Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(112.268673869\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1009}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 252 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{1009})\). 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 | −7.38238 | 0 | 0 | 0 | 49.0000 | 0 | −188.500 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 24.3824 | 0 | 0 | 0 | 49.0000 | 0 | 351.500 | 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 | 700.6.a.g | 2 | |
| 5.b | even | 2 | 1 | 140.6.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 700.6.e.f | 4 | ||
| 20.d | odd | 2 | 1 | 560.6.a.o | 2 | ||
| 35.c | odd | 2 | 1 | 980.6.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.a.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 560.6.a.o | 2 | 20.d | odd | 2 | 1 | ||
| 700.6.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 700.6.e.f | 4 | 5.c | odd | 4 | 2 | ||
| 980.6.a.f | 2 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 17T_{3} - 180 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(700))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 17T - 180 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T - 49)^{2} \)
|
| $11$ |
\( T^{2} + 167T - 13460 \)
|
| $13$ |
\( T^{2} + 445T - 374526 \)
|
| $17$ |
\( T^{2} + 99T - 130990 \)
|
| $19$ |
\( T^{2} + 1334T - 88872 \)
|
| $23$ |
\( T^{2} - 290T - 948624 \)
|
| $29$ |
\( T^{2} + 6959 T - 4814262 \)
|
| $31$ |
\( T^{2} + 10480 T + 24293376 \)
|
| $37$ |
\( T^{2} - 1588 T - 12026460 \)
|
| $41$ |
\( T^{2} + 24110 T + 139946064 \)
|
| $43$ |
\( T^{2} - 20406 T - 43907992 \)
|
| $47$ |
\( T^{2} + 619 T - 137663232 \)
|
| $53$ |
\( T^{2} - 21794 T + 106312720 \)
|
| $59$ |
\( T^{2} - 1976 T - 247254000 \)
|
| $61$ |
\( T^{2} - 22614 T - 379356880 \)
|
| $67$ |
\( T^{2} + 4068 T - 97896960 \)
|
| $71$ |
\( T^{2} + 46528 T + 117530560 \)
|
| $73$ |
\( T^{2} + 27096 T - 211854580 \)
|
| $79$ |
\( T^{2} + 12481 T - 899688752 \)
|
| $83$ |
\( T^{2} + \cdots - 1661783280 \)
|
| $89$ |
\( T^{2} + \cdots - 5444221000 \)
|
| $97$ |
\( T^{2} + \cdots + 3603178714 \)
|
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