Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,7,Mod(48,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.48");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(11.2726500974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 6\nu^{2} + 4\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} - 9\beta_{2} + 7\beta_1 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} - 7 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 48.1 |
|
−0.242641 | − | 37.0395i | −63.9411 | − | 165.776i | 8.98729i | 0 | 31.0437 | −642.926 | 40.2239i | ||||||||||||||||||||||||||||
| 48.2 | −0.242641 | 37.0395i | −63.9411 | 165.776i | − | 8.98729i | 0 | 31.0437 | −642.926 | − | 40.2239i | |||||||||||||||||||||||||||||
| 48.3 | 8.24264 | − | 26.6472i | 3.94113 | − | 79.1732i | − | 219.643i | 0 | −495.044 | 18.9260 | − | 652.596i | |||||||||||||||||||||||||||
| 48.4 | 8.24264 | 26.6472i | 3.94113 | 79.1732i | 219.643i | 0 | −495.044 | 18.9260 | 652.596i | |||||||||||||||||||||||||||||||
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 | 49.7.b.b | 4 | |
| 3.b | odd | 2 | 1 | 441.7.d.b | 4 | ||
| 7.b | odd | 2 | 1 | inner | 49.7.b.b | 4 | |
| 7.c | even | 3 | 1 | 7.7.d.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | 49.7.d.c | 4 | ||
| 7.d | odd | 6 | 1 | 7.7.d.b | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 49.7.d.c | 4 | ||
| 21.c | even | 2 | 1 | 441.7.d.b | 4 | ||
| 21.g | even | 6 | 1 | 63.7.m.b | 4 | ||
| 21.h | odd | 6 | 1 | 63.7.m.b | 4 | ||
| 28.f | even | 6 | 1 | 112.7.s.b | 4 | ||
| 28.g | odd | 6 | 1 | 112.7.s.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.7.d.b | ✓ | 4 | 7.c | even | 3 | 1 | |
| 7.7.d.b | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 49.7.b.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.7.b.b | 4 | 7.b | odd | 2 | 1 | inner | |
| 49.7.d.c | 4 | 7.c | even | 3 | 1 | ||
| 49.7.d.c | 4 | 7.d | odd | 6 | 1 | ||
| 63.7.m.b | 4 | 21.g | even | 6 | 1 | ||
| 63.7.m.b | 4 | 21.h | odd | 6 | 1 | ||
| 112.7.s.b | 4 | 28.f | even | 6 | 1 | ||
| 112.7.s.b | 4 | 28.g | odd | 6 | 1 | ||
| 441.7.d.b | 4 | 3.b | odd | 2 | 1 | ||
| 441.7.d.b | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 8T_{2} - 2 \)
acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T - 2)^{2} \)
|
| $3$ |
\( T^{4} + 2082 T^{2} + 974169 \)
|
| $5$ |
\( T^{4} + 33750 T^{2} + 172265625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 1882 T + 295783)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 37734434122896 \)
|
| $17$ |
\( T^{4} + \cdots + 226702498429449 \)
|
| $19$ |
\( T^{4} + \cdots + 718603078673529 \)
|
| $23$ |
\( (T^{2} + 2470 T - 210366473)^{2} \)
|
| $29$ |
\( (T^{2} + 17272 T - 154827704)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 611468988954201 \)
|
| $37$ |
\( (T^{2} - 43870 T - 225134303)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 26\!\cdots\!84 \)
|
| $43$ |
\( (T^{2} - 160108 T + 6274490716)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 81\!\cdots\!29 \)
|
| $53$ |
\( (T^{2} + 104530 T - 38017784975)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 35\!\cdots\!69 \)
|
| $61$ |
\( T^{4} + \cdots + 29\!\cdots\!29 \)
|
| $67$ |
\( (T^{2} + 432494 T + 37630003759)^{2} \)
|
| $71$ |
\( (T^{2} + 740764 T + 136232520316)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 56\!\cdots\!41 \)
|
| $79$ |
\( (T^{2} + 935990 T + 197161678663)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 10\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots + 85\!\cdots\!21 \)
|
| $97$ |
\( T^{4} + \cdots + 95\!\cdots\!84 \)
|
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