Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1400,4,Mod(1,1400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1400.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1400.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: | \(82.6026740080\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 60x^{2} + 52x + 200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{4} - x^{3} - 60x^{2} + 52x + 200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 52\nu - 25 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 30 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 3\beta_{2} + 52\beta _1 - 5 \)
|
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 | −7.56234 | 0 | 0 | 0 | 7.00000 | 0 | 30.1890 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.39269 | 0 | 0 | 0 | 7.00000 | 0 | −21.2751 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.47836 | 0 | 0 | 0 | 7.00000 | 0 | −24.8145 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 7.47667 | 0 | 0 | 0 | 7.00000 | 0 | 28.9005 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
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 | 1400.4.a.q | ✓ | 4 |
| 5.b | even | 2 | 1 | 1400.4.a.r | yes | 4 | |
| 5.c | odd | 4 | 2 | 1400.4.g.p | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1400.4.a.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1400.4.a.r | yes | 4 | 5.b | even | 2 | 1 | |
| 1400.4.g.p | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1400))\):
|
\( T_{3}^{4} + T_{3}^{3} - 60T_{3}^{2} - 52T_{3} + 200 \)
|
|
\( T_{11}^{4} + 10T_{11}^{3} - 1856T_{11}^{2} + 24030T_{11} + 28935 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 200 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} + 10 T^{3} + \cdots + 28935 \)
|
| $13$ |
\( T^{4} + 28 T^{3} + \cdots + 3007920 \)
|
| $17$ |
\( T^{4} + 15 T^{3} + \cdots + 12814272 \)
|
| $19$ |
\( T^{4} + 125 T^{3} + \cdots - 65461608 \)
|
| $23$ |
\( T^{4} + 73 T^{3} + \cdots - 969120 \)
|
| $29$ |
\( T^{4} - 21 T^{3} + \cdots + 72369666 \)
|
| $31$ |
\( T^{4} + 50 T^{3} + \cdots - 51604128 \)
|
| $37$ |
\( T^{4} + 79 T^{3} + \cdots + 5434056 \)
|
| $41$ |
\( T^{4} + 117 T^{3} + \cdots + 616060120 \)
|
| $43$ |
\( T^{4} + 135 T^{3} + \cdots - 10146720 \)
|
| $47$ |
\( T^{4} + 68 T^{3} + \cdots - 292385232 \)
|
| $53$ |
\( T^{4} + 786 T^{3} + \cdots + 326153760 \)
|
| $59$ |
\( T^{4} + \cdots - 37371767520 \)
|
| $61$ |
\( T^{4} + \cdots + 9606649104 \)
|
| $67$ |
\( T^{4} + \cdots + 2684171355 \)
|
| $71$ |
\( T^{4} + 779 T^{3} + \cdots + 27895950 \)
|
| $73$ |
\( T^{4} + \cdots + 160486015000 \)
|
| $79$ |
\( T^{4} + \cdots + 110517598386 \)
|
| $83$ |
\( T^{4} + \cdots + 9761245272 \)
|
| $89$ |
\( T^{4} + \cdots - 74608463320 \)
|
| $97$ |
\( T^{4} + \cdots + 104808835856 \)
|
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