Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,9,Mod(61,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.61");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.m (of order \(6\), degree \(2\), 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: | \(34.2198032451\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} + \cdots + 15\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{8}\cdot 7^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 38255 x^{8} + 1483053595 x^{6} - 139470625170 x^{5} + 5194605060018 x^{4} + \cdots + 15\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 27\!\cdots\!39 \nu^{9} + \cdots + 25\!\cdots\!80 ) / 19\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 53\!\cdots\!52 \nu^{9} + \cdots - 10\!\cdots\!10 ) / 57\!\cdots\!30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 20\!\cdots\!70 \nu^{9} + \cdots - 25\!\cdots\!70 ) / 19\!\cdots\!10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 42\!\cdots\!33 \nu^{9} + \cdots + 33\!\cdots\!20 ) / 24\!\cdots\!65 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 16\!\cdots\!17 \nu^{9} + \cdots - 42\!\cdots\!50 ) / 57\!\cdots\!30 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 21\!\cdots\!39 \nu^{9} + \cdots - 14\!\cdots\!80 ) / 76\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 31\!\cdots\!97 \nu^{9} + \cdots - 16\!\cdots\!40 ) / 10\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 68\!\cdots\!98 \nu^{9} + \cdots - 49\!\cdots\!50 ) / 19\!\cdots\!10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21\!\cdots\!87 \nu^{9} + \cdots - 30\!\cdots\!00 ) / 58\!\cdots\!70 \)
|
| \(\nu\) | \(=\) |
\( ( -7\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 7\beta_{4} - \beta_{3} - 18\beta_{2} + 10\beta _1 - 2 ) / 252 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 133 \beta_{9} - 25 \beta_{7} + 641 \beta_{6} - 612 \beta_{5} + 308 \beta_{4} + 154 \beta_{3} + \cdots + 1285468 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 98350 \beta_{9} + 9962 \beta_{8} + 19924 \beta_{7} - 44872 \beta_{6} + 204264 \beta_{5} + \cdots - 46106924 ) / 63 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7241731 \beta_{9} - 1087531 \beta_{8} - 1087531 \beta_{7} + 15842271 \beta_{6} - 31426884 \beta_{5} + \cdots + 12762494 ) / 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2674551586 \beta_{9} + 817640630 \beta_{8} + 408820315 \beta_{7} - 1888573513 \beta_{6} + \cdots + 2924353887244 ) / 63 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 17306620562 \beta_{9} - 4718798746 \beta_{8} + 8653310281 \beta_{6} - 55189236609 \beta_{5} + \cdots - 77852302903406 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 38202992333786 \beta_{9} + 17818961417759 \beta_{8} - 17818961417759 \beta_{7} + 62423963401070 \beta_{6} + \cdots + 31\!\cdots\!16 ) / 63 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 51\!\cdots\!84 \beta_{9} + \cdots - 24\!\cdots\!06 ) / 21 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 74\!\cdots\!70 \beta_{9} + \cdots + 81\!\cdots\!52 ) / 63 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
0 | −40.5000 | + | 23.3827i | 0 | −489.814 | − | 282.795i | 0 | −1456.69 | − | 1908.63i | 0 | 1093.50 | − | 1894.00i | 0 | ||||||||||||||||||||||||||||||||||||||||
| 61.2 | 0 | −40.5000 | + | 23.3827i | 0 | −374.307 | − | 216.106i | 0 | 83.7630 | + | 2399.54i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 61.3 | 0 | −40.5000 | + | 23.3827i | 0 | 111.592 | + | 64.4274i | 0 | 2392.99 | + | 195.963i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 61.4 | 0 | −40.5000 | + | 23.3827i | 0 | 656.878 | + | 379.249i | 0 | 1906.96 | − | 1458.87i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 61.5 | 0 | −40.5000 | + | 23.3827i | 0 | 790.151 | + | 456.194i | 0 | −2318.52 | + | 623.902i | 0 | 1093.50 | − | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.1 | 0 | −40.5000 | − | 23.3827i | 0 | −489.814 | + | 282.795i | 0 | −1456.69 | + | 1908.63i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.2 | 0 | −40.5000 | − | 23.3827i | 0 | −374.307 | + | 216.106i | 0 | 83.7630 | − | 2399.54i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.3 | 0 | −40.5000 | − | 23.3827i | 0 | 111.592 | − | 64.4274i | 0 | 2392.99 | − | 195.963i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.4 | 0 | −40.5000 | − | 23.3827i | 0 | 656.878 | − | 379.249i | 0 | 1906.96 | + | 1458.87i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 73.5 | 0 | −40.5000 | − | 23.3827i | 0 | 790.151 | − | 456.194i | 0 | −2318.52 | − | 623.902i | 0 | 1093.50 | + | 1894.00i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.9.m.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 252.9.z.b | 10 | ||
| 7.d | odd | 6 | 1 | inner | 84.9.m.a | ✓ | 10 |
| 21.g | even | 6 | 1 | 252.9.z.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.9.m.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 84.9.m.a | ✓ | 10 | 7.d | odd | 6 | 1 | inner |
| 252.9.z.b | 10 | 3.b | odd | 2 | 1 | ||
| 252.9.z.b | 10 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 1389 T_{5}^{9} - 876 T_{5}^{8} + 894492387 T_{5}^{7} - 19821737736 T_{5}^{6} + \cdots + 47\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T^{2} + 81 T + 2187)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 47\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 63\!\cdots\!01 \)
|
| $11$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $13$ |
\( T^{10} + \cdots + 67\!\cdots\!12 \)
|
| $17$ |
\( T^{10} + \cdots + 77\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots + 22\!\cdots\!28 \)
|
| $23$ |
\( T^{10} + \cdots + 38\!\cdots\!56 \)
|
| $29$ |
\( (T^{5} + \cdots + 37\!\cdots\!32)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 46\!\cdots\!43 \)
|
| $37$ |
\( T^{10} + \cdots + 82\!\cdots\!24 \)
|
| $41$ |
\( T^{10} + \cdots + 31\!\cdots\!92 \)
|
| $43$ |
\( (T^{5} + \cdots + 41\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 51\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 16\!\cdots\!96 \)
|
| $59$ |
\( T^{10} + \cdots + 20\!\cdots\!92 \)
|
| $61$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots + 62\!\cdots\!04 \)
|
| $71$ |
\( (T^{5} + \cdots - 22\!\cdots\!80)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 77\!\cdots\!00 \)
|
| $79$ |
\( T^{10} + \cdots + 95\!\cdots\!25 \)
|
| $83$ |
\( T^{10} + \cdots + 34\!\cdots\!72 \)
|
| $89$ |
\( T^{10} + \cdots + 26\!\cdots\!92 \)
|
| $97$ |
\( T^{10} + \cdots + 12\!\cdots\!00 \)
|
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