Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [539,6,Mod(1,539)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("539.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 539 = 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 539.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: | \(86.4468788792\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 128x^{4} + 140x^{3} + 3963x^{2} - 750x - 30944 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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
\( x^{6} - 2x^{5} - 128x^{4} + 140x^{3} + 3963x^{2} - 750x - 30944 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 5\nu^{4} - 105\nu^{3} + 451\nu^{2} + 1738\nu - 5804 ) / 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 35\nu^{4} + 513\nu^{3} - 3289\nu^{2} - 8266\nu + 44528 ) / 168 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{5} + 65\nu^{4} + 1143\nu^{3} - 5827\nu^{2} - 18862\nu + 72128 ) / 168 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{5} + 107\nu^{4} + 1725\nu^{3} - 9841\nu^{2} - 26362\nu + 129728 ) / 168 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{5} + 3\beta_{4} + 8\beta_{3} - 2\beta_{2} + 70\beta _1 + 38 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + 107\beta_{4} - 77\beta_{3} + 115\beta_{2} + 145\beta _1 + 2941 \)
|
| \(\nu^{5}\) | \(=\) |
\( -555\beta_{5} + 399\beta_{4} + 906\beta_{3} - 58\beta_{2} + 5886\beta _1 + 5106 \)
|
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 |
|
−8.81728 | −13.3565 | 45.7445 | 12.6571 | 117.768 | 0 | −121.189 | −64.6027 | −111.601 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −5.16787 | 26.6951 | −5.29307 | −50.6287 | −137.957 | 0 | 192.726 | 469.630 | 261.643 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.29230 | −14.0815 | −26.7454 | −18.9706 | 32.2791 | 0 | 134.662 | −44.7111 | 43.4862 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 5.10369 | −13.8624 | −5.95240 | 88.5786 | −70.7492 | 0 | −193.697 | −50.8344 | 452.078 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.22843 | 7.19541 | −4.66354 | −37.8254 | 37.6207 | 0 | −191.693 | −191.226 | −197.767 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 9.94534 | 19.4099 | 66.9099 | 68.1889 | 193.038 | 0 | 347.191 | 133.744 | 678.162 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 539.6.a.h | 6 | |
| 7.b | odd | 2 | 1 | 77.6.a.d | ✓ | 6 | |
| 21.c | even | 2 | 1 | 693.6.a.i | 6 | ||
| 77.b | even | 2 | 1 | 847.6.a.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.6.a.d | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 539.6.a.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 693.6.a.i | 6 | 21.c | even | 2 | 1 | ||
| 847.6.a.f | 6 | 77.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(539))\):
|
\( T_{2}^{6} - 4T_{2}^{5} - 123T_{2}^{4} + 372T_{2}^{3} + 3610T_{2}^{2} - 7080T_{2} - 27720 \)
|
|
\( T_{3}^{6} - 12T_{3}^{5} - 783T_{3}^{4} + 3682T_{3}^{3} + 190164T_{3}^{2} + 96576T_{3} - 9720576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots - 27720 \)
|
| $3$ |
\( T^{6} - 12 T^{5} + \cdots - 9720576 \)
|
| $5$ |
\( T^{6} + \cdots - 2777378848 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T + 121)^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 306038598994688 \)
|
| $17$ |
\( T^{6} + \cdots - 26\!\cdots\!12 \)
|
| $19$ |
\( T^{6} + \cdots - 65\!\cdots\!44 \)
|
| $23$ |
\( T^{6} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 41\!\cdots\!12 \)
|
| $31$ |
\( T^{6} + \cdots + 45\!\cdots\!80 \)
|
| $37$ |
\( T^{6} + \cdots - 18\!\cdots\!48 \)
|
| $41$ |
\( T^{6} + \cdots - 40\!\cdots\!60 \)
|
| $43$ |
\( T^{6} + \cdots + 24\!\cdots\!08 \)
|
| $47$ |
\( T^{6} + \cdots + 28\!\cdots\!60 \)
|
| $53$ |
\( T^{6} + \cdots - 66\!\cdots\!60 \)
|
| $59$ |
\( T^{6} + \cdots - 10\!\cdots\!88 \)
|
| $61$ |
\( T^{6} + \cdots - 40\!\cdots\!60 \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!60 \)
|
| $71$ |
\( T^{6} + \cdots + 18\!\cdots\!88 \)
|
| $73$ |
\( T^{6} + \cdots - 11\!\cdots\!92 \)
|
| $79$ |
\( T^{6} + \cdots + 71\!\cdots\!60 \)
|
| $83$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 17\!\cdots\!36 \)
|
| $97$ |
\( T^{6} + \cdots - 63\!\cdots\!52 \)
|
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