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: |
\( x^{8} - 3x^{7} - 15x^{6} + 31x^{5} + 79x^{4} - 85x^{3} - 162x^{2} + 45x + 81 \)
|
| 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
\( x^{8} - 3x^{7} - 15x^{6} + 31x^{5} + 79x^{4} - 85x^{3} - 162x^{2} + 45x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 8\nu^{6} + 63\nu^{5} + 50\nu^{4} - 447\nu^{3} - 106\nu^{2} + 662\nu + 267 ) / 102 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -23\nu^{7} + 54\nu^{6} + 327\nu^{5} - 380\nu^{4} - 1373\nu^{3} + 554\nu^{2} + 1320\nu - 387 ) / 306 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\nu^{7} - 66\nu^{6} - 105\nu^{5} + 727\nu^{4} + 235\nu^{3} - 2311\nu^{2} - 378\nu + 1629 ) / 153 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\nu^{7} - 86\nu^{6} - 177\nu^{5} + 886\nu^{4} + 503\nu^{3} - 2406\nu^{2} - 610\nu + 1353 ) / 102 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{7} + 19\nu^{6} + 31\nu^{5} - 191\nu^{4} - 37\nu^{3} + 511\nu^{2} - 21\nu - 309 ) / 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -77\nu^{7} + 336\nu^{6} + 669\nu^{5} - 3188\nu^{4} - 1541\nu^{3} + 7784\nu^{2} + 1470\nu - 4023 ) / 306 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} + \beta_{6} - 3\beta_{5} + 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -15\beta_{7} - 19\beta_{5} + 7\beta_{4} + 13\beta_{3} - 15\beta_{2} + 17\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( -58\beta_{7} + 9\beta_{6} - 71\beta_{5} + 39\beta_{4} + 40\beta_{3} - 45\beta_{2} + 81\beta _1 + 96 \)
|
| \(\nu^{6}\) | \(=\) |
\( -245\beta_{7} + 5\beta_{6} - 332\beta_{5} + 154\beta_{4} + 186\beta_{3} - 231\beta_{2} + 275\beta _1 + 474 \)
|
| \(\nu^{7}\) | \(=\) |
\( -997\beta_{7} + 80\beta_{6} - 1320\beta_{5} + 681\beta_{4} + 682\beta_{3} - 839\beta_{2} + 1180\beta _1 + 1664 \)
|
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 | −3.09186 | 1.00000 | 1.00000 | 3.09186 | −0.0229325 | −1.00000 | 6.55960 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.49715 | 1.00000 | 1.00000 | 1.49715 | 2.81472 | −1.00000 | −0.758530 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.942199 | 1.00000 | 1.00000 | 0.942199 | 2.99101 | −1.00000 | −2.11226 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.157287 | 1.00000 | 1.00000 | −0.157287 | −4.20500 | −1.00000 | −2.97526 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.86933 | 1.00000 | 1.00000 | −1.86933 | 1.98758 | −1.00000 | 0.494407 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.28275 | 1.00000 | 1.00000 | −2.28275 | 0.279113 | −1.00000 | 2.21094 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.77519 | 1.00000 | 1.00000 | −2.77519 | −3.40676 | −1.00000 | 4.70170 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.44665 | 1.00000 | 1.00000 | −3.44665 | 4.56226 | −1.00000 | 8.87940 | −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.m | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4030.2.a.m | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 5T_{3}^{7} - 8T_{3}^{6} + 66T_{3}^{5} - 26T_{3}^{4} - 192T_{3}^{3} + 107T_{3}^{2} + 166T_{3} - 28 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} - 5 T^{7} + \cdots - 28 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 5 T^{7} + \cdots - 7 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots + 16476 \)
|
| $13$ |
\( (T - 1)^{8} \)
|
| $17$ |
\( T^{8} - 19 T^{7} + \cdots + 138057 \)
|
| $19$ |
\( T^{8} + 14 T^{7} + \cdots - 62431 \)
|
| $23$ |
\( T^{8} - 8 T^{7} + \cdots + 1017 \)
|
| $29$ |
\( T^{8} - 21 T^{7} + \cdots + 21 \)
|
| $31$ |
\( (T - 1)^{8} \)
|
| $37$ |
\( T^{8} + 3 T^{7} + \cdots + 324313 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots - 324 \)
|
| $43$ |
\( T^{8} - 25 T^{7} + \cdots + 1379708 \)
|
| $47$ |
\( T^{8} - 11 T^{7} + \cdots - 440313 \)
|
| $53$ |
\( T^{8} - 2 T^{7} + \cdots + 10318788 \)
|
| $59$ |
\( T^{8} + 22 T^{7} + \cdots - 1821957 \)
|
| $61$ |
\( T^{8} + 2 T^{7} + \cdots - 1164869 \)
|
| $67$ |
\( T^{8} + 14 T^{7} + \cdots - 63281852 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots - 23286228 \)
|
| $73$ |
\( T^{8} - 17 T^{7} + \cdots - 29540 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots - 260588 \)
|
| $83$ |
\( T^{8} - 21 T^{7} + \cdots - 451545 \)
|
| $89$ |
\( T^{8} + 25 T^{7} + \cdots + 12456447 \)
|
| $97$ |
\( T^{8} - 8 T^{7} + \cdots + 417941 \)
|
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