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: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} + \cdots - 63 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 518861 \nu^{18} + 866534 \nu^{17} - 35333866 \nu^{16} + 64448440 \nu^{15} + 439605017 \nu^{14} + \cdots - 367316328 ) / 24433851 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 266872 \nu^{18} - 1733205 \nu^{17} - 1918661 \nu^{16} + 30046194 \nu^{15} - 9244957 \nu^{14} + \cdots - 52348346 ) / 8144617 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1351897 \nu^{18} - 5921985 \nu^{17} - 25217420 \nu^{16} + 131218405 \nu^{15} + 170797761 \nu^{14} + \cdots - 256932042 ) / 24433851 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1733783 \nu^{18} + 12425812 \nu^{17} + 5062687 \nu^{16} - 219039295 \nu^{15} + \cdots - 176479527 ) / 24433851 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1774029 \nu^{18} + 16696447 \nu^{17} - 26659212 \nu^{16} - 188001263 \nu^{15} + \cdots + 11552727 ) / 24433851 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2967169 \nu^{18} + 22214635 \nu^{17} - 9319981 \nu^{16} - 292838892 \nu^{15} + 557989561 \nu^{14} + \cdots - 72057150 ) / 24433851 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2975640 \nu^{18} - 27847177 \nu^{17} + 41511942 \nu^{16} + 340341434 \nu^{15} - 1104351670 \nu^{14} + \cdots + 396383271 ) / 24433851 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 2977490 \nu^{18} + 28229263 \nu^{17} - 39139232 \nu^{16} - 377869468 \nu^{15} + \cdots - 594935301 ) / 24433851 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 3222355 \nu^{18} - 23242464 \nu^{17} - 4157897 \nu^{16} + 370547824 \nu^{15} - 503026266 \nu^{14} + \cdots + 179541513 ) / 24433851 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3242789 \nu^{18} - 28246403 \nu^{17} + 32898422 \nu^{16} + 365202966 \nu^{15} - 1060064717 \nu^{14} + \cdots + 380675985 ) / 24433851 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3916792 \nu^{18} - 35489841 \nu^{17} + 45831970 \nu^{16} + 460096249 \nu^{15} - 1382033019 \nu^{14} + \cdots + 854546172 ) / 24433851 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 4506342 \nu^{18} + 35302225 \nu^{17} - 14137671 \nu^{16} - 521843402 \nu^{15} + \cdots - 538160700 ) / 24433851 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 4982707 \nu^{18} + 40706596 \nu^{17} - 33835177 \nu^{16} - 537568329 \nu^{15} + \cdots - 312204861 ) / 24433851 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 5349330 \nu^{18} + 44295845 \nu^{17} - 42791964 \nu^{16} - 563967892 \nu^{15} + \cdots - 661897668 ) / 24433851 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 5831874 \nu^{18} - 45112259 \nu^{17} + 15470457 \nu^{16} + 666919645 \nu^{15} - 1253274389 \nu^{14} + \cdots + 507806415 ) / 24433851 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 7258260 \nu^{18} - 53818736 \nu^{17} + 10811283 \nu^{16} + 785351959 \nu^{15} - 1378655723 \nu^{14} + \cdots + 532661505 ) / 24433851 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{12} + \beta_{10} + \beta_{8} + \beta_{6} + \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{10} + \beta_{8} + \beta_{6} + \beta_{4} + \beta_{3} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{17} + \beta_{15} - \beta_{14} + 8 \beta_{13} + 9 \beta_{12} + 9 \beta_{10} - \beta_{9} + 