Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,4,Mod(1,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.a (trivial) |
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: | yes |
| Analytic conductor: | \(4.54314707044\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.522072.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 12\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 11\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 13\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 2\beta _1 + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} + 6\beta_{2} + \beta _1 + 9 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.89098 | 6.57251 | 15.9217 | 15.5514 | −32.1460 | −7.00000 | −38.7449 | 16.1978 | −76.0614 | ||||||||||||||||||||||||||||||
| 1.2 | −3.24550 | −5.49244 | 2.53327 | −16.0955 | 17.8257 | −7.00000 | 17.7423 | 3.16692 | 52.2379 | |||||||||||||||||||||||||||||||
| 1.3 | 1.53253 | 10.1459 | −5.65135 | 8.69995 | 15.5490 | −7.00000 | −20.9211 | 75.9402 | 13.3330 | |||||||||||||||||||||||||||||||
| 1.4 | 4.60395 | 2.77399 | 13.1964 | 1.84418 | 12.7713 | −7.00000 | 23.9238 | −19.3050 | 8.49053 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(11\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 77.4.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 693.4.a.l | 4 | ||
| 4.b | odd | 2 | 1 | 1232.4.a.s | 4 | ||
| 5.b | even | 2 | 1 | 1925.4.a.p | 4 | ||
| 7.b | odd | 2 | 1 | 539.4.a.g | 4 | ||
| 11.b | odd | 2 | 1 | 847.4.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 539.4.a.g | 4 | 7.b | odd | 2 | 1 | ||
| 693.4.a.l | 4 | 3.b | odd | 2 | 1 | ||
| 847.4.a.d | 4 | 11.b | odd | 2 | 1 | ||
| 1232.4.a.s | 4 | 4.b | odd | 2 | 1 | ||
| 1925.4.a.p | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 27T_{2}^{2} - 40T_{2} + 112 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(77))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 112 \)
|
| $3$ |
\( T^{4} - 14 T^{3} + \cdots - 1016 \)
|
| $5$ |
\( T^{4} - 10 T^{3} + \cdots - 4016 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( (T + 11)^{4} \)
|
| $13$ |
\( T^{4} - 58 T^{3} + \cdots - 4947656 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots - 2705024 \)
|
| $19$ |
\( T^{4} - 258 T^{3} + \cdots - 14423904 \)
|
| $23$ |
\( T^{4} - 8 T^{3} + \cdots - 17449856 \)
|
| $29$ |
\( T^{4} + 396 T^{3} + \cdots + 22336464 \)
|
| $31$ |
\( T^{4} + 56 T^{3} + \cdots - 11250248 \)
|
| $37$ |
\( T^{4} - 84 T^{3} + \cdots + 11157312 \)
|
| $41$ |
\( T^{4} - 52 T^{3} + \cdots - 659233664 \)
|
| $43$ |
\( T^{4} + \cdots + 1210397376 \)
|
| $47$ |
\( T^{4} - 8 T^{3} + \cdots - 318931592 \)
|
| $53$ |
\( T^{4} - 624 T^{3} + \cdots + 403923072 \)
|
| $59$ |
\( T^{4} + \cdots + 17599820728 \)
|
| $61$ |
\( T^{4} + \cdots + 6668930664 \)
|
| $67$ |
\( T^{4} + \cdots - 140865466496 \)
|
| $71$ |
\( T^{4} + \cdots - 72982082688 \)
|
| $73$ |
\( T^{4} + \cdots - 322052228384 \)
|
| $79$ |
\( T^{4} + \cdots - 59537293568 \)
|
| $83$ |
\( T^{4} + \cdots + 21951092064 \)
|
| $89$ |
\( T^{4} + \cdots - 109303561968 \)
|
| $97$ |
\( T^{4} + \cdots + 868634650768 \)
|
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