Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,3,Mod(11,42)]
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([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.11");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.h (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: | \(1.14441711031\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.22474 | + | 0.707107i | −1.94949 | − | 2.28024i | 1.00000 | − | 1.73205i | 7.34847 | − | 4.24264i | 4.00000 | + | 1.41421i | −3.50000 | − | 6.06218i | 2.82843i | −1.39898 | + | 8.89060i | −6.00000 | + | 10.3923i | ||||||||||||||
| 11.2 | 1.22474 | − | 0.707107i | 2.94949 | + | 0.548188i | 1.00000 | − | 1.73205i | −7.34847 | + | 4.24264i | 4.00000 | − | 1.41421i | −3.50000 | − | 6.06218i | − | 2.82843i | 8.39898 | + | 3.23375i | −6.00000 | + | 10.3923i | ||||||||||||||
| 23.1 | −1.22474 | − | 0.707107i | −1.94949 | + | 2.28024i | 1.00000 | + | 1.73205i | 7.34847 | + | 4.24264i | 4.00000 | − | 1.41421i | −3.50000 | + | 6.06218i | − | 2.82843i | −1.39898 | − | 8.89060i | −6.00000 | − | 10.3923i | ||||||||||||||
| 23.2 | 1.22474 | + | 0.707107i | 2.94949 | − | 0.548188i | 1.00000 | + | 1.73205i | −7.34847 | − | 4.24264i | 4.00000 | + | 1.41421i | −3.50000 | + | 6.06218i | 2.82843i | 8.39898 | − | 3.23375i | −6.00000 | − | 10.3923i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 42.3.h.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 42.3.h.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 336.3.bn.c | 4 | ||
| 7.b | odd | 2 | 1 | 294.3.h.b | 4 | ||
| 7.c | even | 3 | 1 | inner | 42.3.h.a | ✓ | 4 |
| 7.c | even | 3 | 1 | 294.3.b.b | 2 | ||
| 7.d | odd | 6 | 1 | 294.3.b.c | 2 | ||
| 7.d | odd | 6 | 1 | 294.3.h.b | 4 | ||
| 12.b | even | 2 | 1 | 336.3.bn.c | 4 | ||
| 21.c | even | 2 | 1 | 294.3.h.b | 4 | ||
| 21.g | even | 6 | 1 | 294.3.b.c | 2 | ||
| 21.g | even | 6 | 1 | 294.3.h.b | 4 | ||
| 21.h | odd | 6 | 1 | inner | 42.3.h.a | ✓ | 4 |
| 21.h | odd | 6 | 1 | 294.3.b.b | 2 | ||
| 28.g | odd | 6 | 1 | 336.3.bn.c | 4 | ||
| 84.n | even | 6 | 1 | 336.3.bn.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.3.h.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.3.h.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 42.3.h.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 42.3.h.a | ✓ | 4 | 21.h | odd | 6 | 1 | inner |
| 294.3.b.b | 2 | 7.c | even | 3 | 1 | ||
| 294.3.b.b | 2 | 21.h | odd | 6 | 1 | ||
| 294.3.b.c | 2 | 7.d | odd | 6 | 1 | ||
| 294.3.b.c | 2 | 21.g | even | 6 | 1 | ||
| 294.3.h.b | 4 | 7.b | odd | 2 | 1 | ||
| 294.3.h.b | 4 | 7.d | odd | 6 | 1 | ||
| 294.3.h.b | 4 | 21.c | even | 2 | 1 | ||
| 294.3.h.b | 4 | 21.g | even | 6 | 1 | ||
| 336.3.bn.c | 4 | 4.b | odd | 2 | 1 | ||
| 336.3.bn.c | 4 | 12.b | even | 2 | 1 | ||
| 336.3.bn.c | 4 | 28.g | odd | 6 | 1 | ||
| 336.3.bn.c | 4 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 72T_{5}^{2} + 5184 \)
acting on \(S_{3}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $7$ |
\( (T^{2} + 7 T + 49)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T + 1)^{4} \)
|
| $17$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $19$ |
\( (T^{2} - 31 T + 961)^{2} \)
|
| $23$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $29$ |
\( (T^{2} + 288)^{2} \)
|
| $31$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $37$ |
\( (T^{2} - T + 1)^{2} \)
|
| $41$ |
\( (T^{2} + 1152)^{2} \)
|
| $43$ |
\( (T + 31)^{4} \)
|
| $47$ |
\( T^{4} - 1800 T^{2} + 3240000 \)
|
| $53$ |
\( T^{4} - 648 T^{2} + 419904 \)
|
| $59$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $61$ |
\( (T^{2} + 50 T + 2500)^{2} \)
|
| $67$ |
\( (T^{2} + 65 T + 4225)^{2} \)
|
| $71$ |
\( (T^{2} + 3528)^{2} \)
|
| $73$ |
\( (T^{2} - 97 T + 9409)^{2} \)
|
| $79$ |
\( (T^{2} - 103 T + 10609)^{2} \)
|
| $83$ |
\( (T^{2} + 1800)^{2} \)
|
| $89$ |
\( T^{4} - 14112 T^{2} + 199148544 \)
|
| $97$ |
\( (T + 166)^{4} \)
|
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