Newspace parameters
Level: | \( N \) | = | \( 1 \) |
Weight: | \( k \) | = | \( 68 \) |
Character orbit: | \([\chi]\) | = | 1.a (trivial) |
Newform invariants
Self dual: | Yes |
Analytic conductor: | \(28.429035193\) |
Analytic rank: | \(0\) |
Dimension: | \(5\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{40}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 \) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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 \(x^{5} - x^{4} - 939384011925257456 x^{3} + 31046449413968483513911200 x^{2} + 156793504704482691874379743265203200 x + 20916736226052669578405116700517591609696000\):
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( 24 \nu - 5 \) |
\(\beta_{2}\) | \(=\) | \((\)\(-1463 \nu^{4} + 61316688179 \nu^{3} + 1134677465176013610916 \nu^{2} - 175871625402178878144147591024 \nu - 92322819705534579210406721528027067456\)\()/ \)\(37\!\cdots\!28\)\( \) |
\(\beta_{3}\) | \(=\) | \((\)\(-1461537 \nu^{4} + 61255371490821 \nu^{3} + 1493224099349187733125372 \nu^{2} - 157864696669790310577560760925712 \nu - 227382046305944113582206155994463766078912\)\()/ \)\(62\!\cdots\!88\)\( \) |
\(\beta_{4}\) | \(=\) | \((\)\(4738030953 \nu^{4} - 6783896933686506093 \nu^{3} - 6265292406150844870725632988 \nu^{2} + 4428965898625782506113044897501160080 \nu + 1149735030546567532337013572577831183558493120\)\()/ \)\(15\!\cdots\!20\)\( \) |
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \((\)\(\beta_{1} + 5\)\()/24\) |
\(\nu^{2}\) | \(=\) | \((\)\(\beta_{3} - 5994 \beta_{2} - 1189790397 \beta_{1} + 216434076347341357501\)\()/576\) |
\(\nu^{3}\) | \(=\) | \((\)\(-409640 \beta_{4} - 1180151429 \beta_{3} + 3889870865010 \beta_{2} + 43600466505262789881 \beta_{1} - 32188957769729085286587521881\)\()/1728\) |
\(\nu^{4}\) | \(=\) | \((\)\(-17168672690120 \beta_{4} + 2277286000240633339 \beta_{3} - 18208831024879688328462 \beta_{2} - 9596312969973284759959813095 \beta_{1} + 393192791921864844386864518484726790407\)\()/1728\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−1.72055e10 | −1.09955e16 | 1.48456e20 | 4.93311e23 | 1.89183e26 | 1.79351e28 | −1.51842e28 | 2.81916e31 | −8.48767e33 | |||||||||||||||||||||||||||||||||
1.2 | −1.34339e10 | 7.87958e15 | 3.28965e19 | −3.01383e23 | −1.05854e26 | −4.20742e26 | 1.54057e30 | −3.06216e31 | 4.04876e33 | ||||||||||||||||||||||||||||||||||
1.3 | 5.06726e9 | −7.82458e15 | −1.21897e20 | −8.33551e22 | −3.96491e25 | −1.15070e28 | −1.36548e30 | −3.14854e31 | −4.22381e32 | ||||||||||||||||||||||||||||||||||
1.4 | 8.43209e9 | 1.56682e16 | −7.64738e19 | 3.83370e23 | 1.32115e26 | −1.75533e27 | −1.88919e30 | 1.52782e32 | 3.23261e33 | ||||||||||||||||||||||||||||||||||
1.5 | 2.26950e10 | −1.28429e15 | 3.67490e20 | −1.61198e23 | −2.91471e25 | 2.93806e28 | 4.99099e30 | −9.10601e31 | −3.65838e33 |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Hecke kernels
There are no other newforms in \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\).