Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4030,2,Mod(1,4030)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, 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("4030.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4030.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: | \(32.1797120146\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - x^{6} - 12x^{5} + 10x^{4} + 26x^{3} - 6x^{2} - 17x - 4 \)
|
| 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
\( x^{7} - x^{6} - 12x^{5} + 10x^{4} + 26x^{3} - 6x^{2} - 17x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + 2\nu^{5} + 10\nu^{4} - 20\nu^{3} - 5\nu^{2} + 12\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 10\nu^{4} + 20\nu^{3} + 6\nu^{2} - 12\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( 3\nu^{6} - 5\nu^{5} - 32\nu^{4} + 51\nu^{3} + 37\nu^{2} - 40\nu - 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( -4\nu^{6} + 7\nu^{5} + 43\nu^{4} - 72\nu^{3} - 53\nu^{2} + 61\nu + 27 \)
|
| \(\beta_{6}\) | \(=\) |
\( -7\nu^{6} + 13\nu^{5} + 73\nu^{4} - 133\nu^{3} - 70\nu^{2} + 106\nu + 33 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - 2\beta_{3} + \beta_{2} + 9\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + 2\beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 34 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{6} + 13\beta_{5} + 3\beta_{4} - 22\beta_{3} + 12\beta_{2} + 85\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{6} + 26\beta_{5} + 16\beta_{4} + 81\beta_{3} + 108\beta_{2} + 2\beta _1 + 317 \)
|
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 | −3.08527 | 1.00000 | 1.00000 | 3.08527 | −0.803633 | −1.00000 | 6.51887 | −1.00000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.84730 | 1.00000 | 1.00000 | 1.84730 | 3.11865 | −1.00000 | 0.412503 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.09573 | 1.00000 | 1.00000 | 1.09573 | −2.85390 | −1.00000 | −1.79939 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.304183 | 1.00000 | 1.00000 | −0.304183 | 2.10315 | −1.00000 | −2.90747 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.803589 | 1.00000 | 1.00000 | −0.803589 | −2.71080 | −1.00000 | −2.35424 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.854702 | 1.00000 | 1.00000 | −0.854702 | 0.252640 | −1.00000 | −2.26949 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 3.06581 | 1.00000 | 1.00000 | −3.06581 | −3.10610 | −1.00000 | 6.39922 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 4030.2.a.h | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.h | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + T_{3}^{6} - 12T_{3}^{5} - 10T_{3}^{4} + 26T_{3}^{3} + 6T_{3}^{2} - 17T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( T^{7} + T^{6} - 12 T^{5} + \cdots + 4 \)
|
| $5$ |
\( (T - 1)^{7} \)
|
| $7$ |
\( T^{7} + 4 T^{6} + \cdots - 32 \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots + 962 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} + 8 T^{6} + \cdots - 7696 \)
|
| $19$ |
\( T^{7} - T^{6} + \cdots + 2996 \)
|
| $23$ |
\( T^{7} + 5 T^{6} + \cdots - 4606 \)
|
| $29$ |
\( T^{7} + 4 T^{6} + \cdots - 104 \)
|
| $31$ |
\( (T - 1)^{7} \)
|
| $37$ |
\( T^{7} + 2 T^{6} + \cdots - 22 \)
|
| $41$ |
\( T^{7} + 6 T^{6} + \cdots - 18954 \)
|
| $43$ |
\( T^{7} + 5 T^{6} + \cdots - 31612 \)
|
| $47$ |
\( T^{7} + 18 T^{6} + \cdots + 19068 \)
|
| $53$ |
\( T^{7} + 12 T^{6} + \cdots - 3846 \)
|
| $59$ |
\( T^{7} - 3 T^{6} + \cdots + 20508 \)
|
| $61$ |
\( T^{7} + 7 T^{6} + \cdots + 172 \)
|
| $67$ |
\( T^{7} - 6 T^{6} + \cdots - 11796 \)
|
| $71$ |
\( T^{7} - 4 T^{6} + \cdots + 64160 \)
|
| $73$ |
\( T^{7} + 31 T^{6} + \cdots + 484 \)
|
| $79$ |
\( T^{7} + 2 T^{6} + \cdots + 19652 \)
|
| $83$ |
\( T^{7} + 40 T^{6} + \cdots - 161072 \)
|
| $89$ |
\( T^{7} - 195 T^{5} + \cdots - 27582 \)
|
| $97$ |
\( T^{7} + 21 T^{6} + \cdots + 621798 \)
|
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