Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(2.47808022024\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1 + \zeta_{6}\) |
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 |
|
1.00000 | + | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 3.00000 | + | 5.19615i | −6.00000 | −3.50000 | + | 18.1865i | −8.00000 | −4.50000 | − | 7.79423i | −6.00000 | + | 10.3923i | ||||||||||
| 37.1 | 1.00000 | − | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 3.00000 | − | 5.19615i | −6.00000 | −3.50000 | − | 18.1865i | −8.00000 | −4.50000 | + | 7.79423i | −6.00000 | − | 10.3923i | |||||||||||
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 | 42.4.e.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 126.4.g.b | 2 | ||
| 4.b | odd | 2 | 1 | 336.4.q.f | 2 | ||
| 7.b | odd | 2 | 1 | 294.4.e.i | 2 | ||
| 7.c | even | 3 | 1 | inner | 42.4.e.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 294.4.a.d | 1 | ||
| 7.d | odd | 6 | 1 | 294.4.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 294.4.e.i | 2 | ||
| 21.c | even | 2 | 1 | 882.4.g.g | 2 | ||
| 21.g | even | 6 | 1 | 882.4.a.l | 1 | ||
| 21.g | even | 6 | 1 | 882.4.g.g | 2 | ||
| 21.h | odd | 6 | 1 | 126.4.g.b | 2 | ||
| 21.h | odd | 6 | 1 | 882.4.a.o | 1 | ||
| 28.f | even | 6 | 1 | 2352.4.a.bf | 1 | ||
| 28.g | odd | 6 | 1 | 336.4.q.f | 2 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.4.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 42.4.e.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 126.4.g.b | 2 | 3.b | odd | 2 | 1 | ||
| 126.4.g.b | 2 | 21.h | odd | 6 | 1 | ||
| 294.4.a.c | 1 | 7.d | odd | 6 | 1 | ||
| 294.4.a.d | 1 | 7.c | even | 3 | 1 | ||
| 294.4.e.i | 2 | 7.b | odd | 2 | 1 | ||
| 294.4.e.i | 2 | 7.d | odd | 6 | 1 | ||
| 336.4.q.f | 2 | 4.b | odd | 2 | 1 | ||
| 336.4.q.f | 2 | 28.g | odd | 6 | 1 | ||
| 882.4.a.l | 1 | 21.g | even | 6 | 1 | ||
| 882.4.a.o | 1 | 21.h | odd | 6 | 1 | ||
| 882.4.g.g | 2 | 21.c | even | 2 | 1 | ||
| 882.4.g.g | 2 | 21.g | even | 6 | 1 | ||
| 2352.4.a.f | 1 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.bf | 1 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 6T_{5} + 36 \)
acting on \(S_{4}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T + 4 \)
|
| $3$ |
\( T^{2} + 3T + 9 \)
|
| $5$ |
\( T^{2} - 6T + 36 \)
|
| $7$ |
\( T^{2} + 7T + 343 \)
|
| $11$ |
\( T^{2} - 30T + 900 \)
|
| $13$ |
\( (T - 53)^{2} \)
|
| $17$ |
\( T^{2} - 84T + 7056 \)
|
| $19$ |
\( T^{2} - 97T + 9409 \)
|
| $23$ |
\( T^{2} + 84T + 7056 \)
|
| $29$ |
\( (T + 180)^{2} \)
|
| $31$ |
\( T^{2} + 179T + 32041 \)
|
| $37$ |
\( T^{2} - 145T + 21025 \)
|
| $41$ |
\( (T - 126)^{2} \)
|
| $43$ |
\( (T + 325)^{2} \)
|
| $47$ |
\( T^{2} - 366T + 133956 \)
|
| $53$ |
\( T^{2} - 768T + 589824 \)
|
| $59$ |
\( T^{2} - 264T + 69696 \)
|
| $61$ |
\( T^{2} + 818T + 669124 \)
|
| $67$ |
\( T^{2} - 523T + 273529 \)
|
| $71$ |
\( (T + 342)^{2} \)
|
| $73$ |
\( T^{2} - 43T + 1849 \)
|
| $79$ |
\( T^{2} - 1171 T + 1371241 \)
|
| $83$ |
\( (T + 810)^{2} \)
|
| $89$ |
\( T^{2} - 600T + 360000 \)
|
| $97$ |
\( (T - 386)^{2} \)
|
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