Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2601,2,Mod(1,2601)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2601.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2601 = 3^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2601.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: | \(20.7690895657\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3418281.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 867) |
| 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
\( x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 4\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 3\nu^{2} + 14\nu + 2 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + 2\beta_{2} + 8\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{4} + 12\beta_{3} + 18\beta_{2} + 31\beta _1 + 21 \)
|
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 |
|
−2.74819 | 0 | 5.55252 | −0.973936 | 0 | 1.60333 | −9.76298 | 0 | 2.67656 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.44395 | 0 | 3.97290 | 2.87349 | 0 | 1.56252 | −4.82168 | 0 | −7.02266 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.907065 | 0 | −1.17723 | 3.19333 | 0 | −3.56234 | 2.88196 | 0 | −2.89656 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.435433 | 0 | −1.81040 | −4.22078 | 0 | −2.90981 | 1.65917 | 0 | 1.83787 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.43915 | 0 | 0.0711653 | −2.31394 | 0 | 4.44173 | −2.77589 | 0 | −3.33012 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.09548 | 0 | 2.39104 | −1.55815 | 0 | −4.13541 | 0.819422 | 0 | −3.26508 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 2601.2.a.bh | 6 | |
| 3.b | odd | 2 | 1 | 867.2.a.o | ✓ | 6 | |
| 17.b | even | 2 | 1 | 2601.2.a.bi | 6 | ||
| 51.c | odd | 2 | 1 | 867.2.a.p | yes | 6 | |
| 51.f | odd | 4 | 2 | 867.2.d.g | 12 | ||
| 51.g | odd | 8 | 4 | 867.2.e.k | 24 | ||
| 51.i | even | 16 | 8 | 867.2.h.m | 48 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 867.2.a.o | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 867.2.a.p | yes | 6 | 51.c | odd | 2 | 1 | |
| 867.2.d.g | 12 | 51.f | odd | 4 | 2 | ||
| 867.2.e.k | 24 | 51.g | odd | 8 | 4 | ||
| 867.2.h.m | 48 | 51.i | even | 16 | 8 | ||
| 2601.2.a.bh | 6 | 1.a | even | 1 | 1 | trivial | |
| 2601.2.a.bi | 6 | 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):
|
\( T_{2}^{6} + 3T_{2}^{5} - 6T_{2}^{4} - 19T_{2}^{3} + 6T_{2}^{2} + 24T_{2} + 8 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} - 18T_{5}^{4} - 51T_{5}^{3} + 60T_{5}^{2} + 228T_{5} + 136 \)
|
|
\( T_{7}^{6} + 3T_{7}^{5} - 27T_{7}^{4} - 75T_{7}^{3} + 171T_{7}^{2} + 297T_{7} - 477 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 8 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 136 \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots - 477 \)
|
| $11$ |
\( T^{6} + 9 T^{5} + \cdots + 296 \)
|
| $13$ |
\( T^{6} - 9 T^{5} + \cdots - 109 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 333 \)
|
| $23$ |
\( T^{6} + 9 T^{5} + \cdots + 1576 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 136 \)
|
| $31$ |
\( T^{6} + 24 T^{5} + \cdots - 127 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots - 109 \)
|
| $41$ |
\( T^{6} + 18 T^{5} + \cdots - 1592 \)
|
| $43$ |
\( T^{6} - 117 T^{4} + \cdots - 11736 \)
|
| $47$ |
\( T^{6} + 24 T^{5} + \cdots + 20312 \)
|
| $53$ |
\( T^{6} + 24 T^{5} + \cdots - 5608 \)
|
| $59$ |
\( T^{6} - 9 T^{5} + \cdots + 1432 \)
|
| $61$ |
\( T^{6} + 21 T^{5} + \cdots + 31841 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots - 14552 \)
|
| $71$ |
\( T^{6} + 27 T^{5} + \cdots - 8704 \)
|
| $73$ |
\( T^{6} + 18 T^{5} + \cdots + 92429 \)
|
| $79$ |
\( T^{6} + 24 T^{5} + \cdots + 31419 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots - 205336 \)
|
| $89$ |
\( T^{6} - 327 T^{4} + \cdots - 39896 \)
|
| $97$ |
\( T^{6} - 33 T^{5} + \cdots - 17389 \)
|
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