Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4010,2,Mod(1,4010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4010, 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("4010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4010 = 2 \cdot 5 \cdot 401 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4010.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: | \(32.0200112105\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - 2x^{8} - 8x^{7} + 16x^{6} + 17x^{5} - 36x^{4} - 4x^{3} + 17x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{9} - 2x^{8} - 8x^{7} + 16x^{6} + 17x^{5} - 36x^{4} - 4x^{3} + 17x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 5\nu^{3} - 4\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 8\nu^{5} + 8\nu^{4} + 17\nu^{3} - 17\nu^{2} - 5\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{8} - \nu^{7} - 9\nu^{6} + 7\nu^{5} + 25\nu^{4} - 13\nu^{3} - 21\nu^{2} + 4\nu + 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 8\nu^{6} + 16\nu^{5} + 16\nu^{4} - 34\nu^{3} + 9\nu \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 7\nu^{6} + 15\nu^{5} + 10\nu^{4} - 30\nu^{3} + 9\nu^{2} + 8\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4\beta_{3} + 5\beta_{2} + \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{6} + 6\beta_{5} + 5\beta_{4} + \beta_{2} + 17\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{8} + 7\beta_{7} - 8\beta_{6} + 8\beta_{5} + 7\beta_{4} + 15\beta_{3} + 22\beta_{2} + 8\beta _1 + 21 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{8} + 30\beta_{7} - 31\beta_{6} + 32\beta_{5} + 22\beta_{4} + 7\beta_{2} + 73\beta _1 - 19 \)
|
| \(\nu^{8}\) | \(=\) |
\( 10\beta_{8} + 39\beta_{7} - 48\beta_{6} + 50\beta_{5} + 38\beta_{4} + 56\beta_{3} + 94\beta_{2} + 49\beta _1 + 80 \)
|
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 | −2.53021 | 1.00000 | 1.00000 | −2.53021 | 1.34700 | 1.00000 | 3.40195 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.18512 | 1.00000 | 1.00000 | −2.18512 | −0.791922 | 1.00000 | 1.77477 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.84294 | 1.00000 | 1.00000 | −1.84294 | −0.0938987 | 1.00000 | 0.396437 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.922780 | 1.00000 | 1.00000 | −0.922780 | −2.45091 | 1.00000 | −2.14848 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −0.375168 | 1.00000 | 1.00000 | −0.375168 | 3.43840 | 1.00000 | −2.85925 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.453213 | 1.00000 | 1.00000 | 0.453213 | −1.57101 | 1.00000 | −2.79460 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 0.706772 | 1.00000 | 1.00000 | 0.706772 | −4.28122 | 1.00000 | −2.50047 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 1.13041 | 1.00000 | 1.00000 | 1.13041 | −0.772637 | 1.00000 | −1.72217 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 1.56583 | 1.00000 | 1.00000 | 1.56583 | −1.82380 | 1.00000 | −0.548185 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(401\) | \(1\) |
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 | 4010.2.a.h | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4010.2.a.h | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):
|
\( T_{3}^{9} + 4T_{3}^{8} - 2T_{3}^{7} - 20T_{3}^{6} - 3T_{3}^{5} + 32T_{3}^{4} + 4T_{3}^{3} - 17T_{3}^{2} + 2 \)
|
|
\( T_{7}^{9} + 7T_{7}^{8} + 2T_{7}^{7} - 80T_{7}^{6} - 189T_{7}^{5} - 52T_{7}^{4} + 269T_{7}^{3} + 311T_{7}^{2} + 112T_{7} + 8 \)
|
|
\( T_{11}^{9} + 11 T_{11}^{8} + 25 T_{11}^{7} - 97 T_{11}^{6} - 413 T_{11}^{5} - 19 T_{11}^{4} + 1403 T_{11}^{3} + \cdots - 388 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{9} \)
|
| $3$ |
\( T^{9} + 4 T^{8} + \cdots + 2 \)
|
| $5$ |
\( (T - 1)^{9} \)
|
| $7$ |
\( T^{9} + 7 T^{8} + \cdots + 8 \)
|
| $11$ |
\( T^{9} + 11 T^{8} + \cdots - 388 \)
|
| $13$ |
\( T^{9} + 14 T^{8} + \cdots - 27134 \)
|
| $17$ |
\( T^{9} + 13 T^{8} + \cdots - 446 \)
|
| $19$ |
\( T^{9} + 11 T^{8} + \cdots - 6904 \)
|
| $23$ |
\( T^{9} + 9 T^{8} + \cdots - 386 \)
|
| $29$ |
\( T^{9} + 20 T^{8} + \cdots - 86624 \)
|
| $31$ |
\( T^{9} + 11 T^{8} + \cdots + 994328 \)
|
| $37$ |
\( T^{9} + 25 T^{8} + \cdots - 5366 \)
|
| $41$ |
\( T^{9} + 29 T^{8} + \cdots + 1136 \)
|
| $43$ |
\( T^{9} + 11 T^{8} + \cdots - 8416 \)
|
| $47$ |
\( T^{9} + 3 T^{8} + \cdots - 8192 \)
|
| $53$ |
\( T^{9} + 9 T^{8} + \cdots + 52574 \)
|
| $59$ |
\( T^{9} + 10 T^{8} + \cdots + 413048 \)
|
| $61$ |
\( T^{9} + 10 T^{8} + \cdots - 2472404 \)
|
| $67$ |
\( T^{9} + 16 T^{8} + \cdots - 20772566 \)
|
| $71$ |
\( T^{9} + 8 T^{8} + \cdots + 8768 \)
|
| $73$ |
\( T^{9} + 22 T^{8} + \cdots - 18746156 \)
|
| $79$ |
\( T^{9} + 9 T^{8} + \cdots - 76312 \)
|
| $83$ |
\( T^{9} - 11 T^{8} + \cdots - 1556096 \)
|
| $89$ |
\( T^{9} + \cdots - 2051742764 \)
|
| $97$ |
\( T^{9} + 28 T^{8} + \cdots - 12271958 \)
|
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