Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8400,2,Mod(1,8400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 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("8400.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8400.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: | \(67.0743376979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 525) |
| 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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.
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 | 1.00000 | 0 | 0 | 0 | −1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | 0 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 8400.2.a.da | 2 | |
| 4.b | odd | 2 | 1 | 525.2.a.e | ✓ | 2 | |
| 5.b | even | 2 | 1 | 8400.2.a.cy | 2 | ||
| 12.b | even | 2 | 1 | 1575.2.a.v | 2 | ||
| 20.d | odd | 2 | 1 | 525.2.a.i | yes | 2 | |
| 20.e | even | 4 | 2 | 525.2.d.e | 4 | ||
| 28.d | even | 2 | 1 | 3675.2.a.r | 2 | ||
| 60.h | even | 2 | 1 | 1575.2.a.l | 2 | ||
| 60.l | odd | 4 | 2 | 1575.2.d.f | 4 | ||
| 140.c | even | 2 | 1 | 3675.2.a.bh | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.2.a.e | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 525.2.a.i | yes | 2 | 20.d | odd | 2 | 1 | |
| 525.2.d.e | 4 | 20.e | even | 4 | 2 | ||
| 1575.2.a.l | 2 | 60.h | even | 2 | 1 | ||
| 1575.2.a.v | 2 | 12.b | even | 2 | 1 | ||
| 1575.2.d.f | 4 | 60.l | odd | 4 | 2 | ||
| 3675.2.a.r | 2 | 28.d | even | 2 | 1 | ||
| 3675.2.a.bh | 2 | 140.c | even | 2 | 1 | ||
| 8400.2.a.cy | 2 | 5.b | even | 2 | 1 | ||
| 8400.2.a.da | 2 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):
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\( T_{11}^{2} - 2T_{11} - 19 \)
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\( T_{13}^{2} + 6T_{13} + 4 \)
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\( T_{17}^{2} - 2T_{17} - 44 \)
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\( T_{19}^{2} - 2T_{19} - 4 \)
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\( T_{23} - 5 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( (T - 1)^{2} \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( (T + 1)^{2} \)
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| $11$ |
\( T^{2} - 2T - 19 \)
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| $13$ |
\( T^{2} + 6T + 4 \)
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| $17$ |
\( T^{2} - 2T - 44 \)
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| $19$ |
\( T^{2} - 2T - 4 \)
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| $23$ |
\( (T - 5)^{2} \)
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| $29$ |
\( T^{2} + 4T - 41 \)
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| $31$ |
\( T^{2} - 20 \)
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| $37$ |
\( T^{2} + 2T - 19 \)
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| $41$ |
\( (T + 8)^{2} \)
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| $43$ |
\( T^{2} - 12T + 31 \)
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| $47$ |
\( T^{2} - 10T + 20 \)
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| $53$ |
\( T^{2} - 8T - 4 \)
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| $59$ |
\( T^{2} - 6T + 4 \)
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| $61$ |
\( T^{2} + 4T - 176 \)
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| $67$ |
\( T^{2} - 8T - 29 \)
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| $71$ |
\( T^{2} - 10T + 5 \)
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| $73$ |
\( T^{2} - 2T - 4 \)
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| $79$ |
\( T^{2} + 8T + 11 \)
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| $83$ |
\( T^{2} - 16T + 44 \)
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| $89$ |
\( T^{2} - 2T - 44 \)
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| $97$ |
\( T^{2} + 16T + 44 \)
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