Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9801,2,Mod(1,9801)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9801, 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("9801.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9801 = 3^{4} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9801.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: | \(78.2613790211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 99) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 11548 \nu^{17} + 68419 \nu^{16} + 192555 \nu^{15} - 1463017 \nu^{14} - 1096141 \nu^{13} + \cdots + 3900820 ) / 707006 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 36305 \nu^{17} + 364697 \nu^{16} + 372927 \nu^{15} - 7844713 \nu^{14} + 1108403 \nu^{13} + \cdots + 19827174 ) / 707006 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 51510 \nu^{17} + 182921 \nu^{16} + 891526 \nu^{15} - 3748291 \nu^{14} - 5255527 \nu^{13} + \cdots + 6678855 ) / 707006 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51510 \nu^{17} + 182921 \nu^{16} + 891526 \nu^{15} - 3748291 \nu^{14} - 5255527 \nu^{13} + \cdots + 6678855 ) / 707006 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 52328 \nu^{17} + 54362 \nu^{16} + 1236567 \nu^{15} - 1121673 \nu^{14} - 11968853 \nu^{13} + \cdots - 6907610 ) / 707006 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 59165 \nu^{17} - 47697 \nu^{16} - 1215970 \nu^{15} + 687502 \nu^{14} + 9752117 \nu^{13} + \cdots - 2716091 ) / 707006 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 70924 \nu^{17} + 153396 \nu^{16} + 1491909 \nu^{15} - 3171363 \nu^{14} - 12579935 \nu^{13} + \cdots + 4716062 ) / 707006 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 167449 \nu^{17} + 398687 \nu^{16} + 3497726 \nu^{15} - 8323726 \nu^{14} - 29175028 \nu^{13} + \cdots + 13513046 ) / 707006 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 178411 \nu^{17} + 191007 \nu^{16} + 3949586 \nu^{15} - 3625994 \nu^{14} - 35359951 \nu^{13} + \cdots - 2519637 ) / 707006 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 198455 \nu^{17} - 17047 \nu^{16} + 4624697 \nu^{15} + 971301 \nu^{14} - 43612733 \nu^{13} + \cdots - 11324008 ) / 707006 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 219191 \nu^{17} + 176950 \nu^{16} + 4993598 \nu^{15} - 3284650 \nu^{14} - 46232663 \nu^{13} + \cdots - 9793037 ) / 707006 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 371415 \nu^{17} + 37 \nu^{16} - 8688581 \nu^{15} - 1231105 \nu^{14} + 82666238 \nu^{13} + \cdots + 31378057 ) / 707006 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 434156 \nu^{17} - 88071 \nu^{16} - 10089686 \nu^{15} + 620365 \nu^{14} + 95253862 \nu^{13} + \cdots + 30509552 ) / 707006 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 442925 \nu^{17} + 429458 \nu^{16} + 9921185 \nu^{15} - 8101007 \nu^{14} - 90157391 \nu^{13} + \cdots - 7435498 ) / 707006 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 548116 \nu^{17} + 473547 \nu^{16} + 12334333 \nu^{15} - 8863276 \nu^{14} - 112476291 \nu^{13} + \cdots - 5126610 ) / 707006 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} - \beta_{11} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{16} + \beta_{14} + \beta_{13} - \beta_{10} - \beta_{9} - \beta_{7} - 8\beta_{6} + 8\beta_{5} + \beta_{2} + 19\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} + 10 \beta_{13} - 7 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 