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: | \(1\) |
| Dimension: | \(15\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{15} - 2 x^{14} - 242 x^{13} + 347 x^{12} + 22092 x^{11} - 27616 x^{10} - 966502 x^{9} + \cdots - 23850793 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 23^{4} \) |
| Twist minimal: | no (minimal twist has level 46) |
| 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 \(1,\beta_1,\ldots,\beta_{14}\) 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
\( x^{15} - 2 x^{14} - 242 x^{13} + 347 x^{12} + 22092 x^{11} - 27616 x^{10} - 966502 x^{9} + \cdots - 23850793 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 49\!\cdots\!27 \nu^{14} + \cdots + 40\!\cdots\!47 ) / 17\!\cdots\!94 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 23\!\cdots\!76 \nu^{14} + \cdots - 80\!\cdots\!30 ) / 74\!\cdots\!78 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12\!\cdots\!74 \nu^{14} + \cdots - 29\!\cdots\!34 ) / 17\!\cdots\!94 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13\!\cdots\!28 \nu^{14} + \cdots + 30\!\cdots\!10 ) / 17\!\cdots\!94 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\!\cdots\!26 \nu^{14} + \cdots + 48\!\cdots\!75 ) / 17\!\cdots\!94 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 30\!\cdots\!10 \nu^{14} + \cdots - 41\!\cdots\!96 ) / 17\!\cdots\!94 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 59\!\cdots\!46 \nu^{14} + \cdots - 73\!\cdots\!59 ) / 17\!\cdots\!94 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 60\!\cdots\!51 \nu^{14} + \cdots + 37\!\cdots\!84 ) / 17\!\cdots\!94 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 72\!\cdots\!07 \nu^{14} + \cdots + 43\!\cdots\!23 ) / 17\!\cdots\!94 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 72\!\cdots\!15 \nu^{14} + \cdots - 59\!\cdots\!33 ) / 17\!\cdots\!94 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 81\!\cdots\!70 \nu^{14} + \cdots - 20\!\cdots\!82 ) / 17\!\cdots\!94 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\!\cdots\!67 \nu^{14} + \cdots - 44\!\cdots\!92 ) / 17\!\cdots\!94 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 11\!\cdots\!99 \nu^{14} + \cdots + 81\!\cdots\!44 ) / 17\!\cdots\!94 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 23\!\cdots\!35 \nu^{14} + \cdots + 45\!\cdots\!84 ) / 17\!\cdots\!94 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} - \beta_{9} - 2\beta_{8} - \beta_{6} + \beta_{5} - 21\beta_{4} + \beta_{3} + \beta_{2} + 5 ) / 23 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{10} + 23 \beta_{9} - 5 \beta_{8} + 23 \beta_{7} - 23 \beta_{6} + 6 \beta_{5} - 44 \beta_{4} + \cdots + 762 ) / 23 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 60 \beta_{14} - 46 \beta_{13} + 46 \beta_{12} - 37 \beta_{11} - 37 \beta_{10} - 14 \beta_{9} + \cdots + 864 ) / 23 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 404 \beta_{14} - 437 \beta_{13} + 483 \beta_{12} + 311 \beta_{11} + 124 \beta_{10} + 1252 \beta_{9} + \cdots + 47497 ) / 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7023 \beta_{14} - 9522 \beta_{13} + 6785 \beta_{12} - 2788 \beta_{11} - 3527 \beta_{10} + 245 \beta_{9} + \cdots + 153730 ) / 23 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 64905 \beta_{14} - 79879 \beta_{13} + 69299 \beta_{12} + 40123 \beta_{11} - 184 \beta_{10} + \cdots + 3770082 ) / 23 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 832554 \beta_{14} - 1194413 \beta_{13} + 848194 \beta_{12} - 69736 \beta_{11} - 333522 \beta_{10} + \cdots + 21554727 ) / 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 8236215 \beta_{14} - 10714067 \beta_{13} + 8572422 \beta_{12} + 4385023 \beta_{11} - 1284194 \beta_{10} + \cdots + 352322721 ) / 23 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 96214602 \beta_{14} - 136900002 \beta_{13} + 99970742 \beta_{12} + 11785012 \beta_{11} + \cdots + 2722600941 ) / 23 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 979663370 \beta_{14} - 1310442756 \beta_{13} + 1018501870 \beta_{12} + 467364552 \beta_{11} + \cdots + 36252925073 ) / 23 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 10995833389 \beta_{14} - 15443332727 \beta_{13} + 11517978548 \beta_{12} + 2665105105 \beta_{11} + \cdots + 327491025600 ) / 23 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 113945793086 \beta_{14} - 154637270248 \beta_{13} + 118888337427 \beta_{12} + 50261069582 \beta_{11} + \cdots + 3926798360604 ) / 23 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 1252895781187 \beta_{14} - 1743906463611 \beta_{13} + 1316457596745 \beta_{12} + 386855257685 \beta_{11} + \cdots + 38443845721179 ) / 23 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 13134015873262 \beta_{14} - 17957635321576 \beta_{13} + 13747760926390 \beta_{12} + \cdots + 436626541056413 ) / 23 \)
