Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,7,Mod(28,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.28");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.g (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: | \(10.3524337629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{201})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 101x^{2} + 2500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{4} + 101x^{2} + 2500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 51\nu ) / 50 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 51 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 50\nu^{2} + 101\nu - 2550 ) / 50 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + \beta _1 - 102 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -51\beta_{3} - 51\beta_{2} + 151\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−4.58872 | + | 4.58872i | 0 | 21.8872i | 123.831 | − | 17.0564i | 0 | 215.476 | − | 215.476i | −394.113 | − | 394.113i | 0 | −489.959 | + | 646.493i | ||||||||||||||||||||
| 28.2 | 9.58872 | − | 9.58872i | 0 | − | 119.887i | −88.8309 | − | 87.9436i | 0 | 59.5240 | − | 59.5240i | −535.887 | − | 535.887i | 0 | −1695.04 | + | 8.50745i | ||||||||||||||||||||
| 37.1 | −4.58872 | − | 4.58872i | 0 | − | 21.8872i | 123.831 | + | 17.0564i | 0 | 215.476 | + | 215.476i | −394.113 | + | 394.113i | 0 | −489.959 | − | 646.493i | ||||||||||||||||||||
| 37.2 | 9.58872 | + | 9.58872i | 0 | 119.887i | −88.8309 | + | 87.9436i | 0 | 59.5240 | + | 59.5240i | −535.887 | + | 535.887i | 0 | −1695.04 | − | 8.50745i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.7.g.a | 4 | |
| 3.b | odd | 2 | 1 | 5.7.c.a | ✓ | 4 | |
| 5.b | even | 2 | 1 | 225.7.g.c | 4 | ||
| 5.c | odd | 4 | 1 | inner | 45.7.g.a | 4 | |
| 5.c | odd | 4 | 1 | 225.7.g.c | 4 | ||
| 12.b | even | 2 | 1 | 80.7.p.b | 4 | ||
| 15.d | odd | 2 | 1 | 25.7.c.c | 4 | ||
| 15.e | even | 4 | 1 | 5.7.c.a | ✓ | 4 | |
| 15.e | even | 4 | 1 | 25.7.c.c | 4 | ||
| 60.l | odd | 4 | 1 | 80.7.p.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.7.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 5.7.c.a | ✓ | 4 | 15.e | even | 4 | 1 | |
| 25.7.c.c | 4 | 15.d | odd | 2 | 1 | ||
| 25.7.c.c | 4 | 15.e | even | 4 | 1 | ||
| 45.7.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 45.7.g.a | 4 | 5.c | odd | 4 | 1 | inner | |
| 80.7.p.b | 4 | 12.b | even | 2 | 1 | ||
| 80.7.p.b | 4 | 60.l | odd | 4 | 1 | ||
| 225.7.g.c | 4 | 5.b | even | 2 | 1 | ||
| 225.7.g.c | 4 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 10T_{2}^{3} + 50T_{2}^{2} + 880T_{2} + 7744 \)
acting on \(S_{7}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 10 T^{3} + \cdots + 7744 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 70 T^{3} + \cdots + 244140625 \)
|
| $7$ |
\( T^{4} - 550 T^{3} + \cdots + 658025104 \)
|
| $11$ |
\( (T^{2} - 526 T - 1061456)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 1753977086884 \)
|
| $17$ |
\( T^{4} + \cdots + 65054193721924 \)
|
| $19$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 32\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + 16586 T - 529326776)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 36\!\cdots\!64 \)
|
| $41$ |
\( (T^{2} - 106966 T - 2214944336)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 14\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 63\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + 55526 T - 1518731456)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 50\!\cdots\!24 \)
|
| $71$ |
\( (T^{2} + 655094 T + 86681396584)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 21\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots + 72\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
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