Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1503,2,Mod(1,1503)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1503, 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("1503.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1503 = 3^{2} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1503.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: | \(12.0015154238\) |
| 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: |
\( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 501) |
| 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
\( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{7} + 38\nu^{5} + 2\nu^{4} - 147\nu^{3} - 11\nu^{2} + 168\nu + 15 ) / 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 15\nu^{5} + 3\nu^{4} - 70\nu^{3} - 20\nu^{2} + 98\nu + 19 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{7} - 7\nu^{6} - 16\nu^{5} + 64\nu^{4} + 21\nu^{3} - 135\nu^{2} + 21\nu + 25 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 7\nu^{6} + 24\nu^{5} - 61\nu^{4} - 28\nu^{3} + 122\nu^{2} - 42\nu - 27 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 18\nu^{4} - 25\nu^{3} + 37\nu^{2} + 8\nu - 5 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} - 7\nu^{6} - 39\nu^{5} + 58\nu^{4} + 98\nu^{3} - 95\nu^{2} - 56\nu - 13 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{5} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} - \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{7} + 7\beta_{5} + \beta_{4} + 8\beta_{3} - 2\beta_{2} - \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 9\beta_{6} + 8\beta_{4} + 2\beta_{3} - 9\beta_{2} + 28\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37\beta_{7} + \beta_{6} + 45\beta_{5} + 8\beta_{4} + 57\beta_{3} - 21\beta_{2} - 8\beta _1 + 84 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13\beta_{7} + 65\beta_{6} + \beta_{5} + 53\beta_{4} + 27\beta_{3} - 71\beta_{2} + 165\beta _1 + 4 \)
|
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 |
|
−2.63734 | 0 | 4.95555 | 1.29727 | 0 | −2.24503 | −7.79478 | 0 | −3.42135 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.18779 | 0 | 2.78642 | −2.52133 | 0 | −0.307143 | −1.72052 | 0 | 5.51613 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.60389 | 0 | 0.572476 | 0.122001 | 0 | 3.47013 | 2.28960 | 0 | −0.195676 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.23856 | 0 | −0.465971 | −3.06821 | 0 | −2.07562 | 3.05425 | 0 | 3.80015 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0452510 | 0 | −1.99795 | −1.52778 | 0 | 0.428515 | −0.180911 | 0 | −0.0691336 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.459587 | 0 | −1.78878 | 2.63865 | 0 | −0.604857 | −1.74127 | 0 | 1.21269 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.71120 | 0 | 0.928220 | −2.45529 | 0 | 2.43610 | −1.83403 | 0 | −4.20151 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.45154 | 0 | 4.01004 | −1.48532 | 0 | −5.10210 | 4.92768 | 0 | −3.64131 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(167\) | \(-1\) |
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 | 1503.2.a.e | 8 | |
| 3.b | odd | 2 | 1 | 501.2.a.e | ✓ | 8 | |
| 12.b | even | 2 | 1 | 8016.2.a.x | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 501.2.a.e | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 1503.2.a.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 8016.2.a.x | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{7} - 8T_{2}^{6} - 28T_{2}^{5} + 9T_{2}^{4} + 64T_{2}^{3} + 17T_{2}^{2} - 23T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1503))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{7} + \cdots + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 7 T^{7} + \cdots - 18 \)
|
| $7$ |
\( T^{8} + 4 T^{7} + \cdots - 16 \)
|
| $11$ |
\( T^{8} + 13 T^{7} + \cdots - 1596 \)
|
| $13$ |
\( T^{8} - 37 T^{6} + \cdots + 56 \)
|
| $17$ |
\( T^{8} + 11 T^{7} + \cdots + 7822 \)
|
| $19$ |
\( T^{8} - 12 T^{7} + \cdots + 4384 \)
|
| $23$ |
\( T^{8} + 7 T^{7} + \cdots + 384 \)
|
| $29$ |
\( T^{8} + T^{7} + \cdots + 88 \)
|
| $31$ |
\( T^{8} + 2 T^{7} + \cdots - 1017696 \)
|
| $37$ |
\( T^{8} + 9 T^{7} + \cdots - 42872 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots - 859446 \)
|
| $43$ |
\( T^{8} - 2 T^{7} + \cdots + 114 \)
|
| $47$ |
\( T^{8} + 17 T^{7} + \cdots - 68732 \)
|
| $53$ |
\( T^{8} + 9 T^{7} + \cdots + 20326 \)
|
| $59$ |
\( T^{8} + 29 T^{7} + \cdots - 21056 \)
|
| $61$ |
\( T^{8} + 12 T^{7} + \cdots + 51244 \)
|
| $67$ |
\( T^{8} - 177 T^{6} + \cdots - 1226366 \)
|
| $71$ |
\( T^{8} + 13 T^{7} + \cdots + 32816 \)
|
| $73$ |
\( T^{8} + 20 T^{7} + \cdots - 9386184 \)
|
| $79$ |
\( T^{8} - 8 T^{7} + \cdots + 39988958 \)
|
| $83$ |
\( T^{8} + 33 T^{7} + \cdots - 2562224 \)
|
| $89$ |
\( T^{8} + 4 T^{7} + \cdots - 216312 \)
|
| $97$ |
\( T^{8} + 31 T^{7} + \cdots - 24584 \)
|
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