Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,4,Mod(1,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("123.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(7.25723493071\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.994300.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 21x^{2} - 2x + 82 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - 21x^{2} - 2x + 82 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 12\nu + 10 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 12\beta _1 + 1 \)
|
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 |
|
−4.06592 | −3.00000 | 8.53174 | 2.57360 | 12.1978 | −12.8786 | −2.16200 | 9.00000 | −10.4641 | ||||||||||||||||||||||||||||||
| 1.2 | −2.18816 | −3.00000 | −3.21195 | 4.94603 | 6.56449 | 18.0847 | 24.5336 | 9.00000 | −10.8227 | |||||||||||||||||||||||||||||||
| 1.3 | 2.37757 | −3.00000 | −2.34715 | 2.45149 | −7.13272 | −7.77982 | −24.6011 | 9.00000 | 5.82860 | |||||||||||||||||||||||||||||||
| 1.4 | 3.87651 | −3.00000 | 7.02736 | −18.9711 | −11.6295 | −11.4263 | −3.77046 | 9.00000 | −73.5418 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(41\) | \(-1\) |
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 | 123.4.a.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 369.4.a.d | 4 | ||
| 4.b | odd | 2 | 1 | 1968.4.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.4.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 369.4.a.d | 4 | 3.b | odd | 2 | 1 | ||
| 1968.4.a.n | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 21T_{2}^{2} + 2T_{2} + 82 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(123))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 21 T^{2} + \cdots + 82 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} + 9 T^{3} + \cdots - 592 \)
|
| $7$ |
\( T^{4} + 14 T^{3} + \cdots - 20704 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 645144 \)
|
| $13$ |
\( T^{4} + 155 T^{3} + \cdots + 1322712 \)
|
| $17$ |
\( T^{4} + 27 T^{3} + \cdots - 920886 \)
|
| $19$ |
\( T^{4} + 119 T^{3} + \cdots - 34918560 \)
|
| $23$ |
\( T^{4} + 70 T^{3} + \cdots + 273831552 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots - 26685040 \)
|
| $31$ |
\( T^{4} + 163 T^{3} + \cdots + 422741768 \)
|
| $37$ |
\( T^{4} + \cdots + 1255391028 \)
|
| $41$ |
\( (T - 41)^{4} \)
|
| $43$ |
\( T^{4} + 266 T^{3} + \cdots + 747065360 \)
|
| $47$ |
\( T^{4} + 126 T^{3} + \cdots + 23768208 \)
|
| $53$ |
\( T^{4} + 528 T^{3} + \cdots + 642651216 \)
|
| $59$ |
\( T^{4} + \cdots - 26277212160 \)
|
| $61$ |
\( T^{4} + \cdots - 6520286676 \)
|
| $67$ |
\( T^{4} + \cdots - 3914497456 \)
|
| $71$ |
\( T^{4} + \cdots - 135461804292 \)
|
| $73$ |
\( T^{4} + \cdots + 6137343194 \)
|
| $79$ |
\( T^{4} + \cdots + 557704979200 \)
|
| $83$ |
\( T^{4} + \cdots + 478090315936 \)
|
| $89$ |
\( T^{4} + \cdots + 389050136680 \)
|
| $97$ |
\( T^{4} + 1548 T^{3} + \cdots - 301253280 \)
|
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