Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1122,2,Mod(1121,1122)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("1122.1121");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1122.h (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: | \(8.95921510679\) |
| Analytic rank: | \(0\) |
| 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: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1122\mathbb{Z}\right)^\times\).
| \(n\) | \(409\) | \(749\) | \(1057\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1121.1 |
|
−1.00000 | −1.00000 | − | 1.41421i | 1.00000 | 0 | 1.00000 | + | 1.41421i | 2.00000 | −1.00000 | −1.00000 | + | 2.82843i | 0 | ||||||||||||||||||
| 1121.2 | −1.00000 | −1.00000 | + | 1.41421i | 1.00000 | 0 | 1.00000 | − | 1.41421i | 2.00000 | −1.00000 | −1.00000 | − | 2.82843i | 0 | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 561.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1122.2.h.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1122.2.h.d | yes | 2 | |
| 11.b | odd | 2 | 1 | 1122.2.h.c | yes | 2 | |
| 17.b | even | 2 | 1 | 1122.2.h.b | yes | 2 | |
| 33.d | even | 2 | 1 | 1122.2.h.b | yes | 2 | |
| 51.c | odd | 2 | 1 | 1122.2.h.c | yes | 2 | |
| 187.b | odd | 2 | 1 | 1122.2.h.d | yes | 2 | |
| 561.h | even | 2 | 1 | inner | 1122.2.h.a | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1122.2.h.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1122.2.h.a | ✓ | 2 | 561.h | even | 2 | 1 | inner |
| 1122.2.h.b | yes | 2 | 17.b | even | 2 | 1 | |
| 1122.2.h.b | yes | 2 | 33.d | even | 2 | 1 | |
| 1122.2.h.c | yes | 2 | 11.b | odd | 2 | 1 | |
| 1122.2.h.c | yes | 2 | 51.c | odd | 2 | 1 | |
| 1122.2.h.d | yes | 2 | 3.b | odd | 2 | 1 | |
| 1122.2.h.d | yes | 2 | 187.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1122, [\chi])\):
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\( T_{5} \)
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\( T_{7} - 2 \)
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\( T_{23} + 6 \)
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\( T_{83} + 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{2} \)
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| $3$ |
\( T^{2} + 2T + 3 \)
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| $5$ |
\( T^{2} \)
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| $7$ |
\( (T - 2)^{2} \)
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| $11$ |
\( T^{2} + 6T + 11 \)
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| $13$ |
\( T^{2} \)
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| $17$ |
\( T^{2} - 6T + 17 \)
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| $19$ |
\( T^{2} \)
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| $23$ |
\( (T + 6)^{2} \)
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| $29$ |
\( T^{2} + 32 \)
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| $31$ |
\( T^{2} + 72 \)
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| $37$ |
\( T^{2} \)
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| $41$ |
\( T^{2} + 32 \)
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| $43$ |
\( T^{2} \)
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| $47$ |
\( T^{2} + 128 \)
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| $53$ |
\( T^{2} + 32 \)
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| $59$ |
\( T^{2} + 32 \)
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| $61$ |
\( (T - 8)^{2} \)
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| $67$ |
\( (T + 4)^{2} \)
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| $71$ |
\( (T + 6)^{2} \)
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| $73$ |
\( (T + 4)^{2} \)
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| $79$ |
\( (T + 10)^{2} \)
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| $83$ |
\( (T + 12)^{2} \)
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| $89$ |
\( T^{2} + 32 \)
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| $97$ |
\( T^{2} \)
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