Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,3,Mod(19,42)]
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, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.g (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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−0.707107 | + | 1.22474i | 1.50000 | − | 0.866025i | −1.00000 | − | 1.73205i | 7.24264 | + | 4.18154i | 2.44949i | −6.74264 | + | 1.88064i | 2.82843 | 1.50000 | − | 2.59808i | −10.2426 | + | 5.91359i | ||||||||||||||||
| 19.2 | 0.707107 | − | 1.22474i | 1.50000 | − | 0.866025i | −1.00000 | − | 1.73205i | −1.24264 | − | 0.717439i | − | 2.44949i | 1.74264 | + | 6.77962i | −2.82843 | 1.50000 | − | 2.59808i | −1.75736 | + | 1.01461i | ||||||||||||||||
| 31.1 | −0.707107 | − | 1.22474i | 1.50000 | + | 0.866025i | −1.00000 | + | 1.73205i | 7.24264 | − | 4.18154i | − | 2.44949i | −6.74264 | − | 1.88064i | 2.82843 | 1.50000 | + | 2.59808i | −10.2426 | − | 5.91359i | ||||||||||||||||
| 31.2 | 0.707107 | + | 1.22474i | 1.50000 | + | 0.866025i | −1.00000 | + | 1.73205i | −1.24264 | + | 0.717439i | 2.44949i | 1.74264 | − | 6.77962i | −2.82843 | 1.50000 | + | 2.59808i | −1.75736 | − | 1.01461i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | 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.g.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.3.n.a | 4 | ||
| 4.b | odd | 2 | 1 | 336.3.bh.e | 4 | ||
| 5.b | even | 2 | 1 | 1050.3.p.a | 4 | ||
| 5.c | odd | 4 | 2 | 1050.3.q.a | 8 | ||
| 7.b | odd | 2 | 1 | 294.3.g.a | 4 | ||
| 7.c | even | 3 | 1 | 294.3.c.a | 4 | ||
| 7.c | even | 3 | 1 | 294.3.g.a | 4 | ||
| 7.d | odd | 6 | 1 | inner | 42.3.g.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | 294.3.c.a | 4 | ||
| 12.b | even | 2 | 1 | 1008.3.cg.h | 4 | ||
| 21.c | even | 2 | 1 | 882.3.n.e | 4 | ||
| 21.g | even | 6 | 1 | 126.3.n.a | 4 | ||
| 21.g | even | 6 | 1 | 882.3.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 882.3.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 882.3.n.e | 4 | ||
| 28.f | even | 6 | 1 | 336.3.bh.e | 4 | ||
| 28.f | even | 6 | 1 | 2352.3.f.e | 4 | ||
| 28.g | odd | 6 | 1 | 2352.3.f.e | 4 | ||
| 35.i | odd | 6 | 1 | 1050.3.p.a | 4 | ||
| 35.k | even | 12 | 2 | 1050.3.q.a | 8 | ||
| 84.j | odd | 6 | 1 | 1008.3.cg.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.3.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 42.3.g.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 126.3.n.a | 4 | 3.b | odd | 2 | 1 | ||
| 126.3.n.a | 4 | 21.g | even | 6 | 1 | ||
| 294.3.c.a | 4 | 7.c | even | 3 | 1 | ||
| 294.3.c.a | 4 | 7.d | odd | 6 | 1 | ||
| 294.3.g.a | 4 | 7.b | odd | 2 | 1 | ||
| 294.3.g.a | 4 | 7.c | even | 3 | 1 | ||
| 336.3.bh.e | 4 | 4.b | odd | 2 | 1 | ||
| 336.3.bh.e | 4 | 28.f | even | 6 | 1 | ||
| 882.3.c.b | 4 | 21.g | even | 6 | 1 | ||
| 882.3.c.b | 4 | 21.h | odd | 6 | 1 | ||
| 882.3.n.e | 4 | 21.c | even | 2 | 1 | ||
| 882.3.n.e | 4 | 21.h | odd | 6 | 1 | ||
| 1008.3.cg.h | 4 | 12.b | even | 2 | 1 | ||
| 1008.3.cg.h | 4 | 84.j | odd | 6 | 1 | ||
| 1050.3.p.a | 4 | 5.b | even | 2 | 1 | ||
| 1050.3.p.a | 4 | 35.i | odd | 6 | 1 | ||
| 1050.3.q.a | 8 | 5.c | odd | 4 | 2 | ||
| 1050.3.q.a | 8 | 35.k | even | 12 | 2 | ||
| 2352.3.f.e | 4 | 28.f | even | 6 | 1 | ||
| 2352.3.f.e | 4 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2T^{2} + 4 \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 144 \)
|
| $7$ |
\( T^{4} + 10 T^{3} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $13$ |
\( T^{4} + 774 T^{2} + 145161 \)
|
| $17$ |
\( T^{4} + 48 T^{3} + \cdots + 28224 \)
|
| $19$ |
\( T^{4} + 42 T^{3} + \cdots + 15129 \)
|
| $23$ |
\( T^{4} - 24 T^{3} + \cdots + 254016 \)
|
| $29$ |
\( (T^{2} - 1152)^{2} \)
|
| $31$ |
\( T^{4} - 102 T^{3} + \cdots + 423801 \)
|
| $37$ |
\( T^{4} - 22 T^{3} + \cdots + 27889 \)
|
| $41$ |
\( T^{4} + 4248 T^{2} + 3732624 \)
|
| $43$ |
\( (T^{2} - 14 T - 23)^{2} \)
|
| $47$ |
\( T^{4} + 132 T^{3} + \cdots + 2039184 \)
|
| $53$ |
\( T^{4} - 120 T^{3} + \cdots + 8714304 \)
|
| $59$ |
\( T^{4} + 24 T^{3} + \cdots + 1272384 \)
|
| $61$ |
\( T^{4} + 72 T^{3} + \cdots + 2304 \)
|
| $67$ |
\( T^{4} - 110 T^{3} + \cdots + 253009 \)
|
| $71$ |
\( (T^{2} - 156 T + 2556)^{2} \)
|
| $73$ |
\( T^{4} + 66 T^{3} + \cdots + 85322169 \)
|
| $79$ |
\( T^{4} + 10 T^{3} + \cdots + 75463969 \)
|
| $83$ |
\( T^{4} + 17928 T^{2} + 69956496 \)
|
| $89$ |
\( (T^{2} + 36 T + 432)^{2} \)
|
| $97$ |
\( T^{4} + 1056 T^{2} + 112896 \)
|
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