Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,4,Mod(1,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(10.4433380710\) |
| 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: |
\( x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 16\nu + 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} + 30\nu^{5} - 135\nu^{4} - 247\nu^{3} + 842\nu^{2} + 294\nu - 748 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - 30\nu^{5} + 137\nu^{4} + 243\nu^{3} - 886\nu^{2} - 256\nu + 876 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} + 26\nu^{6} + 145\nu^{5} - 707\nu^{4} - 1110\nu^{3} + 4513\nu^{2} + 898\nu - 4128 ) / 16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{7} - 15\nu^{6} - 88\nu^{5} + 403\nu^{4} + 689\nu^{3} - 2520\nu^{2} - 648\nu + 2248 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 18\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 26\beta_{2} + 39\beta _1 + 228 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} + 2\beta_{5} + 8\beta_{4} + 28\beta_{3} + 75\beta_{2} + 387\beta _1 + 554 \)
|
| \(\nu^{6}\) | \(=\) |
\( 20\beta_{7} + 16\beta_{6} + 74\beta_{5} + 84\beta_{4} + 79\beta_{3} + 654\beta_{2} + 1202\beta _1 + 5068 \)
|
| \(\nu^{7}\) | \(=\) |
\( 220\beta_{7} + 80\beta_{6} + 160\beta_{5} + 386\beta_{4} + 718\beta_{3} + 2358\beta_{2} + 9045\beta _1 + 17078 \)
|
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 |
|
−4.26439 | 3.00000 | 10.1850 | 13.9771 | −12.7932 | 21.4669 | −9.31764 | 9.00000 | −59.6039 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.77847 | 3.00000 | 6.27681 | −5.74283 | −11.3354 | −24.4226 | 6.51103 | 9.00000 | 21.6991 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.26905 | 3.00000 | −6.38952 | 14.8820 | −3.80714 | 22.4903 | 18.2610 | 9.00000 | −18.8859 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0357401 | 3.00000 | −7.99872 | −7.80970 | −0.107220 | 4.90513 | 0.571796 | 9.00000 | 0.279120 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.04902 | 3.00000 | −3.80150 | 16.1855 | 6.14707 | −1.13960 | −24.1816 | 9.00000 | 33.1645 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.06139 | 3.00000 | 8.49485 | 16.2722 | 12.1842 | 6.77038 | 2.00979 | 9.00000 | 66.0875 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.08481 | 3.00000 | 8.68564 | −7.45529 | 12.2544 | 34.0237 | 2.80073 | 9.00000 | −30.4534 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.15242 | 3.00000 | 18.5474 | 1.69104 | 15.4573 | −11.0943 | 54.3449 | 9.00000 | 8.71294 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 177.4.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 531.4.a.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.4.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 531.4.a.e | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 6T_{2}^{7} - 31T_{2}^{6} + 209T_{2}^{5} + 214T_{2}^{4} - 2015T_{2}^{3} + 336T_{2}^{2} + 3596T_{2} + 128 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 6 T^{7} + \cdots + 128 \)
|
| $3$ |
\( (T - 3)^{8} \)
|
| $5$ |
\( T^{8} - 42 T^{7} + \cdots - 30976256 \)
|
| $7$ |
\( T^{8} - 53 T^{7} + \cdots - 168443392 \)
|
| $11$ |
\( T^{8} + \cdots - 229585463488 \)
|
| $13$ |
\( T^{8} + \cdots + 1833054248 \)
|
| $17$ |
\( T^{8} + \cdots + 75388834904 \)
|
| $19$ |
\( T^{8} + \cdots - 40302801741824 \)
|
| $23$ |
\( T^{8} + \cdots - 7187882575616 \)
|
| $29$ |
\( T^{8} + \cdots - 19\!\cdots\!20 \)
|
| $31$ |
\( T^{8} + \cdots - 17\!\cdots\!64 \)
|
| $37$ |
\( T^{8} + \cdots - 40\!\cdots\!24 \)
|
| $41$ |
\( T^{8} + \cdots + 14\!\cdots\!60 \)
|
| $43$ |
\( T^{8} + \cdots - 99\!\cdots\!72 \)
|
| $47$ |
\( T^{8} + \cdots + 16\!\cdots\!24 \)
|
| $53$ |
\( T^{8} + \cdots - 29\!\cdots\!24 \)
|
| $59$ |
\( (T - 59)^{8} \)
|
| $61$ |
\( T^{8} + \cdots + 17\!\cdots\!44 \)
|
| $67$ |
\( T^{8} + \cdots + 50\!\cdots\!16 \)
|
| $71$ |
\( T^{8} + \cdots - 14\!\cdots\!12 \)
|
| $73$ |
\( T^{8} + \cdots - 24\!\cdots\!28 \)
|
| $79$ |
\( T^{8} + \cdots + 58\!\cdots\!76 \)
|
| $83$ |
\( T^{8} + \cdots - 31\!\cdots\!44 \)
|
| $89$ |
\( T^{8} + \cdots - 98\!\cdots\!28 \)
|
| $97$ |
\( T^{8} + \cdots + 24\!\cdots\!72 \)
|
| show more | |
| show less |