Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1058,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1058.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1058 = 2 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(62.4240207861\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 75x^{4} + 102x^{3} + 1209x^{2} - 844x - 2078 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 75x^{4} + 102x^{3} + 1209x^{2} - 844x - 2078 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 57\nu^{5} + 1728\nu^{4} - 6104\nu^{3} - 81343\nu^{2} + 98344\nu + 141566 ) / 99613 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 278\nu^{5} + 3185\nu^{4} - 22780\nu^{3} - 251675\nu^{2} + 214008\nu + 2832998 ) / 298839 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -164\nu^{5} + 271\nu^{4} + 10572\nu^{3} - 10624\nu^{2} - 116933\nu + 40072 ) / 99613 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -706\nu^{5} + 4811\nu^{4} + 40652\nu^{3} - 315419\nu^{2} + 110088\nu + 2774591 ) / 298839 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1880\nu^{5} + 677\nu^{4} + 157635\nu^{3} - 41611\nu^{2} - 2778766\nu + 716898 ) / 298839 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 5\beta_{2} + 4\beta _1 + 51 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{5} + 20\beta_{4} - 74\beta_{3} - 23\beta_{2} + 6\beta _1 + 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 24\beta_{5} - 25\beta_{4} - 96\beta_{3} - 263\beta_{2} + 312\beta _1 + 2263 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1200\beta_{5} + 1889\beta_{4} - 9488\beta_{3} - 2363\beta_{2} + 1030\beta _1 + 4337 ) / 2 \)
|
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 |
|
−2.00000 | −5.21166 | 4.00000 | −4.33058 | 10.4233 | 9.04844 | −8.00000 | 0.161433 | 8.66116 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −5.21166 | 4.00000 | 4.33058 | 10.4233 | −9.04844 | −8.00000 | 0.161433 | −8.66116 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 1.36391 | 4.00000 | −3.18284 | −2.72781 | 22.7911 | −8.00000 | −25.1398 | 6.36569 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 1.36391 | 4.00000 | 3.18284 | −2.72781 | −22.7911 | −8.00000 | −25.1398 | −6.36569 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 9.84776 | 4.00000 | −17.2370 | −19.6955 | 26.4706 | −8.00000 | 69.9783 | 34.4741 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 9.84776 | 4.00000 | 17.2370 | −19.6955 | −26.4706 | −8.00000 | 69.9783 | −34.4741 | |||||||||||||||||||||||||||||||||||||
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 | 1058.4.a.p | ✓ | 6 |
| 23.b | odd | 2 | 1 | inner | 1058.4.a.p | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1058.4.a.p | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1058.4.a.p | ✓ | 6 | 23.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):
|
\( T_{3}^{3} - 6T_{3}^{2} - 45T_{3} + 70 \)
|
|
\( T_{5}^{6} - 326T_{5}^{4} + 8772T_{5}^{2} - 56448 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( (T^{3} - 6 T^{2} - 45 T + 70)^{2} \)
|
| $5$ |
\( T^{6} - 326 T^{4} + \cdots - 56448 \)
|
| $7$ |
\( T^{6} - 1302 T^{4} + \cdots - 29799200 \)
|
| $11$ |
\( T^{6} + \cdots - 1627921800 \)
|
| $13$ |
\( (T^{3} - 22 T^{2} + \cdots - 1050)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 3144879432 \)
|
| $19$ |
\( T^{6} - 3334 T^{4} + \cdots - 643256712 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( (T^{3} - 290 T^{2} + \cdots - 394266)^{2} \)
|
| $31$ |
\( (T^{3} + 134 T^{2} + \cdots - 2098368)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 1546154091008 \)
|
| $41$ |
\( (T^{3} + 24 T^{2} + \cdots - 6708528)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 16\!\cdots\!28 \)
|
| $47$ |
\( (T^{3} - 662 T^{2} + \cdots + 42000360)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 74\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} - 322 T^{2} + \cdots - 23390160)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 285973178580000 \)
|
| $67$ |
\( T^{6} + \cdots - 54\!\cdots\!52 \)
|
| $71$ |
\( (T^{3} + 264 T^{2} + \cdots + 118093572)^{2} \)
|
| $73$ |
\( (T^{3} - 634 T^{2} + \cdots + 183848070)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 23\!\cdots\!32 \)
|
| $83$ |
\( T^{6} + \cdots - 97\!\cdots\!92 \)
|
| $89$ |
\( T^{6} + \cdots - 12\!\cdots\!52 \)
|
| $97$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
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