Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,8,Mod(1,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 300.a (trivial) |
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: | yes |
| Analytic conductor: | \(93.7155076452\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1129}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 282 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 60) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 5\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −27.0000 | 0 | 0 | 0 | 284.997 | 0 | 729.000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | 0 | 0 | 621.003 | 0 | 729.000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 300.8.a.j | 2 | |
| 5.b | even | 2 | 1 | 300.8.a.k | 2 | ||
| 5.c | odd | 4 | 2 | 60.8.d.b | ✓ | 4 | |
| 15.e | even | 4 | 2 | 180.8.d.c | 4 | ||
| 20.e | even | 4 | 2 | 240.8.f.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 60.8.d.b | ✓ | 4 | 5.c | odd | 4 | 2 | |
| 180.8.d.c | 4 | 15.e | even | 4 | 2 | ||
| 240.8.f.c | 4 | 20.e | even | 4 | 2 | ||
| 300.8.a.j | 2 | 1.a | even | 1 | 1 | trivial | |
| 300.8.a.k | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(300))\):
|
\( T_{7}^{2} - 906T_{7} + 176984 \)
|
|
\( T_{11}^{2} - 6342T_{11} + 1898216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T + 27)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 906T + 176984 \)
|
| $11$ |
\( T^{2} - 6342 T + 1898216 \)
|
| $13$ |
\( T^{2} - 6690 T + 4838400 \)
|
| $17$ |
\( T^{2} - 31306 T + 31381384 \)
|
| $19$ |
\( T^{2} - 5640 T - 483840000 \)
|
| $23$ |
\( T^{2} + 34976 T - 868781456 \)
|
| $29$ |
\( T^{2} + \cdots - 23936383584 \)
|
| $31$ |
\( T^{2} + \cdots - 52210714784 \)
|
| $37$ |
\( T^{2} + \cdots + 6997356656 \)
|
| $41$ |
\( T^{2} + \cdots + 132110549900 \)
|
| $43$ |
\( T^{2} + \cdots - 5472599536 \)
|
| $47$ |
\( T^{2} + \cdots + 70271336704 \)
|
| $53$ |
\( T^{2} + \cdots + 756658769976 \)
|
| $59$ |
\( T^{2} + \cdots - 1410030476104 \)
|
| $61$ |
\( T^{2} + \cdots + 65990395684 \)
|
| $67$ |
\( T^{2} + \cdots - 13206551822176 \)
|
| $71$ |
\( T^{2} + \cdots - 10803266611200 \)
|
| $73$ |
\( T^{2} + \cdots - 17372893691104 \)
|
| $79$ |
\( T^{2} + \cdots + 2846846202624 \)
|
| $83$ |
\( T^{2} + \cdots - 6944201174576 \)
|
| $89$ |
\( T^{2} + \cdots + 27181827906500 \)
|
| $97$ |
\( T^{2} + \cdots + 200488661648576 \)
|
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