Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [164,6,Mod(1,164)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.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: | \(26.3029464493\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 2x^{5} - 296x^{4} + 358x^{3} + 11628x^{2} + 29880x + 21384 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 296x^{4} + 358x^{3} + 11628x^{2} + 29880x + 21384 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 499\nu^{5} - 1778\nu^{4} - 143960\nu^{3} + 402742\nu^{2} + 5025264\nu + 6953364 ) / 18516 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1375\nu^{5} - 5057\nu^{4} - 397778\nu^{3} + 1169668\nu^{2} + 13791678\nu + 16797834 ) / 27774 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3553\nu^{5} + 12734\nu^{4} + 1032080\nu^{3} - 2908882\nu^{2} - 36852864\nu - 47528820 ) / 55548 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1951\nu^{5} - 6896\nu^{4} - 566828\nu^{3} + 1562218\nu^{2} + 20177196\nu + 27533268 ) / 18516 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 367\nu^{5} - 1453\nu^{4} - 105798\nu^{3} + 336892\nu^{2} + 3609942\nu + 4260746 ) / 3086 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 5 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} - 4\beta_{4} - 5\beta_{3} + 5\beta_{2} - \beta _1 + 402 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} - 32\beta_{4} - 44\beta_{3} - 19\beta_{2} + 60\beta _1 + 276 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -183\beta_{5} - 479\beta_{4} - 527\beta_{3} + 788\beta_{2} - 18\beta _1 + 44189 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -537\beta_{5} - 7646\beta_{4} - 9197\beta_{3} - 3013\beta_{2} + 16045\beta _1 + 38588 ) / 2 \)
|
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 | −27.5314 | 0 | −56.3407 | 0 | 78.5501 | 0 | 514.977 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −18.5709 | 0 | −0.888060 | 0 | 3.63187 | 0 | 101.878 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.81770 | 0 | 78.7010 | 0 | −88.2922 | 0 | −235.061 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.89868 | 0 | 5.50767 | 0 | 121.752 | 0 | −227.800 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 10.1491 | 0 | −73.4338 | 0 | 106.859 | 0 | −139.996 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 21.2368 | 0 | −21.5461 | 0 | −134.501 | 0 | 208.001 | 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 | 164.6.a.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 656.6.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.6.a.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 656.6.a.e | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 8T_{3}^{5} - 808T_{3}^{4} - 1112T_{3}^{3} + 143328T_{3}^{2} - 807264T_{3} + 1210572 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(164))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 8 T^{5} + \cdots + 1210572 \)
|
| $5$ |
\( T^{6} + 68 T^{5} + \cdots + 34314576 \)
|
| $7$ |
\( T^{6} + \cdots + 44077441756 \)
|
| $11$ |
\( T^{6} + \cdots - 325282989702708 \)
|
| $13$ |
\( T^{6} + \cdots + 18\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots + 42\!\cdots\!28 \)
|
| $19$ |
\( T^{6} + \cdots - 58\!\cdots\!88 \)
|
| $23$ |
\( T^{6} + \cdots - 11\!\cdots\!64 \)
|
| $29$ |
\( T^{6} + \cdots - 10\!\cdots\!24 \)
|
| $31$ |
\( T^{6} + \cdots - 11\!\cdots\!48 \)
|
| $37$ |
\( T^{6} + \cdots - 14\!\cdots\!16 \)
|
| $41$ |
\( (T - 1681)^{6} \)
|
| $43$ |
\( T^{6} + \cdots + 18\!\cdots\!88 \)
|
| $47$ |
\( T^{6} + \cdots - 10\!\cdots\!28 \)
|
| $53$ |
\( T^{6} + \cdots - 38\!\cdots\!84 \)
|
| $59$ |
\( T^{6} + \cdots - 56\!\cdots\!32 \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!72 \)
|
| $67$ |
\( T^{6} + \cdots - 44\!\cdots\!72 \)
|
| $71$ |
\( T^{6} + \cdots + 52\!\cdots\!12 \)
|
| $73$ |
\( T^{6} + \cdots - 12\!\cdots\!48 \)
|
| $79$ |
\( T^{6} + \cdots + 11\!\cdots\!52 \)
|
| $83$ |
\( T^{6} + \cdots - 27\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots - 20\!\cdots\!44 \)
|
| $97$ |
\( T^{6} + \cdots + 19\!\cdots\!48 \)
|
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