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: |
\( x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 17\nu + 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} - 161\nu^{5} + 103\nu^{4} + 2404\nu^{3} - 1309\nu^{2} - 8982\nu + 1672 ) / 352 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} - 117\nu^{5} + 15\nu^{4} + 1172\nu^{3} - 121\nu^{2} - 2470\nu + 1232 ) / 88 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} + 34\nu^{6} + 229\nu^{5} - 827\nu^{4} - 2764\nu^{3} + 3553\nu^{2} + 14094\nu + 4312 ) / 352 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} - 26\nu^{6} + 239\nu^{5} + 1119\nu^{4} - 2980\nu^{3} - 11165\nu^{2} + 10570\nu + 15752 ) / 352 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 19\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{3} + 29\beta_{2} + 38\beta _1 + 203 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{6} + 4\beta_{5} - 8\beta_{4} + 34\beta_{3} + 87\beta_{2} + 433\beta _1 + 315 \)
|
| \(\nu^{6}\) | \(=\) |
\( 30\beta_{7} + 42\beta_{6} + 40\beta_{5} - 12\beta_{4} + 133\beta_{3} + 804\beta_{2} + 1299\beta _1 + 4489 \)
|
| \(\nu^{7}\) | \(=\) |
\( 93\beta_{7} + 101\beta_{6} + 207\beta_{5} - 320\beta_{4} + 1009\beta_{3} + 3043\beta_{2} + 11004\beta _1 + 11561 \)
|
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 |
|
−5.31088 | 0 | 20.2055 | 7.25200 | 0 | 7.00000 | −64.8218 | 0 | −38.5145 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.75652 | 0 | 14.6245 | −20.4224 | 0 | 7.00000 | −31.5093 | 0 | 97.1395 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.09275 | 0 | 1.56510 | 0.265436 | 0 | 7.00000 | 19.9015 | 0 | −0.820927 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.68078 | 0 | −5.17496 | 17.3718 | 0 | 7.00000 | 22.1443 | 0 | −29.1983 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.221922 | 0 | −7.95075 | −15.1615 | 0 | 7.00000 | 3.53982 | 0 | 3.36467 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.94273 | 0 | −4.22579 | 4.10121 | 0 | 7.00000 | −23.7514 | 0 | 7.96756 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.66328 | 0 | −0.906920 | 8.78457 | 0 | 7.00000 | −23.7217 | 0 | 23.3958 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.45684 | 0 | 11.8634 | −12.1911 | 0 | 7.00000 | 17.2186 | 0 | −54.3338 | |||||||||||||||||||||||||||||||||||||||||||
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.r | ✓ | 8 |
| 3.b | odd | 2 | 1 | 693.4.a.u | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.4.a.r | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 693.4.a.u | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{8} + 6T_{2}^{7} - 29T_{2}^{6} - 187T_{2}^{5} + 200T_{2}^{4} + 1497T_{2}^{3} - 292T_{2}^{2} - 3164T_{2} - 672 \)
|
|
\( T_{5}^{8} + 10 T_{5}^{7} - 572 T_{5}^{6} - 3572 T_{5}^{5} + 100333 T_{5}^{4} + 133342 T_{5}^{3} + \cdots - 4547664 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 6 T^{7} + \cdots - 672 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 10 T^{7} + \cdots - 4547664 \)
|
| $7$ |
\( (T - 7)^{8} \)
|
| $11$ |
\( (T + 11)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 746331240672 \)
|
| $17$ |
\( T^{8} + \cdots - 6029195028480 \)
|
| $19$ |
\( T^{8} + \cdots - 7962599853072 \)
|
| $23$ |
\( T^{8} + \cdots - 11\!\cdots\!64 \)
|
| $29$ |
\( T^{8} + \cdots - 314767624533120 \)
|
| $31$ |
\( T^{8} + \cdots + 21\!\cdots\!72 \)
|
| $37$ |
\( T^{8} + \cdots + 33\!\cdots\!72 \)
|
| $41$ |
\( T^{8} + \cdots - 20\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots - 99\!\cdots\!04 \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!20 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots - 18\!\cdots\!80 \)
|
| $61$ |
\( T^{8} + \cdots + 28\!\cdots\!80 \)
|
| $67$ |
\( T^{8} + \cdots - 59\!\cdots\!76 \)
|
| $71$ |
\( T^{8} + \cdots + 27\!\cdots\!52 \)
|
| $73$ |
\( T^{8} + \cdots + 29\!\cdots\!44 \)
|
| $79$ |
\( T^{8} + \cdots - 12\!\cdots\!60 \)
|
| $83$ |
\( T^{8} + \cdots - 14\!\cdots\!20 \)
|
| $89$ |
\( T^{8} + \cdots - 86\!\cdots\!40 \)
|
| $97$ |
\( T^{8} + \cdots - 70\!\cdots\!36 \)
|
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