Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [239,2,Mod(1,239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(239, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("239.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 239 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 239.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: | \(1.90842460831\) |
| 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: |
\( x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2861757 \nu^{16} + 2069683 \nu^{15} - 74529028 \nu^{14} - 59373989 \nu^{13} + 768331898 \nu^{12} + \cdots + 128997701 ) / 11107271 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3427187 \nu^{16} + 6181042 \nu^{15} - 92146964 \nu^{14} - 163673784 \nu^{13} + 979124437 \nu^{12} + \cdots + 187483100 ) / 11107271 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 550174 \nu^{16} + 924456 \nu^{15} - 14665692 \nu^{14} - 24610776 \nu^{13} + 154256901 \nu^{12} + \cdots + 12767993 ) / 1586753 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 641596 \nu^{16} - 912943 \nu^{15} + 17130733 \nu^{14} + 24430526 \nu^{13} - 181095707 \nu^{12} + \cdots - 27498884 ) / 1586753 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 6696568 \nu^{16} - 8516319 \nu^{15} + 176926190 \nu^{14} + 230617909 \nu^{13} + \cdots - 261422000 ) / 11107271 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1212120 \nu^{16} - 1662081 \nu^{15} + 32007121 \nu^{14} + 44956652 \nu^{13} - 333745650 \nu^{12} + \cdots - 51585352 ) / 1586753 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11677912 \nu^{16} + 15595482 \nu^{15} - 308905793 \nu^{14} - 422390585 \nu^{13} + \cdots + 462319543 ) / 11107271 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12203491 \nu^{16} - 13792352 \nu^{15} + 320507355 \nu^{14} + 380091963 \nu^{13} + \cdots - 476302848 ) / 11107271 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 16501139 \nu^{16} - 18838666 \nu^{15} + 433888308 \nu^{14} + 516984745 \nu^{13} + \cdots - 531232758 ) / 11107271 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 16771351 \nu^{16} - 20065815 \nu^{15} + 442373694 \nu^{14} + 548454202 \nu^{13} + \cdots - 608340411 ) / 11107271 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 22511799 \nu^{16} - 28856065 \nu^{15} + 595209258 \nu^{14} + 783888478 \nu^{13} + \cdots - 928678597 ) / 11107271 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 28123016 \nu^{16} + 35294287 \nu^{15} - 741907502 \nu^{14} - 960763074 \nu^{13} + \cdots + 1092357616 ) / 11107271 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 31952759 \nu^{16} - 40216701 \nu^{15} + 843349731 \nu^{14} + 1093768675 \nu^{13} + \cdots - 1197538062 ) / 11107271 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 48244866 \nu^{16} + 61658282 \nu^{15} - 1274476952 \nu^{14} - 1675542503 \nu^{13} + \cdots + 1848670313 ) / 11107271 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - \beta_{12} - \beta_{8} - \beta_{7} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{16} + \beta_{14} - \beta_{10} - \beta_{6} + \beta_{4} + 8\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{16} + 11 \beta_{15} - 9 \beta_{12} - \beta_{11} + \beta_{10} - 10 \beta_{8} - 8 \beta_{7} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 13 \beta_{16} - 2 \beta_{15} + 9 \beta_{14} - 3 \beta_{13} + \beta_{12} - 2 \beta_{11} - 11 \beta_{10} + \cdots + 83 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{16} + 94 \beta_{15} - \beta_{14} - 70 \beta_{12} - 12 \beta_{11} + 14 \beta_{10} + 2 \beta_{9} + \cdots + 48 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 124 \beta_{16} - 31 \beta_{15} + 62 \beta_{14} - 45 \beta_{13} + 15 \beta_{12} - 29 \beta_{11} + \cdots + 483 \)
|
| \(\nu^{9}\) | \(=\) |
\( 124 \beta_{16} + 740 \beta_{15} - 19 \beta_{14} - \beta_{13} - 527 \beta_{12} - 105 \beta_{11} + \cdots + 419 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1055 \beta_{16} - 336 \beta_{15} + 386 \beta_{14} - 475 \beta_{13} + 156 \beta_{12} - 298 \beta_{11} + \cdots + 2901 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1050 \beta_{16} + 5616 \beta_{15} - 233 \beta_{14} - 21 \beta_{13} - 3921 \beta_{12} - 823 \beta_{11} + \cdots + 3239 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 8482 \beta_{16} - 3145 \beta_{15} + 2272 \beta_{14} - 4351 \beta_{13} + 1416 \beta_{12} + \cdots + 17820 \)
|
| \(\nu^{13}\) | \(=\) |
\( 8382 \beta_{16} + 41805 \beta_{15} - 2363 \beta_{14} - 277 \beta_{13} - 28983 \beta_{12} - 6155 \beta_{11} + \cdots + 23594 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 66014 \beta_{16} - 27261 \beta_{15} + 12831 \beta_{14} - 36980 \beta_{13} + 12039 \beta_{12} + \cdots + 111347 \)
|
| \(\nu^{15}\) | \(=\) |
