Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.9");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(2.24537347738\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{130})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 130x^{2} + 16900 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 130x^{2} + 16900 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 130 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 65 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 130\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 65\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
2.00000 | − | 3.46410i | −7.90175 | − | 13.6862i | −8.00000 | − | 13.8564i | 10.5000 | − | 18.1865i | −63.2140 | 126.411 | + | 28.7642i | −64.0000 | −3.37544 | + | 5.84643i | −42.0000 | − | 72.7461i | ||||||||||||||||
| 9.2 | 2.00000 | − | 3.46410i | 14.9018 | + | 25.8106i | −8.00000 | − | 13.8564i | 10.5000 | − | 18.1865i | 119.214 | −10.4105 | − | 129.223i | −64.0000 | −322.625 | + | 558.802i | −42.0000 | − | 72.7461i | |||||||||||||||||
| 11.1 | 2.00000 | + | 3.46410i | −7.90175 | + | 13.6862i | −8.00000 | + | 13.8564i | 10.5000 | + | 18.1865i | −63.2140 | 126.411 | − | 28.7642i | −64.0000 | −3.37544 | − | 5.84643i | −42.0000 | + | 72.7461i | |||||||||||||||||
| 11.2 | 2.00000 | + | 3.46410i | 14.9018 | − | 25.8106i | −8.00000 | + | 13.8564i | 10.5000 | + | 18.1865i | 119.214 | −10.4105 | + | 129.223i | −64.0000 | −322.625 | − | 558.802i | −42.0000 | + | 72.7461i | |||||||||||||||||
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 | 14.6.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.6.g.e | 4 | ||
| 4.b | odd | 2 | 1 | 112.6.i.b | 4 | ||
| 7.b | odd | 2 | 1 | 98.6.c.f | 4 | ||
| 7.c | even | 3 | 1 | inner | 14.6.c.b | ✓ | 4 |
| 7.c | even | 3 | 1 | 98.6.a.c | 2 | ||
| 7.d | odd | 6 | 1 | 98.6.a.f | 2 | ||
| 7.d | odd | 6 | 1 | 98.6.c.f | 4 | ||
| 21.g | even | 6 | 1 | 882.6.a.bl | 2 | ||
| 21.h | odd | 6 | 1 | 126.6.g.e | 4 | ||
| 21.h | odd | 6 | 1 | 882.6.a.bt | 2 | ||
| 28.f | even | 6 | 1 | 784.6.a.r | 2 | ||
| 28.g | odd | 6 | 1 | 112.6.i.b | 4 | ||
| 28.g | odd | 6 | 1 | 784.6.a.bc | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.6.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 14.6.c.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 98.6.a.c | 2 | 7.c | even | 3 | 1 | ||
| 98.6.a.f | 2 | 7.d | odd | 6 | 1 | ||
| 98.6.c.f | 4 | 7.b | odd | 2 | 1 | ||
| 98.6.c.f | 4 | 7.d | odd | 6 | 1 | ||
| 112.6.i.b | 4 | 4.b | odd | 2 | 1 | ||
| 112.6.i.b | 4 | 28.g | odd | 6 | 1 | ||
| 126.6.g.e | 4 | 3.b | odd | 2 | 1 | ||
| 126.6.g.e | 4 | 21.h | odd | 6 | 1 | ||
| 784.6.a.r | 2 | 28.f | even | 6 | 1 | ||
| 784.6.a.bc | 2 | 28.g | odd | 6 | 1 | ||
| 882.6.a.bl | 2 | 21.g | even | 6 | 1 | ||
| 882.6.a.bt | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 14T_{3}^{3} + 667T_{3}^{2} + 6594T_{3} + 221841 \)
acting on \(S_{6}^{\mathrm{new}}(14, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} - 14 T^{3} + \cdots + 221841 \)
|
| $5$ |
\( (T^{2} - 21 T + 441)^{2} \)
|
| $7$ |
\( T^{4} - 232 T^{3} + \cdots + 282475249 \)
|
| $11$ |
\( T^{4} + \cdots + 43143859521 \)
|
| $13$ |
\( (T^{2} + 140 T - 13820)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 65188813041 \)
|
| $19$ |
\( T^{4} + \cdots + 693354317041 \)
|
| $23$ |
\( T^{4} + \cdots + 15861668294241 \)
|
| $29$ |
\( (T^{2} - 1668 T - 221724)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 26\!\cdots\!01 \)
|
| $37$ |
\( T^{4} + \cdots + 45\!\cdots\!01 \)
|
| $41$ |
\( (T^{2} + 5124 T - 207761436)^{2} \)
|
| $43$ |
\( (T^{2} + 4520 T - 53598320)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!21 \)
|
| $53$ |
\( T^{4} + \cdots + 98\!\cdots\!21 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!25 \)
|
| $61$ |
\( T^{4} + \cdots + 20\!\cdots\!21 \)
|
| $67$ |
\( T^{4} + \cdots + 24\!\cdots\!61 \)
|
| $71$ |
\( (T^{2} + 89376 T + 1817230464)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 81\!\cdots\!21 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!25 \)
|
| $83$ |
\( (T^{2} - 68376 T + 1030366224)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 56\!\cdots\!61 \)
|
| $97$ |
\( (T^{2} + 43652 T - 8307517724)^{2} \)
|
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