Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [799,2,Mod(1,799)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, 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("799.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 799.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: | \(6.38004712150\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} - 24 x^{18} + 108 x^{17} + 221 x^{16} - 1200 x^{15} - 931 x^{14} + 7128 x^{13} + \cdots + 63 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 4 x^{19} - 24 x^{18} + 108 x^{17} + 221 x^{16} - 1200 x^{15} - 931 x^{14} + 7128 x^{13} + \cdots + 63 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3328601967 \nu^{19} + 9458558925 \nu^{18} + 91164622653 \nu^{17} - 254938130792 \nu^{16} + \cdots + 236845524837 ) / 15967470742 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7381257869 \nu^{19} + 20144750249 \nu^{18} + 203216423239 \nu^{17} - 540607276783 \nu^{16} + \cdots + 236899537813 ) / 31934941484 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 26003449665 \nu^{19} + 72645198981 \nu^{18} + 713701746747 \nu^{17} - 1955375195139 \nu^{16} + \cdots + 1859684269869 ) / 63869882968 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 43755045167 \nu^{19} + 122391131113 \nu^{18} + 1195482203313 \nu^{17} + \cdots + 2248919397473 ) / 63869882968 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 12871876115 \nu^{19} - 36507152062 \nu^{18} - 351275762739 \nu^{17} + 981238216974 \nu^{16} + \cdots - 615148108719 ) / 15967470742 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 52629049555 \nu^{19} - 145361119305 \nu^{18} - 1445401058125 \nu^{17} + 3904766389943 \nu^{16} + \cdots - 2820437728489 ) / 63869882968 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 63110429755 \nu^{19} + 178823258711 \nu^{18} + 1722829448105 \nu^{17} + \cdots + 3566123025275 ) / 63869882968 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 32705668979 \nu^{19} + 91644650578 \nu^{18} + 895142757799 \nu^{17} - 2461675548174 \nu^{16} + \cdots + 1835016759356 ) / 31934941484 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 69669879571 \nu^{19} - 192759890567 \nu^{18} - 1906578622613 \nu^{17} + 5157798342329 \nu^{16} + \cdots - 3339973985187 ) / 63869882968 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 71113163465 \nu^{19} + 199900674391 \nu^{18} + 1940508802507 \nu^{17} + \cdots + 3446748044851 ) / 63869882968 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 37294964537 \nu^{19} + 104774098571 \nu^{18} + 1018385457297 \nu^{17} + \cdots + 1968529630357 ) / 31934941484 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 38991643152 \nu^{19} - 108336021729 \nu^{18} - 1068984852128 \nu^{17} + 2908229758693 \nu^{16} + \cdots - 2178927700769 ) / 31934941484 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 108581508953 \nu^{19} + 308144013427 \nu^{18} + 2961838140367 \nu^{17} + \cdots + 5738604819011 ) / 63869882968 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 112136700593 \nu^{19} + 315193600417 \nu^{18} + 3067905241147 \nu^{17} + \cdots + 6159437381357 ) / 63869882968 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 140656479275 \nu^{19} - 398110717893 \nu^{18} - 3839633777117 \nu^{17} + 10692468132915 \nu^{16} + \cdots - 7616025409853 ) / 63869882968 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 75681607827 \nu^{19} + 212619980208 \nu^{18} + 2069838284109 \nu^{17} + \cdots + 3885370304634 ) / 31934941484 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{19} + \beta_{18} - \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} + \cdots + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{19} + \beta_{17} + \beta_{16} + \beta_{15} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12 \beta_{19} + 11 \beta_{18} - 11 \beta_{17} - 11 \beta_{16} - 12 \beta_{15} + 10 \beta_{14} + \cdots + 135 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 15 \beta_{19} - \beta_{18} + 15 \beta_{17} + 11 \beta_{16} + 15 \beta_{15} + \beta_{14} + \beta_{12} + \cdots - 26 \)
