Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,6,Mod(1,171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, 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("171.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 171.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: | \(27.4256331880\) |
| 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: |
\( x^{6} - 117x^{4} + 2916x^{2} - 1216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 117x^{4} + 2916x^{2} - 1216 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 39 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 101\nu^{3} + 1828\nu ) / 48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 89\nu^{2} + 856 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 117\nu^{3} - 2884\nu ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 39 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 3\beta_{3} + 66\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{4} + 89\beta_{2} + 2615 \)
|
| \(\nu^{5}\) | \(=\) |
\( 101\beta_{5} + 351\beta_{3} + 4838\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9.01831 | 0 | 49.3298 | 69.9349 | 0 | −24.9809 | −156.286 | 0 | −630.694 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −5.93683 | 0 | 3.24596 | −42.7408 | 0 | −84.7970 | 170.708 | 0 | 253.745 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.651309 | 0 | −31.5758 | 26.1790 | 0 | 104.778 | 41.4075 | 0 | −17.0506 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.651309 | 0 | −31.5758 | −26.1790 | 0 | 104.778 | −41.4075 | 0 | −17.0506 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.93683 | 0 | 3.24596 | 42.7408 | 0 | −84.7970 | −170.708 | 0 | 253.745 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 9.01831 | 0 | 49.3298 | −69.9349 | 0 | −24.9809 | 156.286 | 0 | −630.694 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 171.6.a.k | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 171.6.a.k | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 171.6.a.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 171.6.a.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(171))\):
|
\( T_{2}^{6} - 117T_{2}^{4} + 2916T_{2}^{2} - 1216 \)
|
|
\( T_{5}^{6} - 7403T_{5}^{4} + 13538428T_{5}^{2} - 6123211776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 117 T^{4} + \cdots - 1216 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 6123211776 \)
|
| $7$ |
\( (T^{3} + 5 T^{2} + \cdots - 221952)^{2} \)
|
| $11$ |
\( T^{6} + \cdots - 9842050087344 \)
|
| $13$ |
\( (T^{3} + 628 T^{2} + \cdots + 9924240)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 46\!\cdots\!56 \)
|
| $19$ |
\( (T + 361)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 38\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots - 64\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + 9068 T^{2} + \cdots - 144356694192)^{2} \)
|
| $37$ |
\( (T^{3} + 11882 T^{2} + \cdots - 1352836224)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + 12303 T^{2} + \cdots - 99808040816)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 74\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 17\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots - 21\!\cdots\!64 \)
|
| $61$ |
\( (T^{3} + \cdots + 8793063342100)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 7401693606656)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 22\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 36640554081100)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 114153498391680)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 17\!\cdots\!44 \)
|
| $89$ |
\( T^{6} + \cdots - 54\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 297555192322328)^{2} \)
|
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