Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6975,2,Mod(1,6975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6975, 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("6975.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6975.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: | \(55.6956554098\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 a root \(\nu\) of
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
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 |
|
−2.48119 | 0 | 4.15633 | 0 | 0 | −2.80606 | −5.35026 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | −0.688892 | 0 | −1.52543 | 0 | 0 | −4.90321 | 2.42864 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1.17009 | 0 | −0.630898 | 0 | 0 | −0.290725 | −3.07838 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 6975.2.a.z | ✓ | 3 |
| 3.b | odd | 2 | 1 | 6975.2.a.bg | yes | 3 | |
| 5.b | even | 2 | 1 | 6975.2.a.bh | yes | 3 | |
| 15.d | odd | 2 | 1 | 6975.2.a.ba | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6975.2.a.z | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 6975.2.a.ba | yes | 3 | 15.d | odd | 2 | 1 | |
| 6975.2.a.bg | yes | 3 | 3.b | odd | 2 | 1 | |
| 6975.2.a.bh | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{3} + 2T_{2}^{2} - 2T_{2} - 2 \)
|
|
\( T_{7}^{3} + 8T_{7}^{2} + 16T_{7} + 4 \)
|
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\( T_{11}^{3} - 3T_{11}^{2} - T_{11} + 1 \)
|
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\( T_{13}^{3} + 4T_{13}^{2} - 16T_{13} + 10 \)
|
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\( T_{17}^{3} + 7T_{17}^{2} - 7T_{17} - 59 \)
|
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\( T_{29}^{3} - 9T_{29}^{2} - 7T_{29} + 13 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2 T^{2} + \cdots - 2 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} + 8 T^{2} + \cdots + 4 \)
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| $11$ |
\( T^{3} - 3T^{2} - T + 1 \)
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| $13$ |
\( T^{3} + 4 T^{2} + \cdots + 10 \)
|
| $17$ |
\( T^{3} + 7 T^{2} + \cdots - 59 \)
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| $19$ |
\( (T - 3)^{3} \)
|
| $23$ |
\( T^{3} + 9 T^{2} + \cdots + 17 \)
|
| $29$ |
\( T^{3} - 9 T^{2} + \cdots + 13 \)
|
| $31$ |
\( (T - 1)^{3} \)
|
| $37$ |
\( T^{3} + 8 T^{2} + \cdots + 2 \)
|
| $41$ |
\( T^{3} - 2 T^{2} + \cdots - 134 \)
|
| $43$ |
\( T^{3} + 4 T^{2} + \cdots - 64 \)
|
| $47$ |
\( T^{3} - 20 T^{2} + \cdots - 208 \)
|
| $53$ |
\( T^{3} - 25 T^{2} + \cdots + 349 \)
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| $59$ |
\( T^{3} - 2 T^{2} + \cdots + 74 \)
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| $61$ |
\( T^{3} - 2 T^{2} + \cdots + 40 \)
|
| $67$ |
\( T^{3} + 15 T^{2} + \cdots + 1 \)
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| $71$ |
\( T^{3} + 12 T^{2} + \cdots - 1184 \)
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| $73$ |
\( T^{3} + 2 T^{2} + \cdots + 296 \)
|
| $79$ |
\( T^{3} - 12 T^{2} + \cdots - 86 \)
|
| $83$ |
\( T^{3} + 13 T^{2} + \cdots - 871 \)
|
| $89$ |
\( T^{3} + 15 T^{2} + \cdots + 103 \)
|
| $97$ |
\( T^{3} + 7 T^{2} + \cdots - 103 \)
|
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