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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.45769536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} + 72x^{2} - 109 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 15x^{4} + 72x^{2} - 109 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 10\nu^{2} + 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + 23\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 10\beta_{2} + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{3} + 27\beta_1 \)
|
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 |
|
−2.55580 | 0 | 4.53209 | −3.91571 | 0 | 1.46791 | −6.47150 | 0 | 10.0077 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.31242 | 0 | 3.34730 | −0.803096 | 0 | 2.65270 | −3.11552 | 0 | 1.85710 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76653 | 0 | 1.12061 | 3.31998 | 0 | 4.87939 | 1.55346 | 0 | −5.86484 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.76653 | 0 | 1.12061 | −3.31998 | 0 | 4.87939 | −1.55346 | 0 | −5.86484 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.31242 | 0 | 3.34730 | 0.803096 | 0 | 2.65270 | 3.11552 | 0 | 1.85710 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.55580 | 0 | 4.53209 | 3.91571 | 0 | 1.46791 | 6.47150 | 0 | 10.0077 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(17\) | \(-1\) |
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 | 2601.2.a.bk | yes | 6 |
| 3.b | odd | 2 | 1 | inner | 2601.2.a.bk | yes | 6 |
| 17.b | even | 2 | 1 | 2601.2.a.bj | ✓ | 6 | |
| 51.c | odd | 2 | 1 | 2601.2.a.bj | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2601.2.a.bj | ✓ | 6 | 17.b | even | 2 | 1 | |
| 2601.2.a.bj | ✓ | 6 | 51.c | odd | 2 | 1 | |
| 2601.2.a.bk | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 2601.2.a.bk | yes | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):
|
\( T_{2}^{6} - 15T_{2}^{4} + 72T_{2}^{2} - 109 \)
|
|
\( T_{5}^{6} - 27T_{5}^{4} + 186T_{5}^{2} - 109 \)
|
|
\( T_{7}^{3} - 9T_{7}^{2} + 24T_{7} - 19 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 15 T^{4} + \cdots - 109 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 27 T^{4} + \cdots - 109 \)
|
| $7$ |
\( (T^{3} - 9 T^{2} + 24 T - 19)^{2} \)
|
| $11$ |
\( T^{6} - 45 T^{4} + \cdots - 981 \)
|
| $13$ |
\( (T^{3} + 3 T^{2} - 18 T - 37)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} + 9 T^{2} + 6 T - 53)^{2} \)
|
| $23$ |
\( T^{6} - 111 T^{4} + \cdots - 39349 \)
|
| $29$ |
\( T^{6} - 72 T^{4} + \cdots - 8829 \)
|
| $31$ |
\( (T^{3} - 18 T^{2} + \cdots - 199)^{2} \)
|
| $37$ |
\( (T^{3} + 3 T^{2} - 18 T + 17)^{2} \)
|
| $41$ |
\( T^{6} - 162 T^{4} + \cdots - 981 \)
|
| $43$ |
\( (T^{3} - 6 T^{2} + 8)^{2} \)
|
| $47$ |
\( T^{6} - 108 T^{4} + \cdots - 981 \)
|
| $53$ |
\( T^{6} - 180 T^{4} + \cdots - 8829 \)
|
| $59$ |
\( T^{6} - 261 T^{4} + \cdots - 8829 \)
|
| $61$ |
\( (T^{3} - 15 T^{2} + \cdots - 71)^{2} \)
|
| $67$ |
\( (T^{3} - 84 T + 296)^{2} \)
|
| $71$ |
\( T^{6} - 171 T^{4} + \cdots - 149221 \)
|
| $73$ |
\( (T^{3} + 3 T^{2} - 3)^{2} \)
|
| $79$ |
\( (T^{3} - 21 T^{2} + \cdots - 267)^{2} \)
|
| $83$ |
\( T^{6} - 348 T^{4} + \cdots - 580861 \)
|
| $89$ |
\( T^{6} - 156 T^{4} + \cdots - 31501 \)
|
| $97$ |
\( (T^{3} + 3 T^{2} + \cdots + 109)^{2} \)
|
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