Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2738,2,Mod(1,2738)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2738.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2738 = 2 \cdot 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2738.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: | \(21.8630400734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.37902897.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 74) |
| 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
\( x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 15\nu^{3} + \nu^{2} - 44\nu ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 11\nu^{2} - \nu + 20 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} - 59\nu^{3} - 5\nu^{2} + 124\nu - 16 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} - 4\nu^{4} + 59\nu^{3} + 65\nu^{2} - 120\nu - 144 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} + 11\beta_{4} + 15\beta_{3} + \beta _1 + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 16\beta_{4} + \beta_{3} + 59\beta_{2} + 46\beta _1 + 20 \)
|
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 |
|
1.00000 | −3.14945 | 1.00000 | 1.53209 | −3.14945 | −4.82524 | 1.00000 | 6.91903 | 1.53209 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.83388 | 1.00000 | −1.87939 | −1.83388 | 3.44656 | 1.00000 | 0.363102 | −1.87939 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.37564 | 1.00000 | 0.347296 | −1.37564 | −0.477756 | 1.00000 | −1.10761 | 0.347296 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.27006 | 1.00000 | 1.53209 | 1.27006 | 1.94585 | 1.00000 | −1.38694 | 1.53209 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.18117 | 1.00000 | −1.87939 | 2.18117 | −4.09926 | 1.00000 | 1.75751 | −1.87939 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.90773 | 1.00000 | 0.347296 | 2.90773 | 1.00984 | 1.00000 | 5.45490 | 0.347296 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(37\) | \(1\) |
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 | 2738.2.a.t | 6 | |
| 37.b | even | 2 | 1 | 2738.2.a.q | 6 | ||
| 37.f | even | 9 | 2 | 74.2.f.b | ✓ | 12 | |
| 111.p | odd | 18 | 2 | 666.2.x.g | 12 | ||
| 148.p | odd | 18 | 2 | 592.2.bc.d | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.2.f.b | ✓ | 12 | 37.f | even | 9 | 2 | |
| 592.2.bc.d | 12 | 148.p | odd | 18 | 2 | ||
| 666.2.x.g | 12 | 111.p | odd | 18 | 2 | ||
| 2738.2.a.q | 6 | 37.b | even | 2 | 1 | ||
| 2738.2.a.t | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\):
|
\( T_{3}^{6} - 15T_{3}^{4} + T_{3}^{3} + 60T_{3}^{2} - 64 \)
|
|
\( T_{5}^{3} - 3T_{5} + 1 \)
|
|
\( T_{7}^{6} + 3T_{7}^{5} - 24T_{7}^{4} - 37T_{7}^{3} + 168T_{7}^{2} - 48T_{7} - 64 \)
|
|
\( T_{13}^{6} - 42T_{13}^{4} - 43T_{13}^{3} + 324T_{13}^{2} + 183T_{13} - 719 \)
|
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\( T_{17}^{6} - 3T_{17}^{5} - 24T_{17}^{4} + 69T_{17}^{3} + 120T_{17}^{2} - 390T_{17} + 163 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 64 \)
|
| $5$ |
\( (T^{3} - 3 T + 1)^{2} \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots - 64 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots - 64 \)
|
| $13$ |
\( T^{6} - 42 T^{4} + \cdots - 719 \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 163 \)
|
| $19$ |
\( T^{6} - 21 T^{5} + \cdots + 64 \)
|
| $23$ |
\( T^{6} - 21 T^{5} + \cdots + 64 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 37 \)
|
| $31$ |
\( T^{6} - 21 T^{5} + \cdots + 130112 \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} - 18 T^{5} + \cdots + 21608 \)
|
| $43$ |
\( T^{6} - 18 T^{5} + \cdots + 8704 \)
|
| $47$ |
\( T^{6} + 9 T^{5} + \cdots - 83008 \)
|
| $53$ |
\( T^{6} - 18 T^{5} + \cdots + 17 \)
|
| $59$ |
\( T^{6} - 27 T^{5} + \cdots + 1088 \)
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| $61$ |
\( T^{6} + 24 T^{5} + \cdots + 52928 \)
|
| $67$ |
\( T^{6} - 9 T^{5} + \cdots - 512 \)
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| $71$ |
\( T^{6} - 108 T^{4} + \cdots - 512 \)
|
| $73$ |
\( T^{6} - 27 T^{5} + \cdots + 216289 \)
|
| $79$ |
\( T^{6} - 21 T^{5} + \cdots + 9792 \)
|
| $83$ |
\( T^{6} - 27 T^{5} + \cdots + 1088 \)
|
| $89$ |
\( T^{6} + 21 T^{5} + \cdots - 219419 \)
|
| $97$ |
\( T^{6} - 42 T^{5} + \cdots + 2744 \)
|
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