Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,4,Mod(9,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.c (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: | \(0.826026740080\) |
| 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 \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
1.00000 | − | 1.73205i | 0.500000 | + | 0.866025i | −2.00000 | − | 3.46410i | −3.50000 | + | 6.06218i | 2.00000 | −10.0000 | + | 15.5885i | −8.00000 | 13.0000 | − | 22.5167i | 7.00000 | + | 12.1244i | ||||||||||
| 11.1 | 1.00000 | + | 1.73205i | 0.500000 | − | 0.866025i | −2.00000 | + | 3.46410i | −3.50000 | − | 6.06218i | 2.00000 | −10.0000 | − | 15.5885i | −8.00000 | 13.0000 | + | 22.5167i | 7.00000 | − | 12.1244i | |||||||||||
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 | 14.4.c.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 126.4.g.c | 2 | ||
| 4.b | odd | 2 | 1 | 112.4.i.b | 2 | ||
| 5.b | even | 2 | 1 | 350.4.e.b | 2 | ||
| 5.c | odd | 4 | 2 | 350.4.j.d | 4 | ||
| 7.b | odd | 2 | 1 | 98.4.c.e | 2 | ||
| 7.c | even | 3 | 1 | inner | 14.4.c.b | ✓ | 2 |
| 7.c | even | 3 | 1 | 98.4.a.b | 1 | ||
| 7.d | odd | 6 | 1 | 98.4.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 98.4.c.e | 2 | ||
| 8.b | even | 2 | 1 | 448.4.i.c | 2 | ||
| 8.d | odd | 2 | 1 | 448.4.i.d | 2 | ||
| 21.c | even | 2 | 1 | 882.4.g.d | 2 | ||
| 21.g | even | 6 | 1 | 882.4.a.p | 1 | ||
| 21.g | even | 6 | 1 | 882.4.g.d | 2 | ||
| 21.h | odd | 6 | 1 | 126.4.g.c | 2 | ||
| 21.h | odd | 6 | 1 | 882.4.a.k | 1 | ||
| 28.f | even | 6 | 1 | 784.4.a.j | 1 | ||
| 28.g | odd | 6 | 1 | 112.4.i.b | 2 | ||
| 28.g | odd | 6 | 1 | 784.4.a.l | 1 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.bf | 1 | ||
| 35.j | even | 6 | 1 | 350.4.e.b | 2 | ||
| 35.j | even | 6 | 1 | 2450.4.a.bh | 1 | ||
| 35.l | odd | 12 | 2 | 350.4.j.d | 4 | ||
| 56.k | odd | 6 | 1 | 448.4.i.d | 2 | ||
| 56.p | even | 6 | 1 | 448.4.i.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.4.c.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 14.4.c.b | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 98.4.a.b | 1 | 7.c | even | 3 | 1 | ||
| 98.4.a.c | 1 | 7.d | odd | 6 | 1 | ||
| 98.4.c.e | 2 | 7.b | odd | 2 | 1 | ||
| 98.4.c.e | 2 | 7.d | odd | 6 | 1 | ||
| 112.4.i.b | 2 | 4.b | odd | 2 | 1 | ||
| 112.4.i.b | 2 | 28.g | odd | 6 | 1 | ||
| 126.4.g.c | 2 | 3.b | odd | 2 | 1 | ||
| 126.4.g.c | 2 | 21.h | odd | 6 | 1 | ||
| 350.4.e.b | 2 | 5.b | even | 2 | 1 | ||
| 350.4.e.b | 2 | 35.j | even | 6 | 1 | ||
| 350.4.j.d | 4 | 5.c | odd | 4 | 2 | ||
| 350.4.j.d | 4 | 35.l | odd | 12 | 2 | ||
| 448.4.i.c | 2 | 8.b | even | 2 | 1 | ||
| 448.4.i.c | 2 | 56.p | even | 6 | 1 | ||
| 448.4.i.d | 2 | 8.d | odd | 2 | 1 | ||
| 448.4.i.d | 2 | 56.k | odd | 6 | 1 | ||
| 784.4.a.j | 1 | 28.f | even | 6 | 1 | ||
| 784.4.a.l | 1 | 28.g | odd | 6 | 1 | ||
| 882.4.a.k | 1 | 21.h | odd | 6 | 1 | ||
| 882.4.a.p | 1 | 21.g | even | 6 | 1 | ||
| 882.4.g.d | 2 | 21.c | even | 2 | 1 | ||
| 882.4.g.d | 2 | 21.g | even | 6 | 1 | ||
| 2450.4.a.bf | 1 | 35.i | odd | 6 | 1 | ||
| 2450.4.a.bh | 1 | 35.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - T_{3} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T + 4 \)
|
| $3$ |
\( T^{2} - T + 1 \)
|
| $5$ |
\( T^{2} + 7T + 49 \)
|
| $7$ |
\( T^{2} + 20T + 343 \)
|
| $11$ |
\( T^{2} + 35T + 1225 \)
|
| $13$ |
\( (T - 66)^{2} \)
|
| $17$ |
\( T^{2} + 59T + 3481 \)
|
| $19$ |
\( T^{2} + 137T + 18769 \)
|
| $23$ |
\( T^{2} - 7T + 49 \)
|
| $29$ |
\( (T - 106)^{2} \)
|
| $31$ |
\( T^{2} + 75T + 5625 \)
|
| $37$ |
\( T^{2} + 11T + 121 \)
|
| $41$ |
\( (T + 498)^{2} \)
|
| $43$ |
\( (T - 260)^{2} \)
|
| $47$ |
\( T^{2} - 171T + 29241 \)
|
| $53$ |
\( T^{2} - 417T + 173889 \)
|
| $59$ |
\( T^{2} - 17T + 289 \)
|
| $61$ |
\( T^{2} + 51T + 2601 \)
|
| $67$ |
\( T^{2} + 439T + 192721 \)
|
| $71$ |
\( (T + 784)^{2} \)
|
| $73$ |
\( T^{2} + 295T + 87025 \)
|
| $79$ |
\( T^{2} - 495T + 245025 \)
|
| $83$ |
\( (T - 932)^{2} \)
|
| $89$ |
\( T^{2} - 873T + 762129 \)
|
| $97$ |
\( (T + 290)^{2} \)
|
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