Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [765,2,Mod(271,765)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(765, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("765.271");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 765 = 3^{2} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 765.g (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: | \(6.10855575463\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{2} + 5\beta _1 - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/765\mathbb{Z}\right)^\times\).
| \(n\) | \(307\) | \(496\) | \(596\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 271.1 |
|
−1.48119 | 0 | 0.193937 | − | 1.00000i | 0 | 1.28726i | 2.67513 | 0 | 1.48119i | |||||||||||||||||||||||||||||||||||
| 271.2 | −1.48119 | 0 | 0.193937 | 1.00000i | 0 | − | 1.28726i | 2.67513 | 0 | − | 1.48119i | |||||||||||||||||||||||||||||||||||
| 271.3 | 0.311108 | 0 | −1.90321 | − | 1.00000i | 0 | 1.59210i | −1.21432 | 0 | − | 0.311108i | |||||||||||||||||||||||||||||||||||
| 271.4 | 0.311108 | 0 | −1.90321 | 1.00000i | 0 | − | 1.59210i | −1.21432 | 0 | 0.311108i | ||||||||||||||||||||||||||||||||||||
| 271.5 | 2.17009 | 0 | 2.70928 | − | 1.00000i | 0 | − | 4.87936i | 1.53919 | 0 | − | 2.17009i | ||||||||||||||||||||||||||||||||||
| 271.6 | 2.17009 | 0 | 2.70928 | 1.00000i | 0 | 4.87936i | 1.53919 | 0 | 2.17009i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 765.2.g.b | 6 | |
| 3.b | odd | 2 | 1 | 85.2.d.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 1360.2.c.f | 6 | ||
| 15.d | odd | 2 | 1 | 425.2.d.c | 6 | ||
| 15.e | even | 4 | 1 | 425.2.c.a | 6 | ||
| 15.e | even | 4 | 1 | 425.2.c.b | 6 | ||
| 17.b | even | 2 | 1 | inner | 765.2.g.b | 6 | |
| 51.c | odd | 2 | 1 | 85.2.d.a | ✓ | 6 | |
| 51.f | odd | 4 | 1 | 1445.2.a.j | 3 | ||
| 51.f | odd | 4 | 1 | 1445.2.a.k | 3 | ||
| 204.h | even | 2 | 1 | 1360.2.c.f | 6 | ||
| 255.h | odd | 2 | 1 | 425.2.d.c | 6 | ||
| 255.i | odd | 4 | 1 | 7225.2.a.q | 3 | ||
| 255.i | odd | 4 | 1 | 7225.2.a.r | 3 | ||
| 255.o | even | 4 | 1 | 425.2.c.a | 6 | ||
| 255.o | even | 4 | 1 | 425.2.c.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.d.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 85.2.d.a | ✓ | 6 | 51.c | odd | 2 | 1 | |
| 425.2.c.a | 6 | 15.e | even | 4 | 1 | ||
| 425.2.c.a | 6 | 255.o | even | 4 | 1 | ||
| 425.2.c.b | 6 | 15.e | even | 4 | 1 | ||
| 425.2.c.b | 6 | 255.o | even | 4 | 1 | ||
| 425.2.d.c | 6 | 15.d | odd | 2 | 1 | ||
| 425.2.d.c | 6 | 255.h | odd | 2 | 1 | ||
| 765.2.g.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 765.2.g.b | 6 | 17.b | even | 2 | 1 | inner | |
| 1360.2.c.f | 6 | 12.b | even | 2 | 1 | ||
| 1360.2.c.f | 6 | 204.h | even | 2 | 1 | ||
| 1445.2.a.j | 3 | 51.f | odd | 4 | 1 | ||
| 1445.2.a.k | 3 | 51.f | odd | 4 | 1 | ||
| 7225.2.a.q | 3 | 255.i | odd | 4 | 1 | ||
| 7225.2.a.r | 3 | 255.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(765, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 3 T + 1)^{2} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( T^{6} + 28 T^{4} + \cdots + 100 \)
|
| $11$ |
\( T^{6} + 12 T^{4} + \cdots + 4 \)
|
| $13$ |
\( (T^{3} - 4 T^{2} - 4 T + 20)^{2} \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} - 16 T + 16)^{2} \)
|
| $23$ |
\( T^{6} + 120 T^{4} + \cdots + 45796 \)
|
| $29$ |
\( T^{6} + 80 T^{4} + \cdots + 256 \)
|
| $31$ |
\( T^{6} + 80 T^{4} + \cdots + 3364 \)
|
| $37$ |
\( T^{6} + 96 T^{4} + \cdots + 25600 \)
|
| $41$ |
\( T^{6} + 172 T^{4} + \cdots + 53824 \)
|
| $43$ |
\( (T^{3} - 14 T^{2} + \cdots - 68)^{2} \)
|
| $47$ |
\( (T^{3} - 10 T^{2} + 148)^{2} \)
|
| $53$ |
\( (T^{3} - 14 T^{2} + \cdots + 296)^{2} \)
|
| $59$ |
\( (T^{3} + 4 T^{2} - 8 T - 16)^{2} \)
|
| $61$ |
\( T^{6} + 396 T^{4} + \cdots + 287296 \)
|
| $67$ |
\( (T^{3} - 6 T^{2} + \cdots + 460)^{2} \)
|
| $71$ |
\( T^{6} + 340 T^{4} + \cdots + 391876 \)
|
| $73$ |
\( T^{6} + 332 T^{4} + \cdots + 678976 \)
|
| $79$ |
\( T^{6} + 192 T^{4} + \cdots + 244036 \)
|
| $83$ |
\( (T^{3} + 2 T^{2} + \cdots - 796)^{2} \)
|
| $89$ |
\( (T^{3} + 2 T^{2} + \cdots - 1396)^{2} \)
|
| $97$ |
\( T^{6} + 320 T^{4} + \cdots + 215296 \)
|
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