Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7942,2,Mod(1,7942)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7942, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7942.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7942.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: | \(63.4171892853\) |
| 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: |
\( x^{8} - 14x^{6} - 4x^{5} + 38x^{4} + 3x^{3} - 29x^{2} + 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| 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
\( x^{8} - 14x^{6} - 4x^{5} + 38x^{4} + 3x^{3} - 29x^{2} + 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{7} - 9\nu^{6} - 101\nu^{5} + 97\nu^{4} + 479\nu^{3} - 182\nu^{2} - 511\nu + 77 ) / 61 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} + 23\nu^{6} - 74\nu^{5} - 336\nu^{4} + 145\nu^{3} + 804\nu^{2} - 131\nu - 312 ) / 61 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{7} + 17\nu^{6} - 121\nu^{5} - 251\nu^{4} + 200\nu^{3} + 398\nu^{2} - 126\nu - 159 ) / 61 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -18\nu^{7} - 34\nu^{6} + 242\nu^{5} + 502\nu^{4} - 400\nu^{3} - 735\nu^{2} + 191\nu + 74 ) / 61 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -23\nu^{7} + 4\nu^{6} + 316\nu^{5} + 45\nu^{4} - 789\nu^{3} - 14\nu^{2} + 261\nu + 20 ) / 61 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 49\nu^{7} + 18\nu^{6} - 652\nu^{5} - 438\nu^{4} + 1360\nu^{3} + 547\nu^{2} - 625\nu - 154 ) / 122 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 9\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 10\beta_{5} + 23\beta_{4} - 2\beta_{3} + \beta_{2} + 16\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 26\beta_{7} + 25\beta_{6} + 17\beta_{5} + 21\beta_{4} - \beta_{3} + 10\beta_{2} + 98\beta _1 + 41 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{7} + 22\beta_{6} + 108\beta_{5} + 253\beta_{4} - 25\beta_{3} + 17\beta_{2} + 220\beta _1 + 382 \)
|
| \(\nu^{7}\) | \(=\) |
\( 290\beta_{7} + 278\beta_{6} + 237\beta_{5} + 342\beta_{4} - 22\beta_{3} + 108\beta_{2} + 1118\beta _1 + 630 \)
|
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 | −1.87414 | 1.00000 | 1.89996 | −1.87414 | −3.86154 | 1.00000 | 0.512419 | 1.89996 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.86579 | 1.00000 | 0.673675 | −1.86579 | 1.45050 | 1.00000 | 0.481174 | 0.673675 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.13526 | 1.00000 | −1.11348 | −1.13526 | 0.931861 | 1.00000 | −1.71118 | −1.11348 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.416894 | 1.00000 | 3.13364 | 0.416894 | 5.24744 | 1.00000 | −2.82620 | 3.13364 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.724436 | 1.00000 | −3.58061 | 0.724436 | −1.97748 | 1.00000 | −2.47519 | −3.58061 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.97884 | 1.00000 | 2.94620 | 1.97884 | −0.656982 | 1.00000 | 0.915826 | 2.94620 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.35644 | 1.00000 | −0.0660228 | 2.35644 | −0.317755 | 1.00000 | 2.55283 | −0.0660228 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.39858 | 1.00000 | −2.89336 | 3.39858 | 3.18396 | 1.00000 | 8.55033 | −2.89336 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 7942.2.a.br | yes | 8 |
| 19.b | odd | 2 | 1 | 7942.2.a.bm | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7942.2.a.bm | ✓ | 8 | 19.b | odd | 2 | 1 | |
| 7942.2.a.br | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):
|
\( T_{3}^{8} - 4T_{3}^{7} - 7T_{3}^{6} + 34T_{3}^{5} + 13T_{3}^{4} - 87T_{3}^{3} + 2T_{3}^{2} + 58T_{3} - 19 \)
|
|
\( T_{5}^{8} - T_{5}^{7} - 22T_{5}^{6} + 25T_{5}^{5} + 134T_{5}^{4} - 154T_{5}^{3} - 167T_{5}^{2} + 126T_{5} + 9 \)
|
|
\( T_{13}^{8} - 12T_{13}^{7} + 3T_{13}^{6} + 486T_{13}^{5} - 1903T_{13}^{4} - 1281T_{13}^{3} + 19105T_{13}^{2} - 31494T_{13} + 10604 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots - 19 \)
|
| $5$ |
\( T^{8} - T^{7} - 22 T^{6} + \cdots + 9 \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 36 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} - 12 T^{7} + \cdots + 10604 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots - 9424 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots + 119291 \)
|
| $29$ |
\( T^{8} + T^{7} + \cdots + 128304 \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots + 25821 \)
|
| $37$ |
\( T^{8} - 12 T^{7} + \cdots + 12889 \)
|
| $41$ |
\( T^{8} - 18 T^{7} + \cdots - 21796 \)
|
| $43$ |
\( T^{8} + 4 T^{7} + \cdots - 22900 \)
|
| $47$ |
\( T^{8} - 13 T^{7} + \cdots - 1709159 \)
|
| $53$ |
\( T^{8} + 17 T^{7} + \cdots - 1025 \)
|
| $59$ |
\( T^{8} - 26 T^{7} + \cdots + 442775 \)
|
| $61$ |
\( T^{8} + 7 T^{7} + \cdots + 22716 \)
|
| $67$ |
\( T^{8} - 6 T^{7} + \cdots + 13275 \)
|
| $71$ |
\( T^{8} + 3 T^{7} + \cdots + 283376 \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots - 2105476 \)
|
| $79$ |
\( T^{8} - 16 T^{7} + \cdots - 27251316 \)
|
| $83$ |
\( T^{8} + 21 T^{7} + \cdots - 32644 \)
|
| $89$ |
\( T^{8} - 16 T^{7} + \cdots - 251299 \)
|
| $97$ |
\( T^{8} + 15 T^{7} + \cdots + 68364 \)
|
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