Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1690,2,Mod(1,1690)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1690, 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("1690.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1690.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: | \(13.4947179416\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 | −1.24698 | 1.00000 | 1.00000 | 1.24698 | −0.198062 | −1.00000 | −1.44504 | −1.00000 | |||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0.445042 | 1.00000 | 1.00000 | −0.445042 | −3.24698 | −1.00000 | −2.80194 | −1.00000 | ||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.80194 | 1.00000 | 1.00000 | −1.80194 | −1.55496 | −1.00000 | 0.246980 | −1.00000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1690.2.a.q | ✓ | 3 |
| 5.b | even | 2 | 1 | 8450.2.a.bz | 3 | ||
| 13.b | even | 2 | 1 | 1690.2.a.s | yes | 3 | |
| 13.c | even | 3 | 2 | 1690.2.e.q | 6 | ||
| 13.d | odd | 4 | 2 | 1690.2.d.j | 6 | ||
| 13.e | even | 6 | 2 | 1690.2.e.o | 6 | ||
| 13.f | odd | 12 | 4 | 1690.2.l.l | 12 | ||
| 65.d | even | 2 | 1 | 8450.2.a.bo | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1690.2.a.q | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1690.2.a.s | yes | 3 | 13.b | even | 2 | 1 | |
| 1690.2.d.j | 6 | 13.d | odd | 4 | 2 | ||
| 1690.2.e.o | 6 | 13.e | even | 6 | 2 | ||
| 1690.2.e.q | 6 | 13.c | even | 3 | 2 | ||
| 1690.2.l.l | 12 | 13.f | odd | 12 | 4 | ||
| 8450.2.a.bo | 3 | 65.d | even | 2 | 1 | ||
| 8450.2.a.bz | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\):
|
\( T_{3}^{3} - T_{3}^{2} - 2T_{3} + 1 \)
|
|
\( T_{7}^{3} + 5T_{7}^{2} + 6T_{7} + 1 \)
|
|
\( T_{11}^{3} + 4T_{11}^{2} - 4T_{11} - 8 \)
|
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\( T_{31}^{3} + 8T_{31}^{2} - 44T_{31} - 344 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
|
| $3$ |
\( T^{3} - T^{2} - 2T + 1 \)
|
| $5$ |
\( (T - 1)^{3} \)
|
| $7$ |
\( T^{3} + 5 T^{2} + \cdots + 1 \)
|
| $11$ |
\( T^{3} + 4 T^{2} + \cdots - 8 \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} - 10 T^{2} + \cdots - 8 \)
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| $19$ |
\( T^{3} + 8 T^{2} + \cdots - 8 \)
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| $23$ |
\( T^{3} + 3 T^{2} + \cdots - 139 \)
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| $29$ |
\( T^{3} - 3 T^{2} + \cdots - 1 \)
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| $31$ |
\( T^{3} + 8 T^{2} + \cdots - 344 \)
|
| $37$ |
\( T^{3} + 8 T^{2} + \cdots - 64 \)
|
| $41$ |
\( T^{3} + 11 T^{2} + \cdots - 827 \)
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| $43$ |
\( T^{3} + T^{2} + \cdots - 43 \)
|
| $47$ |
\( T^{3} + 15 T^{2} + \cdots - 71 \)
|
| $53$ |
\( T^{3} - 4 T^{2} + \cdots + 568 \)
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| $59$ |
\( T^{3} + 24 T^{2} + \cdots + 344 \)
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| $61$ |
\( T^{3} + 15 T^{2} + \cdots - 533 \)
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| $67$ |
\( T^{3} + 5 T^{2} + \cdots + 169 \)
|
| $71$ |
\( T^{3} + 32 T^{2} + \cdots + 1112 \)
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| $73$ |
\( T^{3} - 26 T^{2} + \cdots + 104 \)
|
| $79$ |
\( T^{3} - 2 T^{2} + \cdots + 344 \)
|
| $83$ |
\( T^{3} + 7 T^{2} + \cdots - 287 \)
|
| $89$ |
\( T^{3} - 17 T^{2} + \cdots + 2197 \)
|
| $97$ |
\( T^{3} - 6 T^{2} + \cdots + 1112 \)
|
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