Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [114,3,Mod(31,114)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(114, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("114.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 114 = 2 \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 114.f (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(3.10627501371\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.520060207104.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} - 44x^{6} + 664x^{4} - 3528x^{2} + 8100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 22\nu^{2} + 12\nu + 90 ) / 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 22\nu^{2} + 12\nu - 90 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{6} - 77\nu^{4} + 1002\nu^{2} - 156\nu - 3378 ) / 312 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 44\nu^{5} - 574\nu^{3} + 1548\nu + 1080 ) / 2160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} - 15\nu^{6} + 263\nu^{5} + 480\nu^{4} - 3208\nu^{3} - 3030\nu^{2} + 8586\nu - 9180 ) / 14040 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 15\nu^{6} + 263\nu^{5} - 480\nu^{4} - 3208\nu^{3} + 3030\nu^{2} + 8586\nu + 9180 ) / 14040 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} + 90\nu^{6} - 664\nu^{5} - 2880\nu^{4} + 14234\nu^{3} + 32220\nu^{2} - 100908\nu - 99360 ) / 28080 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -3\beta_{6} + 3\beta_{5} + \beta_{3} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{6} - 6\beta_{5} + 18\beta_{4} - 3\beta_{3} + 16\beta_{2} + 13\beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( -66\beta_{6} + 66\beta_{5} + 22\beta_{3} - 12\beta_{2} + 34\beta _1 + 152 \)
|
| \(\nu^{5}\) | \(=\) |
\( 108\beta_{7} - 264\beta_{6} - 264\beta_{5} + 660\beta_{4} - 54\beta_{3} + 238\beta_{2} + 184\beta _1 - 330 \)
|
| \(\nu^{6}\) | \(=\) |
\( -1038\beta_{6} + 1038\beta_{5} + 502\beta_{3} - 384\beta_{2} + 886\beta _1 + 2030 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1308\beta_{7} - 8172\beta_{6} - 8172\beta_{5} + 16548\beta_{4} - 654\beta_{3} + 2836\beta_{2} + 2182\beta _1 - 8274 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).
| \(n\) | \(77\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
−1.22474 | + | 0.707107i | 1.50000 | − | 0.866025i | 1.00000 | − | 1.73205i | −3.25884 | − | 5.64448i | −1.22474 | + | 2.12132i | −11.0157 | 2.82843i | 1.50000 | − | 2.59808i | 7.98249 | + | 4.60870i | ||||||||||||||||||||||||||||
| 31.2 | −1.22474 | + | 0.707107i | 1.50000 | − | 0.866025i | 1.00000 | − | 1.73205i | 3.03409 | + | 5.25521i | −1.22474 | + | 2.12132i | −2.33275 | 2.82843i | 1.50000 | − | 2.59808i | −7.43198 | − | 4.29086i | |||||||||||||||||||||||||||||
| 31.3 | 1.22474 | − | 0.707107i | 1.50000 | − | 0.866025i | 1.00000 | − | 1.73205i | −2.40185 | − | 4.16012i | 1.22474 | − | 2.12132i | 2.71177 | − | 2.82843i | 1.50000 | − | 2.59808i | −5.88330 | − | 3.39673i | ||||||||||||||||||||||||||||
| 31.4 | 1.22474 | − | 0.707107i | 1.50000 | − | 0.866025i | 1.00000 | − | 1.73205i | 4.62659 | + | 8.01349i | 1.22474 | − | 2.12132i | −1.36330 | − | 2.82843i | 1.50000 | − | 2.59808i | 11.3328 | + | 6.54299i | ||||||||||||||||||||||||||||
