Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [260,6,Mod(1,260)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(260, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("260.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.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: | \(41.6997931514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{51}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 51 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(\beta = \sqrt{51}\). We also show the integral \(q\)-expansion of the trace form.
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 | −24.4243 | 0 | −25.0000 | 0 | −127.131 | 0 | 353.546 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 18.4243 | 0 | −25.0000 | 0 | −12.8686 | 0 | 96.4543 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 260.6.a.c | ✓ | 2 |
| 4.b | odd | 2 | 1 | 1040.6.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.6.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1040.6.a.f | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 6T_{3} - 450 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(260))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 6T - 450 \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( T^{2} + 140T + 1636 \)
|
| $11$ |
\( T^{2} - 66T - 61386 \)
|
| $13$ |
\( (T - 169)^{2} \)
|
| $17$ |
\( T^{2} + 1872 T + 802452 \)
|
| $19$ |
\( T^{2} - 2134T - 55370 \)
|
| $23$ |
\( T^{2} - 3150 T + 2472006 \)
|
| $29$ |
\( T^{2} + 5100 T - 10535784 \)
|
| $31$ |
\( T^{2} + 5486 T - 4571570 \)
|
| $37$ |
\( T^{2} + 20312 T + 102409936 \)
|
| $41$ |
\( T^{2} - 13776 T + 14313108 \)
|
| $43$ |
\( T^{2} + 2462 T - 65346914 \)
|
| $47$ |
\( T^{2} - 10908 T + 10393860 \)
|
| $53$ |
\( T^{2} + 3600 T - 348449676 \)
|
| $59$ |
\( T^{2} + 6846 T - 10972410 \)
|
| $61$ |
\( T^{2} - 7492 T - 35148008 \)
|
| $67$ |
\( T^{2} + 3944 T - 74781980 \)
|
| $71$ |
\( T^{2} + 62178 T + 895393110 \)
|
| $73$ |
\( T^{2} + \cdots - 1238069168 \)
|
| $79$ |
\( T^{2} + 71084 T - 337268840 \)
|
| $83$ |
\( T^{2} + \cdots + 2668017540 \)
|
| $89$ |
\( T^{2} + \cdots - 5546687436 \)
|
| $97$ |
\( T^{2} + \cdots - 9220783868 \)
|
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