdots\!96 \)
|
| $41$ |
\( T^{8} + \cdots - 24\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots + 13\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots - 35\!\cdots\!44 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!56 \)
|
| $59$ |
\( T^{8} + \cdots - 34\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 14\!\cdots\!76 \)
|
| $67$ |
\( T^{8} + \cdots + 89\!\cdots\!76 \)
|
| $71$ |
\( T^{8} + \cdots - 20\!\cdots\!04 \)
|
| $73$ |
\( T^{8} + \cdots + 57\!\cdots\!36 \)
|
| $79$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 27\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots - 14\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 84\!\cdots\!56 \)
|
| show more | |
| show less |