Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8624,2,Mod(1,8624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("8624.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8624.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: | \(68.8629867032\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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 \(\beta = \sqrt{2}\). 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 | −2.41421 | 0 | 0.585786 | 0 | 0 | 0 | 2.82843 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0.414214 | 0 | 3.41421 | 0 | 0 | 0 | −2.82843 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 8624.2.a.bh | 2 | |
| 4.b | odd | 2 | 1 | 1078.2.a.x | 2 | ||
| 7.b | odd | 2 | 1 | 8624.2.a.cc | 2 | ||
| 7.d | odd | 6 | 2 | 1232.2.q.f | 4 | ||
| 12.b | even | 2 | 1 | 9702.2.a.ch | 2 | ||
| 28.d | even | 2 | 1 | 1078.2.a.t | 2 | ||
| 28.f | even | 6 | 2 | 154.2.e.e | ✓ | 4 | |
| 28.g | odd | 6 | 2 | 1078.2.e.m | 4 | ||
| 84.h | odd | 2 | 1 | 9702.2.a.cx | 2 | ||
| 84.j | odd | 6 | 2 | 1386.2.k.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.2.e.e | ✓ | 4 | 28.f | even | 6 | 2 | |
| 1078.2.a.t | 2 | 28.d | even | 2 | 1 | ||
| 1078.2.a.x | 2 | 4.b | odd | 2 | 1 | ||
| 1078.2.e.m | 4 | 28.g | odd | 6 | 2 | ||
| 1232.2.q.f | 4 | 7.d | odd | 6 | 2 | ||
| 1386.2.k.t | 4 | 84.j | odd | 6 | 2 | ||
| 8624.2.a.bh | 2 | 1.a | even | 1 | 1 | trivial | |
| 8624.2.a.cc | 2 | 7.b | odd | 2 | 1 | ||
| 9702.2.a.ch | 2 | 12.b | even | 2 | 1 | ||
| 9702.2.a.cx | 2 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):
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\( T_{3}^{2} + 2T_{3} - 1 \)
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\( T_{5}^{2} - 4T_{5} + 2 \)
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\( T_{13}^{2} - 2T_{13} - 7 \)
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\( T_{17}^{2} - 4T_{17} - 28 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} + 2T - 1 \)
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| $5$ |
\( T^{2} - 4T + 2 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( (T + 1)^{2} \)
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| $13$ |
\( T^{2} - 2T - 7 \)
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| $17$ |
\( T^{2} - 4T - 28 \)
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| $19$ |
\( T^{2} + 4T + 2 \)
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| $23$ |
\( T^{2} - 4T - 14 \)
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| $29$ |
\( T^{2} + 6T - 23 \)
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| $31$ |
\( (T + 4)^{2} \)
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| $37$ |
\( T^{2} + 16T + 62 \)
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| $41$ |
\( T^{2} - 8T + 14 \)
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| $43$ |
\( T^{2} - 32 \)
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| $47$ |
\( T^{2} + 4T - 68 \)
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| $53$ |
\( T^{2} + 4T - 94 \)
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| $59$ |
\( T^{2} + 14T + 47 \)
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| $61$ |
\( T^{2} + 18T + 73 \)
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| $67$ |
\( T^{2} + 14T + 31 \)
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| $71$ |
\( T^{2} - 8T - 34 \)
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| $73$ |
\( T^{2} - 16T + 62 \)
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| $79$ |
\( T^{2} - 18T + 63 \)
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| $83$ |
\( T^{2} - 4T - 196 \)
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| $89$ |
\( T^{2} + 8T - 56 \)
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| $97$ |
\( T^{2} - 2T - 7 \)
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