|
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 | −8.76968 | 4.00000 | 17.7677 | 17.5394 | −24.1196 | −8.00000 | 49.9074 | −35.5353 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −8.72876 | 4.00000 | 10.5657 | 17.4575 | 19.1436 | −8.00000 | 49.1912 | −21.1314 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −6.63708 | 4.00000 | −7.23654 | 13.2742 | 17.5258 | −8.00000 | 17.0508 | 14.4731 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −4.39473 | 4.00000 | −13.3637 | 8.78946 | −29.0924 | −8.00000 | −7.68636 | 26.7275 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | −4.21331 | 4.00000 | −20.5767 | 8.42662 | −29.7298 | −8.00000 | −9.24801 | 41.1535 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | −2.42114 | 4.00000 | 18.6449 | 4.84229 | −20.0603 | −8.00000 | −21.1381 | −37.2899 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | −1.18755 | 4.00000 | −16.5269 | 2.37510 | −0.451014 | −8.00000 | −25.5897 | 33.0538 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | −0.358528 | 4.00000 | 9.16248 | 0.717057 | 16.3260 | −8.00000 | −26.8715 | −18.3250 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −2.00000 | 0.448706 | 4.00000 | 16.1561 | −0.897413 | −19.2599 | −8.00000 | −26.7987 | −32.3122 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −2.00000 | 2.69808 | 4.00000 | 7.96134 | −5.39616 | 20.0247 | −8.00000 | −19.7204 | −15.9227 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −2.00000 | 5.16469 | 4.00000 | −6.57860 | −10.3294 | 29.3246 | −8.00000 | −0.325939 | 13.1572 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −2.00000 | 5.47758 | 4.00000 | −8.97134 | −10.9552 | −3.67611 | −8.00000 | 3.00387 | 17.9427 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | −2.00000 | 6.26666 | 4.00000 | 10.4231 | −12.5333 | −11.8502 | −8.00000 | 12.2711 | −20.8461 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | −2.00000 | 8.59985 | 4.00000 | −13.7601 | −17.1997 | −33.9854 | −8.00000 | 46.9575 | 27.5202 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | −2.00000 | 9.05521 | 4.00000 | −3.66728 | −18.1104 | −10.1201 | −8.00000 | 54.9968 | 7.33455 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(23\) | \(-1\) |
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 | 1058.4.a.t | 15 | |
| 23.b | odd | 2 | 1 | 1058.4.a.u | 15 | ||
| 23.d | odd | 22 | 2 | 46.4.c.b | ✓ | 30 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.4.c.b | ✓ | 30 | 23.d | odd | 22 | 2 | |
| 1058.4.a.t | 15 | 1.a | even | 1 | 1 | trivial | |
| 1058.4.a.u | 15 | 23.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):
|
\( T_{3}^{15} - T_{3}^{14} - 250 T_{3}^{13} + 204 T_{3}^{12} + 23850 T_{3}^{11} - 13876 T_{3}^{10} + \cdots - 162084473 \)
|
|
\( T_{5}^{15} - 1273 T_{5}^{13} - 80 T_{5}^{12} + 646049 T_{5}^{11} + 193100 T_{5}^{10} + \cdots - 42\!\cdots\!07 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{15} \)
|
| $3$ |
\( T^{15} + \cdots - 162084473 \)
|
| $5$ |
\( T^{15} + \cdots - 42\!\cdots\!07 \)
|
| $7$ |
\( T^{15} + \cdots - 17\!\cdots\!43 \)
|
| $11$ |
\( T^{15} + \cdots - 10\!\cdots\!17 \)
|
| $13$ |
\( T^{15} + \cdots + 10\!\cdots\!01 \)
|
| $17$ |
\( T^{15} + \cdots + 47\!\cdots\!59 \)
|
| $19$ |
\( T^{15} + \cdots + 14\!\cdots\!68 \)
|
| $23$ |
\( T^{15} \)
|
| $29$ |
\( T^{15} + \cdots + 51\!\cdots\!93 \)
|
| $31$ |
\( T^{15} + \cdots + 50\!\cdots\!61 \)
|
| $37$ |
\( T^{15} + \cdots - 32\!\cdots\!91 \)
|
| $41$ |
\( T^{15} + \cdots - 79\!\cdots\!73 \)
|
| $43$ |
\( T^{15} + \cdots + 89\!\cdots\!81 \)
|
| $47$ |
\( T^{15} + \cdots + 86\!\cdots\!68 \)
|
| $53$ |
\( T^{15} + \cdots - 21\!\cdots\!03 \)
|
| $59$ |
\( T^{15} + \cdots + 51\!\cdots\!67 \)
|
| $61$ |
\( T^{15} + \cdots + 12\!\cdots\!93 \)
|
| $67$ |
\( T^{15} + \cdots - 51\!\cdots\!71 \)
|
| $71$ |
\( T^{15} + \cdots - 75\!\cdots\!27 \)
|
| $73$ |
\( T^{15} + \cdots + 11\!\cdots\!37 \)
|
| $79$ |
\( T^{15} + \cdots - 60\!\cdots\!57 \)
|
| $83$ |
\( T^{15} + \cdots + 32\!\cdots\!01 \)
|
| $89$ |
\( T^{15} + \cdots - 14\!\cdots\!99 \)
|
| $97$ |
\( T^{15} + \cdots + 12\!\cdots\!61 \)
|
| show more | |
| show less |