Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,4,Mod(1,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, 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("416.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.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: | \(24.5447945624\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.24965.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 41x + 60 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 41x + 60 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 27 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{2} - \beta _1 + 53 ) / 2 \)
|
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 | −6.76338 | 0 | 15.7433 | 0 | −5.19639 | 0 | 18.7433 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 4.30197 | 0 | −11.4931 | 0 | 16.0803 | 0 | −8.49309 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 6.46141 | 0 | 11.7498 | 0 | −36.8839 | 0 | 14.7498 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 416.4.a.f | yes | 3 |
| 4.b | odd | 2 | 1 | 416.4.a.e | ✓ | 3 | |
| 8.b | even | 2 | 1 | 832.4.a.ba | 3 | ||
| 8.d | odd | 2 | 1 | 832.4.a.bd | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.4.a.e | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 416.4.a.f | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 832.4.a.ba | 3 | 8.b | even | 2 | 1 | ||
| 832.4.a.bd | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 4T_{3}^{2} - 45T_{3} + 188 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(416))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 4 T^{2} + \cdots + 188 \)
|
| $5$ |
\( T^{3} - 16 T^{2} + \cdots + 2126 \)
|
| $7$ |
\( T^{3} + 26 T^{2} + \cdots - 3082 \)
|
| $11$ |
\( T^{3} + 94 T^{2} + \cdots + 7080 \)
|
| $13$ |
\( (T + 13)^{3} \)
|
| $17$ |
\( T^{3} + 176 T^{2} + \cdots - 399142 \)
|
| $19$ |
\( T^{3} + 42 T^{2} + \cdots + 336312 \)
|
| $23$ |
\( T^{3} - 192 T^{2} + \cdots + 414720 \)
|
| $29$ |
\( T^{3} - 270 T^{2} + \cdots + 1047000 \)
|
| $31$ |
\( T^{3} + 462 T^{2} + \cdots - 842880 \)
|
| $37$ |
\( T^{3} + 416 T^{2} + \cdots - 27637826 \)
|
| $41$ |
\( T^{3} + 214 T^{2} + \cdots - 13672448 \)
|
| $43$ |
\( T^{3} - 160 T^{2} + \cdots + 17140424 \)
|
| $47$ |
\( T^{3} + 114 T^{2} + \cdots - 5926770 \)
|
| $53$ |
\( T^{3} - 558 T^{2} + \cdots + 1625760 \)
|
| $59$ |
\( T^{3} + 198 T^{2} + \cdots - 2425464 \)
|
| $61$ |
\( T^{3} - 334 T^{2} + \cdots + 12563968 \)
|
| $67$ |
\( T^{3} + 170 T^{2} + \cdots - 492552 \)
|
| $71$ |
\( T^{3} + 1774 T^{2} + \cdots + 136314162 \)
|
| $73$ |
\( T^{3} + 1386 T^{2} + \cdots + 82276760 \)
|
| $79$ |
\( T^{3} + 964 T^{2} + \cdots - 154301760 \)
|
| $83$ |
\( T^{3} + 1130 T^{2} + \cdots - 129894528 \)
|
| $89$ |
\( T^{3} + 498 T^{2} + \cdots - 39317064 \)
|
| $97$ |
\( T^{3} - 694 T^{2} + \cdots + 963864 \)
|
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