Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1339,2,Mod(1,1339)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1339, 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("1339.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1339 = 13 \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1339.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: | \(10.6919688306\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −2.50015 | −2.63774 | 4.25075 | −1.17684 | 6.59474 | 4.82219 | −5.62720 | 3.95767 | 2.94227 | ||||||||||||||||||
1.2 | −2.31499 | 1.57292 | 3.35919 | −1.84365 | −3.64130 | 3.43971 | −3.14652 | −0.525924 | 4.26804 | ||||||||||||||||||
1.3 | −2.12469 | 0.347667 | 2.51433 | −4.40399 | −0.738685 | −0.0796229 | −1.09279 | −2.87913 | 9.35713 | ||||||||||||||||||
1.4 | −1.94515 | 1.90358 | 1.78363 | 1.71445 | −3.70276 | −2.09421 | 0.420881 | 0.623614 | −3.33487 | ||||||||||||||||||
1.5 | −1.91337 | −1.36048 | 1.66097 | 1.75251 | 2.60311 | −4.42851 | 0.648686 | −1.14908 | −3.35319 | ||||||||||||||||||
1.6 | −1.47455 | −1.76266 | 0.174289 | 0.466891 | 2.59913 | −0.582262 | 2.69210 | 0.106981 | −0.688453 | ||||||||||||||||||
1.7 | −1.10076 | 0.263891 | −0.788329 | 3.16171 | −0.290481 | 1.73698 | 3.06928 | −2.93036 | −3.48028 | ||||||||||||||||||
1.8 | −0.791299 | 0.744814 | −1.37385 | −0.897428 | −0.589370 | −1.66332 | 2.66972 | −2.44525 | 0.710134 | ||||||||||||||||||
1.9 | −0.784481 | −1.61992 | −1.38459 | −4.15260 | 1.27079 | 4.61578 | 2.65515 | −0.375870 | 3.25764 | ||||||||||||||||||
1.10 | −0.615393 | 2.88796 | −1.62129 | −1.40822 | −1.77723 | −0.315755 | 2.22852 | 5.34034 | 0.866607 | ||||||||||||||||||
1.11 | −0.0264721 | −1.87769 | −1.99930 | 2.52751 | 0.0497063 | −1.16160 | 0.105870 | 0.525708 | −0.0669085 | ||||||||||||||||||
1.12 | 0.106842 | −3.24890 | −1.98858 | −2.21706 | −0.347120 | −0.711342 | −0.426149 | 7.55538 | −0.236876 | ||||||||||||||||||
1.13 | 0.267251 | 2.13493 | −1.92858 | −0.961056 | 0.570561 | −2.53524 | −1.04992 | 1.55791 | −0.256843 | ||||||||||||||||||
1.14 | 0.499344 | 0.601343 | −1.75066 | −1.13095 | 0.300277 | 3.15035 | −1.87287 | −2.63839 | −0.564734 | ||||||||||||||||||
1.15 | 1.33589 | 2.00110 | −0.215389 | −3.88298 | 2.67326 | 2.54624 | −2.95952 | 1.00441 | −5.18725 | ||||||||||||||||||
1.16 | 1.40414 | 1.06219 | −0.0283926 | 0.529418 | 1.49147 | −2.26187 | −2.84815 | −1.87175 | 0.743377 | ||||||||||||||||||
1.17 | 1.85697 | −2.04069 | 1.44833 | 0.337566 | −3.78949 | −0.868388 | −1.02443 | 1.16441 | 0.626849 | ||||||||||||||||||
1.18 | 2.08679 | −1.93108 | 2.35469 | 1.87136 | −4.02977 | −0.548741 | 0.740165 | 0.729089 | 3.90513 | ||||||||||||||||||
1.19 | 2.21898 | 0.899770 | 2.92386 | −2.41101 | 1.99657 | −4.38690 | 2.05003 | −2.19041 | −5.34997 | ||||||||||||||||||
1.20 | 2.31727 | −3.32603 | 3.36976 | −2.87188 | −7.70732 | 2.40638 | 3.17410 | 8.06248 | −6.65492 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(1\) |
\(103\) | \(1\) |
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 | 1339.2.a.e | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1339.2.a.e | ✓ | 21 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):
\( T_{2}^{21} + T_{2}^{20} - 28 T_{2}^{19} - 29 T_{2}^{18} + 328 T_{2}^{17} + 353 T_{2}^{16} - 2080 T_{2}^{15} - 2338 T_{2}^{14} + 7701 T_{2}^{13} + 9124 T_{2}^{12} - 16656 T_{2}^{11} - 21190 T_{2}^{10} + 19645 T_{2}^{9} + 28040 T_{2}^{8} + \cdots + 1 \) |
\( T_{3}^{21} + 6 T_{3}^{20} - 19 T_{3}^{19} - 163 T_{3}^{18} + 92 T_{3}^{17} + 1832 T_{3}^{16} + 376 T_{3}^{15} - 11267 T_{3}^{14} - 5517 T_{3}^{13} + 42112 T_{3}^{12} + 22875 T_{3}^{11} - 100182 T_{3}^{10} - 44449 T_{3}^{9} + \cdots - 731 \) |