\( 64775 \beta_{16} + 307691 \beta_{15} - 21616 \beta_{14} - 2954 \beta_{13} - 213234 \beta_{12} + \cdots + 166311 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 503410 \beta_{16} - 225817 \beta_{15} + 69648 \beta_{14} - 300336 \beta_{13} + 98643 \beta_{12} + \cdots + 705176 \)
|
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 |
|
−2.65254 | −2.17464 | 5.03596 | 3.62180 | 5.76831 | −2.98274 | −8.05299 | 1.72905 | −9.60695 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.61209 | −1.08948 | 4.82304 | −3.37708 | 2.84581 | 1.51525 | −7.37403 | −1.81304 | 8.82124 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.49907 | 2.18603 | 4.24535 | 1.73010 | −5.46304 | 5.13009 | −5.61129 | 1.77873 | −4.32364 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.82989 | 2.75561 | 1.34850 | −3.32555 | −5.04246 | −0.206452 | 1.19217 | 4.59337 | 6.08539 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.26402 | 0.451883 | −0.402263 | 1.81204 | −0.571188 | 1.66736 | 3.03650 | −2.79580 | −2.29044 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.25955 | −3.09733 | −0.413530 | −3.10986 | 3.90124 | −4.47710 | 3.03997 | 6.59342 | 3.91703 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.17471 | 2.92477 | −0.620061 | 4.24400 | −3.43575 | −3.49274 | 3.07781 | 5.55426 | −4.98546 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.364980 | −1.75420 | −1.86679 | −2.96035 | 0.640247 | 5.11564 | 1.41130 | 0.0772132 | 1.08047 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.0842488 | −2.39070 | −1.99290 | 2.54226 | 0.201413 | −1.62505 | 0.336397 | 2.71544 | −0.214182 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.535519 | 1.37525 | −1.71322 | 0.933854 | 0.736476 | 4.14877 | −1.98850 | −1.10867 | 0.500097 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.685793 | 3.13266 | −1.52969 | −0.233223 | 2.14835 | 1.65270 | −2.42063 | 6.81355 | −0.159942 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.39112 | 0.714117 | −0.0647804 | 4.19763 | 0.993424 | −0.214893 | −2.87236 | −2.49004 | 5.83941 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.88923 | 2.65562 | 1.56918 | −1.09185 | 5.01707 | −4.47828 | −0.813913 | 4.05230 | −2.06276 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.08085 | −3.39324 | 2.32992 | 2.29912 | −7.06082 | 3.55134 | 0.686506 | 8.51411 | 4.78411 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.12688 | 0.0731401 | 2.52362 | 0.256394 | 0.155560 | 0.961358 | 1.11367 | −2.99465 | 0.545320 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.33063 | 1.45147 | 3.43183 | −3.31496 | 3.38284 | 2.77554 | 3.33707 | −0.893227 | −7.72595 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.70108 | −0.820966 | 5.29584 | 1.77568 | −2.21750 | −4.04078 | 8.90233 | −2.32602 | 4.79626 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(239\) | \(-1\) |
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 | 239.2.a.b | ✓ | 17 |
| 3.b | odd | 2 | 1 | 2151.2.a.i | 17 | ||
| 4.b | odd | 2 | 1 | 3824.2.a.p | 17 | ||
| 5.b | even | 2 | 1 | 5975.2.a.g | 17 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 239.2.a.b | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 2151.2.a.i | 17 | 3.b | odd | 2 | 1 | ||
| 3824.2.a.p | 17 | 4.b | odd | 2 | 1 | ||
| 5975.2.a.g | 17 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{17} - 28 T_{2}^{15} + T_{2}^{14} + 319 T_{2}^{13} - 17 T_{2}^{12} - 1903 T_{2}^{11} + 91 T_{2}^{10} + \cdots + 49 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(239))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - 28 T^{15} + \cdots + 49 \)
|
| $3$ |
\( T^{17} - 3 T^{16} + \cdots + 592 \)
|
| $5$ |
\( T^{17} - 6 T^{16} + \cdots + 43871 \)
|
| $7$ |
\( T^{17} - 5 T^{16} + \cdots - 262144 \)
|
| $11$ |
\( T^{17} + \cdots + 151629817 \)
|
| $13$ |
\( T^{17} - 15 T^{16} + \cdots + 11583488 \)
|
| $17$ |
\( T^{17} + \cdots - 207461296 \)
|
| $19$ |
\( T^{17} + \cdots + 7399800832 \)
|
| $23$ |
\( T^{17} + 9 T^{16} + \cdots + 44744704 \)
|
| $29$ |
\( T^{17} + \cdots - 117964056107 \)
|
| $31$ |
\( T^{17} + \cdots + 2326460584 \)
|
| $37$ |
\( T^{17} + \cdots + 491454464 \)
|
| $41$ |
\( T^{17} + \cdots - 65418838016 \)
|
| $43$ |
\( T^{17} + \cdots - 36073422848 \)
|
| $47$ |
\( T^{17} + \cdots - 170160357376 \)
|
| $53$ |
\( T^{17} + \cdots + 527283519488 \)
|
| $59$ |
\( T^{17} + \cdots + 19084961644544 \)
|
| $61$ |
\( T^{17} + \cdots + 58961351552 \)
|
| $67$ |
\( T^{17} + \cdots - 11993903104 \)
|
| $71$ |
\( T^{17} + \cdots + 30800934780608 \)
|
| $73$ |
\( T^{17} + \cdots - 552452096 \)
|
| $79$ |
\( T^{17} + \cdots - 1721480118272 \)
|
| $83$ |
\( T^{17} + \cdots + 16034570219864 \)
|
| $89$ |
\( T^{17} + \cdots + 6261514240000 \)
|
| $97$ |
\( T^{17} + \cdots + 111392870170624 \)
|
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