Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2639,2,Mod(1,2639)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2639, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2639.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2639 = 7 \cdot 13 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2639.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: | \(21.0725210934\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 13x^{9} - x^{8} + 59x^{7} + 10x^{6} - 112x^{5} - 29x^{4} + 81x^{3} + 26x^{2} - 19x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 13x^{9} - x^{8} + 59x^{7} + 10x^{6} - 112x^{5} - 29x^{4} + 81x^{3} + 26x^{2} - 19x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{8} - \nu^{7} - 10\nu^{6} + 9\nu^{5} + 30\nu^{4} - 21\nu^{3} - 31\nu^{2} + 11\nu + 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{9} - \nu^{8} - 10\nu^{7} + 8\nu^{6} + 30\nu^{5} - 12\nu^{4} - 32\nu^{3} - 10\nu^{2} + 13\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{9} - 2\nu^{8} - 10\nu^{7} + 19\nu^{6} + 30\nu^{5} - 50\nu^{4} - 31\nu^{3} + 36\nu^{2} + 8\nu - 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{9} - 12\nu^{7} - \nu^{6} + 48\nu^{5} + 9\nu^{4} - 73\nu^{3} - 21\nu^{2} + 29\nu + 10 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{10} + \nu^{9} + 11\nu^{8} - 10\nu^{7} - 39\nu^{6} + 30\nu^{5} + 53\nu^{4} - 32\nu^{3} - 24\nu^{2} + 12\nu + 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( -\nu^{10} + 13\nu^{8} - 58\nu^{6} + 103\nu^{4} - 60\nu^{2} + 9 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{10} - 14\nu^{8} + 2\nu^{7} + 68\nu^{6} - 18\nu^{5} - 133\nu^{4} + 42\nu^{3} + 92\nu^{2} - 20\nu - 20 \)
|
| \(\beta_{10}\) | \(=\) |
\( 3 \nu^{10} - 2 \nu^{9} - 36 \nu^{8} + 20 \nu^{7} + 146 \nu^{6} - 59 \nu^{5} - 238 \nu^{4} + 57 \nu^{3} + \cdots - 21 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{7} + \beta_{5} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta_{3} + 5\beta_{2} - \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 9\beta_{8} - 5\beta_{7} + \beta_{6} + 6\beta_{5} - \beta_{2} + 19\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{9} + \beta_{7} - 8\beta_{6} + 8\beta_{5} - \beta_{4} + 18\beta_{3} + 24\beta_{2} - 9\beta _1 + 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9 \beta_{10} + 10 \beta_{9} + 61 \beta_{8} - 24 \beta_{7} + 9 \beta_{6} + 33 \beta_{5} + \beta_{3} + \cdots - 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( 11 \beta_{9} + \beta_{8} + 10 \beta_{7} - 50 \beta_{6} + 50 \beta_{5} - 10 \beta_{4} + 122 \beta_{3} + \cdots + 81 \)
|
| \(\nu^{9}\) | \(=\) |
\( 60 \beta_{10} + 73 \beta_{9} + 373 \beta_{8} - 120 \beta_{7} + 62 \beta_{6} + 180 \beta_{5} - \beta_{4} + \cdots - 73 \)
|
| \(\nu^{10}\) | \(=\) |
\( 85 \beta_{9} + 12 \beta_{8} + 72 \beta_{7} - 289 \beta_{6} + 289 \beta_{5} - 72 \beta_{4} + 748 \beta_{3} + \cdots + 342 \)
|
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 |
|
