Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2006,2,Mod(1,2006)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2006, 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("2006.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2006 = 2 \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2006.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: | \(16.0179906455\) |
| 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: |
\( x^{9} - x^{8} - 23x^{7} + 18x^{6} + 185x^{5} - 91x^{4} - 615x^{3} + 126x^{2} + 668x - 44 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 23x^{7} + 18x^{6} + 185x^{5} - 91x^{4} - 615x^{3} + 126x^{2} + 668x - 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{8} - 4\nu^{7} - 49\nu^{6} + 61\nu^{5} + 208\nu^{4} - 221\nu^{3} - 166\nu^{2} + 200\nu - 60 ) / 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{8} - 2\nu^{7} + 161\nu^{6} + 41\nu^{5} - 834\nu^{4} - 275\nu^{3} + 1114\nu^{2} + 212\nu + 68 ) / 28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{8} + 10\nu^{7} - 301\nu^{6} - 177\nu^{5} + 1566\nu^{4} + 1011\nu^{3} - 2266\nu^{2} - 1256\nu + 332 ) / 56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{8} + 10\nu^{7} - 301\nu^{6} - 177\nu^{5} + 1566\nu^{4} + 1011\nu^{3} - 2210\nu^{2} - 1256\nu + 52 ) / 56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{8} + 30\nu^{7} - 161\nu^{6} - 545\nu^{5} + 834\nu^{4} + 2935\nu^{3} - 1002\nu^{2} - 4048\nu + 268 ) / 56 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{8} - 6\nu^{7} + 17\nu^{6} + 105\nu^{5} - 82\nu^{4} - 543\nu^{3} + 66\nu^{2} + 752\nu - 12 ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3\nu^{8} + 6\nu^{7} - 55\nu^{6} - 107\nu^{5} + 298\nu^{4} + 569\nu^{3} - 414\nu^{2} - 760\nu + 36 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + 2\beta_{6} + 11\beta_{5} - 9\beta_{4} + \beta_{3} - \beta_{2} + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{8} - 2\beta_{7} - 15\beta_{6} - 4\beta_{5} + 3\beta_{4} + 10\beta_{3} + 11\beta_{2} + 67\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -19\beta_{8} + 11\beta_{7} + 30\beta_{6} + 109\beta_{5} - 81\beta_{4} + 10\beta_{3} - 15\beta_{2} - 11\beta _1 + 281 \)
|
| \(\nu^{7}\) | \(=\) |
\( 121\beta_{8} - 36\beta_{7} - 173\beta_{6} - 76\beta_{5} + 58\beta_{4} + 86\beta_{3} + 103\beta_{2} + 583\beta _1 - 32 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 250 \beta_{8} + 103 \beta_{7} + 352 \beta_{6} + 1053 \beta_{5} - 738 \beta_{4} + 79 \beta_{3} + \cdots + 2417 \)
|
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 |
|
−1.00000 | −2.90546 | 1.00000 | 2.56462 | 2.90546 | −4.07426 | −1.00000 | 5.44169 | −2.56462 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.81706 | 1.00000 | 4.05009 | 2.81706 | 2.34757 | −1.00000 | 4.93585 | −4.05009 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.73114 | 1.00000 | −1.91728 | 2.73114 | −0.139698 | −1.00000 | 4.45914 | 1.91728 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.39195 | 1.00000 | −1.67969 | 1.39195 | 2.61106 | −1.00000 | −1.06246 | 1.67969 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.0653222 | 1.00000 | 0.997627 | 0.0653222 | 0.699368 | −1.00000 | −2.99573 | −0.997627 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.59799 | 1.00000 | −2.96953 | −1.59799 | 3.73606 | −1.00000 | −0.446436 | 2.96953 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.94466 | 1.00000 | 4.05667 | −1.94466 | −2.97242 | −1.00000 | 0.781710 | −4.05667 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.19580 | 1.00000 | 2.87903 | −2.19580 | 2.16209 | −1.00000 | 1.82156 | −2.87903 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 3.17249 | 1.00000 | −0.981527 | −3.17249 | −4.36977 | −1.00000 | 7.06469 | 0.981527 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(17\) | \(-1\) |
| \(59\) | \(1\) |
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 | 2006.2.a.u | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2006.2.a.u | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\):
|
\( T_{3}^{9} + T_{3}^{8} - 23T_{3}^{7} - 18T_{3}^{6} + 185T_{3}^{5} + 91T_{3}^{4} - 615T_{3}^{3} - 126T_{3}^{2} + 668T_{3} + 44 \)
|
|
\( T_{5}^{9} - 7T_{5}^{8} - 8T_{5}^{7} + 125T_{5}^{6} - 39T_{5}^{5} - 744T_{5}^{4} + 312T_{5}^{3} + 1772T_{5}^{2} - 272T_{5} - 1136 \)
|
|
\( T_{11}^{9} - 5 T_{11}^{8} - 77 T_{11}^{7} + 372 T_{11}^{6} + 1832 T_{11}^{5} - 8592 T_{11}^{4} + \cdots - 6656 \)
|
|
\( T_{31}^{9} - 6 T_{31}^{8} - 126 T_{31}^{7} + 592 T_{31}^{6} + 4417 T_{31}^{5} - 10254 T_{31}^{4} + \cdots - 10016 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{9} \)
|
| $3$ |
\( T^{9} + T^{8} + \cdots + 44 \)
|
| $5$ |
\( T^{9} - 7 T^{8} + \cdots - 1136 \)
|
| $7$ |
\( T^{9} - 38 T^{7} + \cdots - 256 \)
|
| $11$ |
\( T^{9} - 5 T^{8} + \cdots - 6656 \)
|
| $13$ |
\( T^{9} - 25 T^{8} + \cdots - 209216 \)
|
| $17$ |
\( (T - 1)^{9} \)
|
| $19$ |
\( T^{9} - 14 T^{8} + \cdots + 64000 \)
|
| $23$ |
\( T^{9} - 2 T^{8} + \cdots + 6592 \)
|
| $29$ |
\( T^{9} - 18 T^{8} + \cdots - 740096 \)
|
| $31$ |
\( T^{9} - 6 T^{8} + \cdots - 10016 \)
|
| $37$ |
\( T^{9} - 11 T^{8} + \cdots - 17408 \)
|
| $41$ |
\( T^{9} - 18 T^{8} + \cdots - 247168 \)
|
| $43$ |
\( T^{9} + 10 T^{8} + \cdots - 149504 \)
|
| $47$ |
\( T^{9} + 20 T^{8} + \cdots + 4726784 \)
|
| $53$ |
\( T^{9} + 7 T^{8} + \cdots - 427904 \)
|
| $59$ |
\( (T + 1)^{9} \)
|
| $61$ |
\( T^{9} - 30 T^{8} + \cdots + 555008 \)
|
| $67$ |
\( T^{9} - 6 T^{8} + \cdots - 763712 \)
|
| $71$ |
\( T^{9} - 30 T^{8} + \cdots + 8198144 \)
|
| $73$ |
\( T^{9} - 257 T^{7} + \cdots - 2500352 \)
|
| $79$ |
\( T^{9} - 29 T^{8} + \cdots + 1114304 \)
|
| $83$ |
\( T^{9} + \cdots + 1069391104 \)
|
| $89$ |
\( T^{9} - 8 T^{8} + \cdots - 79266304 \)
|
| $97$ |
\( T^{9} + 13 T^{8} + \cdots + 502635248 \)
|
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