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: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 9\nu^{5} + 24\nu^{4} + 287\nu^{3} + 51\nu^{2} - 1588\nu - 716 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 32\nu^{4} + 47\nu^{3} - 197\nu^{2} - 332\nu + 4 ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{6} + 33\nu^{5} + 232\nu^{4} - 999\nu^{3} - 1915\nu^{2} + 6420\nu + 2156 ) / 256 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -11\nu^{6} - 3\nu^{5} + 392\nu^{4} + 149\nu^{3} - 3119\nu^{2} - 508\nu + 2748 ) / 256 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + 3\beta_{4} - 3\beta_{3} + 17\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{6} - 4\beta_{5} - 8\beta_{3} + 22\beta_{2} + 9\beta _1 + 210 \)
|
| \(\nu^{5}\) | \(=\) |
\( -34\beta_{6} - 26\beta_{5} + 94\beta_{4} - 98\beta_{3} + 9\beta_{2} + 344\beta _1 + 132 \)
|
| \(\nu^{6}\) | \(=\) |
\( 115\beta_{6} - 149\beta_{5} + 15\beta_{4} - 299\beta_{3} + 498\beta_{2} + 411\beta _1 + 4322 \)
|
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 |
|
−5.07078 | −3.00000 | 17.7128 | 5.77165 | 15.2123 | −31.1296 | −49.2517 | 9.00000 | −29.2668 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.06012 | −3.00000 | 1.36432 | 0.675250 | 9.18035 | −3.38755 | 20.3060 | 9.00000 | −2.06635 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.61892 | −3.00000 | −1.14126 | −7.96684 | 7.85676 | 16.8751 | 23.9402 | 9.00000 | 20.8645 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.0253269 | −3.00000 | −7.99936 | 8.85515 | −0.0759806 | 13.8749 | −0.405214 | 9.00000 | 0.224273 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.74916 | −3.00000 | −0.442134 | 13.7745 | −8.24747 | −35.6462 | −23.2088 | 9.00000 | 37.8683 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.39497 | −3.00000 | 3.52580 | −6.09684 | −10.1849 | 1.40309 | −15.1898 | 9.00000 | −20.6986 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.58037 | −3.00000 | 12.9798 | −17.0129 | −13.7411 | −20.9898 | 22.8093 | 9.00000 | −77.9254 | ||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 531.4.a.c | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.4.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 531.4.a.c | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 41T_{2}^{5} + 7T_{2}^{4} + 484T_{2}^{3} - 63T_{2}^{2} - 1736T_{2} + 44 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 41 T^{5} + \cdots + 44 \)
|
| $3$ |
\( (T + 3)^{7} \)
|
| $5$ |
\( T^{7} + 2 T^{6} + \cdots + 392832 \)
|
| $7$ |
\( T^{7} + 59 T^{6} + \cdots - 25920400 \)
|
| $11$ |
\( T^{7} + \cdots - 1989231628 \)
|
| $13$ |
\( T^{7} + \cdots - 31480216830 \)
|
| $17$ |
\( T^{7} + \cdots - 2416315770100 \)
|
| $19$ |
\( T^{7} + \cdots - 4741294131040 \)
|
| $23$ |
\( T^{7} + \cdots + 3139469662768 \)
|
| $29$ |
\( T^{7} + \cdots - 6671473883660 \)
|
| $31$ |
\( T^{7} + \cdots + 161706929046080 \)
|
| $37$ |
\( T^{7} + \cdots - 60\!\cdots\!74 \)
|
| $41$ |
\( T^{7} + \cdots - 50\!\cdots\!80 \)
|
| $43$ |
\( T^{7} + \cdots - 33\!\cdots\!28 \)
|
| $47$ |
\( T^{7} + \cdots - 11\!\cdots\!84 \)
|
| $53$ |
\( T^{7} + \cdots + 26\!\cdots\!04 \)
|
| $59$ |
\( (T - 59)^{7} \)
|
| $61$ |
\( T^{7} + \cdots - 56\!\cdots\!52 \)
|
| $67$ |
\( T^{7} + \cdots + 45\!\cdots\!48 \)
|
| $71$ |
\( T^{7} + \cdots - 30\!\cdots\!42 \)
|
| $73$ |
\( T^{7} + \cdots + 71\!\cdots\!76 \)
|
| $79$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 45\!\cdots\!04 \)
|
| $89$ |
\( T^{7} + \cdots - 42\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 61\!\cdots\!48 \)
|
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