Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,6,Mod(25,42)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.25");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.e (of order \(3\), 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: | \(6.73612043215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{505})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 7 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 125\nu^{3} + 127\nu^{2} - 32131\nu + 47754 ) / 16002 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 379\nu^{3} - 127\nu^{2} + 16129\nu + 63756 ) / 16002 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 2\beta_{2} + 887\beta _1 - 886 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 254\beta_{3} + 127\beta_{2} + 127\beta _1 - 1517 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
2.00000 | + | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | + | 13.8564i | −43.5764 | − | 75.4765i | −36.0000 | −113.236 | + | 63.1236i | −64.0000 | −40.5000 | − | 70.1481i | 174.305 | − | 301.906i | ||||||||||||||||
| 25.2 | 2.00000 | + | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | + | 13.8564i | 35.0764 | + | 60.7540i | −36.0000 | −90.7639 | − | 92.5684i | −64.0000 | −40.5000 | − | 70.1481i | −140.305 | + | 243.016i | |||||||||||||||||
| 37.1 | 2.00000 | − | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | − | 13.8564i | −43.5764 | + | 75.4765i | −36.0000 | −113.236 | − | 63.1236i | −64.0000 | −40.5000 | + | 70.1481i | 174.305 | + | 301.906i | |||||||||||||||||
| 37.2 | 2.00000 | − | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | − | 13.8564i | 35.0764 | − | 60.7540i | −36.0000 | −90.7639 | + | 92.5684i | −64.0000 | −40.5000 | + | 70.1481i | −140.305 | − | 243.016i | |||||||||||||||||
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 | 42.6.e.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.6.g.g | 4 | ||
| 4.b | odd | 2 | 1 | 336.6.q.h | 4 | ||
| 7.b | odd | 2 | 1 | 294.6.e.y | 4 | ||
| 7.c | even | 3 | 1 | inner | 42.6.e.d | ✓ | 4 |
| 7.c | even | 3 | 1 | 294.6.a.p | 2 | ||
| 7.d | odd | 6 | 1 | 294.6.a.o | 2 | ||
| 7.d | odd | 6 | 1 | 294.6.e.y | 4 | ||
| 21.g | even | 6 | 1 | 882.6.a.bs | 2 | ||
| 21.h | odd | 6 | 1 | 126.6.g.g | 4 | ||
| 21.h | odd | 6 | 1 | 882.6.a.bm | 2 | ||
| 28.g | odd | 6 | 1 | 336.6.q.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.6.e.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.6.e.d | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 126.6.g.g | 4 | 3.b | odd | 2 | 1 | ||
| 126.6.g.g | 4 | 21.h | odd | 6 | 1 | ||
| 294.6.a.o | 2 | 7.d | odd | 6 | 1 | ||
| 294.6.a.p | 2 | 7.c | even | 3 | 1 | ||
| 294.6.e.y | 4 | 7.b | odd | 2 | 1 | ||
| 294.6.e.y | 4 | 7.d | odd | 6 | 1 | ||
| 336.6.q.h | 4 | 4.b | odd | 2 | 1 | ||
| 336.6.q.h | 4 | 28.g | odd | 6 | 1 | ||
| 882.6.a.bm | 2 | 21.h | odd | 6 | 1 | ||
| 882.6.a.bs | 2 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 17T_{5}^{3} + 6403T_{5}^{2} - 103938T_{5} + 37380996 \)
acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( (T^{2} + 9 T + 81)^{2} \)
|
| $5$ |
\( T^{4} + 17 T^{3} + \cdots + 37380996 \)
|
| $7$ |
\( T^{4} + 408 T^{3} + \cdots + 282475249 \)
|
| $11$ |
\( T^{4} + \cdots + 88726536900 \)
|
| $13$ |
\( (T^{2} + 715 T + 121620)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 4015631241216 \)
|
| $19$ |
\( T^{4} + \cdots + 17788453428496 \)
|
| $23$ |
\( T^{4} + \cdots + 24815700919296 \)
|
| $29$ |
\( (T^{2} - 7865 T - 6069780)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 331894030125681 \)
|
| $37$ |
\( T^{4} + \cdots + 156377425969216 \)
|
| $41$ |
\( (T^{2} - 7350 T - 13441680)^{2} \)
|
| $43$ |
\( (T^{2} + 5921 T - 15788666)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 24\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 92983900409856 \)
|
| $59$ |
\( T^{4} + \cdots + 76\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 15\!\cdots\!96 \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $71$ |
\( (T^{2} - 91744 T + 1804825884)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 47\!\cdots\!81 \)
|
| $83$ |
\( (T^{2} - 33841 T + 280358334)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 6584027556096 \)
|
| $97$ |
\( (T^{2} + 46671 T - 4062781666)^{2} \)
|
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