Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(131.386683876\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - 2x^{6} - 10491x^{5} + 5390x^{4} + 33206195x^{3} + 155482410x^{2} - 32794886585x - 417193412918 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 10491x^{5} + 5390x^{4} + 33206195x^{3} + 155482410x^{2} - 32794886585x - 417193412918 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 2997 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 148889 \nu^{6} + 20937381 \nu^{5} - 1134316368 \nu^{4} - 166432349378 \nu^{3} + \cdots + 33\!\cdots\!02 ) / 960358106880 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 717377 \nu^{6} - 53915253 \nu^{5} + 4967289864 \nu^{4} + 438918502154 \nu^{3} + \cdots - 11\!\cdots\!46 ) / 1920716213760 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4415 \nu^{6} - 76623 \nu^{5} + 36451068 \nu^{4} + 428397050 \nu^{3} - 73544321895 \nu^{2} + \cdots + 21675510076714 ) / 8002984224 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 488381 \nu^{6} + 3079911 \nu^{5} + 4462450152 \nu^{4} - 21258671518 \nu^{3} + \cdots + 41\!\cdots\!22 ) / 320119368960 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 2997 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{6} - 35\beta_{5} + 24\beta_{4} + 12\beta_{3} - 29\beta_{2} + 4158\beta _1 + 5522 \)
|
| \(\nu^{4}\) | \(=\) |
\( 280\beta_{6} - 293\beta_{5} - 1752\beta_{4} - 2508\beta_{3} + 5668\beta_{2} - 13143\beta _1 + 12488987 \)
|
| \(\nu^{5}\) | \(=\) |
\( 64800 \beta_{6} - 273616 \beta_{5} + 207808 \beta_{4} + 164672 \beta_{3} - 204142 \beta_{2} + \cdots - 47026840 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1963376 \beta_{6} - 2879222 \beta_{5} - 15742608 \beta_{4} - 22400040 \beta_{3} + 30867223 \beta_{2} + \cdots + 59449385689 \)
|
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 |
|
−75.9480 | 0 | 3720.10 | 559.147 | 0 | −42737.5 | −126992. | 0 | −42466.1 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −50.9989 | 0 | 552.887 | 5942.60 | 0 | −7665.48 | 76249.1 | 0 | −303066. | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −33.0563 | 0 | −955.283 | 4184.25 | 0 | −36896.0 | 99277.3 | 0 | −138316. | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −13.6839 | 0 | −1860.75 | −9795.05 | 0 | 62632.5 | 53487.1 | 0 | 134035. | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 52.1639 | 0 | 673.068 | −1532.30 | 0 | 19544.1 | −71721.8 | 0 | −79930.7 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 57.8852 | 0 | 1302.70 | 3115.26 | 0 | −65913.0 | −43142.1 | 0 | 180328. | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 72.6380 | 0 | 3228.29 | 11833.1 | 0 | 68826.3 | 85733.6 | 0 | 859532. | ||||||||||||||||||||||||||||||||||||||||
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 | 171.12.a.a | 7 | |
| 3.b | odd | 2 | 1 | 19.12.a.a | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.12.a.a | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 171.12.a.a | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 9 T_{2}^{6} - 10458 T_{2}^{5} + 57780 T_{2}^{4} + 33079800 T_{2}^{3} + 56001024 T_{2}^{2} + \cdots - 384276234240 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(171))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 384276234240 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + \cdots - 76\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 67\!\cdots\!89 \)
|
| $11$ |
\( T^{7} + \cdots + 82\!\cdots\!84 \)
|
| $13$ |
\( T^{7} + \cdots - 36\!\cdots\!20 \)
|
| $17$ |
\( T^{7} + \cdots - 63\!\cdots\!43 \)
|
| $19$ |
\( (T - 2476099)^{7} \)
|
| $23$ |
\( T^{7} + \cdots - 11\!\cdots\!20 \)
|
| $29$ |
\( T^{7} + \cdots - 34\!\cdots\!68 \)
|
| $31$ |
\( T^{7} + \cdots - 50\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 20\!\cdots\!00 \)
|
| $41$ |
\( T^{7} + \cdots - 34\!\cdots\!00 \)
|
| $43$ |
\( T^{7} + \cdots + 41\!\cdots\!24 \)
|
| $47$ |
\( T^{7} + \cdots + 17\!\cdots\!04 \)
|
| $53$ |
\( T^{7} + \cdots + 14\!\cdots\!00 \)
|
| $59$ |
\( T^{7} + \cdots - 93\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots - 49\!\cdots\!24 \)
|
| $67$ |
\( T^{7} + \cdots + 39\!\cdots\!88 \)
|
| $71$ |
\( T^{7} + \cdots - 55\!\cdots\!04 \)
|
| $73$ |
\( T^{7} + \cdots - 17\!\cdots\!09 \)
|
| $79$ |
\( T^{7} + \cdots - 52\!\cdots\!48 \)
|
| $83$ |
\( T^{7} + \cdots - 26\!\cdots\!80 \)
|
| $89$ |
\( T^{7} + \cdots + 25\!\cdots\!40 \)
|
| $97$ |
\( T^{7} + \cdots - 35\!\cdots\!72 \)
|
| show more | |
| show less |