Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,22,Mod(1,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.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: | \(226.376648873\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{20} - 3 x^{19} - 30906825 x^{18} + 1599806295 x^{17} + 397632537600480 x^{16} + \cdots + 18\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{56}\cdot 3^{135}\cdot 5^{4}\cdot 7^{6} \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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
\( x^{20} - 3 x^{19} - 30906825 x^{18} + 1599806295 x^{17} + 397632537600480 x^{16} + \cdots + 18\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 73\nu - 3090694 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\!\cdots\!39 \nu^{19} + \cdots - 19\!\cdots\!24 ) / 57\!\cdots\!92 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\!\cdots\!43 \nu^{19} + \cdots - 32\!\cdots\!72 ) / 28\!\cdots\!96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!11 \nu^{19} + \cdots + 34\!\cdots\!44 ) / 25\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 12\!\cdots\!71 \nu^{19} + \cdots - 98\!\cdots\!08 ) / 57\!\cdots\!92 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 76\!\cdots\!71 \nu^{19} + \cdots - 80\!\cdots\!32 ) / 57\!\cdots\!92 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 40\!\cdots\!27 \nu^{19} + \cdots - 72\!\cdots\!64 ) / 28\!\cdots\!96 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 10\!\cdots\!21 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 28\!\cdots\!96 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 12\!\cdots\!93 \nu^{19} + \cdots + 54\!\cdots\!76 ) / 57\!\cdots\!92 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 20\!\cdots\!19 \nu^{19} + \cdots + 68\!\cdots\!48 ) / 57\!\cdots\!92 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 28\!\cdots\!99 \nu^{19} + \cdots + 35\!\cdots\!92 ) / 57\!\cdots\!92 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 52\!\cdots\!09 \nu^{19} + \cdots - 91\!\cdots\!64 ) / 89\!\cdots\!28 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 11\!\cdots\!47 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 12\!\cdots\!68 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 78\!\cdots\!67 \nu^{19} + \cdots - 75\!\cdots\!96 ) / 57\!\cdots\!92 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 13\!\cdots\!39 \nu^{19} + \cdots + 62\!\cdots\!76 ) / 57\!\cdots\!92 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 21\!\cdots\!67 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 71\!\cdots\!24 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 92\!\cdots\!83 \nu^{19} + \cdots - 97\!\cdots\!56 ) / 28\!\cdots\!96 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 19\!\cdots\!79 \nu^{19} + \cdots - 71\!\cdots\!48 ) / 57\!\cdots\!92 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 73\beta _1 + 3090694 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 4\beta_{3} - 52\beta_{2} + 5175475\beta _1 - 226839197 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{12} + 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + \beta_{7} - 10 \beta_{6} + \cdots + 15995878320808 \)
|
| \(\nu^{5}\) | \(=\) |
\( 235 \beta_{19} - 83 \beta_{18} + 251 \beta_{17} + 160 \beta_{16} - 96 \beta_{15} + 38 \beta_{14} + \cdots - 21\!\cdots\!69 \)
|
| \(\nu^{6}\) | \(=\) |
\( 143577 \beta_{19} + 234263 \beta_{18} + 146465 \beta_{17} + 561888 \beta_{16} - 219800 \beta_{15} + \cdots + 96\!\cdots\!01 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2938712685 \beta_{19} - 943675925 \beta_{18} + 2907856301 \beta_{17} + 2209623200 \beta_{16} + \cdots - 19\!\cdots\!40 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1866149725699 \beta_{19} + 3241584748477 \beta_{18} + 1484255020843 \beta_{17} + 8333234360416 \beta_{16} + \cdots + 62\!\cdots\!01 \)
|
| \(\nu^{9}\) | \(=\) |
\( 27\!\cdots\!18 \beta_{19} + \cdots - 15\!\cdots\!10 \)
|
| \(\nu^{10}\) | \(=\) |
\( 16\!\cdots\!26 \beta_{19} + \cdots + 41\!\cdots\!76 \)
|
| \(\nu^{11}\) | \(=\) |
\( 23\!\cdots\!94 \beta_{19} + \cdots - 12\!\cdots\!55 \)
|
| \(\nu^{12}\) | \(=\) |
\( 13\!\cdots\!18 \beta_{19} + \cdots + 29\!\cdots\!78 \)
|
| \(\nu^{13}\) | \(=\) |
\( 19\!\cdots\!45 \beta_{19} + \cdots - 98\!\cdots\!11 \)
|
| \(\nu^{14}\) | \(=\) |
\( 93\!\cdots\!15 \beta_{19} + \cdots + 20\!\cdots\!71 \)
|
| \(\nu^{15}\) | \(=\) |
\( 15\!\cdots\!31 \beta_{19} + \cdots - 76\!\cdots\!30 \)
|
| \(\nu^{16}\) | \(=\) |
\( 61\!\cdots\!37 \beta_{19} + \cdots + 14\!\cdots\!91 \)
|
| \(\nu^{17}\) | \(=\) |
\( 12\!\cdots\!92 \beta_{19} + \cdots - 58\!\cdots\!24 \)
|
| \(\nu^{18}\) | \(=\) |
\( 37\!\cdots\!64 \beta_{19} + \cdots + 10\!\cdots\!10 \)
|
