Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,2,Mod(1,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 471.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: | \(3.76095393520\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 39x^{5} - 53x^{4} - 49x^{3} + 45x^{2} + 14x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 2x^{8} - 11x^{7} + 19x^{6} + 39x^{5} - 53x^{4} - 49x^{3} + 45x^{2} + 14x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{8} + 16\nu^{7} + 75\nu^{6} - 164\nu^{5} - 341\nu^{4} + 506\nu^{3} + 586\nu^{2} - 464\nu - 168 ) / 59 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} + 17\nu^{7} - 42\nu^{6} - 189\nu^{5} + 283\nu^{4} + 663\nu^{3} - 491\nu^{2} - 670\nu + 87 ) / 59 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{8} + 41\nu^{7} + 41\nu^{6} - 376\nu^{5} + 37\nu^{4} + 950\nu^{3} - 143\nu^{2} - 599\nu - 106 ) / 59 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -10\nu^{8} + 7\nu^{7} + 125\nu^{6} - 57\nu^{5} - 470\nu^{4} + 96\nu^{3} + 485\nu^{2} + 33\nu + 15 ) / 59 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -19\nu^{8} + 31\nu^{7} + 208\nu^{6} - 244\nu^{5} - 716\nu^{4} + 442\nu^{3} + 774\nu^{2} - 132\nu - 119 ) / 59 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 38\nu^{8} - 62\nu^{7} - 416\nu^{6} + 547\nu^{5} + 1432\nu^{4} - 1356\nu^{3} - 1607\nu^{2} + 972\nu + 238 ) / 59 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} + 7\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} + 10\beta_{7} - 8\beta_{6} - 8\beta_{5} + 8\beta_{4} + 9\beta_{2} + 20\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{8} + 22\beta_{7} + 9\beta_{6} - 11\beta_{5} + 20\beta_{4} + 3\beta_{3} + 47\beta_{2} + 2\beta _1 + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( 16\beta_{8} + 82\beta_{7} - 48\beta_{6} - 55\beta_{5} + 60\beta_{4} + 5\beta_{3} + 71\beta_{2} + 109\beta _1 + 153 \)
|
| \(\nu^{8}\) | \(=\) |
\( 96\beta_{8} + 191\beta_{7} + 62\beta_{6} - 93\beta_{5} + 162\beta_{4} + 41\beta_{3} + 315\beta_{2} + 29\beta _1 + 638 \)
|
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 |
|
−2.29794 | −1.00000 | 3.28055 | 1.95196 | 2.29794 | −3.21152 | −2.94263 | 1.00000 | −4.48549 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.68197 | −1.00000 | 0.829028 | 4.19743 | 1.68197 | 4.46776 | 1.96954 | 1.00000 | −7.05996 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.34370 | −1.00000 | −0.194460 | −3.48167 | 1.34370 | −4.92854 | 2.94870 | 1.00000 | 4.67834 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.316645 | −1.00000 | −1.89974 | 1.69099 | 0.316645 | 0.220418 | 1.23483 | 1.00000 | −0.535444 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0604903 | −1.00000 | −1.99634 | −1.41604 | −0.0604903 | −3.07838 | −0.241740 | 1.00000 | −0.0856568 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.06833 | −1.00000 | −0.858677 | −0.921169 | −1.06833 | 4.41562 | −3.05400 | 1.00000 | −0.984110 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.51659 | −1.00000 | 0.300033 | 4.06961 | −1.51659 | −0.404124 | −2.57815 | 1.00000 | 6.17191 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.31676 | −1.00000 | 3.36737 | −0.520771 | −2.31676 | 2.00784 | 3.16788 | 1.00000 | −1.20650 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.67810 | −1.00000 | 5.17223 | 2.42967 | −2.67810 | −1.48907 | 8.49556 | 1.00000 | 6.50691 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(157\) | \(-1\) |
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 | 471.2.a.d | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1413.2.a.f | 9 | ||
| 4.b | odd | 2 | 1 | 7536.2.a.bj | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 471.2.a.d | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1413.2.a.f | 9 | 3.b | odd | 2 | 1 | ||
| 7536.2.a.bj | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 2T_{2}^{8} - 11T_{2}^{7} + 19T_{2}^{6} + 39T_{2}^{5} - 53T_{2}^{4} - 49T_{2}^{3} + 45T_{2}^{2} + 14T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 2 T^{8} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} - 8 T^{8} + \cdots - 324 \)
|
| $7$ |
\( T^{9} + 2 T^{8} + \cdots + 256 \)
|
| $11$ |
\( T^{9} - 7 T^{8} + \cdots + 5632 \)
|
| $13$ |
\( T^{9} + T^{8} + \cdots + 17536 \)
|
| $17$ |
\( T^{9} - 20 T^{8} + \cdots + 896 \)
|
| $19$ |
\( T^{9} - 2 T^{8} + \cdots - 64 \)
|
| $23$ |
\( T^{9} - 8 T^{8} + \cdots + 44716 \)
|
| $29$ |
\( T^{9} - 19 T^{8} + \cdots + 183636 \)
|
| $31$ |
\( T^{9} - 5 T^{8} + \cdots - 1193456 \)
|
| $37$ |
\( T^{9} + 8 T^{8} + \cdots - 4672 \)
|
| $41$ |
\( T^{9} - 42 T^{8} + \cdots + 3700116 \)
|
| $43$ |
\( T^{9} + 34 T^{8} + \cdots + 22791168 \)
|
| $47$ |
\( T^{9} - 3 T^{8} + \cdots + 34068992 \)
|
| $53$ |
\( T^{9} - 16 T^{8} + \cdots - 1940144 \)
|
| $59$ |
\( T^{9} + 6 T^{8} + \cdots + 3271216 \)
|
| $61$ |
\( T^{9} - 14 T^{8} + \cdots + 253103744 \)
|
| $67$ |
\( T^{9} + 37 T^{8} + \cdots - 931072 \)
|
| $71$ |
\( T^{9} + 21 T^{8} + \cdots - 4146176 \)
|
| $73$ |
\( T^{9} - 23 T^{8} + \cdots - 628096 \)
|
| $79$ |
\( T^{9} + 36 T^{8} + \cdots + 1985792 \)
|
| $83$ |
\( T^{9} + 3 T^{8} + \cdots - 19772404 \)
|
| $89$ |
\( T^{9} - 27 T^{8} + \cdots - 53488512 \)
|
| $97$ |
\( T^{9} + 2 T^{8} + \cdots + 6125184 \)
|
| show more | |
| show less |