Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6825,2,Mod(1,6825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6825.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6825.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: | \(54.4978993795\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.17428.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 4x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{4} - x^{3} - 6x^{2} + 4x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + 2\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 \)
|
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 |
|
−2.61050 | −1.00000 | 4.81471 | 0 | 2.61050 | −1.00000 | −7.34780 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.43986 | −1.00000 | 0.0731828 | 0 | 1.43986 | −1.00000 | 2.77434 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.670843 | −1.00000 | −1.54997 | 0 | −0.670843 | −1.00000 | −2.38147 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.37951 | −1.00000 | 3.66208 | 0 | −2.37951 | −1.00000 | 3.95493 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
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 | 6825.2.a.bg | 4 | |
| 5.b | even | 2 | 1 | 273.2.a.e | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 819.2.a.k | 4 | ||
| 20.d | odd | 2 | 1 | 4368.2.a.br | 4 | ||
| 35.c | odd | 2 | 1 | 1911.2.a.s | 4 | ||
| 65.d | even | 2 | 1 | 3549.2.a.w | 4 | ||
| 105.g | even | 2 | 1 | 5733.2.a.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.a.e | ✓ | 4 | 5.b | even | 2 | 1 | |
| 819.2.a.k | 4 | 15.d | odd | 2 | 1 | ||
| 1911.2.a.s | 4 | 35.c | odd | 2 | 1 | ||
| 3549.2.a.w | 4 | 65.d | even | 2 | 1 | ||
| 4368.2.a.br | 4 | 20.d | odd | 2 | 1 | ||
| 5733.2.a.bf | 4 | 105.g | even | 2 | 1 | ||
| 6825.2.a.bg | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6825))\):
|
\( T_{2}^{4} + T_{2}^{3} - 7T_{2}^{2} - 5T_{2} + 6 \)
|
|
\( T_{11}^{4} + 2T_{11}^{3} - 24T_{11}^{2} - 32T_{11} + 96 \)
|
|
\( T_{17}^{4} - 2T_{17}^{3} - 28T_{17}^{2} + 40T_{17} + 96 \)
|
|
\( T_{19}^{4} - 7T_{19}^{3} - 12T_{19}^{2} + 48T_{19} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} - 7 T^{2} - 5 T + 6 \)
|
| $3$ |
\( (T + 1)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T + 1)^{4} \)
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| $11$ |
\( T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 96 \)
|
| $13$ |
\( (T + 1)^{4} \)
|
| $17$ |
\( T^{4} - 2 T^{3} - 28 T^{2} + 40 T + 96 \)
|
| $19$ |
\( T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64 \)
|
| $23$ |
\( T^{4} + 3 T^{3} - 52 T^{2} - 256 T - 288 \)
|
| $29$ |
\( T^{4} - T^{3} - 30 T^{2} + 52 T + 72 \)
|
| $31$ |
\( T^{4} - 3 T^{3} - 128 T^{2} + \cdots + 3968 \)
|
| $37$ |
\( T^{4} + 10 T^{3} - 84 T^{2} + \cdots - 128 \)
|
| $41$ |
\( T^{4} + 16 T^{3} - 688 T - 1392 \)
|
| $43$ |
\( T^{4} + 3 T^{3} - 44 T^{2} - 112 T - 64 \)
|
| $47$ |
\( T^{4} + 5 T^{3} - 40 T^{2} - 16 T + 144 \)
|
| $53$ |
\( T^{4} + 5 T^{3} - 38 T^{2} - 68 T - 24 \)
|
| $59$ |
\( T^{4} + 20 T^{3} + 80 T^{2} + \cdots - 1536 \)
|
| $61$ |
\( T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 496 \)
|
| $67$ |
\( T^{4} - 22 T^{3} - 40 T^{2} + \cdots - 15488 \)
|
| $71$ |
\( T^{4} - 232 T^{2} + 304 T + 10176 \)
|
| $73$ |
\( T^{4} - 13 T^{3} - 166 T^{2} + \cdots - 11672 \)
|
| $79$ |
\( T^{4} - 11 T^{3} - 120 T^{2} + \cdots - 3456 \)
|
| $83$ |
\( T^{4} + T^{3} - 36 T^{2} + 80 T - 48 \)
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| $89$ |
\( T^{4} + 5 T^{3} - 162 T^{2} + \cdots - 1704 \)
|
| $97$ |
\( T^{4} - 17 T^{3} - 14 T^{2} + \cdots - 1528 \)
|
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