Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.78066700.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 18x^{3} + 7x^{2} + 30x - 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{4} + 3\nu^{3} + 27\nu^{2} - 20\nu + 10 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{4} + 24\nu^{3} - 84\nu^{2} - 320\nu + 70 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{4} - 12\nu^{3} - 138\nu^{2} + 140\nu + 155 ) / 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -32\nu^{4} + 18\nu^{3} + 582\nu^{2} + 100\nu - 785 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 4\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} - 18\beta _1 + 70 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{4} + 23\beta_{3} + 12\beta_{2} + 4\beta _1 + 70 ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 38\beta_{4} + 2\beta_{3} + 53\beta_{2} - 644\beta _1 + 2150 ) / 20 \)
|
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 |
|
−4.88936 | −3.00000 | 15.9059 | 0 | 14.6681 | 7.00000 | −38.6546 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.20666 | −3.00000 | −3.13065 | 0 | 6.61998 | 7.00000 | 24.5616 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.428319 | −3.00000 | −7.81654 | 0 | 1.28496 | 7.00000 | 6.77452 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.33774 | −3.00000 | 3.14050 | 0 | −10.0132 | 7.00000 | −16.2197 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.18660 | −3.00000 | 18.9008 | 0 | −15.5598 | 7.00000 | 56.5383 | 9.00000 | 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 | 525.4.a.x | 5 | |
| 3.b | odd | 2 | 1 | 1575.4.a.bo | 5 | ||
| 5.b | even | 2 | 1 | 525.4.a.w | 5 | ||
| 5.c | odd | 4 | 2 | 105.4.d.b | ✓ | 10 | |
| 15.d | odd | 2 | 1 | 1575.4.a.bp | 5 | ||
| 15.e | even | 4 | 2 | 315.4.d.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.d.b | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 315.4.d.b | 10 | 15.e | even | 4 | 2 | ||
| 525.4.a.w | 5 | 5.b | even | 2 | 1 | ||
| 525.4.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
| 1575.4.a.bo | 5 | 3.b | odd | 2 | 1 | ||
| 1575.4.a.bp | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):
|
\( T_{2}^{5} - T_{2}^{4} - 33T_{2}^{3} + 17T_{2}^{2} + 200T_{2} + 80 \)
|
|
\( T_{11}^{5} - 66T_{11}^{4} - 2100T_{11}^{3} + 140456T_{11}^{2} + 1472448T_{11} - 55852416 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots + 80 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( T^{5} - 66 T^{4} + \cdots - 55852416 \)
|
| $13$ |
\( T^{5} + 2 T^{4} + \cdots - 41380960 \)
|
| $17$ |
\( T^{5} - 108 T^{4} + \cdots - 232018688 \)
|
| $19$ |
\( T^{5} - 174 T^{4} + \cdots - 784374624 \)
|
| $23$ |
\( T^{5} + 116 T^{4} + \cdots + 488160000 \)
|
| $29$ |
\( T^{5} - 370 T^{4} + \cdots + 1150048 \)
|
| $31$ |
\( T^{5} + \cdots + 52737095200 \)
|
| $37$ |
\( T^{5} + \cdots + 315202167808 \)
|
| $41$ |
\( T^{5} + \cdots + 531402107648 \)
|
| $43$ |
\( T^{5} + \cdots - 132088069120 \)
|
| $47$ |
\( T^{5} + \cdots + 65728742400 \)
|
| $53$ |
\( T^{5} + \cdots + 462468251232 \)
|
| $59$ |
\( T^{5} + \cdots - 33724261457920 \)
|
| $61$ |
\( T^{5} + \cdots + 1105143174112 \)
|
| $67$ |
\( T^{5} + \cdots + 1579424171008 \)
|
| $71$ |
\( T^{5} + \cdots + 237519904000 \)
|
| $73$ |
\( T^{5} + \cdots - 1941655936032 \)
|
| $79$ |
\( T^{5} + \cdots + 43229481181184 \)
|
| $83$ |
\( T^{5} + \cdots + 128527499264 \)
|
| $89$ |
\( T^{5} + \cdots + 1125486676224 \)
|
| $97$ |
\( T^{5} + \cdots - 19868737339616 \)
|
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