| 103.1 | −1.22474 | − | 0.707107i | 1.50000 | + | 0.866025i | 1.00000 | + | 1.73205i | −3.25884 | + | 5.64448i | −1.22474 | − | 2.12132i | −11.0157 | − | 2.82843i | 1.50000 | + | 2.59808i | 7.98249 | − | 4.60870i | ||||||||||||||||||||||||||||
| 103.2 | −1.22474 | − | 0.707107i | 1.50000 | + | 0.866025i | 1.00000 | + | 1.73205i | 3.03409 | − | 5.25521i | −1.22474 | − | 2.12132i | −2.33275 | − | 2.82843i | 1.50000 | + | 2.59808i | −7.43198 | + | 4.29086i | ||||||||||||||||||||||||||||
| 103.3 | 1.22474 | + | 0.707107i | 1.50000 | + | 0.866025i | 1.00000 | + | 1.73205i | −2.40185 | + | 4.16012i | 1.22474 | + | 2.12132i | 2.71177 | 2.82843i | 1.50000 | + | 2.59808i | −5.88330 | + | 3.39673i | |||||||||||||||||||||||||||||
| 103.4 | 1.22474 | + | 0.707107i | 1.50000 | + | 0.866025i | 1.00000 | + | 1.73205i | 4.62659 | − | 8.01349i | 1.22474 | + | 2.12132i | −1.36330 | 2.82843i | 1.50000 | + | 2.59808i | 11.3328 | − | 6.54299i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 114.3.f.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 342.3.m.b | 8 | ||
| 4.b | odd | 2 | 1 | 912.3.be.g | 8 | ||
| 19.d | odd | 6 | 1 | inner | 114.3.f.b | ✓ | 8 |
| 57.f | even | 6 | 1 | 342.3.m.b | 8 | ||
| 76.f | even | 6 | 1 | 912.3.be.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.3.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 114.3.f.b | ✓ | 8 | 19.d | odd | 6 | 1 | inner |
| 342.3.m.b | 8 | 3.b | odd | 2 | 1 | ||
| 342.3.m.b | 8 | 57.f | even | 6 | 1 | ||
| 912.3.be.g | 8 | 4.b | odd | 2 | 1 | ||
| 912.3.be.g | 8 | 76.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 4T_{5}^{7} + 102T_{5}^{6} + 32T_{5}^{5} + 6262T_{5}^{4} + 648T_{5}^{3} + 175524T_{5}^{2} + 274248T_{5} + 3090564 \)
acting on \(S_{3}^{\mathrm{new}}(114, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{4} \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 3090564 \)
|
| $7$ |
\( (T^{4} + 12 T^{3} + \cdots - 95)^{2} \)
|
| $11$ |
\( (T^{4} + 4 T^{3} + \cdots + 1902)^{2} \)
|
| $13$ |
\( T^{8} - 24 T^{7} + \cdots + 845588241 \)
|
| $17$ |
\( T^{8} + \cdots + 42571243584 \)
|
| $19$ |
\( T^{8} + \cdots + 16983563041 \)
|
| $23$ |
\( T^{8} - 40 T^{7} + \cdots + 1726596 \)
|
| $29$ |
\( T^{8} + 48 T^{7} + \cdots + 269485056 \)
|
| $31$ |
\( T^{8} + \cdots + 157608206001 \)
|
| $37$ |
\( T^{8} + \cdots + 3853057329 \)
|
| $41$ |
\( T^{8} + \cdots + 4588472453184 \)
|
| $43$ |
\( T^{8} + \cdots + 196975358413681 \)
|
| $47$ |
\( T^{8} + \cdots + 791744040000 \)
|
| $53$ |
\( T^{8} + \cdots + 984988761156 \)
|
| $59$ |
\( T^{8} + \cdots + 76622956930116 \)
|
| $61$ |
\( T^{8} + \cdots + 2354429392225 \)
|
| $67$ |
\( T^{8} + \cdots + 421690488129 \)
|
| $71$ |
\( T^{8} + \cdots + 158714643240000 \)
|
| $73$ |
\( T^{8} + \cdots + 14\!\cdots\!21 \)
|
| $79$ |
\( T^{8} + \cdots + 36966995842401 \)
|
| $83$ |
\( (T^{4} - 212 T^{3} + \cdots - 740232)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 341891268051204 \)
|
| $97$ |
\( T^{8} + \cdots + 312852286224 \)
|
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