Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1305,2,Mod(1,1305)]
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, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1305.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1305.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: | \(10.4204774638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 8\nu^{3} - 7\nu^{2} + 11\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 10\nu^{4} - 7\nu^{3} + 25\nu^{2} + 6\nu - 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 8\beta_{3} + 37\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} - 2\beta_{5} + 11\beta_{4} - \beta_{3} + 45\beta_{2} - \beta _1 + 141 \)
|
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 |
|
−2.66072 | 0 | 5.07943 | 1.00000 | 0 | 4.68271 | −8.19351 | 0 | −2.66072 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16940 | 0 | 2.70628 | 1.00000 | 0 | −0.484966 | −1.53221 | 0 | −2.16940 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.897436 | 0 | −1.19461 | 1.00000 | 0 | 3.83036 | 2.86696 | 0 | −0.897436 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.431560 | 0 | −1.81376 | 1.00000 | 0 | −2.42505 | −1.64586 | 0 | 0.431560 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.28209 | 0 | −0.356256 | 1.00000 | 0 | 3.04834 | −3.02092 | 0 | 1.28209 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.43890 | 0 | 3.94822 | 1.00000 | 0 | 3.59688 | 4.75150 | 0 | 2.43890 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.57501 | 0 | 4.63069 | 1.00000 | 0 | −2.24826 | 6.77405 | 0 | 2.57501 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 1305.2.a.t | yes | 7 |
| 3.b | odd | 2 | 1 | 1305.2.a.s | ✓ | 7 | |
| 5.b | even | 2 | 1 | 6525.2.a.bv | 7 | ||
| 15.d | odd | 2 | 1 | 6525.2.a.bw | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1305.2.a.s | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 1305.2.a.t | yes | 7 | 1.a | even | 1 | 1 | trivial |
| 6525.2.a.bv | 7 | 5.b | even | 2 | 1 | ||
| 6525.2.a.bw | 7 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\):
|
\( T_{2}^{7} - T_{2}^{6} - 13T_{2}^{5} + 12T_{2}^{4} + 47T_{2}^{3} - 37T_{2}^{2} - 35T_{2} + 18 \)
|
|
\( T_{7}^{7} - 10T_{7}^{6} + 15T_{7}^{5} + 114T_{7}^{4} - 280T_{7}^{3} - 400T_{7}^{2} + 956T_{7} + 520 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 18 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} - 10 T^{6} + \cdots + 520 \)
|
| $11$ |
\( T^{7} + 3 T^{6} + \cdots - 12 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots + 1408 \)
|
| $17$ |
\( T^{7} - 8 T^{6} + \cdots + 384 \)
|
| $19$ |
\( T^{7} - 10 T^{6} + \cdots + 6656 \)
|
| $23$ |
\( T^{7} - 11 T^{6} + \cdots - 4752 \)
|
| $29$ |
\( (T + 1)^{7} \)
|
| $31$ |
\( T^{7} - 18 T^{6} + \cdots - 1408 \)
|
| $37$ |
\( T^{7} - 13 T^{6} + \cdots - 3328 \)
|
| $41$ |
\( T^{7} + 13 T^{6} + \cdots - 2496 \)
|
| $43$ |
\( T^{7} - 9 T^{6} + \cdots - 171008 \)
|
| $47$ |
\( T^{7} - 2 T^{6} + \cdots - 23040 \)
|
| $53$ |
\( T^{7} - 5 T^{6} + \cdots + 734208 \)
|
| $59$ |
\( T^{7} + 8 T^{6} + \cdots + 49152 \)
|
| $61$ |
\( T^{7} - 14 T^{6} + \cdots + 2422656 \)
|
| $67$ |
\( T^{7} - 14 T^{6} + \cdots + 366536 \)
|
| $71$ |
\( T^{7} + 8 T^{6} + \cdots + 245760 \)
|
| $73$ |
\( T^{7} - 3 T^{6} + \cdots + 2022656 \)
|
| $79$ |
\( T^{7} - 4 T^{6} + \cdots + 27712 \)
|
| $83$ |
\( T^{7} - 17 T^{6} + \cdots - 52800 \)
|
| $89$ |
\( T^{7} + 20 T^{6} + \cdots - 26400 \)
|
| $97$ |
\( T^{7} - 13 T^{6} + \cdots + 249856 \)
|
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