Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,10,Mod(65,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.65");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.i (of order \(3\), degree \(2\), not 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: | \(57.6840136504\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 1116x^{4} - 3085x^{3} + 1245325x^{2} - 2341500x + 4410000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 1116x^{4} - 3085x^{3} + 1245325x^{2} - 2341500x + 4410000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 1081 \nu^{5} + 1206396 \nu^{4} + 41103264 \nu^{3} + 1346196325 \nu^{2} + \cdots + 962136423000 ) / 10405809000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20739\nu^{5} - 20704\nu^{4} + 23105664\nu^{3} - 20388855\nu^{2} + 25783208800\nu - 48478416000 ) / 48560442000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6283441 \nu^{5} - 5742096 \nu^{4} + 6408179136 \nu^{3} - 25009225165 \nu^{2} + \cdots - 13445117784000 ) / 145681326000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5096423 \nu^{5} + 24024872 \nu^{4} + 5561870848 \nu^{3} + 11292574565 \nu^{2} + \cdots + 12258477336000 ) / 24280221000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1283932 \nu^{5} + 4959877 \nu^{4} - 1488519232 \nu^{3} + 9413731140 \nu^{2} + \cdots + 6114081939000 ) / 6070055250 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} - 15\beta_{3} + 33\beta_{2} - 15\beta _1 + 33 ) / 84 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 34\beta_{5} - 17\beta_{4} - 60\beta_{3} + 31254\beta_{2} ) / 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1081\beta_{5} - 1081\beta_{4} + 16845\beta _1 + 77097 ) / 84 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -38929\beta_{5} + 77858\beta_{4} + 119145\beta_{3} - 69550023\beta_{2} + 119145\beta _1 - 69550023 ) / 84 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1237786\beta_{5} - 618893\beta_{4} + 9324660\beta_{3} + 73839966\beta_{2} ) / 42 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −11.9224 | − | 20.6501i | 0 | −541.184 | + | 937.358i | 0 | 5624.74 | + | 2952.27i | 0 | 9557.22 | − | 16553.6i | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | −6.98734 | − | 12.1024i | 0 | 859.469 | − | 1488.64i | 0 | −1802.93 | − | 6091.23i | 0 | 9743.85 | − | 16876.9i | 0 | |||||||||||||||||||||||||||||
| 65.3 | 0 | 135.410 | + | 234.536i | 0 | −684.785 | + | 1186.08i | 0 | −6327.81 | − | 558.972i | 0 | −26830.1 | + | 46471.0i | 0 | |||||||||||||||||||||||||||||
| 81.1 | 0 | −11.9224 | + | 20.6501i | 0 | −541.184 | − | 937.358i | 0 | 5624.74 | − | 2952.27i | 0 | 9557.22 | + | 16553.6i | 0 | |||||||||||||||||||||||||||||
| 81.2 | 0 | −6.98734 | + | 12.1024i | 0 | 859.469 | + | 1488.64i | 0 | −1802.93 | + | 6091.23i | 0 | 9743.85 | + | 16876.9i | 0 | |||||||||||||||||||||||||||||
| 81.3 | 0 | 135.410 | − | 234.536i | 0 | −684.785 | − | 1186.08i | 0 | −6327.81 | + | 558.972i | 0 | −26830.1 | − | 46471.0i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.10.i.b | 6 | |
| 4.b | odd | 2 | 1 | 14.10.c.a | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 112.10.i.b | 6 | |
| 12.b | even | 2 | 1 | 126.10.g.f | 6 | ||
| 28.d | even | 2 | 1 | 98.10.c.k | 6 | ||
| 28.f | even | 6 | 1 | 98.10.a.i | 3 | ||
| 28.f | even | 6 | 1 | 98.10.c.k | 6 | ||
| 28.g | odd | 6 | 1 | 14.10.c.a | ✓ | 6 | |
| 28.g | odd | 6 | 1 | 98.10.a.j | 3 | ||
| 84.n | even | 6 | 1 | 126.10.g.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.10.c.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 14.10.c.a | ✓ | 6 | 28.g | odd | 6 | 1 | |
| 98.10.a.i | 3 | 28.f | even | 6 | 1 | ||
| 98.10.a.j | 3 | 28.g | odd | 6 | 1 | ||
| 98.10.c.k | 6 | 28.d | even | 2 | 1 | ||
| 98.10.c.k | 6 | 28.f | even | 6 | 1 | ||
| 112.10.i.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 112.10.i.b | 6 | 7.c | even | 3 | 1 | inner | |
| 126.10.g.f | 6 | 12.b | even | 2 | 1 | ||
| 126.10.g.f | 6 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 233T_{3}^{5} + 64198T_{3}^{4} + 2489283T_{3}^{3} + 77161662T_{3}^{2} + 894217887T_{3} + 8143799049 \)
acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 8143799049 \)
|
| $5$ |
\( T^{6} + \cdots + 64\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 65\!\cdots\!43 \)
|
| $11$ |
\( T^{6} + \cdots + 34\!\cdots\!25 \)
|
| $13$ |
\( (T^{3} + \cdots - 62445940634280)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 38\!\cdots\!89 \)
|
| $19$ |
\( T^{6} + \cdots + 13\!\cdots\!29 \)
|
| $23$ |
\( T^{6} + \cdots + 37\!\cdots\!61 \)
|
| $29$ |
\( (T^{3} + \cdots - 12\!\cdots\!84)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 45\!\cdots\!29 \)
|
| $41$ |
\( (T^{3} + \cdots - 89\!\cdots\!08)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 56\!\cdots\!09 \)
|
| $53$ |
\( T^{6} + \cdots + 54\!\cdots\!01 \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!25 \)
|
| $61$ |
\( T^{6} + \cdots + 20\!\cdots\!21 \)
|
| $67$ |
\( T^{6} + \cdots + 13\!\cdots\!89 \)
|
| $71$ |
\( (T^{3} + \cdots - 38\!\cdots\!16)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 70\!\cdots\!25 \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!25 \)
|
| $83$ |
\( (T^{3} + \cdots + 47\!\cdots\!64)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!69 \)
|
| $97$ |
\( (T^{3} + \cdots + 24\!\cdots\!80)^{2} \)
|
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