Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5239,2,Mod(1,5239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, 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("5239.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5239 = 13^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5239.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: | \(41.8336256189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{17} - 4 x^{16} - 19 x^{15} + 90 x^{14} + 116 x^{13} - 776 x^{12} - 146 x^{11} + 3232 x^{10} + \cdots - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 403) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 4 x^{16} - 19 x^{15} + 90 x^{14} + 116 x^{13} - 776 x^{12} - 146 x^{11} + 3232 x^{10} + \cdots - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 176347878 \nu^{16} + 1547321127 \nu^{15} + 697352446 \nu^{14} - 34249732334 \nu^{13} + \cdots + 13148468887 ) / 2911905218 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 121769365 \nu^{16} - 618430856 \nu^{15} - 1851645148 \nu^{14} + 13690502351 \nu^{13} + \cdots - 1314968788 ) / 1455952609 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 277699462 \nu^{16} - 1654324367 \nu^{15} - 3765163820 \nu^{14} + 36953965476 \nu^{13} + \cdots - 320130275 ) / 2911905218 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 288357949 \nu^{16} + 837114472 \nu^{15} + 6477199202 \nu^{14} - 19082687678 \nu^{13} + \cdots - 1913490494 ) / 2911905218 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 498959126 \nu^{16} - 2300629651 \nu^{15} - 8426107392 \nu^{14} + 51480185436 \nu^{13} + \cdots - 2040491815 ) / 2911905218 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 517861878 \nu^{16} - 2042470229 \nu^{15} - 9928219232 \nu^{14} + 45892190880 \nu^{13} + \cdots + 10490654855 ) / 2911905218 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 284139419 \nu^{16} - 1079695125 \nu^{15} - 5588570639 \nu^{14} + 24310795064 \nu^{13} + \cdots - 712278330 ) / 1455952609 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 295207041 \nu^{16} - 618271402 \nu^{15} - 7149014592 \nu^{14} + 14148175876 \nu^{13} + \cdots + 3476240696 ) / 1455952609 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 612684228 \nu^{16} + 2680473007 \nu^{15} + 10949612684 \nu^{14} - 60135840180 \nu^{13} + \cdots - 7263502677 ) / 2911905218 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 721946816 \nu^{16} - 2820688979 \nu^{15} - 13907798692 \nu^{14} + 63158957810 \nu^{13} + \cdots - 12662395123 ) / 2911905218 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 780568670 \nu^{16} + 2966415803 \nu^{15} + 15390738234 \nu^{14} - 66729518802 \nu^{13} + \cdots - 1511400471 ) / 2911905218 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 916015187 \nu^{16} - 3344241710 \nu^{15} - 18033693932 \nu^{14} + 74872502246 \nu^{13} + \cdots - 2870082680 ) / 2911905218 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 460199705 \nu^{16} + 1837923012 \nu^{15} + 8791132436 \nu^{14} - 41417221237 \nu^{13} + \cdots - 4625059463 ) / 1455952609 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 1083899629 \nu^{16} - 3630184506 \nu^{15} - 22474819482 \nu^{14} + 81466180868 \nu^{13} + \cdots - 8622184886 ) / 2911905218 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{16} + \beta_{14} - \beta_{13} + \beta_{11} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{16} + 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - 2 \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 9 \beta_{16} + 11 \beta_{14} - 9 \beta_{13} - \beta_{12} + 10 \beta_{11} - \beta_{10} - \beta_{8} + \cdots + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 12 \beta_{16} + 23 \beta_{14} - 23 \beta_{13} - 10 \beta_{12} + \beta_{11} + \beta_{10} - 23 \beta_{9} + \cdots + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 73 \beta_{16} + 2 \beta_{15} + 100 \beta_{14} - 71 \beta_{13} - 11 \beta_{12} + 84 \beta_{11} + \cdots + 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 114 \beta_{16} + \beta_{15} + 211 \beta_{14} - 213 \beta_{13} - 84 \beta_{12} + 20 \beta_{11} + \cdots + 641 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 582 \beta_{16} + 34 \beta_{15} + 849 \beta_{14} - 545 \beta_{13} - 91 \beta_{12} + 673 \beta_{11} + \cdots + 181 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 998 \beta_{16} + 13 \beta_{15} + 1796 \beta_{14} - 1824 \beta_{13} - 668 \beta_{12} + 259 \beta_{11} + \cdots + 4366 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 4612 \beta_{16} + 394 \beta_{15} + 6969 \beta_{14} - 4175 \beta_{13} - 688 \beta_{12} + 5305 \beta_{11} + \cdots + 1798 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 8411 \beta_{16} + 113 \beta_{15} + 14798 \beta_{14} - 15049 \beta_{13} - 5188 \beta_{12} + \cdots + 30626 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 36432 \beta_{16} + 3888 \beta_{15} + 56168 \beta_{14} - 32182 \beta_{13} - 5053 \beta_{12} + \cdots + 16730 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 69483 \beta_{16} + 853 \beta_{15} + 120030 \beta_{14} - 121753 \beta_{13} - 39857 \beta_{12} + \cdots + 219815 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 287308 \beta_{16} + 35216 \beta_{15} + 448115 \beta_{14} - 250144 \beta_{13} - 37037 \beta_{12} + \cdots + 149765 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 567560 \beta_{16} + 6317 \beta_{15} + 966010 \beta_{14} - 974385 \beta_{13} - 304829 \beta_{12} + \cdots + 1606886 \)
