Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,9,Mod(80,81)]
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([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.80");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(32.9976674150\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{84} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu^{2} + 384 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 16592055498945 \nu^{14} + \cdots + 31\!\cdots\!08 ) / 24\!\cdots\!60 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2497774802274 \nu^{14} + \cdots + 11\!\cdots\!08 ) / 30\!\cdots\!20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25410686450643 \nu^{14} + \cdots - 98\!\cdots\!16 ) / 98\!\cdots\!24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 143265531846591 \nu^{14} + \cdots - 85\!\cdots\!20 ) / 24\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 23\!\cdots\!33 \nu^{14} + \cdots - 14\!\cdots\!52 ) / 49\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 594985394614059 \nu^{14} + \cdots + 28\!\cdots\!08 ) / 12\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 55090601 \nu^{15} - 55395890770 \nu^{13} - 21996486413661 \nu^{11} + \cdots - 34\!\cdots\!12 \nu ) / 83\!\cdots\!04 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 69\!\cdots\!75 \nu^{15} + \cdots - 63\!\cdots\!48 \nu ) / 60\!\cdots\!64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 11\!\cdots\!79 \nu^{15} + \cdots + 35\!\cdots\!56 \nu ) / 50\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 30719884451 \nu^{15} + 30582399292090 \nu^{13} + \cdots + 18\!\cdots\!84 \nu ) / 83\!\cdots\!04 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 11\!\cdots\!63 \nu^{15} + \cdots + 94\!\cdots\!84 \nu ) / 30\!\cdots\!20 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 636078186037275 \nu^{15} + \cdots + 30\!\cdots\!84 \nu ) / 33\!\cdots\!60 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 74\!\cdots\!17 \nu^{15} + \cdots - 39\!\cdots\!36 \nu ) / 30\!\cdots\!20 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 26\!\cdots\!27 \nu^{15} + \cdots - 12\!\cdots\!72 \nu ) / 30\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{15} - 26 \beta_{14} - 27 \beta_{13} - 33 \beta_{12} + 62 \beta_{11} + 4 \beta_{10} + \cdots + 10850 \beta_{8} ) / 177147 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 - 384 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 101 \beta_{15} + 3922 \beta_{14} + 6453 \beta_{13} + 7959 \beta_{12} - 31039 \beta_{11} + \cdots - 3536290 \beta_{8} ) / 177147 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + 8\beta_{3} + \beta_{2} - 852\beta _1 + 235323 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 69514 \beta_{15} - 726014 \beta_{14} - 1602774 \beta_{13} - 1954284 \beta_{12} + \cdots + 1324336535 \beta_{8} ) / 177147 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 29 \beta_{7} - 86 \beta_{6} - 1402 \beta_{5} - 206 \beta_{4} - 15202 \beta_{3} - 1528 \beta_{2} + \cdots - 165481192 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 27108917 \beta_{15} + 145439524 \beta_{14} + 412743735 \beta_{13} + \cdots - 477292582570 \beta_{8} ) / 177147 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 44142 \beta_{7} + 151412 \beta_{6} + 1491915 \beta_{5} + 256900 \beta_{4} + 19087316 \beta_{3} + \cdots + 124022951365 ) / 81 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 8996587270 \beta_{15} - 30640114838 \beta_{14} - 109372233150 \beta_{13} + \cdots + 162516241907297 \beta_{8} ) / 177147 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 15281869 \beta_{7} - 62299606 \beta_{6} - 481409782 \beta_{5} - 66055758 \beta_{4} + \cdots - 32351942692828 ) / 81 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 2819717649515 \beta_{15} + 6750809798956 \beta_{14} + 29650652884593 \beta_{13} + \cdots - 53\!\cdots\!50 \beta_{8} ) / 177147 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 4527415742 \beta_{7} + 22266002740 \beta_{6} + 148886537917 \beta_{5} + 11886560580 \beta_{4} + \cdots + 87\!\cdots\!59 ) / 81 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 862090036459174 \beta_{15} + \cdots + 16\!\cdots\!71 \beta_{8} ) / 177147 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 3713925118737 \beta_{7} - 22225534276430 \beta_{6} - 135217317246954 \beta_{5} + \cdots - 72\!\cdots\!20 ) / 243 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 26\!\cdots\!65 \beta_{15} + \cdots - 52\!\cdots\!02 \beta_{8} ) / 177147 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 80.1 |
