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: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 11\nu^{4} + 2\nu^{3} - 33\nu^{2} - 10\nu + 21 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{6} + \nu^{5} + 20\nu^{4} - 5\nu^{3} - 52\nu^{2} - \nu + 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( -3\nu^{6} + \nu^{5} + 30\nu^{4} - 3\nu^{3} - 78\nu^{2} - 10\nu + 41 \)
|
| \(\beta_{6}\) | \(=\) |
\( 4\nu^{6} - \nu^{5} - 41\nu^{4} + 2\nu^{3} + 111\nu^{2} + 15\nu - 62 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + 7\beta_{5} + 3\beta_{4} + 9\beta_{3} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} - 9\beta_{5} + 11\beta_{4} + 12\beta_{3} + 44\beta_{2} + 11\beta _1 + 76 \)
|
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 |
|
−2.40300 | −1.17935 | 3.77440 | −1.00000 | 2.83397 | −2.15598 | −4.26389 | −1.60914 | 2.40300 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.54354 | 2.58921 | 0.382516 | −1.00000 | −3.99655 | −3.41636 | 2.49665 | 3.70400 | 1.54354 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.08185 | −0.870874 | −0.829591 | −1.00000 | 0.942160 | 4.02863 | 3.06121 | −2.24158 | 1.08185 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.719047 | 0.163114 | −1.48297 | −1.00000 | 0.117287 | 1.05678 | −2.50442 | −2.97339 | −0.719047 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.745312 | −2.05058 | −1.44451 | −1.00000 | −1.52832 | −3.86782 | −2.56724 | 1.20489 | −0.745312 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.97792 | 2.65412 | 1.91218 | −1.00000 | 5.24965 | 1.06653 | −0.173707 | 4.04436 | −1.97792 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.58611 | 1.69436 | 4.68797 | −1.00000 | 4.38180 | 2.28823 | 6.95140 | −0.129144 | −2.58611 | ||||||||||||||||||||||||||||||||||||||||
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.h | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9405.2.a.bd | 7 | ||
| 5.b | even | 2 | 1 | 5225.2.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.2.a.h | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 5225.2.a.m | 7 | 5.b | even | 2 | 1 | ||
| 9405.2.a.bd | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 8T_{2}^{4} + 27T_{2}^{3} - 16T_{2}^{2} - 18T_{2} + 11 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots + 11 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 4 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} + T^{6} + \cdots + 296 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - T^{6} + \cdots - 1184 \)
|
| $17$ |
\( T^{7} - T^{6} + \cdots + 1096 \)
|
| $19$ |
\( (T + 1)^{7} \)
|
| $23$ |
\( T^{7} + 8 T^{6} + \cdots - 7724 \)
|
| $29$ |
\( T^{7} - 11 T^{6} + \cdots - 368 \)
|
| $31$ |
\( T^{7} - 7 T^{6} + \cdots + 8300 \)
|
| $37$ |
\( T^{7} + 17 T^{6} + \cdots - 20896 \)
|
| $41$ |
\( T^{7} - 17 T^{6} + \cdots - 1328 \)
|
| $43$ |
\( T^{7} + 3 T^{6} + \cdots + 2872 \)
|
| $47$ |
\( T^{7} - 14 T^{6} + \cdots + 35972 \)
|
| $53$ |
\( T^{7} - 7 T^{6} + \cdots + 1211488 \)
|
| $59$ |
\( T^{7} - 35 T^{6} + \cdots - 106388 \)
|
| $61$ |
\( T^{7} - 17 T^{6} + \cdots + 1052 \)
|
| $67$ |
\( T^{7} - 4 T^{6} + \cdots - 1748 \)
|
| $71$ |
\( T^{7} - 10 T^{6} + \cdots + 14492 \)
|
| $73$ |
\( T^{7} - 22 T^{6} + \cdots + 85096 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots + 19664 \)
|
| $83$ |
\( T^{7} - 39 T^{6} + \cdots + 2216 \)
|
| $89$ |
\( T^{7} - 18 T^{6} + \cdots - 20716 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots - 6460112 \)
|
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