Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,4,Mod(1,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.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: | \(40.8883236340\) |
| 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: |
\( x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{8} - 2x^{7} - 45x^{6} + 77x^{5} + 540x^{4} - 915x^{3} - 1452x^{2} + 2660x - 672 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{7} - 13\nu^{6} + 338\nu^{5} + 655\nu^{4} - 4289\nu^{3} - 6920\nu^{2} + 13644\nu + 8736 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{7} - 9\nu^{6} - 278\nu^{5} + 483\nu^{4} + 4435\nu^{3} - 7528\nu^{2} - 17828\nu + 23456 ) / 512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17\nu^{7} - 5\nu^{6} - 766\nu^{5} + 55\nu^{4} + 8807\nu^{3} - 1864\nu^{2} - 22420\nu + 9504 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -15\nu^{7} + 27\nu^{6} + 706\nu^{5} - 937\nu^{4} - 8953\nu^{3} + 8632\nu^{2} + 23788\nu - 15072 ) / 512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 46\nu^{5} + 31\nu^{4} + 571\nu^{3} - 344\nu^{2} - 1780\nu + 944 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} + 21\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 30\beta_{2} + 11\beta _1 + 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( 34\beta_{7} + 38\beta_{6} - 26\beta_{5} - 4\beta_{4} + 8\beta_{3} + 45\beta_{2} + 526\beta _1 + 182 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7\beta_{7} + 103\beta_{6} + 81\beta_{5} + 84\beta_{4} + 68\beta_{3} + 863\beta_{2} + 653\beta _1 + 6463 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1016\beta_{7} + 1218\beta_{6} - 606\beta_{5} - 162\beta_{4} + 374\beta_{3} + 1776\beta_{2} + 14297\beta _1 + 9512 \)
|
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 |
|
−5.62357 | 0 | 23.6246 | −7.12019 | 0 | −7.00000 | −87.8658 | 0 | 40.0409 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.50363 | 0 | 4.27544 | 15.3058 | 0 | −7.00000 | 13.0495 | 0 | −53.6258 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.21621 | 0 | −3.08840 | −22.0394 | 0 | −7.00000 | 24.5743 | 0 | 48.8440 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.22364 | 0 | −6.50270 | 11.7152 | 0 | −7.00000 | 17.7461 | 0 | −14.3353 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.315819 | 0 | −7.90026 | −5.90817 | 0 | −7.00000 | 5.02161 | 0 | 1.86591 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09348 | 0 | −3.61736 | 6.99302 | 0 | −7.00000 | −24.3207 | 0 | 14.6397 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.85529 | 0 | 6.86327 | 5.82801 | 0 | −7.00000 | −4.38244 | 0 | 22.4687 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.93411 | 0 | 16.3455 | −14.7743 | 0 | −7.00000 | 41.1774 | 0 | −72.8980 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 693.4.a.s | ✓ | 8 |
| 3.b | odd | 2 | 1 | 693.4.a.t | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.4.a.s | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 693.4.a.t | yes | 8 | 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{8} + 2T_{2}^{7} - 45T_{2}^{6} - 77T_{2}^{5} + 540T_{2}^{4} + 915T_{2}^{3} - 1452T_{2}^{2} - 2660T_{2} - 672 \)
|
|
\( T_{5}^{8} + 10 T_{5}^{7} - 572 T_{5}^{6} - 3132 T_{5}^{5} + 100813 T_{5}^{4} + 218046 T_{5}^{3} + \cdots + 100101776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots - 672 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 10 T^{7} + \cdots + 100101776 \)
|
| $7$ |
\( (T + 7)^{8} \)
|
| $11$ |
\( (T - 11)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 1312663328 \)
|
| $17$ |
\( T^{8} + \cdots - 97682010484736 \)
|
| $19$ |
\( T^{8} + \cdots - 11\!\cdots\!16 \)
|
| $23$ |
\( T^{8} + \cdots + 33\!\cdots\!24 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!24 \)
|
| $31$ |
\( T^{8} + \cdots - 10\!\cdots\!76 \)
|
| $37$ |
\( T^{8} + \cdots + 39\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots - 87\!\cdots\!88 \)
|
| $43$ |
\( T^{8} + \cdots + 85\!\cdots\!12 \)
|
| $47$ |
\( T^{8} + \cdots + 65\!\cdots\!84 \)
|
| $53$ |
\( T^{8} + \cdots + 51\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots + 17\!\cdots\!92 \)
|
| $61$ |
\( T^{8} + \cdots - 10\!\cdots\!12 \)
|
| $67$ |
\( T^{8} + \cdots + 38\!\cdots\!24 \)
|
| $71$ |
\( T^{8} + \cdots + 91\!\cdots\!04 \)
|
| $73$ |
\( T^{8} + \cdots + 51\!\cdots\!32 \)
|
| $79$ |
\( T^{8} + \cdots - 51\!\cdots\!32 \)
|
| $83$ |
\( T^{8} + \cdots + 50\!\cdots\!84 \)
|
| $89$ |
\( T^{8} + \cdots - 99\!\cdots\!72 \)
|
| $97$ |
\( T^{8} + \cdots + 94\!\cdots\!88 \)
|
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