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: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 13\nu^{4} + 13\nu^{3} + 38\nu^{2} - 31\nu - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} + 4\nu^{6} + 25\nu^{5} - 52\nu^{4} - 61\nu^{3} + 139\nu^{2} - 25\nu - 13 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 13\nu^{6} + 58\nu^{5} - 172\nu^{4} - 94\nu^{3} + 484\nu^{2} - 217\nu - 64 ) / 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} + 10\nu^{5} - 66\nu^{4} + 2\nu^{3} + 192\nu^{2} - 87\nu - 30 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -7\nu^{7} + 23\nu^{6} + 80\nu^{5} - 302\nu^{4} - 110\nu^{3} + 854\nu^{2} - 353\nu - 116 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{4} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} + 8\beta_{2} - \beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{7} - 11\beta_{6} - 2\beta_{5} - 12\beta_{4} - 2\beta_{3} - \beta_{2} + 52\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{6} - 15\beta_{5} + 27\beta_{4} + 25\beta_{3} + 65\beta_{2} - 21\beta _1 + 202 \)
|
| \(\nu^{7}\) | \(=\) |
\( 132\beta_{7} - 103\beta_{6} - 29\beta_{5} - 119\beta_{4} - 27\beta_{3} - 21\beta_{2} + 408\beta _1 - 64 \)
|
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 | −3.01481 | 1.00000 | 1.00000 | −3.01481 | 2.60866 | 1.00000 | 6.08910 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.95716 | 1.00000 | 1.00000 | −1.95716 | −0.791637 | 1.00000 | 0.830474 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.256875 | 1.00000 | 1.00000 | −0.256875 | −3.07701 | 1.00000 | −2.93402 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.194747 | 1.00000 | 1.00000 | −0.194747 | 1.49202 | 1.00000 | −2.96207 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.852741 | 1.00000 | 1.00000 | 0.852741 | 3.80351 | 1.00000 | −2.27283 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.25864 | 1.00000 | 1.00000 | 2.25864 | 2.41815 | 1.00000 | 2.10143 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.51811 | 1.00000 | 1.00000 | 2.51811 | −2.11544 | 1.00000 | 3.34088 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.79411 | 1.00000 | 1.00000 | 2.79411 | 2.66175 | 1.00000 | 4.80704 | 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.o | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.o | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 3T_{3}^{7} - 12T_{3}^{6} + 40T_{3}^{5} + 24T_{3}^{4} - 118T_{3}^{3} + 25T_{3}^{2} + 30T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 3 T^{7} + \cdots + 4 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 7 T^{7} + \cdots - 491 \)
|
| $11$ |
\( T^{8} - 10 T^{7} + \cdots + 516 \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} - 11 T^{7} + \cdots + 6177 \)
|
| $19$ |
\( T^{8} - 2 T^{7} + \cdots - 7 \)
|
| $23$ |
\( T^{8} - 12 T^{7} + \cdots - 2371 \)
|
| $29$ |
\( T^{8} - 9 T^{7} + \cdots - 11907 \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} - 19 T^{7} + \cdots - 47007 \)
|
| $41$ |
\( T^{8} - 6 T^{7} + \cdots - 89964 \)
|
| $43$ |
\( T^{8} - 21 T^{7} + \cdots + 284 \)
|
| $47$ |
\( T^{8} - T^{7} + \cdots + 278339 \)
|
| $53$ |
\( T^{8} - 18 T^{7} + \cdots - 31212 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 1791 \)
|
| $61$ |
\( T^{8} - 10 T^{7} + \cdots - 618261 \)
|
| $67$ |
\( T^{8} - 8 T^{7} + \cdots + 2948 \)
|
| $71$ |
\( T^{8} - 18 T^{7} + \cdots - 24924 \)
|
| $73$ |
\( T^{8} - 9 T^{7} + \cdots + 308628 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots + 49252 \)
|
| $83$ |
\( T^{8} - 3 T^{7} + \cdots + 2911951 \)
|
| $89$ |
\( T^{8} + 15 T^{7} + \cdots - 18861477 \)
|
| $97$ |
\( T^{8} - 12 T^{7} + \cdots + 17521 \)
|
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