Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,3,Mod(43,77)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.c (of order \(2\), degree \(1\), 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: | \(2.09809803557\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5x^{9} + 38x^{8} - 122x^{7} + 427x^{6} - 875x^{5} + 1742x^{4} - 2158x^{3} + 2308x^{2} - 1356x + 296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5x^{9} + 38x^{8} - 122x^{7} + 427x^{6} - 875x^{5} + 1742x^{4} - 2158x^{3} + 2308x^{2} - 1356x + 296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} - 30\nu^{6} + 76\nu^{5} - 231\nu^{4} + 340\nu^{3} - 474\nu^{2} + 316\nu - 44 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{8} + 4\nu^{7} - 30\nu^{6} + 76\nu^{5} - 231\nu^{4} + 340\nu^{3} - 478\nu^{2} + 320\nu - 68 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 34 \nu^{9} + 153 \nu^{8} - 1108 \nu^{7} + 3164 \nu^{6} - 9668 \nu^{5} + 16617 \nu^{4} - 25098 \nu^{3} + \cdots + 2584 ) / 172 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 34 \nu^{9} + 153 \nu^{8} - 1108 \nu^{7} + 3164 \nu^{6} - 9668 \nu^{5} + 16617 \nu^{4} - 25098 \nu^{3} + \cdots + 2498 ) / 86 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 30 \nu^{9} - 123 \nu^{8} + 24 \nu^{7} - 4928 \nu^{6} + 10440 \nu^{5} - 45027 \nu^{4} + 62418 \nu^{3} + \cdots - 15092 ) / 172 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 58 \nu^{9} - 261 \nu^{8} + 1880 \nu^{7} - 5362 \nu^{6} + 16194 \nu^{5} - 27689 \nu^{4} + 40472 \nu^{3} + \cdots - 2172 ) / 86 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{8} + 20\nu^{7} - 152\nu^{6} + 386\nu^{5} - 1201\nu^{4} + 1782\nu^{3} - 2586\nu^{2} + 1756\nu - 392 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 30 \nu^{9} - 393 \nu^{8} + 2040 \nu^{7} - 10724 \nu^{6} + 29292 \nu^{5} - 78297 \nu^{4} + 120418 \nu^{3} + \cdots - 30316 ) / 172 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 250 \nu^{9} + 1125 \nu^{8} - 8056 \nu^{7} + 22946 \nu^{6} - 68574 \nu^{5} + 116695 \nu^{4} + \cdots + 6788 ) / 172 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} - 2\beta_{2} + 2\beta _1 - 11 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{8} - 3\beta_{6} + \beta_{5} - 15\beta_{4} + 18\beta_{3} - 3\beta_{2} + 3\beta _1 - 17 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{8} + 4\beta_{7} - 6\beta_{6} - 31\beta_{4} + 38\beta_{3} + 32\beta_{2} - 28\beta _1 + 101 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{9} + 13 \beta_{8} + 10 \beta_{7} + 45 \beta_{6} - 23 \beta_{5} + 197 \beta_{4} - 194 \beta_{3} + \cdots + 281 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 6 \beta_{9} + 90 \beta_{8} - 66 \beta_{7} + 150 \beta_{6} - 28 \beta_{5} + 669 \beta_{4} + \cdots - 1073 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 38 \beta_{9} - 97 \beta_{8} - 266 \beta_{7} - 515 \beta_{6} + 349 \beta_{5} - 2261 \beta_{4} + \cdots - 4759 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 180 \beta_{9} - 1516 \beta_{8} + 752 \beta_{7} - 2774 \beta_{6} + 828 \beta_{5} - 12239 \beta_{4} + \cdots + 10841 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 418 \beta_{9} - 199 \beta_{8} + 5022 \beta_{7} + 4745 \beta_{6} - 4451 \beta_{5} + 21085 \beta_{4} + \cdots + 78553 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
