Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,2,Mod(1,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, 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("1045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1045 = 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1045.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: | \(8.34436701122\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.131947641.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 7\nu - 3 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} - 3\nu^{2} + 7\nu + 9 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 10\nu^{3} - 22\nu + 3 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 6\beta_{3} + 7\beta_{2} + \beta _1 + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{4} + 10\beta_{2} + 28\beta _1 + 3 \)
|
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 |
|
−2.36323 | 2.87321 | 3.58485 | 1.00000 | −6.79006 | −0.269449 | −3.74537 | 5.25535 | −2.36323 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.04201 | −1.09835 | 2.16980 | 1.00000 | 2.24284 | −0.469142 | −0.346728 | −1.79363 | −2.04201 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.577704 | −0.746709 | −1.66626 | 1.00000 | 0.431377 | −4.19297 | 2.11801 | −2.44243 | −0.577704 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.759131 | −2.25829 | −1.42372 | 1.00000 | −1.71434 | 4.95189 | −2.59905 | 2.09989 | 0.759131 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.65636 | 3.32622 | 0.743534 | 1.00000 | 5.50942 | 2.81120 | −2.08116 | 8.06374 | 1.65636 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.56745 | 0.903918 | 4.59179 | 1.00000 | 2.32076 | 2.16848 | 6.65430 | −2.18293 | 2.56745 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(1\) |
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 | 1045.2.a.g | ✓ | 6 |
| 3.b | odd | 2 | 1 | 9405.2.a.w | 6 | ||
| 5.b | even | 2 | 1 | 5225.2.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.g | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5225.2.a.k | 6 | 5.b | even | 2 | 1 | ||
| 9405.2.a.w | 6 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 10T_{2}^{4} + 25T_{2}^{2} - 3T_{2} - 9 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 10 T^{4} + \cdots - 9 \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots - 16 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots - 16 \)
|
| $11$ |
\( (T + 1)^{6} \)
|
| $13$ |
\( T^{6} + 9 T^{5} + \cdots - 14 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots - 6290 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 8 T^{5} + \cdots + 92 \)
|
| $29$ |
\( T^{6} + 5 T^{5} + \cdots + 482 \)
|
| $31$ |
\( T^{6} + T^{5} + \cdots - 3240 \)
|
| $37$ |
\( T^{6} - 9 T^{5} + \cdots + 1906 \)
|
| $41$ |
\( T^{6} - 25 T^{5} + \cdots + 9526 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 4460 \)
|
| $47$ |
\( T^{6} - 24 T^{5} + \cdots - 7268 \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots - 627158 \)
|
| $59$ |
\( T^{6} - 39 T^{5} + \cdots + 35420 \)
|
| $61$ |
\( T^{6} + 11 T^{5} + \cdots - 156150 \)
|
| $67$ |
\( T^{6} - 24 T^{5} + \cdots - 123184 \)
|
| $71$ |
\( T^{6} + 24 T^{5} + \cdots - 56840 \)
|
| $73$ |
\( T^{6} + 26 T^{5} + \cdots + 34902 \)
|
| $79$ |
\( T^{6} - 11 T^{5} + \cdots + 8152 \)
|
| $83$ |
\( T^{6} - 39 T^{5} + \cdots + 2100044 \)
|
| $89$ |
\( T^{6} - 22 T^{5} + \cdots + 554 \)
|
| $97$ |
\( T^{6} - 22 T^{5} + \cdots + 506 \)
|
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