Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,6,Mod(1,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 29.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: | \(4.65113077458\) |
| 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: |
\( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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
\( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 6\nu^{5} - 114\nu^{4} + 618\nu^{3} + 2727\nu^{2} - 10724\nu - 11462 ) / 240 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 119\nu^{3} + 547\nu^{2} + 2890\nu - 11242 ) / 96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 10\nu^{5} - 218\nu^{4} - 1094\nu^{3} + 13231\nu^{2} + 21356\nu - 185766 ) / 960 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 226\nu^{4} - 238\nu^{3} + 14279\nu^{2} + 5916\nu - 208054 ) / 960 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} + 6\beta_{5} - 8\beta_{4} - \beta_{3} + 4\beta_{2} + 71\beta _1 - 41 \)
|
| \(\nu^{4}\) | \(=\) |
\( -16\beta_{6} + 8\beta_{5} + 2\beta_{3} + 105\beta_{2} - 95\beta _1 + 3846 \)
|
| \(\nu^{5}\) | \(=\) |
\( -318\beta_{6} + 754\beta_{5} - 856\beta_{4} - 109\beta_{3} + 454\beta_{2} + 5631\beta _1 - 3945 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2496\beta_{6} + 1728\beta_{5} - 192\beta_{4} + 432\beta_{3} + 9495\beta_{2} - 7471\beta _1 + 304316 \)
|
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 |
|
−8.56883 | 18.3274 | 41.4249 | −84.3249 | −157.045 | 216.816 | −80.7606 | 92.8952 | 722.566 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.83842 | −24.5656 | 14.7640 | −54.3066 | 167.990 | −37.6697 | 117.867 | 360.469 | 371.372 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.90786 | 29.3989 | −16.7287 | 64.0801 | −114.887 | −138.793 | 190.425 | 621.298 | −250.416 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.60554 | −13.2844 | −25.2112 | 69.0035 | 34.6131 | 156.573 | 149.066 | −66.5241 | −179.792 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.83960 | 15.9679 | 2.10095 | 31.5616 | 93.2461 | 106.304 | −174.599 | 11.9736 | 184.307 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 9.92709 | 15.4219 | 66.5471 | −58.0818 | 153.094 | −210.388 | 342.952 | −5.16616 | −576.583 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.1540 | −15.2661 | 71.1029 | 64.0682 | −155.012 | 91.1564 | 397.049 | −9.94539 | 650.545 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(-1\) |
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 | 29.6.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 261.6.a.e | 7 | ||
| 4.b | odd | 2 | 1 | 464.6.a.k | 7 | ||
| 5.b | even | 2 | 1 | 725.6.a.b | 7 | ||
| 29.b | even | 2 | 1 | 841.6.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.6.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 261.6.a.e | 7 | 3.b | odd | 2 | 1 | ||
| 464.6.a.k | 7 | 4.b | odd | 2 | 1 | ||
| 725.6.a.b | 7 | 5.b | even | 2 | 1 | ||
| 841.6.a.b | 7 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 4T_{2}^{6} - 181T_{2}^{5} + 346T_{2}^{4} + 10616T_{2}^{3} + 2416T_{2}^{2} - 186896T_{2} - 351200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots - 351200 \)
|
| $3$ |
\( T^{7} - 26 T^{6} + \cdots + 661023756 \)
|
| $5$ |
\( T^{7} + \cdots + 2378174390186 \)
|
| $7$ |
\( T^{7} + \cdots + 361848785235968 \)
|
| $11$ |
\( T^{7} + \cdots + 51\!\cdots\!44 \)
|
| $13$ |
\( T^{7} + \cdots + 10\!\cdots\!34 \)
|
| $17$ |
\( T^{7} + \cdots - 17\!\cdots\!28 \)
|
| $19$ |
\( T^{7} + \cdots + 15\!\cdots\!92 \)
|
| $23$ |
\( T^{7} + \cdots - 15\!\cdots\!56 \)
|
| $29$ |
\( (T - 841)^{7} \)
|
| $31$ |
\( T^{7} + \cdots + 64\!\cdots\!48 \)
|
| $37$ |
\( T^{7} + \cdots - 16\!\cdots\!56 \)
|
| $41$ |
\( T^{7} + \cdots - 56\!\cdots\!72 \)
|
| $43$ |
\( T^{7} + \cdots - 58\!\cdots\!60 \)
|
| $47$ |
\( T^{7} + \cdots - 14\!\cdots\!36 \)
|
| $53$ |
\( T^{7} + \cdots + 84\!\cdots\!94 \)
|
| $59$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 45\!\cdots\!80 \)
|
| $67$ |
\( T^{7} + \cdots - 11\!\cdots\!60 \)
|
| $71$ |
\( T^{7} + \cdots - 20\!\cdots\!52 \)
|
| $73$ |
\( T^{7} + \cdots + 43\!\cdots\!96 \)
|
| $79$ |
\( T^{7} + \cdots - 28\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 97\!\cdots\!52 \)
|
| $89$ |
\( T^{7} + \cdots - 79\!\cdots\!20 \)
|
| $97$ |
\( T^{7} + \cdots + 62\!\cdots\!52 \)
|
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