Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1728,3,Mod(703,1728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.703");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1728.g (of order \(2\), degree \(1\), not 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: | \(47.0845896815\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.56070144.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 864) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu^{2} - 6\nu + 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24\nu^{7} - 84\nu^{6} + 268\nu^{5} - 460\nu^{4} + 468\nu^{3} - 284\nu^{2} - 164\nu + 116 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{6} - 6\nu^{5} + 24\nu^{4} - 38\nu^{3} + 62\nu^{2} - 44\nu + 25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -48\nu^{7} + 168\nu^{6} - 684\nu^{5} + 1290\nu^{4} - 2268\nu^{3} + 2196\nu^{2} - 1596\nu + 471 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( -8\nu^{6} + 24\nu^{5} - 84\nu^{4} + 128\nu^{3} - 170\nu^{2} + 110\nu - 43 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -100\nu^{7} + 350\nu^{6} - 1314\nu^{5} + 2410\nu^{4} - 3874\nu^{3} + 3576\nu^{2} - 2622\nu + 787 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -112\nu^{7} + 392\nu^{6} - 1448\nu^{5} + 2640\nu^{4} - 4108\nu^{3} + 3718\nu^{2} - 2318\nu + 618 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 2\beta _1 - 24 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{7} + 4\beta_{6} + 4\beta_{4} - 8\beta_{2} + 3\beta _1 - 39 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -16\beta_{7} + 10\beta_{6} + \beta_{5} + 8\beta_{4} + 4\beta_{3} - 17\beta_{2} - 7\beta _1 + 54 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 38\beta_{7} - 14\beta_{6} + 5\beta_{5} - 38\beta_{4} + 20\beta_{3} + 43\beta_{2} - 45\beta _1 + 402 ) / 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 196\beta_{7} - 94\beta_{6} - 9\beta_{5} - 154\beta_{4} - 24\beta_{3} + 215\beta_{2} + 23\beta _1 - 120 ) / 24 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4\beta_{7} - 20\beta_{6} - 49\beta_{5} + 36\beta_{4} - 154\beta_{3} + 7\beta_{2} + 245\beta _1 - 1920 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(703\) | \(1217\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 703.1 |
|
0 | 0 | 0 | −6.59655 | 0 | − | 4.56106i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 703.2 | 0 | 0 | 0 | −6.59655 | 0 | 4.56106i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 703.3 | 0 | 0 | 0 | −0.956810 | 0 | − | 6.34610i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 703.4 | 0 | 0 | 0 | −0.956810 | 0 | 6.34610i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 703.5 | 0 | 0 | 0 | 5.13244 | 0 | − | 4.02516i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 703.6 | 0 | 0 | 0 | 5.13244 | 0 | 4.02516i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 703.7 | 0 | 0 | 0 | 6.42091 | 0 | − | 13.8102i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 703.8 | 0 | 0 | 0 | 6.42091 | 0 | 13.8102i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1728.3.g.m | 8 | |
| 3.b | odd | 2 | 1 | 1728.3.g.j | 8 | ||
| 4.b | odd | 2 | 1 | inner | 1728.3.g.m | 8 | |
| 8.b | even | 2 | 1 | 864.3.g.b | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 864.3.g.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 1728.3.g.j | 8 | ||
| 24.f | even | 2 | 1 | 864.3.g.d | yes | 8 | |
| 24.h | odd | 2 | 1 | 864.3.g.d | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 864.3.g.b | ✓ | 8 | 8.b | even | 2 | 1 | |
| 864.3.g.b | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 864.3.g.d | yes | 8 | 24.f | even | 2 | 1 | |
| 864.3.g.d | yes | 8 | 24.h | odd | 2 | 1 | |
| 1728.3.g.j | 8 | 3.b | odd | 2 | 1 | ||
| 1728.3.g.j | 8 | 12.b | even | 2 | 1 | ||
| 1728.3.g.m | 8 | 1.a | even | 1 | 1 | trivial | |
| 1728.3.g.m | 8 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):
|
\( T_{5}^{4} - 4T_{5}^{3} - 48T_{5}^{2} + 176T_{5} + 208 \)
|
|
\( T_{7}^{8} + 268T_{7}^{6} + 16566T_{7}^{4} + 362092T_{7}^{2} + 2588881 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 4 T^{3} + \cdots + 208)^{2} \)
|
| $7$ |
\( T^{8} + 268 T^{6} + \cdots + 2588881 \)
|
| $11$ |
\( T^{8} + 400 T^{6} + \cdots + 891136 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots - 3167)^{2} \)
|
| $17$ |
\( (T^{4} + 12 T^{3} + \cdots + 5328)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 13809305169 \)
|
| $23$ |
\( T^{8} + 1360 T^{6} + \cdots + 522762496 \)
|
| $29$ |
\( (T^{4} + 64 T^{3} + \cdots - 966656)^{2} \)
|
| $31$ |
\( T^{8} + 1984 T^{6} + \cdots + 7929856 \)
|
| $37$ |
\( (T^{4} + 12 T^{3} + \cdots - 661167)^{2} \)
|
| $41$ |
\( (T^{4} + 80 T^{3} + \cdots - 929792)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 8446345216 \)
|
| $47$ |
\( T^{8} + \cdots + 80138017536256 \)
|
| $53$ |
\( (T^{4} + 24 T^{3} + \cdots + 3624192)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 1539862591744 \)
|
| $61$ |
\( (T^{4} - 68 T^{3} + \cdots - 11656223)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 701937325489 \)
|
| $71$ |
\( T^{8} + \cdots + 121012672331776 \)
|
| $73$ |
\( (T^{4} - 36 T^{3} + \cdots + 1325601)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 40981548721 \)
|
| $83$ |
\( T^{8} + \cdots + 64\!\cdots\!44 \)
|
| $89$ |
\( (T^{4} - 84 T^{3} + \cdots - 146736)^{2} \)
|
| $97$ |
\( (T^{4} - 52 T^{3} + \cdots + 32388241)^{2} \)
|
| show more | |
| show less |