−2.36027 | 0.954473 | 3.57087 | −1.56717 | −2.25281 | 1.00000 | −3.70768 | −2.08898 | 3.69894 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.69380 | −1.38407 | 0.868957 | 2.46333 | 2.34434 | 1.00000 | 1.91576 | −1.08435 | −4.17239 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.66112 | −2.80175 | 0.759319 | 0.189830 | 4.65404 | 1.00000 | 2.06092 | 4.84981 | −0.315330 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.818434 | −0.0746756 | −1.33017 | 0.657812 | 0.0611171 | 1.00000 | 2.72552 | −2.99442 | −0.538376 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.649805 | 1.93731 | −1.57775 | 0.841620 | −1.25887 | 1.00000 | 2.32484 | 0.753159 | −0.546889 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.411300 | −1.52729 | −1.83083 | −2.39965 | 0.628172 | 1.00000 | 1.57562 | −0.667397 | 0.986977 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.706811 | 1.72112 | −1.50042 | −2.67499 | 1.21651 | 1.00000 | −2.47414 | −0.0377474 | −1.89071 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.858512 | −1.12680 | −1.26296 | 0.250837 | −0.967368 | 1.00000 | −2.80129 | −1.73033 | 0.215346 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.78108 | 1.12554 | 1.17224 | −0.987663 | 2.00467 | 1.00000 | −1.47430 | −1.73316 | −1.75911 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.89330 | −0.477495 | 1.58457 | 1.08446 | −0.904040 | 1.00000 | −0.786537 | −2.77200 | 2.05320 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.35503 | −2.34636 | 3.54617 | −2.85842 | −5.52576 | 1.00000 | 3.64127 | 2.50542 | −6.73166 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
| \(13\) | \( -1 \) |
| \(29\) | \( +1 \) |
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 | 2639.2.a.c | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2639.2.a.c | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 13T_{2}^{9} - T_{2}^{8} + 59T_{2}^{7} + 10T_{2}^{6} - 112T_{2}^{5} - 29T_{2}^{4} + 81T_{2}^{3} + 26T_{2}^{2} - 19T_{2} - 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2639))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 13 T^{9} + \cdots - 7 \)
|
| $3$ |
\( T^{11} + 4 T^{10} + \cdots + 2 \)
|
| $5$ |
\( T^{11} + 5 T^{10} + \cdots + 2 \)
|
| $7$ |
\( (T - 1)^{11} \)
|
| $11$ |
\( T^{11} + 4 T^{10} + \cdots - 29428 \)
|
| $13$ |
\( (T - 1)^{11} \)
|
| $17$ |
\( T^{11} - 73 T^{9} + \cdots - 2708 \)
|
| $19$ |
\( T^{11} + 5 T^{10} + \cdots - 14294 \)
|
| $23$ |
\( T^{11} + 9 T^{10} + \cdots + 32 \)
|
| $29$ |
\( (T + 1)^{11} \)
|
| $31$ |
\( T^{11} + 27 T^{10} + \cdots - 16234 \)
|
| $37$ |
\( T^{11} + 22 T^{10} + \cdots - 924398 \)
|
| $41$ |
\( T^{11} - 2 T^{10} + \cdots + 43448 \)
|
| $43$ |
\( T^{11} + 12 T^{10} + \cdots + 1816 \)
|
| $47$ |
\( T^{11} - 5 T^{10} + \cdots - 7442318 \)
|
| $53$ |
\( T^{11} + \cdots - 808435498 \)
|
| $59$ |
\( T^{11} + 11 T^{10} + \cdots + 4589396 \)
|
| $61$ |
\( T^{11} + 21 T^{10} + \cdots + 124948 \)
|
| $67$ |
\( T^{11} + 29 T^{10} + \cdots - 9809044 \)
|
| $71$ |
\( T^{11} + T^{10} + \cdots - 2866648 \)
|
| $73$ |
\( T^{11} + 42 T^{10} + \cdots + 32649124 \)
|
| $79$ |
\( T^{11} + \cdots - 195756356 \)
|
| $83$ |
\( T^{11} + \cdots - 180779248 \)
|
| $89$ |
\( T^{11} + 30 T^{10} + \cdots - 3704956 \)
|
| $97$ |
\( T^{11} + 32 T^{10} + \cdots + 3087076 \)
|
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