|
− | 29.8006i | 0 | −632.074 | 177.778i | 0 | 3390.81 | 11207.2i | 0 | 5297.89 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.2 | − | 25.6809i | 0 | −403.506 | − | 204.369i | 0 | −3702.42 | 3788.08i | 0 | −5248.38 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.3 | − | 24.1178i | 0 | −325.666 | − | 436.510i | 0 | 550.753 | 1680.18i | 0 | −10527.6 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.4 | − | 22.4204i | 0 | −246.676 | 1080.12i | 0 | −1006.64 | − | 209.045i | 0 | 24216.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.5 | − | 14.9418i | 0 | 32.7440 | − | 1089.48i | 0 | 1308.49 | − | 4314.34i | 0 | −16278.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.6 | − | 10.0153i | 0 | 155.694 | 594.387i | 0 | 4409.51 | − | 4123.23i | 0 | 5952.96 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.7 | − | 9.11454i | 0 | 172.925 | − | 181.837i | 0 | −873.072 | − | 3909.46i | 0 | −1657.36 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.8 | − | 5.78284i | 0 | 222.559 | 626.068i | 0 | −2231.44 | − | 2767.43i | 0 | 3620.45 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.9 | 5.78284i | 0 | 222.559 | − | 626.068i | 0 | −2231.44 | 2767.43i | 0 | 3620.45 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.10 | 9.11454i | 0 | 172.925 | 181.837i | 0 | −873.072 | 3909.46i | 0 | −1657.36 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.11 | 10.0153i | 0 | 155.694 | − | 594.387i | 0 | 4409.51 | 4123.23i | 0 | 5952.96 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.12 | 14.9418i | 0 | 32.7440 | 1089.48i | 0 | 1308.49 | 4314.34i | 0 | −16278.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.13 | 22.4204i | 0 | −246.676 | − | 1080.12i | 0 | −1006.64 | 209.045i | 0 | 24216.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.14 | 24.1178i | 0 | −325.666 | 436.510i | 0 | 550.753 | − | 1680.18i | 0 | −10527.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.15 | 25.6809i | 0 | −403.506 | 204.369i | 0 | −3702.42 | − | 3788.08i | 0 | −5248.38 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 80.16 | 29.8006i | 0 | −632.074 | − | 177.778i | 0 | 3390.81 | − | 11207.2i | 0 | 5297.89 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.9.b.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 81.9.b.b | ✓ | 16 |
| 9.c | even | 3 | 2 | 81.9.d.g | 32 | ||
| 9.d | odd | 6 | 2 | 81.9.d.g | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.9.b.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 81.9.b.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 81.9.d.g | 32 | 9.c | even | 3 | 2 | ||
| 81.9.d.g | 32 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 3072 T_{2}^{14} + 3777309 T_{2}^{12} + 2381488884 T_{2}^{10} + 819880359588 T_{2}^{8} + \cdots + 10\!\cdots\!64 \)
acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 10\!\cdots\!64 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + \cdots + 15\!\cdots\!25 \)
|
| $7$ |
\( (T^{8} + \cdots + 78\!\cdots\!56)^{2} \)
|
| $11$ |
\( T^{16} + \cdots + 11\!\cdots\!64 \)
|
| $13$ |
\( (T^{8} + \cdots + 40\!\cdots\!93)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 10\!\cdots\!69 \)
|
| $19$ |
\( (T^{8} + \cdots + 24\!\cdots\!92)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 30\!\cdots\!36 \)
|
| $29$ |
\( T^{16} + \cdots + 11\!\cdots\!09 \)
|
| $31$ |
\( (T^{8} + \cdots - 14\!\cdots\!44)^{2} \)
|
| $37$ |
\( (T^{8} + \cdots - 11\!\cdots\!63)^{2} \)
|
| $41$ |
\( T^{16} + \cdots + 25\!\cdots\!36 \)
|
| $43$ |
\( (T^{8} + \cdots + 24\!\cdots\!52)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 35\!\cdots\!16 \)
|
| $53$ |
\( T^{16} + \cdots + 12\!\cdots\!44 \)
|
| $59$ |
\( T^{16} + \cdots + 46\!\cdots\!44 \)
|
| $61$ |
\( (T^{8} + \cdots + 11\!\cdots\!77)^{2} \)
|
| $67$ |
\( (T^{8} + \cdots + 33\!\cdots\!68)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 19\!\cdots\!16 \)
|
| $73$ |
\( (T^{8} + \cdots - 97\!\cdots\!99)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots - 12\!\cdots\!84)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 28\!\cdots\!76 \)
|
| $89$ |
\( T^{16} + \cdots + 62\!\cdots\!81 \)
|
| $97$ |
\( (T^{8} + \cdots - 17\!\cdots\!84)^{2} \)
|
| show more | |
| show less |