Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8993,2,Mod(1,8993)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8993.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8993 = 17 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8993.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: | \(71.8094665377\) |
| Analytic rank: | \(1\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 20 x^{12} - 3 x^{11} + 154 x^{10} + 40 x^{9} - 575 x^{8} - 187 x^{7} + 1083 x^{6} + 377 x^{5} + \cdots - 51 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 20 x^{12} - 3 x^{11} + 154 x^{10} + 40 x^{9} - 575 x^{8} - 187 x^{7} + 1083 x^{6} + 377 x^{5} + \cdots - 51 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 35 \nu^{13} - 26 \nu^{12} - 682 \nu^{11} + 422 \nu^{10} + 5089 \nu^{9} - 2707 \nu^{8} - 18166 \nu^{7} + \cdots - 2448 ) / 46 \)
|
| \(\beta_{5}\) | \(=\) |
\( - \nu^{13} + \nu^{12} + 19 \nu^{11} - 16 \nu^{10} - 138 \nu^{9} + 98 \nu^{8} + 477 \nu^{7} - 290 \nu^{6} + \cdots + 48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 97 \nu^{13} - 99 \nu^{12} - 1852 \nu^{11} + 1598 \nu^{10} + 13561 \nu^{9} - 9912 \nu^{8} - 47501 \nu^{7} + \cdots - 5781 ) / 46 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 129 \nu^{13} - 107 \nu^{12} - 2490 \nu^{11} + 1696 \nu^{10} + 18407 \nu^{9} - 10384 \nu^{8} + \cdots - 7065 ) / 46 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 166 \nu^{13} - 145 \nu^{12} - 3186 \nu^{11} + 2288 \nu^{10} + 23428 \nu^{9} - 13891 \nu^{8} + \cdots - 9507 ) / 46 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 174 \nu^{13} + 147 \nu^{12} + 3357 \nu^{11} - 2347 \nu^{10} - 24789 \nu^{9} + 14492 \nu^{8} + \cdots + 10058 ) / 46 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 227 \nu^{13} - 212 \nu^{12} - 4349 \nu^{11} + 3379 \nu^{10} + 31934 \nu^{9} - 20696 \nu^{8} + \cdots - 12705 ) / 46 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 227 \nu^{13} + 212 \nu^{12} + 4349 \nu^{11} - 3379 \nu^{10} - 31934 \nu^{9} + 20696 \nu^{8} + \cdots + 12935 ) / 46 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 363 \nu^{13} + 338 \nu^{12} + 6934 \nu^{11} - 5348 \nu^{10} - 50747 \nu^{9} + 32477 \nu^{8} + \cdots + 20278 ) / 46 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 394 \nu^{13} + 363 \nu^{12} + 7542 \nu^{11} - 5752 \nu^{10} - 55328 \nu^{9} + 34987 \nu^{8} + \cdots + 21473 ) / 46 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} + 8\beta_{3} + \beta_{2} + 29\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{13} + 11 \beta_{11} + 11 \beta_{10} + 2 \beta_{8} + \beta_{6} - \beta_{4} + 11 \beta_{3} + \cdots + 77 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{13} + 4 \beta_{11} + 13 \beta_{10} + 11 \beta_{9} + 14 \beta_{8} + 12 \beta_{7} + 3 \beta_{6} + \cdots + 83 \)
|
| \(\nu^{8}\) | \(=\) |
\( 14 \beta_{13} + 2 \beta_{12} + 92 \beta_{11} + 92 \beta_{10} + 2 \beta_{9} + 32 \beta_{8} + \beta_{7} + \cdots + 462 \)
|
| \(\nu^{9}\) | \(=\) |
\( 105 \beta_{13} + 3 \beta_{12} + 66 \beta_{11} + 126 \beta_{10} + 92 \beta_{9} + 143 \beta_{8} + \cdots + 644 \)
|
| \(\nu^{10}\) | \(=\) |
\( 141 \beta_{13} + 34 \beta_{12} + 706 \beta_{11} + 703 \beta_{10} + 37 \beta_{9} + 351 \beta_{8} + \cdots + 2928 \)
|
| \(\nu^{11}\) | \(=\) |
\( 822 \beta_{13} + 57 \beta_{12} + 741 \beta_{11} + 1093 \beta_{10} + 700 \beta_{9} + 1289 \beta_{8} + \cdots + 4836 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1251 \beta_{13} + 382 \beta_{12} + 5246 \beta_{11} + 5180 \beta_{10} + 450 \beta_{9} + 3291 \beta_{8} + \cdots + 19238 \)
|
| \(\nu^{13}\) | \(=\) |
\( 6136 \beta_{13} + 703 \beta_{12} + 7071 \beta_{11} + 8961 \beta_{10} + 5112 \beta_{9} + 10873 \beta_{8} + \cdots + 35710 \)
|
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 |
|
