Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(119,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.119");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 465.t (of order \(6\), 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: | \(3.71304369399\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.3317760000.8 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 4\nu^{5} - 7\nu^{3} - 36\nu ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} - \beta_{5} + 3\beta_{4} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -5\beta_{7} + 7\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} - 7\beta_{2} - 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( -21\beta_{5} - 21\beta_{4} - 29\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(406\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 119.1 |
|
−2.44949 | −1.40294 | + | 1.01575i | 4.00000 | −1.93649 | + | 1.11803i | 3.43649 | − | 2.48808i | 0 | −4.89898 | 0.936492 | − | 2.85008i | 4.74342 | − | 2.73861i | ||||||||||||||||||||||||||||||||
| 119.2 | −2.44949 | 0.178197 | − | 1.72286i | 4.00000 | 1.93649 | − | 1.11803i | −0.436492 | + | 4.22013i | 0 | −4.89898 | −2.93649 | − | 0.614017i | −4.74342 | + | 2.73861i | |||||||||||||||||||||||||||||||||
| 119.3 | 2.44949 | −0.178197 | + | 1.72286i | 4.00000 | 1.93649 | − | 1.11803i | −0.436492 | + | 4.22013i | 0 | 4.89898 | −2.93649 | − | 0.614017i | 4.74342 | − | 2.73861i | |||||||||||||||||||||||||||||||||
| 119.4 | 2.44949 | 1.40294 | − | 1.01575i | 4.00000 | −1.93649 | + | 1.11803i | 3.43649 | − | 2.48808i | 0 | 4.89898 | 0.936492 | − | 2.85008i | −4.74342 | + | 2.73861i | |||||||||||||||||||||||||||||||||
| 254.1 | −2.44949 | −1.40294 | − | 1.01575i | 4.00000 | −1.93649 | − | 1.11803i | 3.43649 | + | 2.48808i | 0 | −4.89898 | 0.936492 | + | 2.85008i | 4.74342 | + | 2.73861i | |||||||||||||||||||||||||||||||||
| 254.2 | −2.44949 | 0.178197 | + | 1.72286i | 4.00000 | 1.93649 | + | 1.11803i | −0.436492 | − | 4.22013i | 0 | −4.89898 | −2.93649 | + | 0.614017i | −4.74342 | − | 2.73861i | |||||||||||||||||||||||||||||||||
| 254.3 | 2.44949 | −0.178197 | − | 1.72286i | 4.00000 | 1.93649 | + | 1.11803i | −0.436492 | − | 4.22013i | 0 | 4.89898 | −2.93649 | + | 0.614017i | 4.74342 | + | 2.73861i | |||||||||||||||||||||||||||||||||
| 254.4 | 2.44949 | 1.40294 | + | 1.01575i | 4.00000 | −1.93649 | − | 1.11803i | 3.43649 | + | 2.48808i | 0 | 4.89898 | 0.936492 | + | 2.85008i | −4.74342 | − | 2.73861i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 31.e | odd | 6 | 1 | inner |
| 93.g | even | 6 | 1 | inner |
| 155.i | odd | 6 | 1 | inner |
| 465.t | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 465.2.t.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 15.d | odd | 2 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 31.e | odd | 6 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 93.g | even | 6 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 155.i | odd | 6 | 1 | inner | 465.2.t.c | ✓ | 8 |
| 465.t | even | 6 | 1 | inner | 465.2.t.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.t.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 465.2.t.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 31.e | odd | 6 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 93.g | even | 6 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 155.i | odd | 6 | 1 | inner |
| 465.2.t.c | ✓ | 8 | 465.t | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 6)^{4} \)
|
| $3$ |
\( T^{8} + 4 T^{6} + \cdots + 81 \)
|
| $5$ |
\( (T^{4} - 5 T^{2} + 25)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 15 T^{2} + 225)^{2} \)
|
| $13$ |
\( (T^{4} + 10 T^{2} + 100)^{2} \)
|
| $17$ |
\( (T^{4} - 32 T^{2} + 1024)^{2} \)
|
| $19$ |
\( (T^{2} + 2 T + 4)^{4} \)
|
| $23$ |
\( (T^{2} + 32)^{4} \)
|
| $29$ |
\( (T^{2} - 15)^{4} \)
|
| $31$ |
\( (T^{2} + 7 T + 31)^{4} \)
|
| $37$ |
\( (T^{4} + 40 T^{2} + 1600)^{2} \)
|
| $41$ |
\( (T^{4} - 125 T^{2} + 15625)^{2} \)
|
| $43$ |
\( (T^{4} + 10 T^{2} + 100)^{2} \)
|
| $47$ |
\( (T^{2} - 96)^{4} \)
|
| $53$ |
\( (T^{4} - 32 T^{2} + 1024)^{2} \)
|
| $59$ |
\( (T^{4} - 5 T^{2} + 25)^{2} \)
|
| $61$ |
\( (T^{2} + 3)^{4} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T^{4} - 5 T^{2} + 25)^{2} \)
|
| $73$ |
\( (T^{4} + 10 T^{2} + 100)^{2} \)
|
| $79$ |
\( (T^{2} - 9 T + 27)^{4} \)
|
| $83$ |
\( (T^{4} - 98 T^{2} + 9604)^{2} \)
|
| $89$ |
\( (T^{2} - 135)^{4} \)
|
| $97$ |
\( (T^{2} + 120)^{4} \)
|
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