Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [464,2,Mod(289,464)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(464, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("464.289");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 464.e (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: | \(3.70505865379\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.4589249536.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 232) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 11x^{6} + 39x^{4} + 49x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 6\nu^{3} - 3\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 7\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 10\nu^{3} - 23\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} + 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 15\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + 8\nu^{4} + 17\nu^{2} + 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{6} - 9\nu^{4} - 21\nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{4} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} - 5\beta_{2} - 4\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{7} - 3\beta_{6} - 2\beta_{4} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6\beta_{3} + 27\beta_{2} + 17\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 31\beta_{7} + 33\beta_{6} + 15\beta_{4} - 108 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4\beta_{5} - 31\beta_{3} - 143\beta_{2} - 76\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).
| \(n\) | \(117\) | \(175\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 289.1 |
|
0 | − | 3.19117i | 0 | 1.80122 | 0 | −4.55683 | 0 | −7.18356 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 289.2 | 0 | − | 1.57671i | 0 | −2.66740 | 0 | 3.78244 | 0 | 0.513978 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.3 | 0 | − | 1.50350i | 0 | 3.26753 | 0 | 1.40925 | 0 | 0.739474 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.4 | 0 | − | 0.264377i | 0 | −1.40135 | 0 | −2.63486 | 0 | 2.93011 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.5 | 0 | 0.264377i | 0 | −1.40135 | 0 | −2.63486 | 0 | 2.93011 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.6 | 0 | 1.50350i | 0 | 3.26753 | 0 | 1.40925 | 0 | 0.739474 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.7 | 0 | 1.57671i | 0 | −2.66740 | 0 | 3.78244 | 0 | 0.513978 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.8 | 0 | 3.19117i | 0 | 1.80122 | 0 | −4.55683 | 0 | −7.18356 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 464.2.e.d | 8 | |
| 3.b | odd | 2 | 1 | 4176.2.o.r | 8 | ||
| 4.b | odd | 2 | 1 | 232.2.e.a | ✓ | 8 | |
| 8.b | even | 2 | 1 | 1856.2.e.i | 8 | ||
| 8.d | odd | 2 | 1 | 1856.2.e.j | 8 | ||
| 12.b | even | 2 | 1 | 2088.2.o.e | 8 | ||
| 29.b | even | 2 | 1 | inner | 464.2.e.d | 8 | |
| 87.d | odd | 2 | 1 | 4176.2.o.r | 8 | ||
| 116.d | odd | 2 | 1 | 232.2.e.a | ✓ | 8 | |
| 116.e | even | 4 | 1 | 6728.2.a.s | 4 | ||
| 116.e | even | 4 | 1 | 6728.2.a.t | 4 | ||
| 232.b | odd | 2 | 1 | 1856.2.e.j | 8 | ||
| 232.g | even | 2 | 1 | 1856.2.e.i | 8 | ||
| 348.b | even | 2 | 1 | 2088.2.o.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.2.e.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 232.2.e.a | ✓ | 8 | 116.d | odd | 2 | 1 | |
| 464.2.e.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 464.2.e.d | 8 | 29.b | even | 2 | 1 | inner | |
| 1856.2.e.i | 8 | 8.b | even | 2 | 1 | ||
| 1856.2.e.i | 8 | 232.g | even | 2 | 1 | ||
| 1856.2.e.j | 8 | 8.d | odd | 2 | 1 | ||
| 1856.2.e.j | 8 | 232.b | odd | 2 | 1 | ||
| 2088.2.o.e | 8 | 12.b | even | 2 | 1 | ||
| 2088.2.o.e | 8 | 348.b | even | 2 | 1 | ||
| 4176.2.o.r | 8 | 3.b | odd | 2 | 1 | ||
| 4176.2.o.r | 8 | 87.d | odd | 2 | 1 | ||
| 6728.2.a.s | 4 | 116.e | even | 4 | 1 | ||
| 6728.2.a.t | 4 | 116.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 15T_{3}^{6} + 55T_{3}^{4} + 61T_{3}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(464, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 15 T^{6} + \cdots + 4 \)
|
| $5$ |
\( (T^{4} - T^{3} - 11 T^{2} + \cdots + 22)^{2} \)
|
| $7$ |
\( (T^{4} + 2 T^{3} - 20 T^{2} + \cdots + 64)^{2} \)
|
| $11$ |
\( T^{8} + 39 T^{6} + \cdots + 4 \)
|
| $13$ |
\( (T^{4} + 5 T^{3} + \cdots - 118)^{2} \)
|
| $17$ |
\( T^{8} + 92 T^{6} + \cdots + 65536 \)
|
| $19$ |
\( T^{8} + 84 T^{6} + \cdots + 1024 \)
|
| $23$ |
\( (T^{4} - 2 T^{3} - 20 T^{2} + \cdots + 64)^{2} \)
|
| $29$ |
\( T^{8} - 4 T^{7} + \cdots + 707281 \)
|
| $31$ |
\( T^{8} + 119 T^{6} + \cdots + 24964 \)
|
| $37$ |
\( T^{8} + 208 T^{6} + \cdots + 262144 \)
|
| $41$ |
\( T^{8} + 220 T^{6} + \cdots + 3444736 \)
|
| $43$ |
\( T^{8} + 79 T^{6} + \cdots + 13924 \)
|
| $47$ |
\( T^{8} + 183 T^{6} + \cdots + 4 \)
|
| $53$ |
\( (T^{4} - 3 T^{3} - 55 T^{2} + \cdots + 2)^{2} \)
|
| $59$ |
\( (T^{4} - 6 T^{3} + \cdots + 1696)^{2} \)
|
| $61$ |
\( T^{8} + 332 T^{6} + \cdots + 37356544 \)
|
| $67$ |
\( (T^{4} - 12 T^{3} + \cdots + 4096)^{2} \)
|
| $71$ |
\( (T^{4} - 192 T^{2} + \cdots - 1024)^{2} \)
|
| $73$ |
\( (T^{2} + 64)^{4} \)
|
| $79$ |
\( T^{8} + 255 T^{6} + \cdots + 6832996 \)
|
| $83$ |
\( (T^{4} - 26 T^{3} + \cdots - 18272)^{2} \)
|
| $89$ |
\( T^{8} + 252 T^{6} + \cdots + 495616 \)
|
| $97$ |
\( T^{8} + 412 T^{6} + \cdots + 16384 \)
|
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