Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1305,2,Mod(784,1305)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1305.784");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1305.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: | \(10.4204774638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.84345856.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 145) |
| 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
\( x^{6} + 13x^{4} + 41x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 14\nu^{3} + 6\nu^{2} + 47\nu - 5 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 14\nu^{3} - 6\nu^{2} + 47\nu + 5 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 14\nu^{3} + 10\nu^{2} - 47\nu + 15 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 12\nu^{3} + 35\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + \beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} - 8\beta_{3} + 2\beta_{2} + 35 \)
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| \(\nu^{5}\) | \(=\) |
\( 14\beta_{5} - 12\beta_{3} - 12\beta_{2} + 37\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).
| \(n\) | \(146\) | \(262\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 784.1 |
|
− | 2.77035i | 0 | −5.67486 | 1.96358 | − | 1.06975i | 0 | 1.86960i | 10.1807i | 0 | −2.96358 | − | 5.43981i | |||||||||||||||||||||||||||||||
| 784.2 | − | 2.30229i | 0 | −3.30056 | −2.17686 | + | 0.511167i | 0 | − | 3.91261i | 2.99427i | 0 | 1.17686 | + | 5.01177i | |||||||||||||||||||||||||||||||
| 784.3 | − | 0.156785i | 0 | 1.97542 | −1.28672 | + | 1.82876i | 0 | − | 1.09364i | − | 0.623285i | 0 | 0.286721 | + | 0.201738i | ||||||||||||||||||||||||||||||
| 784.4 | 0.156785i | 0 | 1.97542 | −1.28672 | − | 1.82876i | 0 | 1.09364i | 0.623285i | 0 | 0.286721 | − | 0.201738i | |||||||||||||||||||||||||||||||||
| 784.5 | 2.30229i | 0 | −3.30056 | −2.17686 | − | 0.511167i | 0 | 3.91261i | − | 2.99427i | 0 | 1.17686 | − | 5.01177i | ||||||||||||||||||||||||||||||||
| 784.6 | 2.77035i | 0 | −5.67486 | 1.96358 | + | 1.06975i | 0 | − | 1.86960i | − | 10.1807i | 0 | −2.96358 | + | 5.43981i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1305.2.c.h | 6 | |
| 3.b | odd | 2 | 1 | 145.2.b.c | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 1305.2.c.h | 6 | |
| 5.c | odd | 4 | 2 | 6525.2.a.bt | 6 | ||
| 12.b | even | 2 | 1 | 2320.2.d.g | 6 | ||
| 15.d | odd | 2 | 1 | 145.2.b.c | ✓ | 6 | |
| 15.e | even | 4 | 2 | 725.2.a.l | 6 | ||
| 60.h | even | 2 | 1 | 2320.2.d.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.2.b.c | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 145.2.b.c | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 725.2.a.l | 6 | 15.e | even | 4 | 2 | ||
| 1305.2.c.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 1305.2.c.h | 6 | 5.b | even | 2 | 1 | inner | |
| 2320.2.d.g | 6 | 12.b | even | 2 | 1 | ||
| 2320.2.d.g | 6 | 60.h | even | 2 | 1 | ||
| 6525.2.a.bt | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):
|
\( T_{2}^{6} + 13T_{2}^{4} + 41T_{2}^{2} + 1 \)
|
|
\( T_{7}^{6} + 20T_{7}^{4} + 76T_{7}^{2} + 64 \)
|
|
\( T_{11}^{3} - 5T_{11}^{2} - 6T_{11} + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 13 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 20 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T^{3} - 5 T^{2} - 6 T + 38)^{2} \)
|
| $13$ |
\( T^{6} + 59 T^{4} + \cdots + 5776 \)
|
| $17$ |
\( T^{6} + 48 T^{4} + \cdots + 784 \)
|
| $19$ |
\( (T^{3} + 8 T^{2} + 6 T - 8)^{2} \)
|
| $23$ |
\( T^{6} + 56 T^{4} + \cdots + 16 \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( (T^{3} - 11 T^{2} + \cdots + 22)^{2} \)
|
| $37$ |
\( T^{6} + 20 T^{4} + \cdots + 64 \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( T^{6} + 127 T^{4} + \cdots + 196 \)
|
| $47$ |
\( T^{6} + 63 T^{4} + \cdots + 8836 \)
|
| $53$ |
\( T^{6} + 187 T^{4} + \cdots + 5776 \)
|
| $59$ |
\( (T - 10)^{6} \)
|
| $61$ |
\( (T^{3} - 6 T^{2} - 64 T - 64)^{2} \)
|
| $67$ |
\( T^{6} + 140 T^{4} + \cdots + 92416 \)
|
| $71$ |
\( (T + 2)^{6} \)
|
| $73$ |
\( T^{6} + 188 T^{4} + \cdots + 3136 \)
|
| $79$ |
\( (T^{3} + T^{2} - 54 T + 98)^{2} \)
|
| $83$ |
\( T^{6} + 48 T^{4} + \cdots + 784 \)
|
| $89$ |
\( (T^{3} + 16 T^{2} + \cdots - 304)^{2} \)
|
| $97$ |
\( T^{6} + 476 T^{4} + \cdots + 7744 \)
|
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