−2.69759 | −1.61035 | 5.27699 | 3.26954 | 4.34407 | 3.45908 | −8.83996 | −0.406769 | −8.81987 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.42353 | 2.51073 | 3.87349 | 2.00175 | −6.08482 | −1.97006 | −4.54045 | 3.30376 | −4.85131 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.86376 | −0.155338 | 1.47358 | 1.12917 | 0.289511 | −0.632613 | 0.981110 | −2.97587 | −2.10450 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.60698 | 1.82746 | 0.582399 | −3.95769 | −2.93670 | −4.72762 | 2.27806 | 0.339610 | 6.35994 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.08002 | −2.90025 | −0.833557 | −0.478982 | 3.13232 | −4.67720 | 3.06030 | 5.41143 | 0.517310 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.688105 | 1.35793 | −1.52651 | −2.29003 | −0.934396 | 3.74343 | 2.42661 | −1.15603 | 1.57578 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.345734 | −1.45523 | −1.88047 | 2.85821 | 0.503123 | 0.862693 | 1.34161 | −0.882301 | −0.988180 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.716126 | −0.453180 | −1.48716 | −2.88714 | −0.324534 | −1.54935 | −2.49725 | −2.79463 | −2.06756 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.909888 | 1.62550 | −1.17210 | 0.367872 | 1.47902 | 0.669483 | −2.88626 | −0.357755 | 0.334722 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.918544 | −2.69486 | −1.15628 | −2.04415 | −2.47535 | 3.47907 | −2.89918 | 4.26227 | −1.87764 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.80038 | 2.94862 | 1.24138 | −2.12361 | 5.30863 | −4.36369 | −1.36581 | 5.69433 | −3.82331 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.93678 | 0.774487 | 1.75113 | 3.97930 | 1.50001 | −3.49931 | −0.482003 | −2.40017 | 7.70705 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.02323 | −0.697191 | 2.09345 | 1.54387 | −1.41058 | 3.58653 | 0.189071 | −2.51392 | 3.12360 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.40076 | −3.07832 | 3.76367 | −3.36812 | −7.39032 | −2.38043 | 4.23415 | 6.47605 | −8.08605 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 8993.2.a.k | ✓ | 14 |
| 23.b | odd | 2 | 1 | 8993.2.a.l | yes | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8993.2.a.k | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 8993.2.a.l | yes | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
|
\( T_{2}^{14} - 20 T_{2}^{12} + 3 T_{2}^{11} + 154 T_{2}^{10} - 40 T_{2}^{9} - 575 T_{2}^{8} + 187 T_{2}^{7} + \cdots - 51 \)
|
|
\( T_{5}^{14} + 2 T_{5}^{13} - 44 T_{5}^{12} - 84 T_{5}^{11} + 739 T_{5}^{10} + 1308 T_{5}^{9} + \cdots - 8748 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 20 T^{12} + \cdots - 51 \)
|
| $3$ |
\( T^{14} + 2 T^{13} + \cdots + 64 \)
|
| $5$ |
\( T^{14} + 2 T^{13} + \cdots - 8748 \)
|
| $7$ |
\( T^{14} + 8 T^{13} + \cdots + 144828 \)
|
| $11$ |
\( T^{14} + 4 T^{13} + \cdots + 2916 \)
|
| $13$ |
\( T^{14} - 10 T^{13} + \cdots + 116449 \)
|
| $17$ |
\( (T - 1)^{14} \)
|
| $19$ |
\( T^{14} + \cdots + 131632128 \)
|
| $23$ |
\( T^{14} \)
|
| $29$ |
\( T^{14} + 6 T^{13} + \cdots - 10458624 \)
|
| $31$ |
\( T^{14} - 6 T^{13} + \cdots - 1866816 \)
|
| $37$ |
\( T^{14} + \cdots + 887123988 \)
|
| $41$ |
\( T^{14} + \cdots - 7638867456 \)
|
| $43$ |
\( T^{14} + \cdots + 4676113152 \)
|
| $47$ |
\( T^{14} - 2 T^{13} + \cdots + 241953 \)
|
| $53$ |
\( T^{14} + 4 T^{13} + \cdots + 8211456 \)
|
| $59$ |
\( T^{14} - 6 T^{13} + \cdots + 10674288 \)
|
| $61$ |
\( T^{14} + \cdots + 5680352448 \)
|
| $67$ |
\( T^{14} + \cdots - 1217846016 \)
|
| $71$ |
\( T^{14} + \cdots - 15275516736 \)
|
| $73$ |
\( T^{14} + \cdots + 4444236224 \)
|
| $79$ |
\( T^{14} + \cdots + 153779963508 \)
|
| $83$ |
\( T^{14} + \cdots - 18116078208768 \)
|
| $89$ |
\( T^{14} + \cdots - 42642091008 \)
|
| $97$ |
\( T^{14} + \cdots + 412406364276 \)
|
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