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: | 6.6.75968016.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2325) |
| 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
\( x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 9\nu^{2} + 6\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 17\nu^{2} - \nu - 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 16\nu^{2} - 4\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 17\nu^{2} + 4\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 8\beta_{4} + \beta_{2} - 2\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{5} - \beta_{4} + 8\beta_{3} + 2\beta_{2} + 32\beta _1 - 14 \)
|
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 |
|
−2.46251 | 0 | 4.06395 | 0 | 0 | 3.91086 | −5.08250 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.89294 | 0 | 1.58320 | 0 | 0 | 1.92485 | 0.788968 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.864597 | 0 | −1.25247 | 0 | 0 | −4.28308 | 2.81208 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.667396 | 0 | −1.55458 | 0 | 0 | 3.64222 | −2.37231 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.814748 | 0 | −1.33619 | 0 | 0 | −3.06134 | −2.71815 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.73790 | 0 | 5.49608 | 0 | 0 | −0.133518 | 9.57192 | 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.ca | 6 | |
| 3.b | odd | 2 | 1 | 2325.2.a.bb | yes | 6 | |
| 5.b | even | 2 | 1 | 6975.2.a.cc | 6 | ||
| 15.d | odd | 2 | 1 | 2325.2.a.y | ✓ | 6 | |
| 15.e | even | 4 | 2 | 2325.2.c.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2325.2.a.y | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 2325.2.a.bb | yes | 6 | 3.b | odd | 2 | 1 | |
| 2325.2.c.r | 12 | 15.e | even | 4 | 2 | ||
| 6975.2.a.ca | 6 | 1.a | even | 1 | 1 | trivial | |
| 6975.2.a.cc | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6975))\):
|
\( T_{2}^{6} + T_{2}^{5} - 9T_{2}^{4} - 9T_{2}^{3} + 14T_{2}^{2} + 6T_{2} - 6 \)
|
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\( T_{7}^{6} - 2T_{7}^{5} - 28T_{7}^{4} + 56T_{7}^{3} + 184T_{7}^{2} - 336T_{7} - 48 \)
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\( T_{11}^{6} + 7T_{11}^{5} - 5T_{11}^{4} - 53T_{11}^{3} + 48T_{11}^{2} + 12T_{11} - 4 \)
|
|
\( T_{13}^{6} - 4T_{13}^{5} - 38T_{13}^{4} + 122T_{13}^{3} + 400T_{13}^{2} - 720T_{13} - 1632 \)
|
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\( T_{17}^{6} - 67T_{17}^{4} + 40T_{17}^{3} + 899T_{17}^{2} + 240T_{17} - 1909 \)
|
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\( T_{29}^{6} - 8T_{29}^{5} - 65T_{29}^{4} + 640T_{29}^{3} + 291T_{29}^{2} - 11364T_{29} + 19277 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 9 T^{4} + \cdots - 6 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 48 \)
|
| $11$ |
\( T^{6} + 7 T^{5} + \cdots - 4 \)
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| $13$ |
\( T^{6} - 4 T^{5} + \cdots - 1632 \)
|
| $17$ |
\( T^{6} - 67 T^{4} + \cdots - 1909 \)
|
| $19$ |
\( T^{6} - 17 T^{5} + \cdots - 596 \)
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| $23$ |
\( T^{6} - T^{5} + \cdots - 184 \)
|
| $29$ |
\( T^{6} - 8 T^{5} + \cdots + 19277 \)
|
| $31$ |
\( (T + 1)^{6} \)
|
| $37$ |
\( T^{6} - 14 T^{5} + \cdots + 1048 \)
|
| $41$ |
\( T^{6} + 18 T^{5} + \cdots + 24 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 12592 \)
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| $47$ |
\( T^{6} - 13 T^{5} + \cdots + 72 \)
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| $53$ |
\( T^{6} - 13 T^{5} + \cdots - 8396 \)
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| $59$ |
\( T^{6} + 32 T^{5} + \cdots + 1784 \)
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| $61$ |
\( T^{6} - 288 T^{4} + \cdots + 39168 \)
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| $67$ |
\( T^{6} - 9 T^{5} + \cdots + 807124 \)
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| $71$ |
\( T^{6} + T^{5} + \cdots + 19368 \)
|
| $73$ |
\( T^{6} - 14 T^{5} + \cdots + 6528 \)
|
| $79$ |
\( T^{6} - 13 T^{5} + \cdots - 266974 \)
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| $83$ |
\( T^{6} + 9 T^{5} + \cdots + 1536 \)
|
| $89$ |
\( T^{6} + 34 T^{5} + \cdots + 6351 \)
|
| $97$ |
\( T^{6} - 30 T^{5} + \cdots - 4663 \)
|
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