Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6724,2,Mod(1,6724)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6724, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6724.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6724 = 2^{2} \cdot 41^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6724.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: | \(53.6914103191\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 16x^{6} + 16x^{5} + 73x^{4} - 58x^{3} - 116x^{2} + 48x + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 164) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 16x^{6} + 16x^{5} + 73x^{4} - 58x^{3} - 116x^{2} + 48x + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 16\nu^{5} + 16\nu^{4} + 73\nu^{3} - 58\nu^{2} - 100\nu + 32 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 14\nu^{5} + 44\nu^{4} + 41\nu^{3} - 148\nu^{2} - 12\nu + 96 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 14\nu^{5} - 44\nu^{4} - 41\nu^{3} + 152\nu^{2} + 12\nu - 112 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} + 11\nu^{6} + 100\nu^{5} - 168\nu^{4} - 335\nu^{3} + 570\nu^{2} + 204\nu - 368 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -13\nu^{7} + 29\nu^{6} + 184\nu^{5} - 440\nu^{4} - 581\nu^{3} + 1554\nu^{2} + 252\nu - 1136 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} - 9\nu^{6} - 72\nu^{5} + 136\nu^{4} + 241\nu^{3} - 466\nu^{2} - 136\nu + 320 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} + 12\beta_{4} + 9\beta_{3} - 3\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{7} - 30\beta_{5} - 16\beta_{4} - 3\beta_{3} + 18\beta_{2} + 58\beta _1 + 18 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} - 28\beta_{6} + 30\beta_{5} + 123\beta_{4} + 76\beta_{3} - 6\beta_{2} - 52\beta _1 + 174 \)
|
| \(\nu^{7}\) | \(=\) |
\( -149\beta_{7} + 4\beta_{6} - 336\beta_{5} - 194\beta_{4} - 58\beta_{3} + 152\beta_{2} + 513\beta _1 + 132 \)
|
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 |
|
0 | −3.16493 | 0 | 1.79297 | 0 | 2.52236 | 0 | 7.01678 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.95118 | 0 | −1.45807 | 0 | 1.70016 | 0 | 0.807096 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.935703 | 0 | −1.44568 | 0 | −2.53614 | 0 | −2.12446 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.895327 | 0 | 4.16209 | 0 | −2.07196 | 0 | −2.19839 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.16621 | 0 | 2.01709 | 0 | 4.12814 | 0 | −1.63996 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.59697 | 0 | −3.08683 | 0 | −4.50363 | 0 | −0.449675 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.31639 | 0 | 0.253661 | 0 | −4.11436 | 0 | 2.36567 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 2.86757 | 0 | 0.764771 | 0 | 4.87543 | 0 | 5.22293 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(1\) |
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 | 6724.2.a.g | 8 | |
| 41.b | even | 2 | 1 | 6724.2.a.f | 8 | ||
| 41.f | even | 10 | 2 | 164.2.g.a | ✓ | 16 | |
| 123.l | odd | 10 | 2 | 1476.2.n.f | 16 | ||
| 164.l | odd | 10 | 2 | 656.2.u.g | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.2.g.a | ✓ | 16 | 41.f | even | 10 | 2 | |
| 656.2.u.g | 16 | 164.l | odd | 10 | 2 | ||
| 1476.2.n.f | 16 | 123.l | odd | 10 | 2 | ||
| 6724.2.a.f | 8 | 41.b | even | 2 | 1 | ||
| 6724.2.a.g | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 16T_{3}^{5} + 73T_{3}^{4} - 58T_{3}^{3} - 116T_{3}^{2} + 48T_{3} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - T^{7} + \cdots + 64 \)
|
| $5$ |
\( T^{8} - 3 T^{7} + \cdots - 19 \)
|
| $7$ |
\( T^{8} - 49 T^{6} + \cdots + 8404 \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots - 7524 \)
|
| $13$ |
\( T^{8} - 10 T^{7} + \cdots + 7201 \)
|
| $17$ |
\( T^{8} + 7 T^{7} + \cdots + 23899 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots - 4 \)
|
| $23$ |
\( T^{8} - 6 T^{7} + \cdots + 1444 \)
|
| $29$ |
\( T^{8} + 17 T^{7} + \cdots - 22284 \)
|
| $31$ |
\( T^{8} + T^{7} + \cdots - 12224 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots - 110659 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} - 18 T^{7} + \cdots - 2201436 \)
|
| $47$ |
\( T^{8} + 3 T^{7} + \cdots - 404956 \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots - 1199 \)
|
| $59$ |
\( T^{8} - 44 T^{7} + \cdots + 2258864 \)
|
| $61$ |
\( T^{8} + 7 T^{7} + \cdots - 177991 \)
|
| $67$ |
\( T^{8} - 19 T^{7} + \cdots - 12044 \)
|
| $71$ |
\( T^{8} - 14 T^{7} + \cdots + 2137796 \)
|
| $73$ |
\( T^{8} - 21 T^{7} + \cdots - 44 \)
|
| $79$ |
\( T^{8} + 9 T^{7} + \cdots + 25344 \)
|
| $83$ |
\( T^{8} - 26 T^{7} + \cdots - 99584 \)
|
| $89$ |
\( T^{8} + 13 T^{7} + \cdots - 26078356 \)
|
| $97$ |
\( T^{8} + 12 T^{7} + \cdots - 53559 \)
|
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