Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,4,Mod(1,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.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: | \(41.3013370040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 140) |
| 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 \(\beta = \sqrt{6}\). 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 | −2.44949 | 0 | 0 | 0 | −7.00000 | 0 | −21.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.44949 | 0 | 0 | 0 | −7.00000 | 0 | −21.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 700.4.a.p | 2 | |
| 5.b | even | 2 | 1 | 700.4.a.q | 2 | ||
| 5.c | odd | 4 | 2 | 140.4.e.c | ✓ | 4 | |
| 15.e | even | 4 | 2 | 1260.4.k.d | 4 | ||
| 20.e | even | 4 | 2 | 560.4.g.d | 4 | ||
| 35.f | even | 4 | 2 | 980.4.e.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.4.e.c | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 560.4.g.d | 4 | 20.e | even | 4 | 2 | ||
| 700.4.a.p | 2 | 1.a | even | 1 | 1 | trivial | |
| 700.4.a.q | 2 | 5.b | even | 2 | 1 | ||
| 980.4.e.d | 4 | 35.f | even | 4 | 2 | ||
| 1260.4.k.d | 4 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(700))\):
|
\( T_{3}^{2} - 6 \)
|
|
\( T_{11}^{2} - 48T_{11} - 600 \)
|
|
\( T_{13}^{2} - 28T_{13} - 2978 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 6 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T + 7)^{2} \)
|
| $11$ |
\( T^{2} - 48T - 600 \)
|
| $13$ |
\( T^{2} - 28T - 2978 \)
|
| $17$ |
\( T^{2} - 28T - 3260 \)
|
| $19$ |
\( T^{2} + 88T + 1642 \)
|
| $23$ |
\( T^{2} + 28T + 172 \)
|
| $29$ |
\( T^{2} + 140T + 3724 \)
|
| $31$ |
\( T^{2} - 104T + 1528 \)
|
| $37$ |
\( T^{2} - 364T + 19300 \)
|
| $41$ |
\( T^{2} - 180T - 134196 \)
|
| $43$ |
\( T^{2} - 392T - 14600 \)
|
| $47$ |
\( T^{2} - 56T - 28616 \)
|
| $53$ |
\( T^{2} - 924T + 198444 \)
|
| $59$ |
\( T^{2} - 568T - 74870 \)
|
| $61$ |
\( T^{2} + 468T - 100770 \)
|
| $67$ |
\( T^{2} - 224T - 485120 \)
|
| $71$ |
\( T^{2} + 972T + 235020 \)
|
| $73$ |
\( T^{2} - 1316 T + 375340 \)
|
| $79$ |
\( T^{2} - 964T - 502676 \)
|
| $83$ |
\( T^{2} - 672T - 381318 \)
|
| $89$ |
\( T^{2} + 1652 T + 512932 \)
|
| $97$ |
\( T^{2} - 1428 T + 507396 \)
|
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