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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.509800.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} + 5x + 15 \)
|
| 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
\( x^{4} - x^{3} - 10x^{2} + 5x + 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 7\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.05706 | −9.64416 | 17.5738 | 12.9427 | 48.7711 | −7.00000 | −48.4155 | 66.0098 | −65.4521 | ||||||||||||||||||||||||||||||
| 1.2 | −3.65527 | 5.17115 | 5.36103 | −10.0822 | −18.9020 | −7.00000 | 9.64616 | −0.259212 | 36.8533 | |||||||||||||||||||||||||||||||
| 1.3 | 0.948670 | −0.163384 | −7.10002 | −5.36789 | −0.154997 | −7.00000 | −14.3249 | −26.9733 | −5.09236 | |||||||||||||||||||||||||||||||
| 1.4 | 3.76366 | −7.36360 | 6.16515 | −15.4926 | −27.7141 | −7.00000 | −6.90574 | 27.2227 | −58.3089 | |||||||||||||||||||||||||||||||
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.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 693.4.a.m | 4 | ||
| 4.b | odd | 2 | 1 | 1232.4.a.w | 4 | ||
| 5.b | even | 2 | 1 | 1925.4.a.q | 4 | ||
| 7.b | odd | 2 | 1 | 539.4.a.f | 4 | ||
| 11.b | odd | 2 | 1 | 847.4.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 539.4.a.f | 4 | 7.b | odd | 2 | 1 | ||
| 693.4.a.m | 4 | 3.b | odd | 2 | 1 | ||
| 847.4.a.e | 4 | 11.b | odd | 2 | 1 | ||
| 1232.4.a.w | 4 | 4.b | odd | 2 | 1 | ||
| 1925.4.a.q | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 19T_{2}^{2} - 56T_{2} + 66 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(77))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 66 \)
|
| $3$ |
\( T^{4} + 12 T^{3} + \cdots - 60 \)
|
| $5$ |
\( T^{4} + 18 T^{3} + \cdots - 10852 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} + 134 T^{3} + \cdots - 9904192 \)
|
| $17$ |
\( T^{4} + 74 T^{3} + \cdots - 4708304 \)
|
| $19$ |
\( T^{4} + 164 T^{3} + \cdots - 85552320 \)
|
| $23$ |
\( T^{4} - 194 T^{3} + \cdots - 39720496 \)
|
| $29$ |
\( T^{4} + 108 T^{3} + \cdots - 365881040 \)
|
| $31$ |
\( T^{4} + \cdots - 5207968724 \)
|
| $37$ |
\( T^{4} + \cdots + 1094639996 \)
|
| $41$ |
\( T^{4} + 18 T^{3} + \cdots - 410971280 \)
|
| $43$ |
\( T^{4} + 496 T^{3} + \cdots - 998066176 \)
|
| $47$ |
\( T^{4} - 62 T^{3} + \cdots + 463480064 \)
|
| $53$ |
\( T^{4} + 828 T^{3} + \cdots - 394495824 \)
|
| $59$ |
\( T^{4} + \cdots + 1674727140 \)
|
| $61$ |
\( T^{4} + \cdots - 3730099088 \)
|
| $67$ |
\( T^{4} + \cdots - 57482107536 \)
|
| $71$ |
\( T^{4} + \cdots + 109860635344 \)
|
| $73$ |
\( T^{4} + \cdots - 34532794928 \)
|
| $79$ |
\( T^{4} + \cdots - 48577598400 \)
|
| $83$ |
\( T^{4} + \cdots + 42421669632 \)
|
| $89$ |
\( T^{4} + \cdots - 926653158300 \)
|
| $97$ |
\( T^{4} + \cdots - 78194289572 \)
|
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