Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1058,2,Mod(1,1058)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1058.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1058 = 2 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1058.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: | \(8.44817253385\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.819879542784.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 151x^{4} - 440x^{3} - 298x^{2} + 532x - 146 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 13\nu^{2} + 14\nu + 19 ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{7} - 21\nu^{6} - 131\nu^{5} + 380\nu^{4} + 889\nu^{3} - 1724\nu^{2} - 1845\nu + 1223 ) / 299 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} + 29\nu^{6} + 90\nu^{5} - 562\nu^{4} - 553\nu^{3} + 2855\nu^{2} + 1572\nu - 2196 ) / 299 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{7} + 28\nu^{6} + 190\nu^{5} - 545\nu^{4} - 1469\nu^{3} + 2912\nu^{2} + 3541\nu - 3371 ) / 299 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\nu^{7} - 56\nu^{6} - 380\nu^{5} + 1090\nu^{4} + 2938\nu^{3} - 5525\nu^{2} - 7082\nu + 4649 ) / 299 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\nu^{7} - 56\nu^{6} - 380\nu^{5} + 1090\nu^{4} + 2938\nu^{3} - 5525\nu^{2} - 7381\nu + 4649 ) / 299 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 42\nu^{7} - 147\nu^{6} - 986\nu^{5} + 2821\nu^{4} + 7672\nu^{3} - 14230\nu^{2} - 19884\nu + 12287 ) / 299 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{6} + \beta_{5} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 13\beta_{6} + 10\beta_{5} + 3\beta_{4} - 2\beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} - 12\beta_{6} + 19\beta_{5} + 32\beta_{4} - 4\beta_{2} + 15\beta _1 + 82 \)
|
| \(\nu^{5}\) | \(=\) |
\( 47\beta_{7} - 200\beta_{6} + 122\beta_{5} + 75\beta_{4} - 21\beta_{2} + 56\beta _1 + 131 \)
|
| \(\nu^{6}\) | \(=\) |
\( 131\beta_{7} - 374\beta_{6} + 319\beta_{5} + 512\beta_{4} + 26\beta_{3} - 66\beta_{2} + 397\beta _1 + 1125 \)
|
| \(\nu^{7}\) | \(=\) |
\( 935\beta_{7} - 3297\beta_{6} + 1690\beta_{5} + 1533\beta_{4} + 91\beta_{3} - 90\beta_{2} + 1514\beta _1 + 2671 \)
|
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 |
|
1.00000 | −2.14236 | 1.00000 | −3.04082 | −2.14236 | −4.44396 | 1.00000 | 1.58969 | −3.04082 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.14236 | 1.00000 | 3.04082 | −2.14236 | 4.44396 | 1.00000 | 1.58969 | 3.04082 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.37562 | 1.00000 | −3.17513 | −1.37562 | 3.35963 | 1.00000 | −1.10767 | −3.17513 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.37562 | 1.00000 | 3.17513 | −1.37562 | −3.35963 | 1.00000 | −1.10767 | 3.17513 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.37562 | 1.00000 | −4.07171 | 2.37562 | 1.94542 | 1.00000 | 2.64357 | −4.07171 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.37562 | 1.00000 | 4.07171 | 2.37562 | −1.94542 | 1.00000 | 2.64357 | 4.07171 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.14236 | 1.00000 | −0.305248 | 3.14236 | 3.02975 | 1.00000 | 6.87441 | −0.305248 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.14236 | 1.00000 | 0.305248 | 3.14236 | −3.02975 | 1.00000 | 6.87441 | 0.305248 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1058.2.a.n | ✓ | 8 |
| 3.b | odd | 2 | 1 | 9522.2.a.ce | 8 | ||
| 4.b | odd | 2 | 1 | 8464.2.a.cb | 8 | ||
| 23.b | odd | 2 | 1 | inner | 1058.2.a.n | ✓ | 8 |
| 69.c | even | 2 | 1 | 9522.2.a.ce | 8 | ||
| 92.b | even | 2 | 1 | 8464.2.a.cb | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1058.2.a.n | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1058.2.a.n | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
| 8464.2.a.cb | 8 | 4.b | odd | 2 | 1 | ||
| 8464.2.a.cb | 8 | 92.b | even | 2 | 1 | ||
| 9522.2.a.ce | 8 | 3.b | odd | 2 | 1 | ||
| 9522.2.a.ce | 8 | 69.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1058))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 9T_{3}^{2} + 10T_{3} + 22 \)
|
|
\( T_{5}^{8} - 36T_{5}^{6} + 417T_{5}^{4} - 1584T_{5}^{2} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T^{4} - 2 T^{3} - 9 T^{2} + \cdots + 22)^{2} \)
|
| $5$ |
\( T^{8} - 36 T^{6} + \cdots + 144 \)
|
| $7$ |
\( T^{8} - 44 T^{6} + \cdots + 7744 \)
|
| $11$ |
\( (T^{2} - 6)^{4} \)
|
| $13$ |
\( (T^{4} - 6 T^{3} - 3 T^{2} + \cdots - 48)^{2} \)
|
| $17$ |
\( T^{8} - 96 T^{6} + \cdots + 125316 \)
|
| $19$ |
\( T^{8} - 56 T^{6} + \cdots + 21316 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( (T^{4} - 63 T^{2} + \cdots - 264)^{2} \)
|
| $31$ |
\( (T^{2} - 12)^{4} \)
|
| $37$ |
\( T^{8} - 44 T^{6} + \cdots + 7744 \)
|
| $41$ |
\( (T^{4} - 6 T^{3} + \cdots + 384)^{2} \)
|
| $43$ |
\( T^{8} - 164 T^{6} + \cdots + 30976 \)
|
| $47$ |
\( (T^{4} + 6 T^{3} + \cdots - 552)^{2} \)
|
| $53$ |
\( T^{8} - 132 T^{6} + \cdots + 82944 \)
|
| $59$ |
\( (T^{4} - 12 T^{3} - 33 T^{2} + \cdots + 6)^{2} \)
|
| $61$ |
\( T^{8} - 572 T^{6} + \cdots + 183439936 \)
|
| $67$ |
\( T^{8} - 116 T^{6} + \cdots + 38416 \)
|
| $71$ |
\( (T^{4} + 6 T^{3} + \cdots + 15864)^{2} \)
|
| $73$ |
\( (T^{4} - 6 T^{3} + \cdots + 1293)^{2} \)
|
| $79$ |
\( T^{8} - 464 T^{6} + \cdots + 22429696 \)
|
| $83$ |
\( T^{8} - 84 T^{6} + \cdots + 69696 \)
|
| $89$ |
\( T^{8} - 360 T^{6} + \cdots + 527076 \)
|
| $97$ |
\( T^{8} - 224 T^{6} + \cdots + 662596 \)
|
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