Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,9,Mod(6,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.6");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.b (of order \(2\), degree \(1\), 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: | \(2.85165027043\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1016x^{2} + 51570 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3\cdot 7 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 1016x^{2} + 51570 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{3} + 3506\nu ) / 201 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{3} + 15454\nu ) / 201 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{2} + 1016 ) / 67 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 67\beta_{3} - 1016 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1753\beta_{2} + 7727\beta_1 ) / 84 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
−5.56466 | − | 53.0420i | −225.035 | − | 1090.79i | 295.161i | −1257.19 | + | 2045.55i | 2676.79 | 3747.55 | 6069.88i | ||||||||||||||||||||||||||
| 6.2 | −5.56466 | 53.0420i | −225.035 | 1090.79i | − | 295.161i | −1257.19 | − | 2045.55i | 2676.79 | 3747.55 | − | 6069.88i | |||||||||||||||||||||||||||
| 6.3 | 21.5647 | − | 119.877i | 209.035 | 786.953i | − | 2585.11i | 1971.19 | + | 1370.84i | −1012.79 | −7809.55 | 16970.4i | |||||||||||||||||||||||||||
| 6.4 | 21.5647 | 119.877i | 209.035 | − | 786.953i | 2585.11i | 1971.19 | − | 1370.84i | −1012.79 | −7809.55 | − | 16970.4i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.9.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.9.d.c | 4 | ||
| 4.b | odd | 2 | 1 | 112.9.c.b | 4 | ||
| 5.b | even | 2 | 1 | 175.9.d.e | 4 | ||
| 5.c | odd | 4 | 2 | 175.9.c.c | 8 | ||
| 7.b | odd | 2 | 1 | inner | 7.9.b.b | ✓ | 4 |
| 7.c | even | 3 | 2 | 49.9.d.b | 8 | ||
| 7.d | odd | 6 | 2 | 49.9.d.b | 8 | ||
| 21.c | even | 2 | 1 | 63.9.d.c | 4 | ||
| 28.d | even | 2 | 1 | 112.9.c.b | 4 | ||
| 35.c | odd | 2 | 1 | 175.9.d.e | 4 | ||
| 35.f | even | 4 | 2 | 175.9.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.9.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 7.9.b.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 49.9.d.b | 8 | 7.c | even | 3 | 2 | ||
| 49.9.d.b | 8 | 7.d | odd | 6 | 2 | ||
| 63.9.d.c | 4 | 3.b | odd | 2 | 1 | ||
| 63.9.d.c | 4 | 21.c | even | 2 | 1 | ||
| 112.9.c.b | 4 | 4.b | odd | 2 | 1 | ||
| 112.9.c.b | 4 | 28.d | even | 2 | 1 | ||
| 175.9.c.c | 8 | 5.c | odd | 4 | 2 | ||
| 175.9.c.c | 8 | 35.f | even | 4 | 2 | ||
| 175.9.d.e | 4 | 5.b | even | 2 | 1 | ||
| 175.9.d.e | 4 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 16T_{2} - 120 \)
acting on \(S_{9}^{\mathrm{new}}(7, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T - 120)^{2} \)
|
| $3$ |
\( T^{4} + 17184 T^{2} + 40430880 \)
|
| $5$ |
\( T^{4} + \cdots + 736852788000 \)
|
| $7$ |
\( T^{4} + \cdots + 33232930569601 \)
|
| $11$ |
\( (T^{2} + 11084 T + 29862948)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 65\!\cdots\!80 \)
|
| $17$ |
\( T^{4} + \cdots + 10\!\cdots\!20 \)
|
| $19$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( (T^{2} - 454036 T + 44948453220)^{2} \)
|
| $29$ |
\( (T^{2} + 736508 T + 35513356932)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots + 2655950396420)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 3483758033500)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 34\!\cdots\!80 \)
|
| $53$ |
\( (T^{2} + \cdots - 23698813561980)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} + \cdots + 244439322987940)^{2} \)
|
| $71$ |
\( (T^{2} + \cdots - 149529370875132)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 54\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 217412488545412)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 61\!\cdots\!20 \)
|
| $89$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 22\!\cdots\!20 \)
|
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