Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1725,2,Mod(1174,1725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1725, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1725.1174");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1725 = 3 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1725.b (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: | \(13.7741943487\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 345) |
| 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).
| \(n\) | \(277\) | \(1151\) | \(1201\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1174.1 |
|
− | 2.34292i | − | 1.00000i | −3.48929 | 0 | −2.34292 | 4.48929i | 3.48929i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1174.2 | − | 1.81361i | 1.00000i | −1.28917 | 0 | 1.81361 | − | 2.28917i | − | 1.28917i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1174.3 | − | 0.470683i | − | 1.00000i | 1.77846 | 0 | −0.470683 | − | 0.778457i | − | 1.77846i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1174.4 | 0.470683i | 1.00000i | 1.77846 | 0 | −0.470683 | 0.778457i | 1.77846i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1174.5 | 1.81361i | − | 1.00000i | −1.28917 | 0 | 1.81361 | 2.28917i | 1.28917i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1174.6 | 2.34292i | 1.00000i | −3.48929 | 0 | −2.34292 | − | 4.48929i | − | 3.48929i | −1.00000 | 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 | 1725.2.b.u | 6 | |
| 5.b | even | 2 | 1 | inner | 1725.2.b.u | 6 | |
| 5.c | odd | 4 | 1 | 345.2.a.j | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 1725.2.a.bi | 3 | ||
| 15.e | even | 4 | 1 | 1035.2.a.n | 3 | ||
| 15.e | even | 4 | 1 | 5175.2.a.br | 3 | ||
| 20.e | even | 4 | 1 | 5520.2.a.by | 3 | ||
| 115.e | even | 4 | 1 | 7935.2.a.u | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 345.2.a.j | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1035.2.a.n | 3 | 15.e | even | 4 | 1 | ||
| 1725.2.a.bi | 3 | 5.c | odd | 4 | 1 | ||
| 1725.2.b.u | 6 | 1.a | even | 1 | 1 | trivial | |
| 1725.2.b.u | 6 | 5.b | even | 2 | 1 | inner | |
| 5175.2.a.br | 3 | 15.e | even | 4 | 1 | ||
| 5520.2.a.by | 3 | 20.e | even | 4 | 1 | ||
| 7935.2.a.u | 3 | 115.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1725, [\chi])\):
|
\( T_{2}^{6} + 9T_{2}^{4} + 20T_{2}^{2} + 4 \)
|
|
\( T_{7}^{6} + 26T_{7}^{4} + 121T_{7}^{2} + 64 \)
|
|
\( T_{11}^{3} + 2T_{11}^{2} - 6T_{11} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 9 T^{4} + \cdots + 4 \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 26 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T^{3} + 2 T^{2} - 6 T - 8)^{2} \)
|
| $13$ |
\( T^{6} + 20 T^{4} + \cdots + 16 \)
|
| $17$ |
\( T^{6} + 10 T^{4} + \cdots + 4 \)
|
| $19$ |
\( (T^{3} + 2 T^{2} - 14 T - 32)^{2} \)
|
| $23$ |
\( (T^{2} + 1)^{3} \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 27 T + 86)^{2} \)
|
| $31$ |
\( (T^{3} - 2 T^{2} - 15 T + 32)^{2} \)
|
| $37$ |
\( T^{6} + 14 T^{4} + \cdots + 4 \)
|
| $41$ |
\( (T^{3} + 2 T^{2} + \cdots - 218)^{2} \)
|
| $43$ |
\( T^{6} + 168 T^{4} + \cdots + 30976 \)
|
| $47$ |
\( T^{6} + 80 T^{4} + \cdots + 256 \)
|
| $53$ |
\( T^{6} + 90 T^{4} + \cdots + 2916 \)
|
| $59$ |
\( (T^{3} - 12 T^{2} + \cdots - 32)^{2} \)
|
| $61$ |
\( (T^{3} + 4 T^{2} - 10 T + 4)^{2} \)
|
| $67$ |
\( T^{6} + 290 T^{4} + \cdots + 524176 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} + \cdots - 148)^{2} \)
|
| $73$ |
\( T^{6} + 340 T^{4} + \cdots + 868624 \)
|
| $79$ |
\( (T^{3} - 4 T^{2} - 112 T - 64)^{2} \)
|
| $83$ |
\( T^{6} + 70 T^{4} + \cdots + 8464 \)
|
| $89$ |
\( (T^{3} + 6 T^{2} - 160 T + 16)^{2} \)
|
| $97$ |
\( T^{6} + 516 T^{4} + \cdots + 4734976 \)
|
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