Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,10,Mod(1,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("289.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 289.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: | \(148.845356651\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 5 x^{11} - 4542 x^{10} + 30079 x^{9} + 7481825 x^{8} - 58373994 x^{7} - 5553197164 x^{6} + \cdots + 245077910606835 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{5}\cdot 17 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 5 x^{11} - 4542 x^{10} + 30079 x^{9} + 7481825 x^{8} - 58373994 x^{7} - 5553197164 x^{6} + \cdots + 245077910606835 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3\nu - 760 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 80\!\cdots\!39 \nu^{11} + \cdots + 70\!\cdots\!59 ) / 56\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!17 \nu^{11} + \cdots + 52\!\cdots\!15 ) / 94\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 90\!\cdots\!89 \nu^{11} + \cdots - 18\!\cdots\!05 ) / 70\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 67\!\cdots\!89 \nu^{11} + \cdots + 23\!\cdots\!65 ) / 35\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 83\!\cdots\!91 \nu^{11} + \cdots + 86\!\cdots\!63 ) / 28\!\cdots\!56 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 40\!\cdots\!85 \nu^{11} + \cdots - 27\!\cdots\!05 ) / 28\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 27\!\cdots\!29 \nu^{11} + \cdots - 82\!\cdots\!05 ) / 14\!\cdots\!40 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 31\!\cdots\!57 \nu^{11} + \cdots - 52\!\cdots\!85 ) / 14\!\cdots\!80 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 41\!\cdots\!63 \nu^{11} + \cdots + 27\!\cdots\!05 ) / 94\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 + 760 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} + 1230\beta _1 - 2344 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + \beta_{8} - 8 \beta_{7} - \beta_{6} + 23 \beta_{5} + \cdots + 934757 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 28 \beta_{11} - 8 \beta_{10} - 30 \beta_{9} + 98 \beta_{8} - 220 \beta_{7} + 2351 \beta_{6} + \cdots - 6048323 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15382 \beta_{11} - 16994 \beta_{10} - 6447 \beta_{9} + 749 \beta_{8} - 32188 \beta_{7} + \cdots + 1389690995 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 123158 \beta_{11} + 35494 \beta_{10} - 121637 \beta_{9} + 299351 \beta_{8} - 619944 \beta_{7} + \cdots - 12695977074 \)
|
| \(\nu^{8}\) | \(=\) |
\( 31949910 \beta_{11} - 37552994 \beta_{10} - 11776623 \beta_{9} - 855251 \beta_{8} - 79699916 \beta_{7} + \cdots + 2298052582854 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 353044934 \beta_{11} + 198388582 \beta_{10} - 307497373 \beta_{9} + 681811423 \beta_{8} + \cdots - 25858085245810 \)
|
| \(\nu^{10}\) | \(=\) |
\( 62576155124 \beta_{11} - 76179819744 \beta_{10} - 20611068286 \beta_{9} - 4985190606 \beta_{8} + \cdots + 40\!\cdots\!31 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 858432302130 \beta_{11} + 620020240718 \beta_{10} - 653127262579 \beta_{9} + 1401239006697 \beta_{8} + \cdots - 53\!\cdots\!41 \)
|
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 |
|
