Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,6,Mod(1,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("525.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.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: | \(84.2015054018\) |
| 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: |
\( x^{6} - 2x^{5} - 159x^{4} + 185x^{3} + 6300x^{2} + 125x - 39450 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 5^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 159x^{4} + 185x^{3} + 6300x^{2} + 125x - 39450 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 53 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 82\nu - 41 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 110\nu^{2} - 8\nu + 1500 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 128\nu^{3} + 313\nu^{2} + 3015\nu - 3366 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 53 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 82\beta _1 + 41 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{4} + 110\beta_{2} + 118\beta _1 + 4330 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 9\beta_{4} + 128\beta_{3} + 17\beta_{2} + 7522\beta _1 + 5015 \)
|
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 |
|
−9.23147 | 9.00000 | 53.2201 | 0 | −83.0833 | −49.0000 | −195.893 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −7.14446 | 9.00000 | 19.0433 | 0 | −64.3001 | −49.0000 | 92.5687 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.61865 | 9.00000 | −29.3800 | 0 | −14.5678 | −49.0000 | 99.3526 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.98787 | 9.00000 | −16.0969 | 0 | 35.8908 | −49.0000 | −191.804 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 7.12018 | 9.00000 | 18.6970 | 0 | 64.0817 | −49.0000 | −94.7196 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.8865 | 9.00000 | 86.5165 | 0 | 97.9787 | −49.0000 | 593.495 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 525.6.a.s | yes | 6 |
| 5.b | even | 2 | 1 | 525.6.a.r | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 525.6.d.p | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.6.a.r | ✓ | 6 | 5.b | even | 2 | 1 | |
| 525.6.a.s | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 525.6.d.p | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 4T_{2}^{5} - 154T_{2}^{4} + 451T_{2}^{3} + 5896T_{2}^{2} - 12640T_{2} - 33000 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(525))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots - 33000 \)
|
| $3$ |
\( (T - 9)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T + 49)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 846458085475044 \)
|
| $13$ |
\( T^{6} + \cdots + 619063177016400 \)
|
| $17$ |
\( T^{6} + \cdots + 40\!\cdots\!04 \)
|
| $19$ |
\( T^{6} + \cdots + 25\!\cdots\!84 \)
|
| $23$ |
\( T^{6} + \cdots - 11\!\cdots\!25 \)
|
| $29$ |
\( T^{6} + \cdots - 40\!\cdots\!91 \)
|
| $31$ |
\( T^{6} + \cdots + 53\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 37\!\cdots\!36 \)
|
| $41$ |
\( T^{6} + \cdots - 47\!\cdots\!16 \)
|
| $43$ |
\( T^{6} + \cdots - 41\!\cdots\!25 \)
|
| $47$ |
\( T^{6} + \cdots + 62\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 58\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 34\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 68\!\cdots\!56 \)
|
| $71$ |
\( T^{6} + \cdots + 46\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 71\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots - 27\!\cdots\!44 \)
|
| $83$ |
\( T^{6} + \cdots + 38\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots + 21\!\cdots\!24 \)
|
| $97$ |
\( T^{6} + \cdots - 10\!\cdots\!24 \)
|
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