Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,4,Mod(6,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 9]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.6");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.l (of order \(10\), degree \(4\), 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: | \(4.54314707044\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.37515625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - \nu^{5} + 3\nu^{4} - \nu^{3} + 6\nu^{2} - 4\nu - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 3\nu^{5} - \nu^{4} + 3\nu^{3} - 4\nu^{2} - 8\nu + 16 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{2} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 7\nu^{2} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{5} - 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 3\nu^{6} + 3\nu^{5} - \nu^{4} + 3\nu^{3} - 18\nu^{2} + 12\nu + 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 3\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{5} - 2\beta_{3} - 5\beta _1 - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{5} - 7\beta_{4} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{3} + \beta_{5} - \beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 6.1 |
|
0.759389 | − | 0.246740i | 0 | −5.95635 | + | 4.32754i | 0 | 0 | 10.8859 | + | 14.9832i | −7.21003 | + | 9.92375i | 8.34346 | + | 25.6785i | 0 | ||||||||||||||||||||||||||||||||
| 6.2 | 4.83078 | − | 1.56962i | 0 | 14.4006 | − | 10.4627i | 0 | 0 | −10.8859 | − | 14.9832i | 29.2592 | − | 40.2718i | 8.34346 | + | 25.6785i | 0 | |||||||||||||||||||||||||||||||||
| 13.1 | 0.759389 | + | 0.246740i | 0 | −5.95635 | − | 4.32754i | 0 | 0 | 10.8859 | − | 14.9832i | −7.21003 | − | 9.92375i | 8.34346 | − | 25.6785i | 0 | |||||||||||||||||||||||||||||||||
| 13.2 | 4.83078 | + | 1.56962i | 0 | 14.4006 | + | 10.4627i | 0 | 0 | −10.8859 | + | 14.9832i | 29.2592 | + | 40.2718i | 8.34346 | − | 25.6785i | 0 | |||||||||||||||||||||||||||||||||
| 41.1 | −3.27565 | − | 4.50854i | 0 | −7.12495 | + | 21.9283i | 0 | 0 | 17.6138 | + | 5.72307i | 79.8028 | − | 25.9295i | −21.8435 | + | 15.8702i | 0 | |||||||||||||||||||||||||||||||||
| 41.2 | −2.31452 | − | 3.18567i | 0 | −2.31932 | + | 7.13814i | 0 | 0 | −17.6138 | − | 5.72307i | −1.85195 | + | 0.601734i | −21.8435 | + | 15.8702i | 0 | |||||||||||||||||||||||||||||||||
| 62.1 | −3.27565 | + | 4.50854i | 0 | −7.12495 | − | 21.9283i | 0 | 0 | 17.6138 | − | 5.72307i | 79.8028 | + | 25.9295i | −21.8435 | − | 15.8702i | 0 | |||||||||||||||||||||||||||||||||
| 62.2 | −2.31452 | + | 3.18567i | 0 | −2.31932 | − | 7.13814i | 0 | 0 | −17.6138 | + | 5.72307i | −1.85195 | − | 0.601734i | −21.8435 | − | 15.8702i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 11.d | odd | 10 | 1 | inner |
| 77.l | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 77.4.l.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | CM | 77.4.l.a | ✓ | 8 |
| 11.d | odd | 10 | 1 | inner | 77.4.l.a | ✓ | 8 |
| 77.l | even | 10 | 1 | inner | 77.4.l.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 77.4.l.a | ✓ | 8 | 7.b | odd | 2 | 1 | CM |
| 77.4.l.a | ✓ | 8 | 11.d | odd | 10 | 1 | inner |
| 77.4.l.a | ✓ | 8 | 77.l | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 7T_{2}^{6} - 200T_{2}^{5} + 409T_{2}^{4} + 1400T_{2}^{3} + 9937T_{2}^{2} - 17800T_{2} + 7921 \)
acting on \(S_{4}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 7 T^{6} + \cdots + 7921 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 13841287201 \)
|
| $11$ |
\( T^{8} + \cdots + 3138428376721 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{4} + 40 T^{3} + \cdots + 645403445)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 43\!\cdots\!25 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + \cdots + 64\!\cdots\!25 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} + \cdots + 12\!\cdots\!01 \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} + \cdots + 73\!\cdots\!25 \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{4} + 740 T^{3} + \cdots - 71331423155)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 30\!\cdots\!41 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!25 \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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