Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,6,Mod(1,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, 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("750.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 750.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: | \(120.287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.33625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 19x^{2} + 14x + 76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 19x^{2} + 14x + 76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5\nu^{3} - 5\nu^{2} + 55\nu + 52 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 7\nu^{2} - \nu + 71 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 11\nu^{2} - 37\nu - 110 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} - 2\beta_{2} - \beta _1 + 1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} + 4\beta _1 + 99 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{3} - 21\beta_{2} - 27\beta _1 + 16 ) / 10 \)
|
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 |
|
−4.00000 | −9.00000 | 16.0000 | 0 | 36.0000 | −124.927 | −64.0000 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −9.00000 | 16.0000 | 0 | 36.0000 | −89.0490 | −64.0000 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −9.00000 | 16.0000 | 0 | 36.0000 | 89.9244 | −64.0000 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | −9.00000 | 16.0000 | 0 | 36.0000 | 166.052 | −64.0000 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 750.6.a.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 750.6.a.h | yes | 4 | |
| 5.c | odd | 4 | 2 | 750.6.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 750.6.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 750.6.a.h | yes | 4 | 5.b | even | 2 | 1 | |
| 750.6.c.c | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 42T_{7}^{3} - 28716T_{7}^{2} + 347472T_{7} + 166113856 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(750))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 42 T^{3} + \cdots + 166113856 \)
|
| $11$ |
\( T^{4} + \cdots - 3668594624 \)
|
| $13$ |
\( T^{4} + \cdots - 112628766704 \)
|
| $17$ |
\( T^{4} + \cdots - 1552877386559 \)
|
| $19$ |
\( T^{4} + \cdots + 10103065458775 \)
|
| $23$ |
\( T^{4} + \cdots - 8584915423769 \)
|
| $29$ |
\( T^{4} + \cdots - 142429670315600 \)
|
| $31$ |
\( T^{4} + \cdots - 62975346422249 \)
|
| $37$ |
\( T^{4} + \cdots - 39\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots + 46\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots - 38\!\cdots\!69 \)
|
| $53$ |
\( T^{4} + \cdots - 33\!\cdots\!99 \)
|
| $59$ |
\( T^{4} + \cdots - 74\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!21 \)
|
| $67$ |
\( T^{4} + \cdots - 93\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots - 23\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots - 21\!\cdots\!04 \)
|
| $79$ |
\( T^{4} + \cdots - 18\!\cdots\!25 \)
|
| $83$ |
\( T^{4} + \cdots + 47\!\cdots\!31 \)
|
| $89$ |
\( T^{4} + \cdots - 55\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 53\!\cdots\!24 \)
|
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