Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,9,Mod(7,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.c (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: | \(10.1844652515\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{141})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 71x^{2} + 1225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 71x^{2} + 1225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 36\nu ) / 35 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 70\nu^{2} + 106\nu + 2485 ) / 35 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 70\nu^{2} + 106\nu - 2485 ) / 35 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 142 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -9\beta_{3} - 9\beta_{2} + 53\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−11.8743 | − | 11.8743i | −11.8743 | + | 11.8743i | 26.0000i | 0 | 282.000 | −2742.97 | − | 2742.97i | −2731.10 | + | 2731.10i | 6279.00i | 0 | ||||||||||||||||||||||
| 7.2 | 11.8743 | + | 11.8743i | 11.8743 | − | 11.8743i | 26.0000i | 0 | 282.000 | 2742.97 | + | 2742.97i | 2731.10 | − | 2731.10i | 6279.00i | 0 | |||||||||||||||||||||||
| 18.1 | −11.8743 | + | 11.8743i | −11.8743 | − | 11.8743i | − | 26.0000i | 0 | 282.000 | −2742.97 | + | 2742.97i | −2731.10 | − | 2731.10i | − | 6279.00i | 0 | |||||||||||||||||||||
| 18.2 | 11.8743 | − | 11.8743i | 11.8743 | + | 11.8743i | − | 26.0000i | 0 | 282.000 | 2742.97 | − | 2742.97i | 2731.10 | + | 2731.10i | − | 6279.00i | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.9.c.a | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 25.9.c.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | inner | 25.9.c.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.9.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 25.9.c.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 25.9.c.a | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 79524 \)
acting on \(S_{9}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 79524 \)
|
| $3$ |
\( T^{4} + 79524 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 226436345031204 \)
|
| $11$ |
\( (T - 12132)^{4} \)
|
| $13$ |
\( T^{4} + 137726936146944 \)
|
| $17$ |
\( T^{4} + 14\!\cdots\!64 \)
|
| $19$ |
\( (T^{2} + 28351824400)^{2} \)
|
| $23$ |
\( T^{4} + 96\!\cdots\!64 \)
|
| $29$ |
\( (T^{2} + 444395556900)^{2} \)
|
| $31$ |
\( (T + 1042808)^{4} \)
|
| $37$ |
\( T^{4} + 74\!\cdots\!84 \)
|
| $41$ |
\( (T + 1321128)^{4} \)
|
| $43$ |
\( T^{4} + 23\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + 69\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + 38\!\cdots\!24 \)
|
| $59$ |
\( (T^{2} + 42231022131600)^{2} \)
|
| $61$ |
\( (T + 14393968)^{4} \)
|
| $67$ |
\( T^{4} + 68\!\cdots\!64 \)
|
| $71$ |
\( (T + 23065488)^{4} \)
|
| $73$ |
\( T^{4} + 36\!\cdots\!64 \)
|
| $79$ |
\( (T^{2} + 7621354062400)^{2} \)
|
| $83$ |
\( T^{4} + 70\!\cdots\!84 \)
|
| $89$ |
\( (T^{2} + 682803552860100)^{2} \)
|
| $97$ |
\( T^{4} + 17\!\cdots\!44 \)
|
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