Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [700,2,Mod(401,700)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("700.401");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 700.i (of order \(3\), degree \(2\), 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: | \(5.58952814149\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.1783323.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{5} + 15\nu^{4} + 8\nu^{3} + 57\nu^{2} + 47\nu + 180 ) / 83 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 131\nu - 240 ) / 83 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} + 25\nu^{4} - 42\nu^{3} + 95\nu^{2} - 60\nu + 300 ) / 83 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu - 45 ) / 249 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 16\nu^{5} + 3\nu^{4} + 68\nu^{3} + 28\nu^{2} + 358\nu + 36 ) / 83 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} - 9\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 9 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{5} + 12\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 18\beta_{5} + 9\beta_{4} + 5\beta_{3} - 23\beta_{2} - 46\beta _1 + 9 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(477\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 401.1 |
|
0 | −1.60220 | − | 2.77509i | 0 | 0 | 0 | 1.53189 | − | 2.15715i | 0 | −3.63409 | + | 6.29444i | 0 | ||||||||||||||||||||||||||||||
| 401.2 | 0 | −0.182224 | − | 0.315621i | 0 | 0 | 0 | −2.11581 | + | 1.58850i | 0 | 1.43359 | − | 2.48305i | 0 | |||||||||||||||||||||||||||||||
| 401.3 | 0 | 1.28442 | + | 2.22469i | 0 | 0 | 0 | 2.58392 | + | 0.568650i | 0 | −1.79949 | + | 3.11682i | 0 | |||||||||||||||||||||||||||||||
| 501.1 | 0 | −1.60220 | + | 2.77509i | 0 | 0 | 0 | 1.53189 | + | 2.15715i | 0 | −3.63409 | − | 6.29444i | 0 | |||||||||||||||||||||||||||||||
| 501.2 | 0 | −0.182224 | + | 0.315621i | 0 | 0 | 0 | −2.11581 | − | 1.58850i | 0 | 1.43359 | + | 2.48305i | 0 | |||||||||||||||||||||||||||||||
| 501.3 | 0 | 1.28442 | − | 2.22469i | 0 | 0 | 0 | 2.58392 | − | 0.568650i | 0 | −1.79949 | − | 3.11682i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 700.2.i.d | ✓ | 6 |
| 5.b | even | 2 | 1 | 700.2.i.e | yes | 6 | |
| 5.c | odd | 4 | 2 | 700.2.r.d | 12 | ||
| 7.c | even | 3 | 1 | inner | 700.2.i.d | ✓ | 6 |
| 7.c | even | 3 | 1 | 4900.2.a.bc | 3 | ||
| 7.d | odd | 6 | 1 | 4900.2.a.bb | 3 | ||
| 35.i | odd | 6 | 1 | 4900.2.a.bd | 3 | ||
| 35.j | even | 6 | 1 | 700.2.i.e | yes | 6 | |
| 35.j | even | 6 | 1 | 4900.2.a.ba | 3 | ||
| 35.k | even | 12 | 2 | 4900.2.e.s | 6 | ||
| 35.l | odd | 12 | 2 | 700.2.r.d | 12 | ||
| 35.l | odd | 12 | 2 | 4900.2.e.t | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 700.2.i.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 700.2.i.d | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 700.2.i.e | yes | 6 | 5.b | even | 2 | 1 | |
| 700.2.i.e | yes | 6 | 35.j | even | 6 | 1 | |
| 700.2.r.d | 12 | 5.c | odd | 4 | 2 | ||
| 700.2.r.d | 12 | 35.l | odd | 12 | 2 | ||
| 4900.2.a.ba | 3 | 35.j | even | 6 | 1 | ||
| 4900.2.a.bb | 3 | 7.d | odd | 6 | 1 | ||
| 4900.2.a.bc | 3 | 7.c | even | 3 | 1 | ||
| 4900.2.a.bd | 3 | 35.i | odd | 6 | 1 | ||
| 4900.2.e.s | 6 | 35.k | even | 12 | 2 | ||
| 4900.2.e.t | 6 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\):
|
\( T_{3}^{6} + T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} + 67T_{3}^{2} + 24T_{3} + 9 \)
|
|
\( T_{11}^{6} + 4T_{11}^{5} + 19T_{11}^{4} + 6T_{11}^{3} + 45T_{11}^{2} + 27T_{11} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + 9 T^{4} + \cdots + 9 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 4 T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 81 \)
|
| $13$ |
\( (T^{3} - 8 T^{2} + 13 T + 3)^{2} \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 6561 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots + 441 \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 6561 \)
|
| $29$ |
\( (T^{3} - 45 T + 81)^{2} \)
|
| $31$ |
\( T^{6} - 3 T^{5} + \cdots + 1369 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 22201 \)
|
| $41$ |
\( (T^{3} - 11 T^{2} + \cdots + 873)^{2} \)
|
| $43$ |
\( (T^{3} + 11 T^{2} + \cdots - 71)^{2} \)
|
| $47$ |
\( T^{6} + 4 T^{5} + \cdots + 81 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots + 301401 \)
|
| $59$ |
\( T^{6} - 5 T^{5} + \cdots + 81 \)
|
| $61$ |
\( T^{6} + 17 T^{5} + \cdots + 1 \)
|
| $67$ |
\( T^{6} + 20 T^{5} + \cdots + 5184 \)
|
| $71$ |
\( (T^{3} + 19 T^{2} + \cdots + 45)^{2} \)
|
| $73$ |
\( (T^{2} + 4 T + 16)^{3} \)
|
| $79$ |
\( T^{6} - T^{5} + \cdots + 201601 \)
|
| $83$ |
\( (T^{3} - 28 T^{2} + \cdots + 981)^{2} \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 2025 \)
|
| $97$ |
\( (T^{3} - 15 T^{2} - 18 T + 5)^{2} \)
|
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