|
| \(\nu^{8}\) | \(=\) |
\( 106 \beta_{19} + 93 \beta_{18} - 95 \beta_{17} - 99 \beta_{16} - 110 \beta_{15} + 82 \beta_{14} + \cdots + 985 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 162 \beta_{19} - 19 \beta_{18} + 166 \beta_{17} + 88 \beta_{16} + 161 \beta_{15} + 15 \beta_{14} + \cdots - 262 \)
|
| \(\nu^{10}\) | \(=\) |
\( 835 \beta_{19} + 721 \beta_{18} - 743 \beta_{17} - 834 \beta_{16} - 907 \beta_{15} + 641 \beta_{14} + \cdots + 7252 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1540 \beta_{19} - 237 \beta_{18} + 1617 \beta_{17} + 622 \beta_{16} + 1511 \beta_{15} + 164 \beta_{14} + \cdots - 2395 \)
|
| \(\nu^{12}\) | \(=\) |
\( 6218 \beta_{19} + 5394 \beta_{18} - 5492 \beta_{17} - 6805 \beta_{16} - 7081 \beta_{15} + 4946 \beta_{14} + \cdots + 53643 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 13726 \beta_{19} - 2474 \beta_{18} + 14705 \beta_{17} + 4115 \beta_{16} + 13250 \beta_{15} + \cdots - 20756 \)
|
| \(\nu^{14}\) | \(=\) |
\( 44844 \beta_{19} + 39736 \beta_{18} - 39083 \beta_{17} - 54505 \beta_{16} - 53552 \beta_{15} + \cdots + 398122 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 117830 \beta_{19} - 23465 \beta_{18} + 128286 \beta_{17} + 25999 \beta_{16} + 111822 \beta_{15} + \cdots - 173799 \)
|
| \(\nu^{16}\) | \(=\) |
\( 316875 \beta_{19} + 290902 \beta_{18} - 270169 \beta_{17} - 431412 \beta_{16} - 397061 \beta_{15} + \cdots + 2962926 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 987399 \beta_{19} - 209912 \beta_{18} + 1089296 \beta_{17} + 157375 \beta_{16} + 922042 \beta_{15} + \cdots - 1420529 \)
|
| \(\nu^{18}\) | \(=\) |
\( 2206817 \beta_{19} + 2125835 \beta_{18} - 1820993 \beta_{17} - 3387388 \beta_{16} - 2906151 \beta_{15} + \cdots + 22104611 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 8137953 \beta_{19} - 1806728 \beta_{18} + 9078576 \beta_{17} + 903505 \beta_{16} + 7490145 \beta_{15} + \cdots - 11403787 \)
|
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 |
|
−2.74861 | 0.350871 | 5.55484 | 4.31399 | −0.964407 | −1.95626 | −9.77085 | −2.87689 | −11.8575 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.18123 | −3.27609 | 2.75774 | 2.59809 | 7.14589 | −1.91867 | −1.65281 | 7.73278 | −5.66702 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.13233 | 0.687233 | 2.54682 | −2.96401 | −1.46540 | −3.49828 | −1.16599 | −2.52771 | 6.32025 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.89867 | 2.39331 | 1.60496 | 2.00968 | −4.54411 | 2.01850 | 0.750053 | 2.72793 | −3.81573 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.86207 | −2.20619 | 1.46730 | −0.859163 | 4.10808 | 3.41505 | 0.991931 | 1.86729 | 1.59982 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.18073 | −1.34483 | −0.605885 | 0.271325 | 1.58788 | −4.77084 | 3.07684 | −1.19142 | −0.320360 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.974245 | 1.87987 | −1.05085 | 1.90979 | −1.83145 | 4.07233 | 2.97227 | 0.533896 | −1.86060 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.533961 | 3.38011 | −1.71489 | −0.716647 | −1.80485 | −4.66840 | 1.98361 | 8.42512 | 0.382662 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.351266 | −1.36638 | −1.87661 | −3.75140 | 0.479962 | −1.09877 | 1.36172 | −1.13301 | 1.31774 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.0537041 | −1.20296 | −1.99712 | 1.78333 | 0.0646037 | 2.05218 | 0.214661 | −1.55290 | −0.0957722 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.530018 | 3.23709 | −1.71908 | 4.03960 | 1.71572 | 0.377707 | −1.97118 | 7.47878 | 2.14106 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.17120 | −3.05359 | −0.628298 | −2.22954 | −3.57635 | −4.41304 | −3.07825 | 6.32439 | −2.61123 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.18897 | −3.18705 | −0.586361 | 3.90754 | −3.78929 | 3.98525 | −3.07509 | 7.15727 | 4.64592 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.22248 | 1.90981 | −0.505532 | 1.00510 | 2.33471 | 1.73367 | −3.06297 | 0.647363 | 1.22872 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.53706 | −1.07444 | 0.362560 | −2.28440 | −1.65149 | 2.77363 | −2.51685 | −1.84557 | −3.51127 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.12739 | 2.71255 | 2.52577 | −1.05415 | 5.77065 | 4.69022 | 1.11852 | 4.35795 | −2.24259 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.27507 | 1.37358 | 3.17596 | 3.49208 | 3.12499 | −3.93958 | 2.67540 | −1.11329 | 7.94474 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.42235 | 0.760154 | 3.86777 | 0.950403 | 1.84136 | 1.38568 | 4.52440 | −2.42217 | 2.30221 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2.64443 | 2.73641 | 4.99303 | −3.13323 | 7.23626 | −1.81022 | 7.91488 | 4.48794 | −8.28563 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 2.79783 | −1.70945 | 5.82786 | 3.71164 | −4.78276 | −1.43018 | 10.7097 | −0.0777701 | 10.3845 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
| \(47\) | \(-1\) |
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 | 799.2.a.g | ✓ | 20 |
| 3.b | odd | 2 | 1 | 7191.2.a.bb | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 799.2.a.g | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 7191.2.a.bb | 20 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(799))\):
|
\( T_{2}^{20} - 4 T_{2}^{19} - 24 T_{2}^{18} + 108 T_{2}^{17} + 221 T_{2}^{16} - 1200 T_{2}^{15} + \cdots + 63 \)
|
|
\( T_{5}^{20} - 13 T_{5}^{19} + 14 T_{5}^{18} + 482 T_{5}^{17} - 1818 T_{5}^{16} - 5729 T_{5}^{15} + \cdots + 468504 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 4 T^{19} + \cdots + 63 \)
|
| $3$ |
\( T^{20} - 3 T^{19} + \cdots - 50176 \)
|
| $5$ |
\( T^{20} - 13 T^{19} + \cdots + 468504 \)
|
| $7$ |
\( T^{20} + 3 T^{19} + \cdots + 39194624 \)
|
| $11$ |
\( T^{20} - 10 T^{19} + \cdots - 840672 \)
|
| $13$ |
\( T^{20} + \cdots + 341855232 \)
|
| $17$ |
\( (T + 1)^{20} \)
|
| $19$ |
\( T^{20} + \cdots - 212426752 \)
|
| $23$ |
\( T^{20} + \cdots + 28273749216 \)
|
| $29$ |
\( T^{20} + \cdots - 18834750816 \)
|
| $31$ |
\( T^{20} + \cdots + 53989720704 \)
|
| $37$ |
\( T^{20} + \cdots - 24561028045824 \)
|
| $41$ |
\( T^{20} + \cdots - 11\!\cdots\!68 \)
|
| $43$ |
\( T^{20} + \cdots - 256040410619904 \)
|
| $47$ |
\( (T - 1)^{20} \)
|
| $53$ |
\( T^{20} + \cdots + 292510322688 \)
|
| $59$ |
\( T^{20} + \cdots + 9255700325376 \)
|
| $61$ |
\( T^{20} + \cdots - 10\!\cdots\!76 \)
|
| $67$ |
\( T^{20} + \cdots - 10549877018624 \)
|
| $71$ |
\( T^{20} + \cdots + 11040633234432 \)
|
| $73$ |
\( T^{20} + \cdots + 33\!\cdots\!48 \)
|
| $79$ |
\( T^{20} + \cdots + 847768509211648 \)
|
| $83$ |
\( T^{20} + \cdots + 294085023744 \)
|
| $89$ |
\( T^{20} + \cdots + 60\!\cdots\!24 \)
|
| $97$ |
\( T^{20} + \cdots + 69\!\cdots\!12 \)
|
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