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: |
\( x^{8} - x^{7} - 16x^{6} + 18x^{5} + 64x^{4} - 84x^{3} - 19x^{2} + 22x + 4 \)
|
| 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
\( x^{8} - x^{7} - 16x^{6} + 18x^{5} + 64x^{4} - 84x^{3} - 19x^{2} + 22x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} + 7\nu^{6} - 44\nu^{5} - 112\nu^{4} + 186\nu^{3} + 426\nu^{2} - 319\nu - 108 ) / 58 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{7} + 5\nu^{6} + 39\nu^{5} - 80\nu^{4} - 211\nu^{3} + 325\nu^{2} + 203\nu - 102 ) / 29 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} - 3\nu^{6} - 122\nu^{5} + 48\nu^{4} + 608\nu^{3} - 166\nu^{2} - 725\nu - 136 ) / 58 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{7} + 20\nu^{6} + 127\nu^{5} - 320\nu^{4} - 467\nu^{3} + 1271\nu^{2} - 203\nu - 205 ) / 29 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -25\nu^{7} + 19\nu^{6} + 386\nu^{5} - 362\nu^{4} - 1434\nu^{3} + 1786\nu^{2} + 145\nu - 318 ) / 58 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 16\nu^{5} - 18\nu^{4} - 62\nu^{3} + 82\nu^{2} + 3\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} + 8\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + \beta_{5} + 10\beta_{4} + 9\beta_{3} - 13\beta_{2} + \beta _1 + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{7} - 13\beta_{6} - \beta_{5} + 12\beta_{4} + 3\beta_{3} - 12\beta_{2} + 69\beta _1 - 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{7} - 28\beta_{6} + 17\beta_{5} + 94\beta_{4} + 81\beta_{3} - 138\beta_{2} + 20\beta _1 + 263 \)
|
| \(\nu^{7}\) | \(=\) |
\( 138\beta_{7} - 138\beta_{6} - 17\beta_{5} + 126\beta_{4} + 49\beta_{3} - 116\beta_{2} + 613\beta _1 - 73 \)
|
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 | −3.08271 | 1.00000 | −1.00000 | −3.08271 | −2.62527 | 1.00000 | 6.50312 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.04806 | 1.00000 | −1.00000 | −2.04806 | 3.38208 | 1.00000 | 1.19454 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.54913 | 1.00000 | −1.00000 | −1.54913 | −2.44745 | 1.00000 | −0.600186 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.635022 | 1.00000 | −1.00000 | −0.635022 | 1.23180 | 1.00000 | −2.59675 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.178838 | 1.00000 | −1.00000 | 0.178838 | −4.63233 | 1.00000 | −2.96802 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0.447568 | 1.00000 | −1.00000 | 0.447568 | 1.86515 | 1.00000 | −2.79968 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.63528 | 1.00000 | −1.00000 | 2.63528 | 0.203142 | 1.00000 | 3.94469 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.05324 | 1.00000 | −1.00000 | 3.05324 | 4.02288 | 1.00000 | 6.32229 | −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.n | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.n | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} - 16T_{3}^{6} - 18T_{3}^{5} + 64T_{3}^{4} + 84T_{3}^{3} - 19T_{3}^{2} - 22T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} + T^{7} - 16 T^{6} + \cdots + 4 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} - T^{7} + \cdots - 189 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots - 52 \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} + 5 T^{7} + \cdots - 19 \)
|
| $19$ |
\( T^{8} - 2 T^{7} + \cdots + 19 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 12343 \)
|
| $29$ |
\( T^{8} - 11 T^{7} + \cdots + 625 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} - 19 T^{7} + \cdots + 1300393 \)
|
| $41$ |
\( T^{8} - 10 T^{7} + \cdots - 1765844 \)
|
| $43$ |
\( T^{8} - 19 T^{7} + \cdots - 2588 \)
|
| $47$ |
\( T^{8} - 11 T^{7} + \cdots - 18599 \)
|
| $53$ |
\( T^{8} - 8 T^{7} + \cdots + 1792116 \)
|
| $59$ |
\( T^{8} - 28 T^{7} + \cdots + 2185429 \)
|
| $61$ |
\( T^{8} + 12 T^{7} + \cdots + 154847 \)
|
| $67$ |
\( T^{8} - 24 T^{7} + \cdots + 468 \)
|
| $71$ |
\( T^{8} - 18 T^{7} + \cdots + 2850148 \)
|
| $73$ |
\( T^{8} + 3 T^{7} + \cdots - 90396 \)
|
| $79$ |
\( T^{8} - 22 T^{7} + \cdots - 11685836 \)
|
| $83$ |
\( T^{8} - 17 T^{7} + \cdots - 24687 \)
|
| $89$ |
\( T^{8} - 17 T^{7} + \cdots - 13637 \)
|
| $97$ |
\( T^{8} + 24 T^{7} + \cdots - 592303 \)
|
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