Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3020,2,Mod(1,3020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3020, 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("3020.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3020 = 2^{2} \cdot 5 \cdot 151 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3020.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: | \(24.1148214104\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.40310669.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 11x^{2} - 11x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 11x^{2} - 11x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 7\nu^{2} + 8\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 9\nu^{2} + 12\nu - 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + 5\nu^{2} + 17\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + 2\beta_{3} + 8\beta_{2} + 11\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 12\beta_{4} + 11\beta_{3} + 21\beta_{2} + 43\beta _1 + 26 \)
|
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 |
|
0 | −2.63522 | 0 | 1.00000 | 0 | 2.09536 | 0 | 3.94440 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.60047 | 0 | 1.00000 | 0 | 3.31076 | 0 | 3.76242 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.871166 | 0 | 1.00000 | 0 | −3.29199 | 0 | −2.24107 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.66122 | 0 | 1.00000 | 0 | 2.25909 | 0 | −0.240356 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.19566 | 0 | 1.00000 | 0 | 0.489165 | 0 | 7.21223 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 3.24998 | 0 | 1.00000 | 0 | −1.86237 | 0 | 7.56237 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(151\) | \(-1\) |
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 | 3020.2.a.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3020.2.a.f | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3020))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 17T_{3}^{4} + 25T_{3}^{3} + 87T_{3}^{2} - 71T_{3} - 103 \)
|
|
\( T_{7}^{6} - 3T_{7}^{5} - 13T_{7}^{4} + 43T_{7}^{3} + 19T_{7}^{2} - 114T_{7} + 47 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 103 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots + 47 \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots - 128 \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 698 \)
|
| $17$ |
\( T^{6} - 9 T^{5} + \cdots + 1120 \)
|
| $19$ |
\( T^{6} - 16 T^{5} + \cdots + 257 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 1504 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots - 1202 \)
|
| $31$ |
\( T^{6} - 15 T^{5} + \cdots - 23464 \)
|
| $37$ |
\( T^{6} - 9 T^{5} + \cdots + 14512 \)
|
| $41$ |
\( T^{6} + 16 T^{5} + \cdots + 238360 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 119488 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 1648 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 25180 \)
|
| $59$ |
\( T^{6} + 20 T^{5} + \cdots - 436 \)
|
| $61$ |
\( T^{6} - 17 T^{5} + \cdots + 17744 \)
|
| $67$ |
\( T^{6} - 29 T^{5} + \cdots - 23468 \)
|
| $71$ |
\( T^{6} - 13 T^{5} + \cdots - 62840 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots + 206 \)
|
| $79$ |
\( T^{6} + 8 T^{5} + \cdots + 58288 \)
|
| $83$ |
\( T^{6} - 36 T^{5} + \cdots + 239764 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots + 189568 \)
|
| $97$ |
\( T^{6} + 23 T^{5} + \cdots - 263072 \)
|
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