9 \beta_{8} + \cdots + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{17} + 9 \beta_{15} + 21 \beta_{13} + 21 \beta_{12} + 21 \beta_{10} - 2 \beta_{9} + 14 \beta_{8} + \cdots + 42 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{17} - \beta_{16} + 12 \beta_{15} - 7 \beta_{14} + 63 \beta_{13} + 69 \beta_{12} + 71 \beta_{10} + \cdots + 67 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{18} + 34 \beta_{17} - 2 \beta_{16} + 65 \beta_{15} + 5 \beta_{14} + 177 \beta_{13} + 170 \beta_{12} + \cdots + 248 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{18} + 152 \beta_{17} - 18 \beta_{16} + 107 \beta_{15} - 23 \beta_{14} + 490 \beta_{13} + \cdots + 472 \)
|
| \(\nu^{10}\) | \(=\) |
\( 19 \beta_{18} + 404 \beta_{17} - 40 \beta_{16} + 447 \beta_{15} + 100 \beta_{14} + 1393 \beta_{13} + \cdots + 1547 \)
|
| \(\nu^{11}\) | \(=\) |
\( 41 \beta_{18} + 1491 \beta_{17} - 218 \beta_{16} + 857 \beta_{15} + 106 \beta_{14} + 3768 \beta_{13} + \cdots + 3260 \)
|
| \(\nu^{12}\) | \(=\) |
\( 234 \beta_{18} + 4133 \beta_{17} - 522 \beta_{16} + 3060 \beta_{15} + 1298 \beta_{14} + 10683 \beta_{13} + \cdots + 9941 \)
|
| \(\nu^{13}\) | \(=\) |
\( 545 \beta_{18} + 13798 \beta_{17} - 2231 \beta_{16} + 6543 \beta_{15} + 2887 \beta_{14} + 28779 \beta_{13} + \cdots + 22354 \)
|
| \(\nu^{14}\) | \(=\) |
\( 2419 \beta_{18} + 38988 \beta_{17} - 5631 \beta_{16} + 21149 \beta_{15} + 13943 \beta_{14} + \cdots + 65089 \)
|
| \(\nu^{15}\) | \(=\) |
\( 6000 \beta_{18} + 122834 \beta_{17} - 20876 \beta_{16} + 48855 \beta_{15} + 36192 \beta_{14} + \cdots + 152983 \)
|
| \(\nu^{16}\) | \(=\) |
\( 22926 \beta_{18} + 349895 \beta_{17} - 54661 \beta_{16} + 148178 \beta_{15} + 135002 \beta_{14} + \cdots + 431911 \)
|
| \(\nu^{17}\) | \(=\) |
\( 59620 \beta_{18} + 1063691 \beta_{17} - 185221 \beta_{16} + 361372 \beta_{15} + 369551 \beta_{14} + \cdots + 1047559 \)
|
| \(\nu^{18}\) | \(=\) |
\( 206988 \beta_{18} + 3037776 \beta_{17} - 497337 \beta_{16} + 1052762 \beta_{15} + 1226490 \beta_{14} + \cdots + 2896095 \)
|
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 |
|
−2.78640 | −0.0654083 | 5.76405 | −3.53830 | 0.182254 | 1.33123 | −10.4882 | −2.99572 | 9.85914 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.64132 | −0.762312 | 4.97659 | 1.72196 | 2.01351 | 1.20730 | −7.86215 | −2.41888 | −4.54826 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.49720 | 1.20166 | 4.23600 | 1.95194 | −3.00079 | −3.56552 | −5.58375 | −1.55601 | −4.87439 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.47938 | 2.80838 | 4.14734 | −2.79508 | −6.96305 | 0.439709 | −5.32409 | 4.88699 | 6.93007 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.93727 | −2.42478 | 1.75303 | 2.12296 | 4.69746 | 2.19023 | 0.478451 | 2.87955 | −4.11276 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.68626 | 0.556240 | 0.843466 | −0.525039 | −0.937964 | 4.12488 | 1.95021 | −2.69060 | 0.885350 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.49342 | 2.98467 | 0.230315 | −3.68204 | −4.45737 | −3.54305 | 2.64289 | 5.90824 | 5.49884 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.26598 | −1.64384 | −0.397300 | −2.39473 | 2.08107 | −1.52311 | 3.03493 | −0.297791 | 3.03168 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.979850 | 0.723336 | −1.03989 | −1.69086 | −0.708761 | 1.13749 | 2.97864 | −2.47678 | 1.65679 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.550251 | −2.20461 | −1.69722 | −3.11514 | 1.21309 | −3.99436 | 2.03440 | 1.86029 | 1.71411 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −0.493077 | 0.177153 | −1.75687 | 2.02635 | −0.0873502 | −2.21543 | 1.85243 | −2.96862 | −0.999149 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.234290 | 1.28880 | −1.94511 | 2.43495 | 0.301953 | −2.52468 | −0.924301 | −1.33900 | 0.570486 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0.427526 | −2.84080 | −1.81722 | −0.938760 | −1.21451 | 1.20823 | −1.63196 | 5.07012 | −0.401345 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 0.794128 | −2.28908 | −1.36936 | −3.96939 | −1.81782 | 4.40368 | −2.67570 | 2.23989 | −3.15220 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.14213 | −2.60408 | −0.695533 | 2.07586 | −2.97420 | −0.495396 | −3.07866 | 3.78123 | 2.37091 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 1.16528 | 1.72020 | −0.642117 | −0.583086 | 2.00452 | −1.01939 | −3.07881 | −0.0409067 | −0.679460 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 1.63365 | 2.37722 | 0.668797 | −3.50821 | 3.88353 | −4.23045 | −2.17471 | 2.65117 | −5.73118 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.17587 | −0.456273 | 2.73443 | −0.352848 | −0.992793 | −3.48073 | 1.59803 | −2.79181 | −0.767754 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.23754 | −0.546477 | 3.00660 | −3.24055 | −1.22277 | 2.54937 | 2.25232 | −2.70136 | −7.25087 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 19 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1339.2.a.d | ✓ | 19 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1339))\):
|
\( T_{2}^{19} + 9 T_{2}^{18} + 13 T_{2}^{17} - 106 T_{2}^{16} - 351 T_{2}^{15} + 279 T_{2}^{14} + 2337 T_{2}^{13} + \cdots + 63 \)
|
|
\( T_{3}^{19} + 2 T_{3}^{18} - 30 T_{3}^{17} - 59 T_{3}^{16} + 358 T_{3}^{15} + 677 T_{3}^{14} - 2194 T_{3}^{13} + \cdots - 7 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} + 9 T^{18} + \cdots + 63 \)
|
| $3$ |
\( T^{19} + 2 T^{18} + \cdots - 7 \)
|
| $5$ |
\( T^{19} + 18 T^{18} + \cdots + 153621 \)
|
| $7$ |
\( T^{19} + 8 T^{18} + \cdots - 314899 \)
|
| $11$ |
\( T^{19} + \cdots + 376582824 \)
|
| $13$ |
\( (T - 1)^{19} \)
|
| $17$ |
\( T^{19} + 16 T^{18} + \cdots - 7321896 \)
|
| $19$ |
\( T^{19} + 10 T^{18} + \cdots - 8428579 \)
|
| $23$ |
\( T^{19} + 14 T^{18} + \cdots + 40504113 \)
|
| $29$ |
\( T^{19} + \cdots - 352233297 \)
|
| $31$ |
\( T^{19} + \cdots - 2728685702893 \)
|
| $37$ |
\( T^{19} + \cdots + 115637719463 \)
|
| $41$ |
\( T^{19} + \cdots + 1454522503176 \)
|
| $43$ |
\( T^{19} + \cdots - 39048730111 \)
|
| $47$ |
\( T^{19} + \cdots - 489280247991 \)
|
| $53$ |
\( T^{19} + \cdots + 450049177778184 \)
|
| $59$ |
\( T^{19} + \cdots - 66585825126153 \)
|
| $61$ |
\( T^{19} + \cdots - 2939483125577 \)
|
| $67$ |
\( T^{19} + \cdots - 2929193309 \)
|
| $71$ |
\( T^{19} + \cdots + 21571585219287 \)
|
| $73$ |
\( T^{19} + \cdots - 25\!\cdots\!53 \)
|
| $79$ |
\( T^{19} + \cdots - 59\!\cdots\!92 \)
|
| $83$ |
\( T^{19} + \cdots + 29\!\cdots\!37 \)
|
| $89$ |
\( T^{19} + \cdots - 3032454681 \)
|
| $97$ |
\( T^{19} + \cdots - 182478376652167 \)
|
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