− | 3.65934i | −3.17023 | −9.39078 | 1.46115 | 11.6009i | − | 2.64575i | 19.7267i | 1.05033 | − | 5.34686i | |||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | − | 3.18138i | 4.75782 | −6.12119 | −4.46186 | − | 15.1364i | − | 2.64575i | 6.74832i | 13.6368 | 14.1949i | ||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | − | 2.48081i | −2.18660 | −2.15443 | −8.53144 | 5.42455i | 2.64575i | − | 4.57852i | −4.21877 | 21.1649i | |||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | − | 1.44039i | −1.46470 | 1.92528 | 5.20089 | 2.10973i | − | 2.64575i | − | 8.53471i | −6.85467 | − | 7.49130i | |||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | − | 0.508799i | 3.06370 | 3.74112 | −2.66874 | − | 1.55881i | 2.64575i | − | 3.93868i | 0.386286 | 1.35785i | ||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 0.508799i | 3.06370 | 3.74112 | −2.66874 | 1.55881i | − | 2.64575i | 3.93868i | 0.386286 | − | 1.35785i | |||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 1.44039i | −1.46470 | 1.92528 | 5.20089 | − | 2.10973i | 2.64575i | 8.53471i | −6.85467 | 7.49130i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 2.48081i | −2.18660 | −2.15443 | −8.53144 | − | 5.42455i | − | 2.64575i | 4.57852i | −4.21877 | − | 21.1649i | ||||||||||||||||||||||||||||||||||||||||||||||
| 43.9 | 3.18138i | 4.75782 | −6.12119 | −4.46186 | 15.1364i | 2.64575i | − | 6.74832i | 13.6368 | − | 14.1949i | |||||||||||||||||||||||||||||||||||||||||||||||
| 43.10 | 3.65934i | −3.17023 | −9.39078 | 1.46115 | − | 11.6009i | 2.64575i | − | 19.7267i | 1.05033 | 5.34686i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 77.3.c.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 693.3.d.b | 10 | ||
| 4.b | odd | 2 | 1 | 1232.3.n.b | 10 | ||
| 7.b | odd | 2 | 1 | 539.3.c.i | 10 | ||
| 11.b | odd | 2 | 1 | inner | 77.3.c.b | ✓ | 10 |
| 33.d | even | 2 | 1 | 693.3.d.b | 10 | ||
| 44.c | even | 2 | 1 | 1232.3.n.b | 10 | ||
| 77.b | even | 2 | 1 | 539.3.c.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.3.c.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 77.3.c.b | ✓ | 10 | 11.b | odd | 2 | 1 | inner |
| 539.3.c.i | 10 | 7.b | odd | 2 | 1 | ||
| 539.3.c.i | 10 | 77.b | even | 2 | 1 | ||
| 693.3.d.b | 10 | 3.b | odd | 2 | 1 | ||
| 693.3.d.b | 10 | 33.d | even | 2 | 1 | ||
| 1232.3.n.b | 10 | 4.b | odd | 2 | 1 | ||
| 1232.3.n.b | 10 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 32T_{2}^{8} + 350T_{2}^{6} + 1504T_{2}^{4} + 2097T_{2}^{2} + 448 \)
acting on \(S_{3}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 32 T^{8} + \cdots + 448 \)
|
| $3$ |
\( (T^{5} - T^{4} - 24 T^{3} + \cdots + 148)^{2} \)
|
| $5$ |
\( (T^{5} + 9 T^{4} + \cdots + 772)^{2} \)
|
| $7$ |
\( (T^{2} + 7)^{5} \)
|
| $11$ |
\( T^{10} + \cdots + 25937424601 \)
|
| $13$ |
\( T^{10} + \cdots + 106688512 \)
|
| $17$ |
\( T^{10} + \cdots + 18718916608 \)
|
| $19$ |
\( T^{10} + \cdots + 20062723072 \)
|
| $23$ |
\( (T^{5} + 49 T^{4} + \cdots - 5262896)^{2} \)
|
| $29$ |
\( T^{10} + \cdots + 30\!\cdots\!12 \)
|
| $31$ |
\( (T^{5} + 3 T^{4} + \cdots - 1436972)^{2} \)
|
| $37$ |
\( (T^{5} - 39 T^{4} + \cdots - 429546240)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 19\!\cdots\!72 \)
|
| $43$ |
\( T^{10} + \cdots + 692814694039552 \)
|
| $47$ |
\( (T^{5} - 3782 T^{3} + \cdots + 3390424)^{2} \)
|
| $53$ |
\( (T^{5} - 82 T^{4} + \cdots + 221056)^{2} \)
|
| $59$ |
\( (T^{5} + 63 T^{4} + \cdots + 44187796)^{2} \)
|
| $61$ |
\( T^{10} + \cdots + 106809226574848 \)
|
| $67$ |
\( (T^{5} - 63 T^{4} + \cdots + 15866000)^{2} \)
|
| $71$ |
\( (T^{5} + 121 T^{4} + \cdots - 373569904)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 27\!\cdots\!48 \)
|
| $79$ |
\( T^{10} + \cdots + 29\!\cdots\!12 \)
|
| $83$ |
\( T^{10} + \cdots + 11\!\cdots\!72 \)
|
| $89$ |
\( (T^{5} + 33 T^{4} + \cdots + 6402512)^{2} \)
|
| $97$ |
\( (T^{5} - 247 T^{4} + \cdots + 164767376)^{2} \)
|
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