Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [150,12,Mod(49,150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("150.49");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.c (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: | \(115.251477084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1161865x^{2} + 337481988624 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 1161865x^{2} + 337481988624 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 580933\nu ) / 580932 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 8713985\nu ) / 96822 \)
|
| \(\beta_{3}\) | \(=\) |
\( 60\nu^{2} + 34855950 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 30\beta_1 ) / 60 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 34855950 ) / 60 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -580933\beta_{2} + 52283910\beta_1 ) / 60 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 32.0000i | − | 243.000i | −1024.00 | 0 | −7776.00 | − | 27124.3i | 32768.0i | −59049.0 | 0 | |||||||||||||||||||||||||||
| 49.2 | − | 32.0000i | − | 243.000i | −1024.00 | 0 | −7776.00 | 64338.3i | 32768.0i | −59049.0 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 32.0000i | 243.000i | −1024.00 | 0 | −7776.00 | − | 64338.3i | − | 32768.0i | −59049.0 | 0 | |||||||||||||||||||||||||||||
| 49.4 | 32.0000i | 243.000i | −1024.00 | 0 | −7776.00 | 27124.3i | − | 32768.0i | −59049.0 | 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 | 150.12.c.k | 4 | |
| 5.b | even | 2 | 1 | inner | 150.12.c.k | 4 | |
| 5.c | odd | 4 | 1 | 150.12.a.n | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 150.12.a.o | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 150.12.a.n | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 150.12.a.o | yes | 2 | 5.c | odd | 4 | 1 | |
| 150.12.c.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 150.12.c.k | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 4875153098T_{7}^{2} + 3045498440391193801 \)
acting on \(S_{12}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1024)^{2} \)
|
| $3$ |
\( (T^{2} + 59049)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 30\!\cdots\!01 \)
|
| $11$ |
\( (T^{2} - 218880 T - 241076974500)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 45\!\cdots\!01 \)
|
| $17$ |
\( T^{4} + \cdots + 56\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots - 42641719789859)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 47\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 821192138631109)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 40\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + \cdots + 58\!\cdots\!76)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 24\!\cdots\!61 \)
|
| $47$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 94\!\cdots\!75)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 17\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} + \cdots - 47\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 63\!\cdots\!16 \)
|
| $79$ |
\( (T^{2} + \cdots + 70\!\cdots\!44)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 87\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + \cdots + 59\!\cdots\!04)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 53\!\cdots\!41 \)
|
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