Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [539,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("539.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 539 = 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(31.8020294931\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1536100192.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 23x^{4} + 162x^{2} - 352 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 23x^{4} + 162x^{2} - 352 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 19\nu^{3} + 78\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 15\nu^{3} - 42\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 17\nu^{3} + 68\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 13\nu^{2} + 32 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 17\nu^{2} + 62 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} - 2\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 15 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 18\beta_{3} + 5\beta_{2} - 26\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -13\beta_{5} + 17\beta_{4} + 131 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 186\beta_{3} + 17\beta_{2} - 306\beta_1 ) / 8 \)
|
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 |
|
−3.44055 | −7.39211 | 3.83740 | 20.8420 | 25.4329 | 0 | 14.3217 | 27.6433 | −71.7078 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.44055 | 7.39211 | 3.83740 | −20.8420 | −25.4329 | 0 | 14.3217 | 27.6433 | 71.7078 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.309984 | −5.98831 | −7.90391 | −5.38039 | 1.85628 | 0 | 4.92995 | 8.85990 | 1.66783 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.309984 | 5.98831 | −7.90391 | 5.38039 | −1.85628 | 0 | 4.92995 | 8.85990 | −1.66783 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.75054 | −3.39069 | 6.06651 | 5.35388 | −12.7169 | 0 | −7.25161 | −15.5032 | 20.0799 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 3.75054 | 3.39069 | 6.06651 | −5.35388 | 12.7169 | 0 | −7.25161 | −15.5032 | −20.0799 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 539.4.a.i | ✓ | 6 |
| 7.b | odd | 2 | 1 | inner | 539.4.a.i | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 539.4.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 539.4.a.i | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):
|
\( T_{2}^{3} - 13T_{2} - 4 \)
|
|
\( T_{3}^{6} - 102T_{3}^{4} + 3000T_{3}^{2} - 22528 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 13 T - 4)^{2} \)
|
| $3$ |
\( T^{6} - 102 T^{4} + \cdots - 22528 \)
|
| $5$ |
\( T^{6} - 492 T^{4} + \cdots - 360448 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} - 7910 T^{4} + \cdots - 282591232 \)
|
| $17$ |
\( T^{6} + \cdots - 4955281408 \)
|
| $19$ |
\( T^{6} + \cdots - 155468634112 \)
|
| $23$ |
\( (T^{3} - 108 T^{2} + \cdots + 746496)^{2} \)
|
| $29$ |
\( (T^{3} + 198 T^{2} + \cdots + 3496)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 4725563392 \)
|
| $37$ |
\( (T^{3} + 344 T^{2} + \cdots - 9888464)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 536614922770432 \)
|
| $43$ |
\( (T^{3} + 194 T^{2} + \cdots - 9686944)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 7892989468672 \)
|
| $53$ |
\( (T^{3} + 1136 T^{2} + \cdots + 42964208)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 34\!\cdots\!68 \)
|
| $61$ |
\( T^{6} + \cdots - 92\!\cdots\!08 \)
|
| $67$ |
\( (T^{3} + 904 T^{2} + \cdots + 12413312)^{2} \)
|
| $71$ |
\( (T^{3} + 252 T^{2} + \cdots + 25621504)^{2} \)
|
| $73$ |
\( T^{6} + \cdots - 39\!\cdots\!12 \)
|
| $79$ |
\( (T^{3} - 90 T^{2} + \cdots + 1955584)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 46\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots - 382505895067648 \)
|
| $97$ |
\( T^{6} + \cdots - 450576845897728 \)
|
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