Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1734,2,Mod(1,1734)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, 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("1734.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1734.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: | \(13.8460597105\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{16})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 102) |
| 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 \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
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 | 1.00000 | 1.00000 | −4.17958 | −1.00000 | −2.28130 | −1.00000 | 1.00000 | 4.17958 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | −2.64885 | −1.00000 | 5.10973 | −1.00000 | 1.00000 | 2.64885 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | −2.43355 | −1.00000 | 0.116520 | −1.00000 | 1.00000 | 2.43355 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | 1.26197 | −1.00000 | −2.94495 | −1.00000 | 1.00000 | −1.26197 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 1734.2.a.u | 4 | |
| 3.b | odd | 2 | 1 | 5202.2.a.bx | 4 | ||
| 17.b | even | 2 | 1 | 1734.2.a.t | 4 | ||
| 17.c | even | 4 | 2 | 1734.2.b.l | 8 | ||
| 17.d | even | 8 | 2 | 1734.2.f.k | 8 | ||
| 17.d | even | 8 | 2 | 1734.2.f.l | 8 | ||
| 17.e | odd | 16 | 2 | 102.2.h.b | ✓ | 8 | |
| 51.c | odd | 2 | 1 | 5202.2.a.bu | 4 | ||
| 51.i | even | 16 | 2 | 306.2.l.e | 8 | ||
| 68.i | even | 16 | 2 | 816.2.bq.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.2.h.b | ✓ | 8 | 17.e | odd | 16 | 2 | |
| 306.2.l.e | 8 | 51.i | even | 16 | 2 | ||
| 816.2.bq.c | 8 | 68.i | even | 16 | 2 | ||
| 1734.2.a.t | 4 | 17.b | even | 2 | 1 | ||
| 1734.2.a.u | 4 | 1.a | even | 1 | 1 | trivial | |
| 1734.2.b.l | 8 | 17.c | even | 4 | 2 | ||
| 1734.2.f.k | 8 | 17.d | even | 8 | 2 | ||
| 1734.2.f.l | 8 | 17.d | even | 8 | 2 | ||
| 5202.2.a.bu | 4 | 51.c | odd | 2 | 1 | ||
| 5202.2.a.bx | 4 | 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\):
|
\( T_{5}^{4} + 8T_{5}^{3} + 16T_{5}^{2} - 8T_{5} - 34 \)
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\( T_{7}^{4} - 20T_{7}^{2} - 32T_{7} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} + 8 T^{3} + \cdots - 34 \)
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| $7$ |
\( T^{4} - 20 T^{2} + \cdots + 4 \)
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| $11$ |
\( T^{4} + 8 T^{3} + \cdots - 272 \)
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| $13$ |
\( T^{4} - 44 T^{2} + \cdots + 316 \)
|
| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} - 8 T^{3} + \cdots - 8 \)
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| $23$ |
\( (T^{2} - 2)^{2} \)
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| $29$ |
\( T^{4} + 16 T^{3} + \cdots - 34 \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
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| $37$ |
\( T^{4} + 16 T^{3} + \cdots - 98 \)
|
| $41$ |
\( T^{4} + 8 T^{3} + \cdots - 62 \)
|
| $43$ |
\( T^{4} - 32 T^{2} + \cdots - 32 \)
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| $47$ |
\( T^{4} + 16 T^{3} + \cdots - 376 \)
|
| $53$ |
\( T^{4} + 16 T^{3} + \cdots + 316 \)
|
| $59$ |
\( T^{4} + 24 T^{3} + \cdots - 392 \)
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| $61$ |
\( T^{4} + 8 T^{3} + \cdots + 94 \)
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| $67$ |
\( T^{4} - 8 T^{3} + \cdots - 136 \)
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| $71$ |
\( T^{4} + 16 T^{3} + \cdots - 10364 \)
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| $73$ |
\( T^{4} + 16 T^{3} + \cdots - 94 \)
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| $79$ |
\( T^{4} + 32 T^{3} + \cdots + 2564 \)
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| $83$ |
\( T^{4} + 8 T^{3} + \cdots - 2696 \)
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| $89$ |
\( T^{4} + 16 T^{3} + \cdots + 6596 \)
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| $97$ |
\( T^{4} - 228 T^{2} + \cdots + 9026 \)
|
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