Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7935,2,Mod(1,7935)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7935, 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("7935.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7935 = 3 \cdot 5 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7935.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: | \(63.3612940039\) |
| 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: |
\( x^{8} - 12x^{6} - 2x^{5} + 44x^{4} + 12x^{3} - 50x^{2} - 8x + 16 \)
|
| 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
\( x^{8} - 12x^{6} - 2x^{5} + 44x^{4} + 12x^{3} - 50x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 12\nu^{5} - 2\nu^{4} + 40\nu^{3} + 12\nu^{2} - 26\nu - 4 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 10\nu^{4} - 2\nu^{3} + 26\nu^{2} + 10\nu - 12 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 8\nu^{4} + 28\nu^{3} - 16\nu^{2} - 20\nu + 8 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} - 12\nu^{5} - 22\nu^{4} + 40\nu^{3} + 64\nu^{2} - 26\nu - 32 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - \beta_{5} - \beta_{4} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + \beta_{6} - 6\beta_{5} - 9\beta_{4} + \beta_{2} + 27\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} - 2\beta_{4} + 10\beta_{3} + 34\beta_{2} + 10\beta _1 + 76 \)
|
| \(\nu^{7}\) | \(=\) |
\( 44\beta_{7} + 12\beta_{6} - 32\beta_{5} - 64\beta_{4} + 2\beta_{3} + 12\beta_{2} + 152\beta _1 + 76 \)
|
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 |
|
−2.36527 | 1.00000 | 3.59451 | 1.00000 | −2.36527 | 0.0275916 | −3.77146 | 1.00000 | −2.36527 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.80545 | 1.00000 | 1.25965 | 1.00000 | −1.80545 | 4.01090 | 1.33666 | 1.00000 | −1.80545 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.37125 | 1.00000 | −0.119679 | 1.00000 | −1.37125 | −3.47202 | 2.90661 | 1.00000 | −1.37125 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.798063 | 1.00000 | −1.36310 | 1.00000 | −0.798063 | −0.435759 | 2.68396 | 1.00000 | −0.798063 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.635899 | 1.00000 | −1.59563 | 1.00000 | 0.635899 | 4.78252 | −2.28646 | 1.00000 | 0.635899 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.963105 | 1.00000 | −1.07243 | 1.00000 | 0.963105 | 2.92900 | −2.95907 | 1.00000 | 0.963105 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.20023 | 1.00000 | 2.84101 | 1.00000 | 2.20023 | −2.98493 | 1.85042 | 1.00000 | 2.20023 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.54080 | 1.00000 | 4.45566 | 1.00000 | 2.54080 | 1.14270 | 6.23934 | 1.00000 | 2.54080 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 7935.2.a.bi | yes | 8 |
| 23.b | odd | 2 | 1 | 7935.2.a.bh | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7935.2.a.bh | ✓ | 8 | 23.b | odd | 2 | 1 | |
| 7935.2.a.bi | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):
|
\( T_{2}^{8} - 12T_{2}^{6} - 2T_{2}^{5} + 44T_{2}^{4} + 12T_{2}^{3} - 50T_{2}^{2} - 8T_{2} + 16 \)
|
|
\( T_{7}^{8} - 6T_{7}^{7} - 17T_{7}^{6} + 130T_{7}^{5} + 30T_{7}^{4} - 712T_{7}^{3} + 380T_{7}^{2} + 280T_{7} - 8 \)
|
|
\( T_{11}^{8} - 12T_{11}^{7} + 17T_{11}^{6} + 242T_{11}^{5} - 769T_{11}^{4} - 526T_{11}^{3} + 3197T_{11}^{2} - 1364T_{11} + 148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 12 T^{6} + \cdots + 16 \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} - 6 T^{7} + \cdots - 8 \)
|
| $11$ |
\( T^{8} - 12 T^{7} + \cdots + 148 \)
|
| $13$ |
\( T^{8} - 4 T^{7} + \cdots - 5898 \)
|
| $17$ |
\( T^{8} - 20 T^{7} + \cdots + 49176 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots + 4 \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} - 120 T^{6} + \cdots - 172224 \)
|
| $31$ |
\( T^{8} - 2 T^{7} + \cdots + 13792 \)
|
| $37$ |
\( T^{8} - 2 T^{7} + \cdots - 329522 \)
|
| $41$ |
\( T^{8} - 28 T^{7} + \cdots - 63792 \)
|
| $43$ |
\( T^{8} - 4 T^{7} + \cdots - 2402 \)
|
| $47$ |
\( T^{8} + 12 T^{7} + \cdots + 100608 \)
|
| $53$ |
\( T^{8} - 6 T^{7} + \cdots - 45032 \)
|
| $59$ |
\( T^{8} - 2 T^{7} + \cdots + 357216 \)
|
| $61$ |
\( T^{8} - 32 T^{7} + \cdots + 1120717 \)
|
| $67$ |
\( T^{8} - 32 T^{7} + \cdots + 117312 \)
|
| $71$ |
\( T^{8} - 2 T^{7} + \cdots - 2015432 \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots - 395552 \)
|
| $79$ |
\( T^{8} + 36 T^{7} + \cdots + 876868 \)
|
| $83$ |
\( T^{8} - 10 T^{7} + \cdots - 118559816 \)
|
| $89$ |
\( T^{8} - 42 T^{7} + \cdots + 25463904 \)
|
| $97$ |
\( T^{8} - 16 T^{7} + \cdots + 91744 \)
|
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