Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,112,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, 112, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 112);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 112 \) |
| 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: | \(78.0257547452\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} + \cdots + 83\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{135}\cdot 3^{56}\cdot 5^{16}\cdot 7^{7}\cdot 11^{3}\cdot 13\cdot 19\cdot 37^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} + \cdots + 83\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 72\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\!\cdots\!23 \nu^{8} + \cdots - 73\!\cdots\!80 ) / 38\!\cdots\!52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 24\!\cdots\!99 \nu^{8} + \cdots - 36\!\cdots\!00 ) / 95\!\cdots\!88 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 39\!\cdots\!89 \nu^{8} + \cdots - 12\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 44\!\cdots\!81 \nu^{8} + \cdots + 43\!\cdots\!00 ) / 18\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 10\!\cdots\!03 \nu^{8} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 12\!\cdots\!89 \nu^{8} + \cdots + 88\!\cdots\!00 ) / 52\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 22\!\cdots\!33 \nu^{8} + \cdots - 16\!\cdots\!00 ) / 19\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 72 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 439252\beta_{2} + 8323470902756930\beta _1 + 3819942678867500861095870306183680 ) / 5184 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 544 \beta_{6} + 1158411 \beta_{5} + 36618818906 \beta_{4} + \cdots + 31\!\cdots\!60 ) / 373248 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2151791746368 \beta_{8} + \cdots + 32\!\cdots\!40 ) / 3359232 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 17\!\cdots\!60 \beta_{8} + \cdots + 30\!\cdots\!80 ) / 15116544 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 17\!\cdots\!20 \beta_{8} + \cdots + 21\!\cdots\!80 ) / 15116544 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 14\!\cdots\!40 \beta_{8} + \cdots + 13\!\cdots\!60 ) / 30233088 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 16\!\cdots\!96 \beta_{8} + \cdots + 23\!\cdots\!20 ) / 10077696 \)
|
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 |
|
−8.96393e16 | 3.53927e26 | 5.43906e33 | 4.17124e38 | −3.17258e43 | 1.21017e47 | −2.54837e50 | 3.39667e52 | −3.73907e55 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.33726e16 | −2.81001e26 | 2.78739e33 | 1.03167e38 | 2.06178e43 | −1.16535e47 | −1.40317e49 | −1.23360e52 | −7.56964e54 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.15853e16 | −6.11584e25 | −8.66814e32 | −6.51103e38 | 2.54329e42 | 1.10096e47 | 1.44008e50 | −8.75572e52 | 2.70763e55 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.48662e16 | 5.20768e26 | −1.97782e33 | −7.34308e38 | −1.29495e43 | −1.22695e47 | 1.13737e50 | 1.79902e53 | 1.82594e55 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.13057e15 | 1.34574e26 | −2.59487e33 | 1.00336e39 | −1.52145e41 | −8.67417e45 | 5.86879e48 | −7.31873e52 | −1.13437e54 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 8.60376e15 | −5.90178e26 | −2.52212e33 | 1.16842e38 | −5.07775e42 | 5.43178e46 | −4.40364e49 | 2.57013e53 | 1.00528e54 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.34436e16 | −3.57964e25 | 2.60068e32 | −4.56192e38 | −1.91309e42 | −2.39634e46 | −1.24849e50 | −9.00162e52 | −2.43805e55 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 7.72845e16 | 5.46172e26 | 3.37675e33 | 2.64030e38 | 4.22106e43 | 1.15348e47 | 6.03282e49 | 2.07006e53 | 2.04054e55 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 9.85626e16 | −3.53632e26 | 7.11844e33 | 7.51271e38 | −3.48549e43 | −5.05213e46 | 4.45728e50 | 3.37583e52 | 7.40472e55 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 1.112.a.a | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.112.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace is the entire newspace \(S_{112}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 26\!\cdots\!24 \)
|
| $3$ |
\( T^{9} + \cdots + 17\!\cdots\!88 \)
|
| $5$ |
\( T^{9} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots + 12\!\cdots\!04 \)
|
| $11$ |
\( T^{9} + \cdots + 50\!\cdots\!48 \)
|
| $13$ |
\( T^{9} + \cdots + 35\!\cdots\!48 \)
|
| $17$ |
\( T^{9} + \cdots - 14\!\cdots\!36 \)
|
| $19$ |
\( T^{9} + \cdots + 80\!\cdots\!00 \)
|
| $23$ |
\( T^{9} + \cdots - 39\!\cdots\!92 \)
|
| $29$ |
\( T^{9} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( T^{9} + \cdots - 20\!\cdots\!32 \)
|
| $37$ |
\( T^{9} + \cdots + 27\!\cdots\!84 \)
|
| $41$ |
\( T^{9} + \cdots + 54\!\cdots\!28 \)
|
| $43$ |
\( T^{9} + \cdots - 84\!\cdots\!72 \)
|
| $47$ |
\( T^{9} + \cdots + 92\!\cdots\!44 \)
|
| $53$ |
\( T^{9} + \cdots - 49\!\cdots\!12 \)
|
| $59$ |
\( T^{9} + \cdots + 84\!\cdots\!00 \)
|
| $61$ |
\( T^{9} + \cdots + 69\!\cdots\!48 \)
|
| $67$ |
\( T^{9} + \cdots - 24\!\cdots\!36 \)
|
| $71$ |
\( T^{9} + \cdots + 11\!\cdots\!08 \)
|
| $73$ |
\( T^{9} + \cdots - 60\!\cdots\!92 \)
|
| $79$ |
\( T^{9} + \cdots - 25\!\cdots\!00 \)
|
| $83$ |
\( T^{9} + \cdots + 74\!\cdots\!68 \)
|
| $89$ |
\( T^{9} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots + 15\!\cdots\!44 \)
|
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