Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8450,2,Mod(1,8450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8450.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8450.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: | \(67.4735897080\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4752.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 650) |
| 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
\( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
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 |
|
1.00000 | −1.49551 | 1.00000 | 0 | −1.49551 | 1.40072 | 1.00000 | −0.763457 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −0.219687 | 1.00000 | 0 | −0.219687 | 1.81988 | 1.00000 | −2.95174 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.21969 | 1.00000 | 0 | 1.21969 | −3.55193 | 1.00000 | −1.51236 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 2.49551 | 1.00000 | 0 | 2.49551 | 0.331331 | 1.00000 | 3.22756 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 8450.2.a.co | 4 | |
| 5.b | even | 2 | 1 | 8450.2.a.ch | 4 | ||
| 13.b | even | 2 | 1 | 8450.2.a.ck | 4 | ||
| 13.f | odd | 12 | 2 | 650.2.m.b | ✓ | 8 | |
| 65.d | even | 2 | 1 | 8450.2.a.cl | 4 | ||
| 65.o | even | 12 | 2 | 650.2.n.c | 8 | ||
| 65.s | odd | 12 | 2 | 650.2.m.d | yes | 8 | |
| 65.t | even | 12 | 2 | 650.2.n.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.2.m.b | ✓ | 8 | 13.f | odd | 12 | 2 | |
| 650.2.m.d | yes | 8 | 65.s | odd | 12 | 2 | |
| 650.2.n.c | 8 | 65.o | even | 12 | 2 | ||
| 650.2.n.f | 8 | 65.t | even | 12 | 2 | ||
| 8450.2.a.ch | 4 | 5.b | even | 2 | 1 | ||
| 8450.2.a.ck | 4 | 13.b | even | 2 | 1 | ||
| 8450.2.a.cl | 4 | 65.d | even | 2 | 1 | ||
| 8450.2.a.co | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 3T_{3}^{2} + 4T_{3} + 1 \)
|
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\( T_{7}^{4} - 9T_{7}^{2} + 12T_{7} - 3 \)
|
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\( T_{11}^{4} - 6T_{11}^{3} - 15T_{11}^{2} + 144T_{11} - 207 \)
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\( T_{17}^{4} - 6T_{17}^{3} - 9T_{17}^{2} + 54T_{17} - 27 \)
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\( T_{31}^{4} - 12T_{31}^{3} + 12T_{31}^{2} + 240T_{31} - 624 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 9 T^{2} + \cdots - 3 \)
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| $11$ |
\( T^{4} - 6 T^{3} + \cdots - 207 \)
|
| $13$ |
\( T^{4} \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots - 27 \)
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| $19$ |
\( T^{4} - 6 T^{3} + \cdots - 291 \)
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| $23$ |
\( T^{4} - 42 T^{2} + \cdots - 39 \)
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| $29$ |
\( T^{4} - 12 T^{3} + \cdots - 759 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots - 624 \)
|
| $37$ |
\( T^{4} - 15 T^{2} + \cdots + 9 \)
|
| $41$ |
\( T^{4} - 12 T^{3} + \cdots - 747 \)
|
| $43$ |
\( T^{4} + 8 T^{3} + \cdots - 11 \)
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| $47$ |
\( T^{4} - 12 T^{3} + \cdots - 531 \)
|
| $53$ |
\( T^{4} - 6 T^{3} + \cdots - 507 \)
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| $59$ |
\( T^{4} - 18 T^{3} + \cdots - 423 \)
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| $61$ |
\( T^{4} - 10 T^{3} + \cdots - 2243 \)
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| $67$ |
\( T^{4} - 6 T^{3} + \cdots + 2841 \)
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| $71$ |
\( T^{4} + 12 T^{3} + \cdots + 1296 \)
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| $73$ |
\( T^{4} + 12 T^{3} + \cdots + 144 \)
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| $79$ |
\( T^{4} + 2 T^{3} + \cdots + 169 \)
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| $83$ |
\( T^{4} + 6 T^{3} + \cdots - 99 \)
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| $89$ |
\( T^{4} - 36 T^{3} + \cdots - 3447 \)
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| $97$ |
\( T^{4} + 36 T^{3} + \cdots + 2733 \)
|
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