Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1450,2,Mod(157,1450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1450, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1450.157");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1450 = 2 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1450.j (of order \(4\), 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: | \(11.5783082931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.6420496384.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 22x^{6} + 155x^{4} + 406x^{2} + 289 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 290) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 22x^{6} + 155x^{4} + 406x^{2} + 289 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{7} + 17\nu^{6} + 66\nu^{5} + 289\nu^{4} + 414\nu^{3} + 1139\nu^{2} + 657\nu + 646 ) / 408 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{7} - 17\nu^{6} + 66\nu^{5} - 289\nu^{4} + 414\nu^{3} - 1139\nu^{2} + 657\nu - 646 ) / 408 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 17\nu^{4} + 73\nu^{2} + 74 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{7} + 71\nu^{5} + 331\nu^{3} + 383\nu ) / 102 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 17\nu^{5} - 3\nu^{4} - 67\nu^{3} - 33\nu^{2} - 38\nu - 51 ) / 24 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 17\nu^{5} + 3\nu^{4} - 67\nu^{3} + 33\nu^{2} - 38\nu + 51 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} - 2\beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 2\beta_{3} - 2\beta_{2} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} - 4\beta_{6} - 11\beta_{4} - 22\beta_{3} + 22\beta_{2} + 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( -26\beta_{7} - 26\beta_{6} - 71\beta_{5} + 42\beta_{3} + 42\beta_{2} + 49\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -68\beta_{7} + 68\beta_{6} + 120\beta_{4} + 228\beta_{3} - 228\beta_{2} - 469 \)
|
| \(\nu^{7}\) | \(=\) |
\( 296\beta_{7} + 296\beta_{6} + 872\beta_{5} - 580\beta_{3} - 580\beta_{2} - 469\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).
| \(n\) | \(901\) | \(1277\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
1.00000i | −2.34226 | −1.00000 | 0 | − | 2.34226i | 0.978925 | + | 0.978925i | − | 1.00000i | 2.48617 | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.2 | 1.00000i | −1.05614 | −1.00000 | 0 | − | 1.05614i | 3.51005 | + | 3.51005i | − | 1.00000i | −1.88457 | 0 | |||||||||||||||||||||||||||||||||||||||
| 157.3 | 1.00000i | 2.05614 | −1.00000 | 0 | 2.05614i | −1.80294 | − | 1.80294i | − | 1.00000i | 1.22771 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 157.4 | 1.00000i | 3.34226 | −1.00000 | 0 | 3.34226i | −0.686031 | − | 0.686031i | − | 1.00000i | 8.17068 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1293.1 | − | 1.00000i | −2.34226 | −1.00000 | 0 | 2.34226i | 0.978925 | − | 0.978925i | 1.00000i | 2.48617 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1293.2 | − | 1.00000i | −1.05614 | −1.00000 | 0 | 1.05614i | 3.51005 | − | 3.51005i | 1.00000i | −1.88457 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1293.3 | − | 1.00000i | 2.05614 | −1.00000 | 0 | − | 2.05614i | −1.80294 | + | 1.80294i | 1.00000i | 1.22771 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1293.4 | − | 1.00000i | 3.34226 | −1.00000 | 0 | − | 3.34226i | −0.686031 | + | 0.686031i | 1.00000i | 8.17068 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 145.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1450.2.j.e | 8 | |
| 5.b | even | 2 | 1 | 290.2.j.e | yes | 8 | |
| 5.c | odd | 4 | 1 | 290.2.e.e | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 1450.2.e.e | 8 | ||
| 29.c | odd | 4 | 1 | 1450.2.e.e | 8 | ||
| 145.e | even | 4 | 1 | inner | 1450.2.j.e | 8 | |
| 145.f | odd | 4 | 1 | 290.2.e.e | ✓ | 8 | |
| 145.j | even | 4 | 1 | 290.2.j.e | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 290.2.e.e | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 290.2.e.e | ✓ | 8 | 145.f | odd | 4 | 1 | |
| 290.2.j.e | yes | 8 | 5.b | even | 2 | 1 | |
| 290.2.j.e | yes | 8 | 145.j | even | 4 | 1 | |
| 1450.2.e.e | 8 | 5.c | odd | 4 | 1 | ||
| 1450.2.e.e | 8 | 29.c | odd | 4 | 1 | ||
| 1450.2.j.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 1450.2.j.e | 8 | 145.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 9T_{3}^{2} + 10T_{3} + 17 \)
|
|
\( T_{11}^{4} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{4} - 2 T^{3} - 9 T^{2} + \cdots + 17)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 289 \)
|
| $11$ |
\( (T^{4} + 16)^{2} \)
|
| $13$ |
\( T^{8} + 8 T^{5} + \cdots + 289 \)
|
| $17$ |
\( T^{8} + 50 T^{6} + \cdots + 3969 \)
|
| $19$ |
\( T^{8} - 20 T^{7} + \cdots + 126736 \)
|
| $23$ |
\( T^{8} - 4 T^{7} + \cdots + 693889 \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots + 841)^{2} \)
|
| $31$ |
\( T^{8} - 40 T^{5} + \cdots + 49 \)
|
| $37$ |
\( (T^{4} - 16 T^{3} + \cdots - 576)^{2} \)
|
| $41$ |
\( T^{8} + 16 T^{7} + \cdots + 614656 \)
|
| $43$ |
\( (T^{4} - 18 T^{3} + 55 T^{2} + \cdots + 9)^{2} \)
|
| $47$ |
\( (T^{4} + 16 T^{3} + \cdots - 92)^{2} \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 961 \)
|
| $59$ |
\( T^{8} + 82 T^{6} + \cdots + 10609 \)
|
| $61$ |
\( T^{8} - 4 T^{7} + \cdots + 96059601 \)
|
| $67$ |
\( T^{8} + 20 T^{7} + \cdots + 1245456 \)
|
| $71$ |
\( T^{8} + 396 T^{6} + \cdots + 26173456 \)
|
| $73$ |
\( T^{8} + 242 T^{6} + \cdots + 564001 \)
|
| $79$ |
\( T^{8} + 24 T^{7} + \cdots + 677329 \)
|
| $83$ |
\( T^{8} + 32 T^{7} + \cdots + 169744 \)
|
| $89$ |
\( T^{8} + 20 T^{7} + \cdots + 3341584 \)
|
| $97$ |
\( (T^{4} + 14 T^{3} + 49 T^{2} + \cdots + 9)^{2} \)
|
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