Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [432,7,Mod(161,432)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(432, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("432.161");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 432 = 2^{4} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 432.e (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: | \(99.3833641238\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 75x^{4} - 90x^{3} + 1861x^{2} + 7864x + 10098 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 216) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 75x^{4} - 90x^{3} + 1861x^{2} + 7864x + 10098 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2776\nu^{5} - 1430\nu^{4} + 317212\nu^{3} + 128208\nu^{2} - 8139426\nu - 20624084 ) / 34873 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -37888\nu^{5} + 173440\nu^{4} + 2168320\nu^{3} - 1877760\nu^{2} - 59184768\nu - 128123648 ) / 174365 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -86352\nu^{5} + 480120\nu^{4} + 4828800\nu^{3} - 10896120\nu^{2} - 112072032\nu - 197251232 ) / 174365 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 24408\nu^{5} - 176766\nu^{4} - 1170660\nu^{3} + 2934480\nu^{2} + 38209686\nu + 73032924 ) / 34873 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 867936\nu^{5} - 5160720\nu^{4} - 48088320\nu^{3} + 135645840\nu^{2} + 1237214016\nu + 1541448256 ) / 174365 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 14\beta_{3} - 9\beta_{2} + 384 ) / 1152 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{5} - 16\beta_{4} + 50\beta_{3} - 11\beta_{2} + 16\beta _1 + 29568 ) / 1152 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 28\beta_{5} - 64\beta_{4} + 360\beta_{3} - 479\beta_{2} + 256\beta _1 + 93184 ) / 768 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 333\beta_{5} - 1376\beta_{4} + 3222\beta_{3} - 4792\beta_{2} + 1760\beta _1 + 1102464 ) / 1152 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4038\beta_{5} - 22000\beta_{4} + 42612\beta_{3} - 107507\beta_{2} + 58480\beta _1 + 14170368 ) / 2304 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).
| \(n\) | \(271\) | \(325\) | \(353\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0 | 0 | 0 | − | 146.431i | 0 | −472.768 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 161.2 | 0 | 0 | 0 | − | 132.847i | 0 | 32.8868 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 161.3 | 0 | 0 | 0 | − | 62.7833i | 0 | 496.882 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 161.4 | 0 | 0 | 0 | 62.7833i | 0 | 496.882 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 161.5 | 0 | 0 | 0 | 132.847i | 0 | 32.8868 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 161.6 | 0 | 0 | 0 | 146.431i | 0 | −472.768 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
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 | 432.7.e.k | 6 | |
| 3.b | odd | 2 | 1 | inner | 432.7.e.k | 6 | |
| 4.b | odd | 2 | 1 | 216.7.e.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 216.7.e.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 216.7.e.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 216.7.e.b | ✓ | 6 | 12.b | even | 2 | 1 | |
| 432.7.e.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 432.7.e.k | 6 | 3.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 \(S_{7}^{\mathrm{new}}(432, [\chi])\):
|
\( T_{5}^{6} + 43032T_{5}^{4} + 532497600T_{5}^{2} + 1491609920000 \)
|
|
\( T_{7}^{3} - 57T_{7}^{2} - 234117T_{7} + 7725429 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 1491609920000 \)
|
| $7$ |
\( (T^{3} - 57 T^{2} + \cdots + 7725429)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 15\!\cdots\!28 \)
|
| $13$ |
\( (T^{3} - 2013 T^{2} + \cdots + 10536080225)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 79\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + 927 T^{2} + \cdots + 98503985549)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 93\!\cdots\!12 \)
|
| $29$ |
\( T^{6} + \cdots + 61\!\cdots\!92 \)
|
| $31$ |
\( (T^{3} + \cdots + 2612099672600)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 210270634515639)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 78\!\cdots\!48 \)
|
| $43$ |
\( (T^{3} + \cdots + 137146758845560)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 53\!\cdots\!68 \)
|
| $59$ |
\( T^{6} + \cdots + 95\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 45\!\cdots\!73)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 22\!\cdots\!17)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 14\!\cdots\!32 \)
|
| $73$ |
\( (T^{3} + \cdots - 687796520330719)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 14\!\cdots\!53)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 12\!\cdots\!31)^{2} \)
|
| show more | |
| show less |