Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6032,2,Mod(1,6032)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6032, 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("6032.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6032 = 2^{4} \cdot 13 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6032.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: | \(48.1657624992\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.226964648.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 22x^{4} + 17x^{3} + 131x^{2} - 82x - 148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 754) |
| 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
\( x^{6} - x^{5} - 22x^{4} + 17x^{3} + 131x^{2} - 82x - 148 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 12\nu^{4} + 62\nu^{3} - 99\nu^{2} - 90\nu - 62 ) / 26 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 12\nu^{4} - 62\nu^{3} + 125\nu^{2} + 90\nu - 146 ) / 26 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + 10\nu^{4} + 17\nu^{3} - 115\nu^{2} + 29\nu + 152 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 15\nu^{4} + 19\nu^{3} - 166\nu^{2} + 102\nu + 176 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} + 7\beta_{4} + 16\beta_{3} + 14\beta_{2} + \beta _1 + 82 \)
|
| \(\nu^{5}\) | \(=\) |
\( -32\beta_{5} + 54\beta_{4} + 31\beta_{3} + 21\beta_{2} + 96\beta _1 + 26 \)
|
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 | −2.68133 | 0 | −2.76342 | 0 | −0.871698 | 0 | 4.18953 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.68133 | 0 | 3.50932 | 0 | 3.80713 | 0 | 4.18953 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.642074 | 0 | 0.777642 | 0 | −3.16546 | 0 | −2.58774 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.642074 | 0 | 2.33727 | 0 | 1.69262 | 0 | −2.58774 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.32340 | 0 | −3.79270 | 0 | 2.66299 | 0 | 2.39821 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.32340 | 0 | 2.93190 | 0 | −3.12559 | 0 | 2.39821 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 6032.2.a.s | 6 | |
| 4.b | odd | 2 | 1 | 754.2.a.j | ✓ | 6 | |
| 12.b | even | 2 | 1 | 6786.2.a.bq | 6 | ||
| 52.b | odd | 2 | 1 | 9802.2.a.z | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 754.2.a.j | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 6032.2.a.s | 6 | 1.a | even | 1 | 1 | trivial | |
| 6786.2.a.bq | 6 | 12.b | even | 2 | 1 | ||
| 9802.2.a.z | 6 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):
|
\( T_{3}^{3} + T_{3}^{2} - 6T_{3} - 4 \)
|
|
\( T_{5}^{6} - 3T_{5}^{5} - 20T_{5}^{4} + 67T_{5}^{3} + 69T_{5}^{2} - 336T_{5} + 196 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{3} + T^{2} - 6 T - 4)^{2} \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 196 \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots - 148 \)
|
| $11$ |
\( T^{6} + 7 T^{5} + \cdots + 1184 \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 112 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 32 \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 896 \)
|
| $29$ |
\( (T - 1)^{6} \)
|
| $31$ |
\( (T^{3} - 2 T^{2} + \cdots + 128)^{2} \)
|
| $37$ |
\( T^{6} - 22 T^{5} + \cdots + 82352 \)
|
| $41$ |
\( T^{6} - T^{5} + \cdots - 49672 \)
|
| $43$ |
\( T^{6} + 4 T^{5} + \cdots - 1024 \)
|
| $47$ |
\( T^{6} - 24 T^{5} + \cdots + 98048 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 49024 \)
|
| $59$ |
\( T^{6} - 10 T^{5} + \cdots + 224 \)
|
| $61$ |
\( T^{6} + 3 T^{5} + \cdots - 615592 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 363712 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 18688 \)
|
| $73$ |
\( T^{6} - T^{5} + \cdots - 68296 \)
|
| $79$ |
\( T^{6} - 39 T^{5} + \cdots - 188744 \)
|
| $83$ |
\( T^{6} + 8 T^{5} + \cdots + 10976 \)
|
| $89$ |
\( T^{6} - 5 T^{5} + \cdots + 149552 \)
|
| $97$ |
\( T^{6} - T^{5} + \cdots + 5488 \)
|
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