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: | \(1\) |
| 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: |
\( x^{8} - x^{7} - 19x^{6} + 14x^{5} + 116x^{4} - 65x^{3} - 235x^{2} + 120x + 80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 5 \) |
| 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
\( x^{8} - x^{7} - 19x^{6} + 14x^{5} + 116x^{4} - 65x^{3} - 235x^{2} + 120x + 80 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 21\nu^{7} + 23\nu^{6} - 427\nu^{5} - 334\nu^{4} + 2460\nu^{3} + 1275\nu^{2} - 3635\nu - 220 ) / 800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} - 59\nu^{6} - 9\nu^{5} + 822\nu^{4} - 580\nu^{3} - 3175\nu^{2} + 1855\nu + 2060 ) / 200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{7} + 7\nu^{6} + 157\nu^{5} - 6\nu^{4} - 460\nu^{3} - 525\nu^{2} - 515\nu + 1220 ) / 200 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\nu^{7} - 7\nu^{6} - 157\nu^{5} + 6\nu^{4} + 660\nu^{3} + 325\nu^{2} - 885\nu - 420 ) / 200 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 89\nu^{7} - 93\nu^{6} - 1543\nu^{5} + 794\nu^{4} + 8140\nu^{3} - 425\nu^{2} - 11215\nu - 1580 ) / 800 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + \beta_{4} + 5\beta_{3} + 10\beta_{2} + 2\beta _1 + 33 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{7} + 15\beta_{6} + 11\beta_{5} + \beta_{4} + 3\beta_{3} + 14\beta_{2} + 57\beta _1 + 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( -20\beta_{7} + 9\beta_{6} + 15\beta_{5} + 13\beta_{4} + 80\beta_{3} + 93\beta_{2} + 41\beta _1 + 241 \)
|
| \(\nu^{7}\) | \(=\) |
\( -55\beta_{7} + 178\beta_{6} + 106\beta_{5} + 22\beta_{4} + 91\beta_{3} + 164\beta_{2} + 499\beta _1 + 237 \)
|
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 | −3.24873 | 1.00000 | 1.63070 | 3.24873 | −1.84467 | −1.00000 | 7.55426 | −1.63070 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.70544 | 1.00000 | 3.32348 | 2.70544 | 3.11031 | −1.00000 | 4.31943 | −3.32348 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.52604 | 1.00000 | −0.0919939 | 1.52604 | 3.95565 | −1.00000 | −0.671202 | 0.0919939 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.10057 | 1.00000 | 1.71860 | 1.10057 | −2.91474 | −1.00000 | −1.78875 | −1.71860 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.405236 | 1.00000 | −2.02327 | −0.405236 | −4.23745 | −1.00000 | −2.83578 | 2.02327 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.03149 | 1.00000 | −1.41345 | −2.03149 | 3.91988 | −1.00000 | 1.12694 | 1.41345 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.39256 | 1.00000 | −1.77453 | −2.39256 | −4.11545 | −1.00000 | 2.72434 | 1.77453 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.75150 | 1.00000 | −4.36954 | −2.75150 | 2.12647 | −1.00000 | 4.57076 | 4.36954 | |||||||||||||||||||||||||||||||||||||||||||
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.bn | ✓ | 8 |
| 19.b | odd | 2 | 1 | 7942.2.a.bq | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7942.2.a.bn | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 7942.2.a.bq | yes | 8 | 19.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):
|
\( T_{3}^{8} + T_{3}^{7} - 19T_{3}^{6} - 14T_{3}^{5} + 116T_{3}^{4} + 65T_{3}^{3} - 235T_{3}^{2} - 120T_{3} + 80 \)
|
|
\( T_{5}^{8} + 3T_{5}^{7} - 18T_{5}^{6} - 45T_{5}^{5} + 76T_{5}^{4} + 180T_{5}^{3} - 87T_{5}^{2} - 216T_{5} - 19 \)
|
|
\( T_{13}^{8} + 3T_{13}^{7} - 48T_{13}^{6} - 45T_{13}^{5} + 606T_{13}^{4} - 220T_{13}^{3} - 1387T_{13}^{2} + 994T_{13} + 121 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 80 \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots - 19 \)
|
| $7$ |
\( T^{8} - 46 T^{6} + \cdots + 9616 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} + 3 T^{7} + \cdots + 121 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots + 5111 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} + T^{7} + \cdots - 304 \)
|
| $29$ |
\( T^{8} - 4 T^{7} + \cdots + 625 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots + 3856 \)
|
| $37$ |
\( T^{8} - 26 T^{7} + \cdots + 1737401 \)
|
| $41$ |
\( T^{8} + 17 T^{7} + \cdots - 8002819 \)
|
| $43$ |
\( T^{8} + 6 T^{7} + \cdots + 2349776 \)
|
| $47$ |
\( T^{8} - 111 T^{6} + \cdots - 11824 \)
|
| $53$ |
\( T^{8} + 49 T^{7} + \cdots + 193351 \)
|
| $59$ |
\( T^{8} + 15 T^{7} + \cdots + 14440000 \)
|
| $61$ |
\( T^{8} + 14 T^{7} + \cdots + 186481 \)
|
| $67$ |
\( T^{8} + 5 T^{7} + \cdots + 147856 \)
|
| $71$ |
\( T^{8} - T^{7} + \cdots + 42736 \)
|
| $73$ |
\( T^{8} + 10 T^{7} + \cdots - 336859 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots + 942400 \)
|
| $83$ |
\( T^{8} + 7 T^{7} + \cdots + 535616 \)
|
| $89$ |
\( T^{8} + 8 T^{7} + \cdots + 3006125 \)
|
| $97$ |
\( T^{8} + 54 T^{7} + \cdots + 117211 \)
|
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