9 \beta_{7} - 11 \beta_{6} + \cdots + 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{17} + 9 \beta_{16} - 2 \beta_{15} + 11 \beta_{14} + 13 \beta_{13} - \beta_{11} - 11 \beta_{10} + \cdots - 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{17} + \beta_{16} + 14 \beta_{14} + 77 \beta_{13} - \beta_{12} - 40 \beta_{11} - 27 \beta_{10} + \cdots + 337 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 9 \beta_{17} + 63 \beta_{16} - 25 \beta_{15} + 92 \beta_{14} + 116 \beta_{13} - 5 \beta_{12} + \cdots + 17 \)
|
| \(\nu^{10}\) | \(=\) |
\( 20 \beta_{17} + 16 \beta_{16} - 2 \beta_{15} + 139 \beta_{14} + 541 \beta_{13} - 23 \beta_{12} + \cdots + 1858 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 41 \beta_{17} + 408 \beta_{16} - 217 \beta_{15} + 701 \beta_{14} + 900 \beta_{13} - 93 \beta_{12} + \cdots + 439 \)
|
| \(\nu^{12}\) | \(=\) |
\( 252 \beta_{17} + 168 \beta_{16} - 42 \beta_{15} + 1203 \beta_{14} + 3649 \beta_{13} - 312 \beta_{12} + \cdots + 10615 \)
|
| \(\nu^{13}\) | \(=\) |
\( 13 \beta_{17} + 2563 \beta_{16} - 1636 \beta_{15} + 5129 \beta_{14} + 6544 \beta_{13} - 1116 \beta_{12} + \cdots + 4958 \)
|
| \(\nu^{14}\) | \(=\) |
\( 2569 \beta_{17} + 1479 \beta_{16} - 552 \beta_{15} + 9689 \beta_{14} + 24145 \beta_{13} - 3334 \beta_{12} + \cdots + 62435 \)
|
| \(\nu^{15}\) | \(=\) |
\( 2588 \beta_{17} + 15905 \beta_{16} - 11532 \beta_{15} + 36767 \beta_{14} + 46015 \beta_{13} + \cdots + 44867 \)
|
| \(\nu^{16}\) | \(=\) |
\( 23236 \beta_{17} + 11872 \beta_{16} - 5832 \beta_{15} + 74711 \beta_{14} + 158427 \beta_{13} + \cdots + 376246 \)
|
| \(\nu^{17}\) | \(=\) |
\( 35645 \beta_{17} + 98350 \beta_{16} - 78522 \beta_{15} + 260484 \beta_{14} + 317661 \beta_{13} + \cdots + 365667 \)
|
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 |
|
−2.32900 | 0 | 3.42425 | 0.317020 | 0 | −1.41849 | −3.31708 | 0 | −0.738340 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.21517 | 0 | 2.90698 | 0.622473 | 0 | 2.57923 | −2.00911 | 0 | −1.37888 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.13886 | 0 | 2.57474 | −1.67599 | 0 | −1.29563 | −1.22928 | 0 | 3.58471 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.54614 | 0 | 0.390538 | −0.592032 | 0 | 0.721951 | 2.48845 | 0 | 0.915362 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.43863 | 0 | 0.0696440 | 3.48854 | 0 | −0.337842 | 2.77706 | 0 | −5.01870 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00657 | 0 | −0.986807 | 3.99113 | 0 | 2.75628 | 3.00644 | 0 | −4.01737 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.677669 | 0 | −1.54077 | −0.290795 | 0 | 3.35549 | 2.39947 | 0 | 0.197063 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.577812 | 0 | −1.66613 | −3.00443 | 0 | −1.16432 | 2.11834 | 0 | 1.73599 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.285148 | 0 | −1.91869 | −2.70873 | 0 | −4.07334 | 1.11741 | 0 | 0.772391 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.490494 | 0 | −1.75942 | −1.70818 | 0 | 3.41052 | −1.84397 | 0 | −0.837851 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.664320 | 0 | −1.55868 | −1.11662 | 0 | −3.90454 | −2.36410 | 0 | −0.741796 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.16886 | 0 | −0.633765 | 3.16693 | 0 | 0.248873 | −3.07850 | 0 | 3.70170 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.45739 | 0 | 0.123972 | 3.00603 | 0 | −1.65018 | −2.73410 | 0 | 4.38094 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.48351 | 0 | 0.200801 | 0.0723272 | 0 | 1.95022 | −2.66913 | 0 | 0.107298 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.89468 | 0 | 1.58980 | −3.51301 | 0 | −3.43200 | −0.777197 | 0 | −6.65603 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.00864 | 0 | 2.03462 | −0.00545179 | 0 | −3.82781 | 0.0695356 | 0 | −0.0109507 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.44543 | 0 | 3.98014 | 3.00287 | 0 | 3.89197 | 4.84231 | 0 | 7.34333 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.60169 | 0 | 4.76877 | −2.05208 | 0 | 3.18962 | 7.20347 | 0 | −5.33887 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
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 | 9801.2.a.cp | 18 | |
| 3.b | odd | 2 | 1 | 9801.2.a.cm | 18 | ||
| 9.d | odd | 6 | 2 | 1089.2.e.p | 36 | ||
| 11.b | odd | 2 | 1 | 9801.2.a.cn | 18 | ||
| 11.c | even | 5 | 2 | 891.2.f.e | 36 | ||
| 33.d | even | 2 | 1 | 9801.2.a.co | 18 | ||
| 33.h | odd | 10 | 2 | 891.2.f.f | 36 | ||
| 99.g | even | 6 | 2 | 1089.2.e.o | 36 | ||
| 99.m | even | 15 | 4 | 297.2.n.b | 72 | ||
| 99.n | odd | 30 | 4 | 99.2.m.b | ✓ | 72 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.2.m.b | ✓ | 72 | 99.n | odd | 30 | 4 | |
| 297.2.n.b | 72 | 99.m | even | 15 | 4 | ||
| 891.2.f.e | 36 | 11.c | even | 5 | 2 | ||
| 891.2.f.f | 36 | 33.h | odd | 10 | 2 | ||
| 1089.2.e.o | 36 | 99.g | even | 6 | 2 | ||
| 1089.2.e.p | 36 | 9.d | odd | 6 | 2 | ||
| 9801.2.a.cm | 18 | 3.b | odd | 2 | 1 | ||
| 9801.2.a.cn | 18 | 11.b | odd | 2 | 1 | ||
| 9801.2.a.co | 18 | 33.d | even | 2 | 1 | ||
| 9801.2.a.cp | 18 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):
|
\( T_{2}^{18} - 2 T_{2}^{17} - 22 T_{2}^{16} + 42 T_{2}^{15} + 198 T_{2}^{14} - 357 T_{2}^{13} - 944 T_{2}^{12} + \cdots - 55 \)
|
|
\( T_{5}^{18} - T_{5}^{17} - 48 T_{5}^{16} + 21 T_{5}^{15} + 932 T_{5}^{14} + 122 T_{5}^{13} - 9148 T_{5}^{12} + \cdots + 1 \)
|
|
\( T_{7}^{18} - T_{7}^{17} - 66 T_{7}^{16} + 73 T_{7}^{15} + 1774 T_{7}^{14} - 2055 T_{7}^{13} + \cdots - 88209 \)
|
|
\( T_{17}^{18} - 20 T_{17}^{17} + 69 T_{17}^{16} + 1099 T_{17}^{15} - 9169 T_{17}^{14} - 2805 T_{17}^{13} + \cdots - 1111869 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 2 T^{17} + \cdots - 55 \)
|
| $3$ |
\( T^{18} \)
|
| $5$ |
\( T^{18} - T^{17} + \cdots + 1 \)
|
| $7$ |
\( T^{18} - T^{17} + \cdots - 88209 \)
|
| $11$ |
\( T^{18} \)
|
| $13$ |
\( T^{18} - 3 T^{17} + \cdots - 45859 \)
|
| $17$ |
\( T^{18} - 20 T^{17} + \cdots - 1111869 \)
|
| $19$ |
\( T^{18} + \cdots - 105205991 \)
|
| $23$ |
\( T^{18} - 10 T^{17} + \cdots + 48738591 \)
|
| $29$ |
\( T^{18} + \cdots - 259142400 \)
|
| $31$ |
\( T^{18} + \cdots + 1830183139 \)
|
| $37$ |
\( T^{18} + \cdots + 9622779851 \)
|
| $41$ |
\( T^{18} - 20 T^{17} + \cdots + 3455155 \)
|
| $43$ |
\( T^{18} + \cdots - 2225775519 \)
|
| $47$ |
\( T^{18} + \cdots - 420016249 \)
|
| $53$ |
\( T^{18} + \cdots - 110996849659 \)
|
| $59$ |
\( T^{18} + \cdots - 8937327603625 \)
|
| $61$ |
\( T^{18} + \cdots + 11452862540295 \)
|
| $67$ |
\( T^{18} + \cdots + 71071176604221 \)
|
| $71$ |
\( T^{18} + \cdots + 833307334875 \)
|
| $73$ |
\( T^{18} + \cdots - 3190859600269 \)
|
| $79$ |
\( T^{18} + \cdots - 427188073345 \)
|
| $83$ |
\( T^{18} + \cdots - 869294927259 \)
|
| $89$ |
\( T^{18} + \cdots + 7354793820429 \)
|
| $97$ |
\( T^{18} + \cdots + 8456262291 \)
|
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