| \(\nu^{19}\) | \(=\) |
\( 96\!\cdots\!08 \beta_{19} + \cdots - 45\!\cdots\!01 \)
|
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 |
|
−2776.32 | 0 | 5.61078e6 | −3.15307e7 | 0 | 5.45002e8 | −9.75494e9 | 0 | 8.75393e10 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2581.48 | 0 | 4.56687e6 | 1.39157e7 | 0 | 6.00320e8 | −6.37551e9 | 0 | −3.59230e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2267.10 | 0 | 3.04258e6 | 3.23490e7 | 0 | −7.54213e8 | −2.14339e9 | 0 | −7.33384e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1982.83 | 0 | 1.83448e6 | −1.02434e7 | 0 | −9.51290e8 | 5.20838e8 | 0 | 2.03109e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1852.20 | 0 | 1.33350e6 | −2.91973e7 | 0 | 2.00050e8 | 1.41443e9 | 0 | 5.40794e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1647.98 | 0 | 618687. | 2.35690e7 | 0 | 6.24083e8 | 2.43648e9 | 0 | −3.88413e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1087.31 | 0 | −914902. | −1.49670e6 | 0 | −7.63950e7 | 3.27505e9 | 0 | 1.62738e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −741.831 | 0 | −1.54684e6 | 4.52810e6 | 0 | 1.01033e9 | 2.70323e9 | 0 | −3.35909e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −684.166 | 0 | −1.62907e6 | −1.91616e7 | 0 | −1.44219e9 | 2.54935e9 | 0 | 1.31097e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −312.580 | 0 | −1.99945e6 | −3.51534e7 | 0 | 1.24694e9 | 1.28051e9 | 0 | 1.09883e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 11.8846 | 0 | −2.09701e6 | 3.62474e7 | 0 | −2.43499e8 | −4.98461e7 | 0 | 4.30788e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 256.638 | 0 | −2.03129e6 | 1.54683e7 | 0 | −1.93722e8 | −1.05952e9 | 0 | 3.96975e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 830.025 | 0 | −1.40821e6 | −937473. | 0 | −4.93697e7 | −2.90954e9 | 0 | −7.78126e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1252.83 | 0 | −527567. | −3.03735e7 | 0 | −1.16597e9 | −3.28833e9 | 0 | −3.80528e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1318.30 | 0 | −359236. | −2.83647e7 | 0 | 3.10086e8 | −3.23826e9 | 0 | −3.73932e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 1775.46 | 0 | 1.05511e6 | 3.60516e7 | 0 | −2.31218e8 | −1.85011e9 | 0 | 6.40082e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 1918.93 | 0 | 1.58515e6 | 1.00885e7 | 0 | 1.43926e9 | −9.82502e8 | 0 | 1.93592e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2274.08 | 0 | 3.07430e6 | 6.72878e6 | 0 | −2.34917e8 | 2.22212e9 | 0 | 1.53018e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 2569.34 | 0 | 4.50437e6 | −2.83332e7 | 0 | 6.46830e8 | 6.18497e9 | 0 | −7.27978e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 2703.30 | 0 | 5.21069e6 | 3.61083e6 | 0 | −1.09050e9 | 8.41682e9 | 0 | 9.76117e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
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 | 81.22.a.c | 20 | |
| 3.b | odd | 2 | 1 | 81.22.a.d | 20 | ||
| 9.c | even | 3 | 2 | 9.22.c.a | ✓ | 40 | |
| 9.d | odd | 6 | 2 | 27.22.c.a | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.22.c.a | ✓ | 40 | 9.c | even | 3 | 2 | |
| 27.22.c.a | 40 | 9.d | odd | 6 | 2 | ||
| 81.22.a.c | 20 | 1.a | even | 1 | 1 | trivial | |
| 81.22.a.d | 20 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} + 1023 T_{2}^{19} - 30409728 T_{2}^{18} - 29819715192 T_{2}^{17} + 383979204828792 T_{2}^{16} + \cdots + 38\!\cdots\!48 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 38\!\cdots\!48 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 51\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots - 18\!\cdots\!84 \)
|
| $11$ |
\( T^{20} + \cdots - 27\!\cdots\!89 \)
|
| $13$ |
\( T^{20} + \cdots - 94\!\cdots\!08 \)
|
| $17$ |
\( T^{20} + \cdots - 17\!\cdots\!48 \)
|
| $19$ |
\( T^{20} + \cdots - 39\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 52\!\cdots\!68 \)
|
| $29$ |
\( T^{20} + \cdots + 72\!\cdots\!16 \)
|
| $31$ |
\( T^{20} + \cdots - 44\!\cdots\!00 \)
|
| $37$ |
\( T^{20} + \cdots - 37\!\cdots\!08 \)
|
| $41$ |
\( T^{20} + \cdots - 74\!\cdots\!25 \)
|
| $43$ |
\( T^{20} + \cdots - 13\!\cdots\!23 \)
|
| $47$ |
\( T^{20} + \cdots + 18\!\cdots\!76 \)
|
| $53$ |
\( T^{20} + \cdots + 22\!\cdots\!16 \)
|
| $59$ |
\( T^{20} + \cdots - 76\!\cdots\!17 \)
|
| $61$ |
\( T^{20} + \cdots - 68\!\cdots\!84 \)
|
| $67$ |
\( T^{20} + \cdots + 95\!\cdots\!97 \)
|
| $71$ |
\( T^{20} + \cdots - 48\!\cdots\!64 \)
|
| $73$ |
\( T^{20} + \cdots + 19\!\cdots\!48 \)
|
| $79$ |
\( T^{20} + \cdots + 23\!\cdots\!68 \)
|
| $83$ |
\( T^{20} + \cdots + 90\!\cdots\!44 \)
|
| $89$ |
\( T^{20} + \cdots + 27\!\cdots\!00 \)
|
| $97$ |
\( T^{20} + \cdots - 36\!\cdots\!93 \)
|
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