Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1001,2,Mod(1,1001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, 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("1001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1001 = 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1001.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: | \(7.99302524233\) |
| 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: |
\( x^{7} - x^{6} - 13x^{5} + 44x^{3} + 23x^{2} - 17x + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 13x^{5} + 44x^{3} + 23x^{2} - 17x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 14\nu^{3} + 15\nu^{2} - 6\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 14\nu^{3} + 16\nu^{2} - 8\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 7\nu^{3} + 29\nu^{2} + 9\nu - 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{6} + 6\nu^{5} + 15\nu^{4} - 32\nu^{3} - 33\nu^{2} + 28\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -3\nu^{6} + 8\nu^{5} + 25\nu^{4} - 40\nu^{3} - 60\nu^{2} + 24\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} + 9\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{6} + 5\beta_{5} - 4\beta_{4} + 11\beta_{3} - 9\beta_{2} + 26\beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( -19\beta_{6} + 22\beta_{5} - 18\beta_{4} + 33\beta_{3} - 28\beta_{2} + 98\beta _1 + 76 \)
|
| \(\nu^{6}\) | \(=\) |
\( -71\beta_{6} + 87\beta_{5} - 68\beta_{4} + 133\beta_{3} - 103\beta_{2} + 326\beta _1 + 306 \)
|
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 |
|
−2.76722 | −1.22927 | 5.65749 | 3.95461 | 3.40166 | −1.00000 | −10.1211 | −1.48889 | −10.9433 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.70845 | 2.49892 | 5.33572 | −2.74784 | −6.76822 | −1.00000 | −9.03464 | 3.24462 | 7.44240 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.83609 | 2.89780 | 1.37123 | 2.48944 | −5.32062 | −1.00000 | 1.15448 | 5.39723 | −4.57084 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.320581 | 0.934684 | −1.89723 | 2.67587 | −0.299642 | −1.00000 | 1.24938 | −2.12637 | −0.857833 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.30747 | 2.83044 | 3.32440 | −0.944299 | 6.53114 | −1.00000 | 3.05602 | 5.01137 | −2.17894 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.53745 | 0.629743 | 4.43864 | 3.39558 | 1.59794 | −1.00000 | 6.18792 | −2.60342 | 8.61609 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.78743 | −2.56231 | 5.76975 | 0.176644 | −7.14226 | −1.00000 | 10.5079 | 3.56545 | 0.492382 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1001.2.a.l | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9009.2.a.bk | 7 | ||
| 7.b | odd | 2 | 1 | 7007.2.a.s | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1001.2.a.l | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 7007.2.a.s | 7 | 7.b | odd | 2 | 1 | ||
| 9009.2.a.bk | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\):
|
\( T_{2}^{7} - 19T_{2}^{5} - T_{2}^{4} + 117T_{2}^{3} + 17T_{2}^{2} - 231T_{2} - 72 \)
|
|
\( T_{3}^{7} - 6T_{3}^{6} + 2T_{3}^{5} + 45T_{3}^{4} - 71T_{3}^{3} - 31T_{3}^{2} + 97T_{3} - 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 19 T^{5} + \cdots - 72 \)
|
| $3$ |
\( T^{7} - 6 T^{6} + \cdots - 38 \)
|
| $5$ |
\( T^{7} - 9 T^{6} + \cdots - 41 \)
|
| $7$ |
\( (T + 1)^{7} \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} - 4 T^{6} + \cdots + 2 \)
|
| $19$ |
\( T^{7} - 3 T^{6} + \cdots + 17807 \)
|
| $23$ |
\( T^{7} - 22 T^{6} + \cdots + 54644 \)
|
| $29$ |
\( T^{7} - 16 T^{6} + \cdots - 27828 \)
|
| $31$ |
\( T^{7} - 17 T^{6} + \cdots + 110804 \)
|
| $37$ |
\( T^{7} + 12 T^{6} + \cdots + 1678208 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots - 3584 \)
|
| $43$ |
\( T^{7} - 23 T^{6} + \cdots - 953257 \)
|
| $47$ |
\( T^{7} - 6 T^{6} + \cdots - 92 \)
|
| $53$ |
\( T^{7} - 12 T^{6} + \cdots + 893681 \)
|
| $59$ |
\( T^{7} - 19 T^{6} + \cdots - 378784 \)
|
| $61$ |
\( T^{7} + 11 T^{6} + \cdots - 1164886 \)
|
| $67$ |
\( T^{7} + 17 T^{6} + \cdots - 141238 \)
|
| $71$ |
\( T^{7} - 25 T^{6} + \cdots + 277856 \)
|
| $73$ |
\( T^{7} + 27 T^{6} + \cdots - 286448 \)
|
| $79$ |
\( T^{7} - 31 T^{6} + \cdots - 1372617 \)
|
| $83$ |
\( T^{7} - 4 T^{6} + \cdots + 26149 \)
|
| $89$ |
\( T^{7} - 5 T^{6} + \cdots - 227251 \)
|
| $97$ |
\( T^{7} + 7 T^{6} + \cdots - 1604 \)
|
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