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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8468.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 465) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 + 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} - 3\beta_{2} + 5\beta _1 + 2 ) / 2 \)
|
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 |
|
−2.46793 | 0 | 4.09069 | 0 | 0 | −2.31451 | −5.15968 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0.0872450 | 0 | −1.99239 | 0 | 0 | 3.46958 | −0.348316 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.79888 | 0 | 1.23597 | 0 | 0 | −4.20813 | −1.37440 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.58181 | 0 | 4.66573 | 0 | 0 | −0.946946 | 6.88240 | 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.bo | 4 | |
| 3.b | odd | 2 | 1 | 2325.2.a.v | 4 | ||
| 5.b | even | 2 | 1 | 1395.2.a.k | 4 | ||
| 15.d | odd | 2 | 1 | 465.2.a.h | ✓ | 4 | |
| 15.e | even | 4 | 2 | 2325.2.c.p | 8 | ||
| 60.h | even | 2 | 1 | 7440.2.a.bz | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.a.h | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 1395.2.a.k | 4 | 5.b | even | 2 | 1 | ||
| 2325.2.a.v | 4 | 3.b | odd | 2 | 1 | ||
| 2325.2.c.p | 8 | 15.e | even | 4 | 2 | ||
| 6975.2.a.bo | 4 | 1.a | even | 1 | 1 | trivial | |
| 7440.2.a.bz | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 12T_{2} - 1 \)
|
|
\( T_{7}^{4} + 4T_{7}^{3} - 10T_{7}^{2} - 46T_{7} - 32 \)
|
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\( T_{11}^{4} + 6T_{11}^{3} - 12T_{11}^{2} - 56T_{11} + 32 \)
|
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\( T_{13}^{4} + 2T_{13}^{3} - 50T_{13}^{2} - 94T_{13} + 4 \)
|
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\( T_{17}^{4} + 10T_{17}^{3} + 20T_{17}^{2} - 52T_{17} - 136 \)
|
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\( T_{29}^{4} - 12T_{29}^{2} - 14T_{29} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 4 T^{3} + \cdots - 32 \)
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| $11$ |
\( T^{4} + 6 T^{3} + \cdots + 32 \)
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| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $17$ |
\( T^{4} + 10 T^{3} + \cdots - 136 \)
|
| $19$ |
\( (T - 4)^{4} \)
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| $23$ |
\( T^{4} - 6 T^{3} + \cdots + 2048 \)
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| $29$ |
\( T^{4} - 12 T^{2} + \cdots - 4 \)
|
| $31$ |
\( (T + 1)^{4} \)
|
| $37$ |
\( T^{4} + 8 T^{3} + \cdots + 388 \)
|
| $41$ |
\( T^{4} - 6 T^{3} + \cdots - 704 \)
|
| $43$ |
\( T^{4} + 2 T^{3} + \cdots + 928 \)
|
| $47$ |
\( T^{4} + 2 T^{3} + \cdots - 736 \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots - 8 \)
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| $59$ |
\( T^{4} + 4 T^{3} + \cdots - 808 \)
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| $61$ |
\( T^{4} - 168 T^{2} + \cdots + 848 \)
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| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 184 \)
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| $71$ |
\( T^{4} + 22 T^{3} + \cdots - 1808 \)
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| $73$ |
\( T^{4} - 32 T^{3} + \cdots + 788 \)
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| $79$ |
\( T^{4} - 24 T^{3} + \cdots - 24064 \)
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| $83$ |
\( T^{4} + 6 T^{3} + \cdots - 464 \)
|
| $89$ |
\( T^{4} - 14 T^{3} + \cdots - 92 \)
|
| $97$ |
\( T^{4} - 14 T^{3} + \cdots - 64 \)
|
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