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: | \(6\) |
| Coefficient field: | 6.6.361944768.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 12x^{4} + 40x^{2} - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 279) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 12x^{4} + 40x^{2} - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{3} + \nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} + 16\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{4} + 9\beta_{2} + 29\beta_1 \)
|
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 |
|
−2.53758 | 0 | 4.43931 | 0 | 0 | −4.38955 | −6.18995 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.15896 | 0 | 2.66112 | 0 | 0 | 2.90180 | −1.42733 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.948456 | 0 | −1.10043 | 0 | 0 | −2.51225 | 2.94062 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.948456 | 0 | −1.10043 | 0 | 0 | −2.51225 | −2.94062 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.15896 | 0 | 2.66112 | 0 | 0 | 2.90180 | 1.42733 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.53758 | 0 | 4.43931 | 0 | 0 | −4.38955 | 6.18995 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(31\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6975.2.a.cb | 6 | |
| 3.b | odd | 2 | 1 | inner | 6975.2.a.cb | 6 | |
| 5.b | even | 2 | 1 | 279.2.a.d | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 279.2.a.d | ✓ | 6 | |
| 20.d | odd | 2 | 1 | 4464.2.a.bt | 6 | ||
| 60.h | even | 2 | 1 | 4464.2.a.bt | 6 | ||
| 155.c | odd | 2 | 1 | 8649.2.a.bb | 6 | ||
| 465.g | even | 2 | 1 | 8649.2.a.bb | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 279.2.a.d | ✓ | 6 | 5.b | even | 2 | 1 | |
| 279.2.a.d | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 4464.2.a.bt | 6 | 20.d | odd | 2 | 1 | ||
| 4464.2.a.bt | 6 | 60.h | even | 2 | 1 | ||
| 6975.2.a.cb | 6 | 1.a | even | 1 | 1 | trivial | |
| 6975.2.a.cb | 6 | 3.b | odd | 2 | 1 | inner | |
| 8649.2.a.bb | 6 | 155.c | odd | 2 | 1 | ||
| 8649.2.a.bb | 6 | 465.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{6} - 12T_{2}^{4} + 40T_{2}^{2} - 27 \)
|
|
\( T_{7}^{3} + 4T_{7}^{2} - 9T_{7} - 32 \)
|
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\( T_{11}^{6} - 68T_{11}^{4} + 1168T_{11}^{2} - 768 \)
|
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\( T_{13}^{3} - 32T_{13} + 40 \)
|
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\( T_{17}^{6} - 76T_{17}^{4} + 1216T_{17}^{2} - 3072 \)
|
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\( T_{29}^{6} - 48T_{29}^{4} + 640T_{29}^{2} - 1728 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 12 T^{4} + \cdots - 27 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{3} + 4 T^{2} - 9 T - 32)^{2} \)
|
| $11$ |
\( T^{6} - 68 T^{4} + \cdots - 768 \)
|
| $13$ |
\( (T^{3} - 32 T + 40)^{2} \)
|
| $17$ |
\( T^{6} - 76 T^{4} + \cdots - 3072 \)
|
| $19$ |
\( (T^{3} - 8 T^{2} + 7 T + 4)^{2} \)
|
| $23$ |
\( T^{6} - 116 T^{4} + \cdots - 6912 \)
|
| $29$ |
\( T^{6} - 48 T^{4} + \cdots - 1728 \)
|
| $31$ |
\( (T - 1)^{6} \)
|
| $37$ |
\( (T^{3} + 4 T^{2} + \cdots - 424)^{2} \)
|
| $41$ |
\( T^{6} - 26 T^{4} + \cdots - 192 \)
|
| $43$ |
\( (T^{3} - 6 T^{2} - 20 T + 16)^{2} \)
|
| $47$ |
\( T^{6} - 132 T^{4} + \cdots - 15552 \)
|
| $53$ |
\( T^{6} - 76 T^{4} + \cdots - 3072 \)
|
| $59$ |
\( T^{6} - 70 T^{4} + \cdots - 12 \)
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| $61$ |
\( (T^{3} - 2 T^{2} + \cdots + 160)^{2} \)
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| $67$ |
\( (T - 4)^{6} \)
|
| $71$ |
\( T^{6} - 78 T^{4} + \cdots - 2028 \)
|
| $73$ |
\( (T^{3} + 16 T^{2} - 200)^{2} \)
|
| $79$ |
\( (T^{3} - 4 T^{2} + \cdots + 832)^{2} \)
|
| $83$ |
\( T^{6} - 280 T^{4} + \cdots - 768 \)
|
| $89$ |
\( T^{6} - 452 T^{4} + \cdots - 2365632 \)
|
| $97$ |
\( (T^{3} + 12 T^{2} + \cdots - 1142)^{2} \)
|
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