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: |
\( x^{8} - 14x^{6} - 6x^{5} + 54x^{4} + 46x^{3} - 32x^{2} - 43x - 11 \)
|
| 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
\( x^{8} - 14x^{6} - 6x^{5} + 54x^{4} + 46x^{3} - 32x^{2} - 43x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 13\nu^{5} - 7\nu^{4} - 47\nu^{3} + 2\nu^{2} + 30\nu + 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 13\nu^{5} + 7\nu^{4} + 47\nu^{3} - \nu^{2} - 31\nu - 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{7} - 7\nu^{6} - 120\nu^{5} + 41\nu^{4} + 448\nu^{3} + 50\nu^{2} - 318\nu - 118 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{7} + 4\nu^{6} + 110\nu^{5} - 7\nu^{4} - 426\nu^{3} - 155\nu^{2} + 316\nu + 176 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\nu^{7} - 9\nu^{6} - 175\nu^{5} + 42\nu^{4} + 666\nu^{3} + 145\nu^{2} - 501\nu - 221 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -24\nu^{7} + 17\nu^{6} + 325\nu^{5} - 86\nu^{4} - 1243\nu^{3} - 230\nu^{2} + 933\nu + 388 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 7\beta_{3} + 9\beta_{2} + 11\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 11\beta_{6} + 9\beta_{5} + 3\beta_{4} + 2\beta_{3} + 12\beta_{2} + 53\beta _1 + 23 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3\beta_{7} + 4\beta_{6} + 12\beta_{5} + 29\beta_{4} + 46\beta_{3} + 75\beta_{2} + 103\beta _1 + 187 \)
|
| \(\nu^{7}\) | \(=\) |
\( 29\beta_{7} + 100\beta_{6} + 75\beta_{5} + 54\beta_{4} + 25\beta_{3} + 122\beta_{2} + 418\beta _1 + 225 \)
|
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 | −3.08723 | 1.00000 | 1.00000 | 3.08723 | 4.56306 | −1.00000 | 6.53101 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.80420 | 1.00000 | 1.00000 | 1.80420 | 1.26189 | −1.00000 | 0.255155 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.43624 | 1.00000 | 1.00000 | 1.43624 | −2.74822 | −1.00000 | −0.937215 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.229462 | 1.00000 | 1.00000 | −0.229462 | 2.13623 | −1.00000 | −2.94735 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.244211 | 1.00000 | 1.00000 | −0.244211 | −1.97655 | −1.00000 | −2.94036 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 1.08982 | 1.00000 | 1.00000 | −1.08982 | 4.96118 | −1.00000 | −1.81229 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.53701 | 1.00000 | 1.00000 | −2.53701 | 1.50878 | −1.00000 | 3.43644 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.22717 | 1.00000 | 1.00000 | −3.22717 | 1.29362 | −1.00000 | 7.41461 | −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.l | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.l | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - T_{3}^{7} - 16T_{3}^{6} + 12T_{3}^{5} + 68T_{3}^{4} - 30T_{3}^{3} - 69T_{3}^{2} + 34T_{3} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} - T^{7} - 16 T^{6} + \cdots - 4 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 11 T^{7} + \cdots + 647 \)
|
| $11$ |
\( T^{8} - 43 T^{6} + \cdots + 340 \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 1085 \)
|
| $19$ |
\( T^{8} + 2 T^{7} + \cdots - 14627 \)
|
| $23$ |
\( T^{8} - 8 T^{7} + \cdots + 75709 \)
|
| $29$ |
\( T^{8} + T^{7} + \cdots - 797 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} - T^{7} + \cdots + 404425 \)
|
| $41$ |
\( T^{8} - 16 T^{7} + \cdots + 257500 \)
|
| $43$ |
\( T^{8} - 3 T^{7} + \cdots - 2912836 \)
|
| $47$ |
\( T^{8} - 29 T^{7} + \cdots + 5433193 \)
|
| $53$ |
\( T^{8} - 22 T^{7} + \cdots - 59836 \)
|
| $59$ |
\( T^{8} + 8 T^{7} + \cdots + 1049527 \)
|
| $61$ |
\( T^{8} - 4 T^{7} + \cdots + 5566625 \)
|
| $67$ |
\( T^{8} - 28 T^{7} + \cdots - 231100 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots + 196292 \)
|
| $73$ |
\( T^{8} - 39 T^{7} + \cdots + 2945060 \)
|
| $79$ |
\( T^{8} + 16 T^{7} + \cdots + 52444 \)
|
| $83$ |
\( T^{8} - 25 T^{7} + \cdots + 3539687 \)
|
| $89$ |
\( T^{8} - 21 T^{7} + \cdots - 864715 \)
|
| $97$ |
\( T^{8} - 28 T^{7} + \cdots - 304948249 \)
|
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