Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1617,2,Mod(1,1617)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1617.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1617.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: | \(12.9118100068\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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 a root \(\nu\) of
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
|
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 |
|
−1.51658 | 1.00000 | 0.300014 | −2.33317 | −1.51658 | 0 | 2.57816 | 1.00000 | 3.53844 | ||||||||||||||||||||||||||||||
| 1.2 | −0.552409 | 1.00000 | −1.69484 | 1.59002 | −0.552409 | 0 | 2.04107 | 1.00000 | −0.878345 | |||||||||||||||||||||||||||||||
| 1.3 | 1.28734 | 1.00000 | −0.342766 | 3.91744 | 1.28734 | 0 | −3.01593 | 1.00000 | 5.04306 | |||||||||||||||||||||||||||||||
| 1.4 | 2.78165 | 1.00000 | 5.73760 | 0.825711 | 2.78165 | 0 | 10.3967 | 1.00000 | 2.29684 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 1617.2.a.z | 4 | |
| 3.b | odd | 2 | 1 | 4851.2.a.bt | 4 | ||
| 7.b | odd | 2 | 1 | 1617.2.a.x | 4 | ||
| 7.c | even | 3 | 2 | 231.2.i.e | ✓ | 8 | |
| 21.c | even | 2 | 1 | 4851.2.a.bu | 4 | ||
| 21.h | odd | 6 | 2 | 693.2.i.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.i.e | ✓ | 8 | 7.c | even | 3 | 2 | |
| 693.2.i.i | 8 | 21.h | odd | 6 | 2 | ||
| 1617.2.a.x | 4 | 7.b | odd | 2 | 1 | ||
| 1617.2.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 4851.2.a.bt | 4 | 3.b | odd | 2 | 1 | ||
| 4851.2.a.bu | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):
|
\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 3 \)
|
|
\( T_{5}^{4} - 4T_{5}^{3} - 4T_{5}^{2} + 20T_{5} - 12 \)
|
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\( T_{13}^{4} + 2T_{13}^{3} - 8T_{13}^{2} - 16T_{13} - 4 \)
|
|
\( T_{17}^{4} - 2T_{17}^{3} - 40T_{17}^{2} + 124T_{17} + 15 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 3 \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots - 12 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} + 2 T^{3} + \cdots - 4 \)
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| $17$ |
\( T^{4} - 2 T^{3} + \cdots + 15 \)
|
| $19$ |
\( T^{4} - 40 T^{2} + \cdots - 89 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 465 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 3 \)
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| $31$ |
\( T^{4} + 12 T^{3} + \cdots - 1396 \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
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| $41$ |
\( T^{4} - 2 T^{3} + \cdots + 60 \)
|
| $43$ |
\( T^{4} - 18 T^{3} + \cdots - 1385 \)
|
| $47$ |
\( T^{4} - 12 T^{3} + \cdots + 9 \)
|
| $53$ |
\( T^{4} + 12 T^{3} + \cdots + 6252 \)
|
| $59$ |
\( T^{4} - 12 T^{3} + \cdots + 249 \)
|
| $61$ |
\( T^{4} - 2 T^{3} + \cdots + 20 \)
|
| $67$ |
\( T^{4} - 28 T^{3} + \cdots - 1460 \)
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| $71$ |
\( T^{4} - 12 T^{3} + \cdots - 699 \)
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| $73$ |
\( T^{4} - 6 T^{3} + \cdots + 17156 \)
|
| $79$ |
\( T^{4} - 2 T^{3} + \cdots + 388 \)
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| $83$ |
\( T^{4} + 12 T^{3} + \cdots + 2592 \)
|
| $89$ |
\( T^{4} - 8 T^{3} + \cdots + 12048 \)
|
| $97$ |
\( T^{4} + 44 T^{3} + \cdots + 8501 \)
|
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