Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,6,Mod(449,700)]
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, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.449");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.e (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(112.268673869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 625x^{4} + 100824x^{2} + 3027600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 140) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} + 625x^{4} + 100824x^{2} + 3027600 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 313\nu^{2} + 1740 ) / 1428 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 1115\nu^{3} + 443796\nu ) / 2484720 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 517\nu^{2} + 44784 ) / 204 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1529\nu^{5} + 779885\nu^{3} + 79275516\nu ) / 2484720 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 7\beta_{2} - 211 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 1529\beta_{3} - 305\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -313\beta_{4} + 3619\beta_{2} + 64303 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1115\beta_{5} - 779885\beta_{3} + 103721\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(477\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | − | 17.3069i | 0 | 0 | 0 | 49.0000i | 0 | −56.5286 | 0 | |||||||||||||||||||||||||||||||||||
| 449.2 | 0 | − | 16.5332i | 0 | 0 | 0 | − | 49.0000i | 0 | −30.3467 | 0 | |||||||||||||||||||||||||||||||||||
| 449.3 | 0 | − | 9.22631i | 0 | 0 | 0 | 49.0000i | 0 | 157.875 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.4 | 0 | 9.22631i | 0 | 0 | 0 | − | 49.0000i | 0 | 157.875 | 0 | ||||||||||||||||||||||||||||||||||||
| 449.5 | 0 | 16.5332i | 0 | 0 | 0 | 49.0000i | 0 | −30.3467 | 0 | |||||||||||||||||||||||||||||||||||||
| 449.6 | 0 | 17.3069i | 0 | 0 | 0 | − | 49.0000i | 0 | −56.5286 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 700.6.e.h | 6 | |
| 5.b | even | 2 | 1 | inner | 700.6.e.h | 6 | |
| 5.c | odd | 4 | 1 | 140.6.a.c | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 700.6.a.j | 3 | ||
| 20.e | even | 4 | 1 | 560.6.a.u | 3 | ||
| 35.f | even | 4 | 1 | 980.6.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.a.c | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 560.6.a.u | 3 | 20.e | even | 4 | 1 | ||
| 700.6.a.j | 3 | 5.c | odd | 4 | 1 | ||
| 700.6.e.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 700.6.e.h | 6 | 5.b | even | 2 | 1 | inner | |
| 980.6.a.i | 3 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 658T_{3}^{4} + 130641T_{3}^{2} + 6969600 \)
acting on \(S_{6}^{\mathrm{new}}(700, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 658 T^{4} + \cdots + 6969600 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} + 2401)^{3} \)
|
| $11$ |
\( (T^{3} - 58 T^{2} + \cdots - 14472500)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 71\!\cdots\!44 \)
|
| $17$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + 1692 T^{2} + \cdots - 8250023232)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 12\!\cdots\!76 \)
|
| $29$ |
\( (T^{3} + 5572 T^{2} + \cdots - 4810314762)^{2} \)
|
| $31$ |
\( (T^{3} - 2816 T^{2} + \cdots + 19217358464)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} - 7706 T^{2} + \cdots + 285705250976)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 53\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 12\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + \cdots + 11147794362880)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 27664467900480)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 61\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + \cdots - 17256845706240)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots - 5100296528456)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 62\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 102863482208800)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 16\!\cdots\!24 \)
|
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