Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(623,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.623");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.n (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(4.98266508613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 623.1 |
|
0 | − | 1.73205i | 0 | 0 | 0 | 4.00000 | 0 | −3.00000 | 0 | |||||||||||||||||||||||
| 623.2 | 0 | 1.73205i | 0 | 0 | 0 | 4.00000 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 52.b | odd | 2 | 1 | inner |
| 156.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.2.n.b | yes | 2 |
| 3.b | odd | 2 | 1 | CM | 624.2.n.b | yes | 2 |
| 4.b | odd | 2 | 1 | 624.2.n.a | ✓ | 2 | |
| 12.b | even | 2 | 1 | 624.2.n.a | ✓ | 2 | |
| 13.b | even | 2 | 1 | 624.2.n.a | ✓ | 2 | |
| 39.d | odd | 2 | 1 | 624.2.n.a | ✓ | 2 | |
| 52.b | odd | 2 | 1 | inner | 624.2.n.b | yes | 2 |
| 156.h | even | 2 | 1 | inner | 624.2.n.b | yes | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 624.2.n.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 624.2.n.a | ✓ | 2 | 12.b | even | 2 | 1 | |
| 624.2.n.a | ✓ | 2 | 13.b | even | 2 | 1 | |
| 624.2.n.a | ✓ | 2 | 39.d | odd | 2 | 1 | |
| 624.2.n.b | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 624.2.n.b | yes | 2 | 3.b | odd | 2 | 1 | CM |
| 624.2.n.b | yes | 2 | 52.b | odd | 2 | 1 | inner |
| 624.2.n.b | yes | 2 | 156.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5} \)
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\( T_{7} - 4 \)
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\( T_{23} \)
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\( T_{107} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 3 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T - 4)^{2} \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} - 2T + 13 \)
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| $17$ |
\( T^{2} \)
|
| $19$ |
\( (T - 8)^{2} \)
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| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( (T + 4)^{2} \)
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| $37$ |
\( T^{2} + 48 \)
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| $41$ |
\( T^{2} \)
|
| $43$ |
\( T^{2} + 108 \)
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| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( (T + 14)^{2} \)
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| $67$ |
\( (T - 16)^{2} \)
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| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} + 192 \)
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| $79$ |
\( T^{2} + 300 \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( T^{2} + 192 \)
|
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