−43.0937 | 38.5560 | 1345.07 | 1118.08 | −1661.52 | 12272.9 | −35899.9 | −18196.4 | −48182.3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −32.3505 | 71.1986 | 534.552 | −1755.25 | −2303.31 | −7326.34 | −729.582 | −14613.8 | 56783.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −30.5756 | −194.921 | 422.865 | 941.135 | 5959.81 | −4513.44 | 2725.35 | 18311.1 | −28775.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −19.4468 | 193.562 | −133.821 | 1266.43 | −3764.16 | 1846.43 | 12559.2 | 17783.1 | −24628.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −13.2275 | −162.576 | −337.033 | −1690.93 | 2150.48 | 10426.0 | 11230.6 | 6748.01 | 22366.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.779367 | −32.1989 | −511.393 | 894.723 | −25.0947 | −2726.00 | −797.598 | −18646.2 | 697.318 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.74483 | 214.211 | −478.997 | −2392.81 | 1230.60 | 2771.45 | −5693.11 | 26203.3 | −13746.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 22.0274 | −192.112 | −26.7930 | −2093.86 | −4231.72 | −11187.9 | −11868.2 | 17223.9 | −46122.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 26.3822 | 66.0249 | 184.023 | −164.991 | 1741.89 | 5776.54 | −8652.77 | −15323.7 | −4352.83 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 26.6779 | −248.720 | 199.710 | 2486.89 | −6635.33 | 7643.33 | −8331.23 | 42178.6 | 66345.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 31.1791 | 218.085 | 460.136 | 1545.34 | 6799.71 | −9980.37 | −1617.07 | 27878.3 | 48182.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 42.9032 | −45.1099 | 1328.68 | −608.765 | −1935.36 | 521.377 | 35038.3 | −17648.1 | −26118.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
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 | 289.10.a.d | ✓ | 12 |
| 17.b | even | 2 | 1 | 289.10.a.e | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 289.10.a.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 289.10.a.e | yes | 12 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):
|
\( T_{2}^{12} - 17 T_{2}^{11} - 4421 T_{2}^{10} + 75004 T_{2}^{9} + 7008044 T_{2}^{8} + \cdots - 10\!\cdots\!00 \)
|
|
\( T_{3}^{12} + 74 T_{3}^{11} - 151309 T_{3}^{10} - 9283700 T_{3}^{9} + 8307631332 T_{3}^{8} + \cdots + 36\!\cdots\!39 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 10\!\cdots\!00 \)
|
| $3$ |
\( T^{12} + \cdots + 36\!\cdots\!39 \)
|
| $5$ |
\( T^{12} + \cdots + 68\!\cdots\!25 \)
|
| $7$ |
\( T^{12} + \cdots - 15\!\cdots\!85 \)
|
| $11$ |
\( T^{12} + \cdots - 12\!\cdots\!69 \)
|
| $13$ |
\( T^{12} + \cdots - 16\!\cdots\!47 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} + \cdots - 42\!\cdots\!37 \)
|
| $23$ |
\( T^{12} + \cdots - 67\!\cdots\!33 \)
|
| $29$ |
\( T^{12} + \cdots + 59\!\cdots\!37 \)
|
| $31$ |
\( T^{12} + \cdots + 61\!\cdots\!75 \)
|
| $37$ |
\( T^{12} + \cdots - 24\!\cdots\!00 \)
|
| $41$ |
\( T^{12} + \cdots + 10\!\cdots\!36 \)
|
| $43$ |
\( T^{12} + \cdots + 86\!\cdots\!59 \)
|
| $47$ |
\( T^{12} + \cdots - 69\!\cdots\!65 \)
|
| $53$ |
\( T^{12} + \cdots - 52\!\cdots\!95 \)
|
| $59$ |
\( T^{12} + \cdots + 21\!\cdots\!88 \)
|
| $61$ |
\( T^{12} + \cdots - 36\!\cdots\!87 \)
|
| $67$ |
\( T^{12} + \cdots + 41\!\cdots\!16 \)
|
| $71$ |
\( T^{12} + \cdots + 52\!\cdots\!80 \)
|
| $73$ |
\( T^{12} + \cdots - 47\!\cdots\!83 \)
|
| $79$ |
\( T^{12} + \cdots + 21\!\cdots\!92 \)
|
| $83$ |
\( T^{12} + \cdots - 11\!\cdots\!73 \)
|
| $89$ |
\( T^{12} + \cdots + 48\!\cdots\!08 \)
|
| $97$ |
\( T^{12} + \cdots - 78\!\cdots\!39 \)
|
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