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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−3.82843 | −2.00000 | 6.65685 | 6.48528 | 7.65685 | 7.00000 | 5.14214 | −23.0000 | −24.8284 | ||||||||||||||||||||||||
| 1.2 | 1.82843 | −2.00000 | −4.65685 | −10.4853 | −3.65685 | 7.00000 | −23.1421 | −23.0000 | −19.1716 | |||||||||||||||||||||||||
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.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 693.4.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 1232.4.a.m | 2 | ||
| 5.b | even | 2 | 1 | 1925.4.a.l | 2 | ||
| 7.b | odd | 2 | 1 | 539.4.a.d | 2 | ||
| 11.b | odd | 2 | 1 | 847.4.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 539.4.a.d | 2 | 7.b | odd | 2 | 1 | ||
| 693.4.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 847.4.a.c | 2 | 11.b | odd | 2 | 1 | ||
| 1232.4.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 1925.4.a.l | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 7 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(77))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 7 \)
|
| $3$ |
\( (T + 2)^{2} \)
|
| $5$ |
\( T^{2} + 4T - 68 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( (T + 11)^{2} \)
|
| $13$ |
\( T^{2} + 80T + 1568 \)
|
| $17$ |
\( T^{2} - 48T - 7112 \)
|
| $19$ |
\( T^{2} + 96T + 2232 \)
|
| $23$ |
\( T^{2} + 56T - 1808 \)
|
| $29$ |
\( T^{2} - 4T - 64796 \)
|
| $31$ |
\( T^{2} - 60T - 39428 \)
|
| $37$ |
\( T^{2} + 380T + 34948 \)
|
| $41$ |
\( T^{2} - 112T - 60232 \)
|
| $43$ |
\( (T + 316)^{2} \)
|
| $47$ |
\( T^{2} + 604T + 15932 \)
|
| $53$ |
\( T^{2} - 676T + 87332 \)
|
| $59$ |
\( T^{2} + 516T + 35812 \)
|
| $61$ |
\( T^{2} + 296T + 14704 \)
|
| $67$ |
\( T^{2} + 1152 T + 280576 \)
|
| $71$ |
\( T^{2} + 1064 T + 112496 \)
|
| $73$ |
\( T^{2} - 168T - 232376 \)
|
| $79$ |
\( T^{2} + 368T - 532192 \)
|
| $83$ |
\( T^{2} - 1728 T + 621496 \)
|
| $89$ |
\( T^{2} - 1524 T + 472996 \)
|
| $97$ |
\( T^{2} - 1404 T + 399492 \)
|
| show more | |
| show less |