|
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 |
|
−2.72325 | −2.85287 | 5.41608 | 1.48160 | 7.76907 | 2.10121 | −9.30282 | 5.13886 | −4.03477 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43732 | −0.331454 | 3.94051 | 0.0487430 | 0.807858 | 0.899705 | −4.72962 | −2.89014 | −0.118802 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.08606 | 1.06940 | 2.35166 | 1.36190 | −2.23085 | −4.48500 | −0.733593 | −1.85637 | −2.84101 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.61260 | 3.21473 | 0.600487 | 1.40674 | −5.18408 | 2.73528 | 2.25686 | 7.33450 | −2.26851 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.15831 | 1.72567 | −0.658309 | −1.50607 | −1.99887 | 1.44419 | 3.07916 | −0.0220652 | 1.74450 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.799756 | −1.21585 | −1.36039 | −2.33311 | 0.972384 | −0.175776 | 2.68749 | −1.52171 | 1.86592 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.182983 | −1.75533 | −1.96652 | 3.62872 | 0.321197 | 3.69449 | 0.725806 | 0.0811940 | −0.663994 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.0723729 | −3.29089 | −1.99476 | −2.88575 | 0.238171 | −1.08155 | 0.289113 | 7.82996 | 0.208850 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.0410842 | 2.09330 | −1.99831 | −1.98920 | 0.0860016 | 2.68785 | −0.164267 | 1.38192 | −0.0817244 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.933824 | −2.71846 | −1.12797 | 3.82118 | −2.53856 | −4.23678 | −2.92098 | 4.39002 | 3.56831 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.15793 | 0.471915 | −0.659198 | 3.25594 | 0.546444 | 0.0266118 | −3.07917 | −2.77730 | 3.77015 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.20115 | −0.0605410 | −0.557240 | −2.34059 | −0.0727188 | −1.65772 | −3.07163 | −2.99633 | −2.81139 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.89716 | 2.58453 | 1.59922 | 2.49083 | 4.90328 | −2.96856 | −0.760339 | 3.67982 | 4.72551 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.22595 | 0.674927 | 2.95488 | −1.70140 | 1.50236 | 4.90728 | 2.12551 | −2.54447 | −3.78724 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.33291 | −1.57374 | 3.44249 | −2.09117 | −3.67140 | −1.06530 | 3.36520 | −0.523340 | −4.87852 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.46207 | 2.76636 | 4.06181 | 1.87256 | 6.81098 | 2.37866 | 5.07632 | 4.65274 | 4.61037 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.82056 | −0.801704 | 5.95558 | 2.47906 | −2.26126 | 0.795418 | 11.1570 | −2.35727 | 6.99235 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 5239.2.a.n | 17 | |
| 13.b | even | 2 | 1 | 5239.2.a.m | 17 | ||
| 13.e | even | 6 | 2 | 403.2.f.b | ✓ | 34 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.f.b | ✓ | 34 | 13.e | even | 6 | 2 | |
| 5239.2.a.m | 17 | 13.b | even | 2 | 1 | ||
| 5239.2.a.n | 17 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):
|
\( T_{2}^{17} - 4 T_{2}^{16} - 19 T_{2}^{15} + 90 T_{2}^{14} + 116 T_{2}^{13} - 776 T_{2}^{12} - 146 T_{2}^{11} + \cdots - 1 \)
|
|
\( T_{5}^{17} - 7 T_{5}^{16} - 22 T_{5}^{15} + 232 T_{5}^{14} + 102 T_{5}^{13} - 3212 T_{5}^{12} + \cdots + 12132 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 4 T^{16} + \cdots - 1 \)
|
| $3$ |
\( T^{17} - 34 T^{15} + \cdots + 39 \)
|
| $5$ |
\( T^{17} - 7 T^{16} + \cdots + 12132 \)
|
| $7$ |
\( T^{17} - 6 T^{16} + \cdots + 347 \)
|
| $11$ |
\( T^{17} - 13 T^{16} + \cdots - 9284067 \)
|
| $13$ |
\( T^{17} \)
|
| $17$ |
\( T^{17} + 6 T^{16} + \cdots + 8080832 \)
|
| $19$ |
\( T^{17} - 4 T^{16} + \cdots + 44521 \)
|
| $23$ |
\( T^{17} + \cdots - 90482045711 \)
|
| $29$ |
\( T^{17} + 6 T^{16} + \cdots - 8054337 \)
|
| $31$ |
\( (T - 1)^{17} \)
|
| $37$ |
\( T^{17} + \cdots - 19098677817 \)
|
| $41$ |
\( T^{17} - 43 T^{16} + \cdots + 33047843 \)
|
| $43$ |
\( T^{17} + \cdots + 2551522052623 \)
|
| $47$ |
\( T^{17} + \cdots + 4197492216576 \)
|
| $53$ |
\( T^{17} + \cdots - 18874853583372 \)
|
| $59$ |
\( T^{17} + \cdots + 3736418595849 \)
|
| $61$ |
\( T^{17} + \cdots - 798980232829 \)
|
| $67$ |
\( T^{17} + \cdots - 64199519597 \)
|
| $71$ |
\( T^{17} + \cdots + 885046062062979 \)
|
| $73$ |
\( T^{17} + \cdots + 174124332 \)
|
| $79$ |
\( T^{17} + \cdots - 25\!\cdots\!12 \)
|
| $83$ |
\( T^{17} + \cdots - 22474701056 \)
|
| $89$ |
\( T^{17} + \cdots + 12514195539273 \)
|
| $97$ |
\( T^{17} + \cdots - 80\!\cdots\!83 \)
|
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