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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 34\nu^{2} + 34\nu + 152 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 5\nu^{3} + 38\nu^{2} - 138\nu - 264 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 17 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} - \beta_{2} + 25\beta _1 + 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 10\beta_{3} + 33\beta_{2} + 25\beta _1 + 437 \)
|
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 |
|
−5.44291 | 0.176620 | 21.6253 | −13.0534 | −0.961327 | 7.00000 | −74.1613 | −26.9688 | 71.0487 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.18888 | 6.48496 | −3.20880 | 7.60736 | −14.1948 | 7.00000 | 24.5347 | 15.0547 | −16.6516 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.22767 | −7.89221 | −6.49284 | −2.21191 | 9.68899 | 7.00000 | 17.7924 | 35.2869 | 2.71549 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.44399 | 8.26395 | 11.7491 | −22.0150 | 36.7249 | 7.00000 | 16.6609 | 41.2928 | −97.8345 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.41547 | −5.03332 | 21.3273 | 5.67299 | −27.2578 | 7.00000 | 72.1734 | −1.66567 | 30.7219 | ||||||||||||||||||||||||||||||||||
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.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 693.4.a.o | 5 | ||
| 4.b | odd | 2 | 1 | 1232.4.a.y | 5 | ||
| 5.b | even | 2 | 1 | 1925.4.a.r | 5 | ||
| 7.b | odd | 2 | 1 | 539.4.a.h | 5 | ||
| 11.b | odd | 2 | 1 | 847.4.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 539.4.a.h | 5 | 7.b | odd | 2 | 1 | ||
| 693.4.a.o | 5 | 3.b | odd | 2 | 1 | ||
| 847.4.a.f | 5 | 11.b | odd | 2 | 1 | ||
| 1232.4.a.y | 5 | 4.b | odd | 2 | 1 | ||
| 1925.4.a.r | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - T_{2}^{4} - 42T_{2}^{3} + 18T_{2}^{2} + 368T_{2} + 352 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(77))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots + 352 \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots - 376 \)
|
| $5$ |
\( T^{5} + 24 T^{4} + \cdots + 27432 \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} + 50 T^{4} + \cdots - 592704 \)
|
| $17$ |
\( T^{5} - 222 T^{4} + \cdots - 848713296 \)
|
| $19$ |
\( T^{5} - 160 T^{4} + \cdots - 231728000 \)
|
| $23$ |
\( T^{5} + \cdots - 11393959488 \)
|
| $29$ |
\( T^{5} - 14 T^{4} + \cdots - 217521440 \)
|
| $31$ |
\( T^{5} + \cdots + 5076110528 \)
|
| $37$ |
\( T^{5} + \cdots - 338018607168 \)
|
| $41$ |
\( T^{5} + \cdots - 97487626768 \)
|
| $43$ |
\( T^{5} + \cdots - 326743954944 \)
|
| $47$ |
\( T^{5} + \cdots + 414600941568 \)
|
| $53$ |
\( T^{5} + \cdots + 10851403442304 \)
|
| $59$ |
\( T^{5} + \cdots - 360770783496 \)
|
| $61$ |
\( T^{5} + \cdots + 2506564965968 \)
|
| $67$ |
\( T^{5} + \cdots - 4961616838944 \)
|
| $71$ |
\( T^{5} + \cdots + 1058966690112 \)
|
| $73$ |
\( T^{5} + \cdots + 15144540953200 \)
|
| $79$ |
\( T^{5} + \cdots - 841495667968 \)
|
| $83$ |
\( T^{5} + \cdots + 729734179328 \)
|
| $89$ |
\( T^{5} + \cdots + 75449135393496 \)
|
| $97$ |
\( T^{5} + \cdots - 94604344400216 \)
|
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