Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(88.0711279840\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - 3x^{5} - 2124x^{4} - 384x^{3} + 1071312x^{2} + 1260144x - 135644992 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 19) |
| 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
\( x^{6} - 3x^{5} - 2124x^{4} - 384x^{3} + 1071312x^{2} + 1260144x - 135644992 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -61\nu^{5} + 2813\nu^{4} + 99542\nu^{3} - 5193940\nu^{2} - 26359096\nu + 1606478048 ) / 3181056 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{5} - 107\nu^{4} - 15034\nu^{3} + 104524\nu^{2} + 594440\nu - 28812320 ) / 167424 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -61\nu^{5} + 2813\nu^{4} + 99542\nu^{3} - 4133588\nu^{2} - 31660856\nu + 857869536 ) / 1060352 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} + 553\nu^{4} + 116662\nu^{3} - 538052\nu^{2} - 34726712\nu + 98477920 ) / 397632 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5\beta _1 + 706 \)
|
| \(\nu^{3}\) | \(=\) |
\( 14\beta_{5} + 3\beta_{4} + 36\beta_{3} - 5\beta_{2} + 1143\beta _1 + 2826 \)
|
| \(\nu^{4}\) | \(=\) |
\( -102\beta_{5} + 2057\beta_{4} + 156\beta_{3} - 4767\beta_{2} + 12457\beta _1 + 795310 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18142\beta_{5} + 14607\beta_{4} + 65940\beta_{3} - 24697\beta_{2} + 1581791\beta _1 + 7509330 \)
|
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 |
|
−34.3588 | 0 | 668.525 | 2543.32 | 0 | 994.457 | −5378.01 | 0 | −87385.2 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −16.4279 | 0 | −242.124 | −702.910 | 0 | −3562.17 | 12388.7 | 0 | 11547.3 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −6.67485 | 0 | −467.446 | 2305.86 | 0 | −8934.31 | 6537.66 | 0 | −15391.2 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 21.8741 | 0 | −33.5233 | 698.718 | 0 | 5865.75 | −11932.8 | 0 | 15283.8 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 26.8288 | 0 | 207.783 | −2247.10 | 0 | −1771.60 | −8161.78 | 0 | −60287.0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 41.7587 | 0 | 1231.79 | 1014.12 | 0 | 11492.9 | 30057.3 | 0 | 42348.3 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 171.10.a.c | 6 | |
| 3.b | odd | 2 | 1 | 19.10.a.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 304.10.a.f | 6 | ||
| 57.d | even | 2 | 1 | 361.10.a.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.10.a.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 171.10.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 304.10.a.f | 6 | 12.b | even | 2 | 1 | ||
| 361.10.a.b | 6 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 33T_{2}^{5} - 1674T_{2}^{4} + 48120T_{2}^{3} + 618576T_{2}^{2} - 12266496T_{2} - 92329216 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(171))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 33 T^{5} + \cdots - 92329216 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 65\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 37\!\cdots\!12 \)
|
| $11$ |
\( T^{6} + \cdots + 36\!\cdots\!88 \)
|
| $13$ |
\( T^{6} + \cdots + 72\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots + 25\!\cdots\!22 \)
|
| $19$ |
\( (T + 130321)^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!72 \)
|
| $29$ |
\( T^{6} + \cdots - 11\!\cdots\!60 \)
|
| $31$ |
\( T^{6} + \cdots - 84\!\cdots\!48 \)
|
| $37$ |
\( T^{6} + \cdots + 51\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots + 11\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots - 43\!\cdots\!84 \)
|
| $47$ |
\( T^{6} + \cdots + 47\!\cdots\!12 \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots - 26\!\cdots\!40 \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!52 \)
|
| $67$ |
\( T^{6} + \cdots - 76\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots - 50\!\cdots\!72 \)
|
| $73$ |
\( T^{6} + \cdots - 69\!\cdots\!46 \)
|
| $79$ |
\( T^{6} + \cdots - 38\!\cdots\!80 \)
|
| $83$ |
\( T^{6} + \cdots - 13\!\cdots\!48 \)
|
| $89$ |
\( T^{6} + \cdots + 13\!\cdots\!20 \)
|
| $97$ |
\( T^{6} + \cdots + 99\!\cdots\!08 \)
|
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