Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,4,Mod(1,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.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: | \(42.5993790241\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6719782761.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 75x^{4} + 135x^{3} + 1857x^{2} - 1425x - 14797 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 19 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 75x^{4} + 135x^{3} + 1857x^{2} - 1425x - 14797 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -129\nu^{5} + 1065\nu^{4} + 4651\nu^{3} - 44727\nu^{2} - 50925\nu + 474415 ) / 12329 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 354\nu^{5} - 3496\nu^{4} - 9896\nu^{3} + 144530\nu^{2} + 30507\nu - 1467321 ) / 12329 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 430\nu^{5} - 3550\nu^{4} - 19613\nu^{3} + 173748\nu^{2} + 219066\nu - 2029337 ) / 12329 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -501\nu^{5} + 3276\nu^{4} + 22364\nu^{3} - 141021\nu^{2} - 244514\nu + 1477213 ) / 12329 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -969\nu^{5} + 6853\nu^{4} + 40671\nu^{3} - 280062\nu^{2} - 428405\nu + 2727264 ) / 12329 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{4} - 3\beta_{2} - 16\beta _1 + 19 ) / 19 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{5} - 28\beta_{4} + 4\beta_{2} - 23\beta _1 + 513 ) / 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 114\beta_{5} - 144\beta_{4} - 57\beta_{3} - 12\beta_{2} - 520\beta _1 + 1235 ) / 19 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1292\beta_{5} - 2165\beta_{4} - 285\beta_{3} + 255\beta_{2} - 1547\beta _1 + 16568 ) / 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 8189\beta_{5} - 14147\beta_{4} - 4408\beta_{3} + 1470\beta_{2} - 19045\beta _1 + 65816 ) / 19 \)
|
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 |
|
−2.00000 | −6.57964 | 4.00000 | −11.3499 | 13.1593 | −8.71985 | −8.00000 | 16.2917 | 22.6998 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.82555 | 4.00000 | −15.1983 | 7.65110 | 1.61327 | −8.00000 | −12.3652 | 30.3966 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −0.235037 | 4.00000 | 11.0362 | 0.470075 | −18.5336 | −8.00000 | −26.9448 | −22.0725 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 4.98337 | 4.00000 | 2.34989 | −9.96675 | 2.84316 | −8.00000 | −2.16599 | −4.69977 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 5.78366 | 4.00000 | 6.19832 | −11.5673 | 11.2225 | −8.00000 | 6.45071 | −12.3966 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 8.87319 | 4.00000 | −20.0362 | −17.7464 | −9.42541 | −8.00000 | 51.7336 | 40.0725 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
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 | 722.4.a.o | 6 | |
| 19.b | odd | 2 | 1 | 722.4.a.p | 6 | ||
| 19.f | odd | 18 | 2 | 38.4.e.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.e.a | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 722.4.a.o | 6 | 1.a | even | 1 | 1 | trivial | |
| 722.4.a.p | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):
|
\( T_{3}^{6} - 9T_{3}^{5} - 57T_{3}^{4} + 531T_{3}^{3} + 597T_{3}^{2} - 6327T_{3} - 1513 \)
|
|
\( T_{5}^{6} + 27T_{5}^{5} - 99T_{5}^{4} - 5427T_{5}^{3} + 1539T_{5}^{2} + 263169T_{5} - 555579 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 9 T^{5} + \cdots - 1513 \)
|
| $5$ |
\( T^{6} + 27 T^{5} + \cdots - 555579 \)
|
| $7$ |
\( T^{6} + 21 T^{5} + \cdots - 78409 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 236331747 \)
|
| $13$ |
\( T^{6} - 24 T^{5} + \cdots - 50265519 \)
|
| $17$ |
\( T^{6} + \cdots + 6425825733 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 1288970568 \)
|
| $29$ |
\( T^{6} + \cdots - 655147831257 \)
|
| $31$ |
\( T^{6} + \cdots - 3586719447 \)
|
| $37$ |
\( T^{6} + \cdots - 20292669795592 \)
|
| $41$ |
\( T^{6} + \cdots - 6229329066963 \)
|
| $43$ |
\( T^{6} + \cdots + 711315853703439 \)
|
| $47$ |
\( T^{6} + \cdots - 108448526272959 \)
|
| $53$ |
\( T^{6} + \cdots + 32216889916737 \)
|
| $59$ |
\( T^{6} + \cdots - 19\!\cdots\!53 \)
|
| $61$ |
\( T^{6} + \cdots + 16\!\cdots\!93 \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!68 \)
|
| $71$ |
\( T^{6} + \cdots + 90\!\cdots\!52 \)
|
| $73$ |
\( T^{6} + \cdots - 45\!\cdots\!92 \)
|
| $79$ |
\( T^{6} + \cdots - 79\!\cdots\!17 \)
|
| $83$ |
\( T^{6} + \cdots + 65\!\cdots\!57 \)
|
| $89$ |
\( T^{6} + \cdots + 89\!\cdots\!13 \)
|
| $97$ |
\( T^{6} + \cdots - 10\!\cdots\!17 \)
|
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