Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,4,Mod(1,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, 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("460.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 460.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: | \(27.1408786026\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 44\nu^{2} - 155\nu - 246 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 4\nu^{3} + 50\nu^{2} + 155\nu + 54 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} + 3\nu^{2} - 52\nu - 103 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{3} - 6\beta_{2} + 52\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{4} + 56\beta_{3} + 124\beta_{2} - 53\beta _1 + 1626 \)
|
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 |
|
0 | −8.88744 | 0 | −5.00000 | 0 | −15.9808 | 0 | 51.9866 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.72746 | 0 | −5.00000 | 0 | 11.8626 | 0 | −13.1061 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.04116 | 0 | −5.00000 | 0 | 21.6113 | 0 | −22.8337 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.72467 | 0 | −5.00000 | 0 | −12.6137 | 0 | 5.77185 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.93138 | 0 | −5.00000 | 0 | 3.12066 | 0 | 8.18131 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(1\) |
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 | 460.4.a.b | ✓ | 5 |
| 4.b | odd | 2 | 1 | 1840.4.a.o | 5 | ||
| 5.b | even | 2 | 1 | 2300.4.a.c | 5 | ||
| 5.c | odd | 4 | 2 | 2300.4.c.d | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.4.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1840.4.a.o | 5 | 4.b | odd | 2 | 1 | ||
| 2300.4.a.c | 5 | 5.b | even | 2 | 1 | ||
| 2300.4.c.d | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 3T_{3}^{4} - 78T_{3}^{3} - 121T_{3}^{2} + 1211T_{3} + 2296 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(460))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 2296 \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} - 8 T^{4} + \cdots - 161268 \)
|
| $11$ |
\( T^{5} - 7 T^{4} + \cdots + 5888920 \)
|
| $13$ |
\( T^{5} - 5 T^{4} + \cdots - 566705734 \)
|
| $17$ |
\( T^{5} + 24 T^{4} + \cdots - 308713360 \)
|
| $19$ |
\( T^{5} + 13 T^{4} + \cdots + 75549672 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 3021093156 \)
|
| $31$ |
\( T^{5} + \cdots + 7527219975 \)
|
| $37$ |
\( T^{5} + 435 T^{4} + \cdots + 100253888 \)
|
| $41$ |
\( T^{5} + \cdots - 691331956043 \)
|
| $43$ |
\( T^{5} + \cdots - 11201296896 \)
|
| $47$ |
\( T^{5} + 572 T^{4} + \cdots - 201337632 \)
|
| $53$ |
\( T^{5} + \cdots - 55648798000 \)
|
| $59$ |
\( T^{5} + \cdots + 17454672233664 \)
|
| $61$ |
\( T^{5} + \cdots - 62953038747840 \)
|
| $67$ |
\( T^{5} + \cdots - 387904064832 \)
|
| $71$ |
\( T^{5} + \cdots + 8400024289995 \)
|
| $73$ |
\( T^{5} + \cdots - 7235588459208 \)
|
| $79$ |
\( T^{5} + \cdots - 21668449786880 \)
|
| $83$ |
\( T^{5} + \cdots + 67480737623744 \)
|
| $89$ |
\( T^{5} + \cdots - 60037186156800 \)
|
| $97$ |
\( T^{5} + \cdots + 6799511647880 \)
|
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