Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.65");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(6.60821392064\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.11163123.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 14x^{4} + 49x^{2} + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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
\( x^{6} + 14x^{4} + 49x^{2} + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 7\nu + 3 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} + 8\nu^{2} - 27 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{4} + 20\nu^{2} + 27 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 2\nu^{4} + 23\nu^{3} - 8\nu^{2} + 81\nu + 27 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} + 2\nu^{4} + 23\nu^{3} + 20\nu^{2} + 45\nu + 27 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - 18 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 72\beta _1 - 36 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{3} + 5\beta_{2} + 63 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -40\beta_{5} + 58\beta_{4} + 20\beta_{3} + 29\beta_{2} - 414\beta _1 + 207 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) |
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 | −4.77144 | − | 8.26438i | 0 | 7.90468 | − | 13.6913i | 0 | 0.861792 | − | 18.5002i | 0 | −32.0333 | + | 55.4834i | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | −0.130780 | − | 0.226518i | 0 | −9.75047 | + | 16.8883i | 0 | −7.51203 | − | 16.9284i | 0 | 13.4658 | − | 23.3234i | 0 | |||||||||||||||||||||||||||||
| 65.3 | 0 | 1.40222 | + | 2.42872i | 0 | 3.34580 | − | 5.79509i | 0 | 8.65024 | + | 16.3760i | 0 | 9.56754 | − | 16.5715i | 0 | |||||||||||||||||||||||||||||
| 81.1 | 0 | −4.77144 | + | 8.26438i | 0 | 7.90468 | + | 13.6913i | 0 | 0.861792 | + | 18.5002i | 0 | −32.0333 | − | 55.4834i | 0 | |||||||||||||||||||||||||||||
| 81.2 | 0 | −0.130780 | + | 0.226518i | 0 | −9.75047 | − | 16.8883i | 0 | −7.51203 | + | 16.9284i | 0 | 13.4658 | + | 23.3234i | 0 | |||||||||||||||||||||||||||||
| 81.3 | 0 | 1.40222 | − | 2.42872i | 0 | 3.34580 | + | 5.79509i | 0 | 8.65024 | − | 16.3760i | 0 | 9.56754 | + | 16.5715i | 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 | 112.4.i.e | 6 | |
| 4.b | odd | 2 | 1 | 56.4.i.b | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 112.4.i.e | 6 | |
| 7.c | even | 3 | 1 | 784.4.a.be | 3 | ||
| 7.d | odd | 6 | 1 | 784.4.a.bb | 3 | ||
| 8.b | even | 2 | 1 | 448.4.i.m | 6 | ||
| 8.d | odd | 2 | 1 | 448.4.i.j | 6 | ||
| 12.b | even | 2 | 1 | 504.4.s.h | 6 | ||
| 28.d | even | 2 | 1 | 392.4.i.m | 6 | ||
| 28.f | even | 6 | 1 | 392.4.a.l | 3 | ||
| 28.f | even | 6 | 1 | 392.4.i.m | 6 | ||
| 28.g | odd | 6 | 1 | 56.4.i.b | ✓ | 6 | |
| 28.g | odd | 6 | 1 | 392.4.a.i | 3 | ||
| 56.k | odd | 6 | 1 | 448.4.i.j | 6 | ||
| 56.p | even | 6 | 1 | 448.4.i.m | 6 | ||
| 84.n | even | 6 | 1 | 504.4.s.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.i.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 56.4.i.b | ✓ | 6 | 28.g | odd | 6 | 1 | |
| 112.4.i.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 112.4.i.e | 6 | 7.c | even | 3 | 1 | inner | |
| 392.4.a.i | 3 | 28.g | odd | 6 | 1 | ||
| 392.4.a.l | 3 | 28.f | even | 6 | 1 | ||
| 392.4.i.m | 6 | 28.d | even | 2 | 1 | ||
| 392.4.i.m | 6 | 28.f | even | 6 | 1 | ||
| 448.4.i.j | 6 | 8.d | odd | 2 | 1 | ||
| 448.4.i.j | 6 | 56.k | odd | 6 | 1 | ||
| 448.4.i.m | 6 | 8.b | even | 2 | 1 | ||
| 448.4.i.m | 6 | 56.p | even | 6 | 1 | ||
| 504.4.s.h | 6 | 12.b | even | 2 | 1 | ||
| 504.4.s.h | 6 | 84.n | even | 6 | 1 | ||
| 784.4.a.bb | 3 | 7.d | odd | 6 | 1 | ||
| 784.4.a.be | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 7T_{3}^{5} + 74T_{3}^{4} - 161T_{3}^{3} + 674T_{3}^{2} + 175T_{3} + 49 \)
acting on \(S_{4}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 7 T^{5} + \cdots + 49 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 4255969 \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + \cdots + 1492817769 \)
|
| $13$ |
\( (T^{3} + 26 T^{2} + \cdots + 26328)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 15768076041 \)
|
| $19$ |
\( T^{6} + 89 T^{5} + \cdots + 507555841 \)
|
| $23$ |
\( T^{6} + \cdots + 7342261969 \)
|
| $29$ |
\( (T^{3} - 190 T^{2} + \cdots + 48504)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 6524157303049 \)
|
| $37$ |
\( T^{6} + \cdots + 783910139769 \)
|
| $41$ |
\( (T^{3} - 58 T^{2} + \cdots + 3860392)^{2} \)
|
| $43$ |
\( (T^{3} + 268 T^{2} + \cdots - 24343488)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 126279990328969 \)
|
| $53$ |
\( T^{6} + \cdots + 55\!\cdots\!81 \)
|
| $59$ |
\( T^{6} + \cdots + 38\!\cdots\!89 \)
|
| $61$ |
\( T^{6} + \cdots + 273487351727209 \)
|
| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!41 \)
|
| $71$ |
\( (T^{3} + 640 T^{2} + \cdots + 7291392)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!09 \)
|
| $79$ |
\( T^{6} + \cdots + 21\!\cdots\!41 \)
|
| $83$ |
\( (T^{3} - 2372 T^{2} + \cdots - 266787264)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 93\!\cdots\!01 \)
|
| $97$ |
\( (T^{3} + 342 T^{2} + \cdots + 217321448)^{2} \)
|
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