Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [464,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("464.289");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 464 = 2^{4} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(27.3768862427\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 199x^{6} + 11895x^{4} + 218821x^{2} + 1196836 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 58) |
| 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
\( x^{8} + 199x^{6} + 11895x^{4} + 218821x^{2} + 1196836 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 144\nu^{4} + 3363\nu^{2} - 42950 ) / 3978 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 246\nu^{4} + 15501\nu^{2} + 159724 ) / 3978 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 178\nu^{4} + 8735\nu^{2} + 89582 ) / 1326 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 73\nu^{7} + 13980\nu^{5} + 733773\nu^{3} + 7494886\nu ) / 2175966 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 124\nu^{7} + 24129\nu^{5} + 1396212\nu^{3} + 22030294\nu ) / 1087983 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 122\nu^{7} + 22090\nu^{5} + 1089623\nu^{3} + 10903725\nu ) / 362661 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 2\beta_{2} - 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 10\beta_{6} - 44\beta_{5} - 81\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -119\beta_{4} + 158\beta_{3} + 199\beta_{2} + 3844 \)
|
| \(\nu^{5}\) | \(=\) |
\( -392\beta_{7} - 1073\beta_{6} + 7576\beta_{5} + 7418\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13773\beta_{4} - 19389\beta_{3} - 17952\beta_{2} - 345799 \)
|
| \(\nu^{7}\) | \(=\) |
\( 65019\beta_{7} + 104970\beta_{6} - 978774\beta_{5} - 709081\beta_1 \)
|
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 | − | 10.1579i | 0 | 8.43433 | 0 | −12.6526 | 0 | −76.1822 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 289.2 | 0 | − | 8.32034i | 0 | −7.65080 | 0 | 29.0613 | 0 | −42.2280 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.3 | 0 | − | 4.04085i | 0 | 4.49818 | 0 | −32.0403 | 0 | 10.6715 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.4 | 0 | − | 3.20332i | 0 | −16.2817 | 0 | 9.63157 | 0 | 16.7387 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 289.5 | 0 | 3.20332i | 0 | −16.2817 | 0 | 9.63157 | 0 | 16.7387 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.6 | 0 | 4.04085i | 0 | 4.49818 | 0 | −32.0403 | 0 | 10.6715 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.7 | 0 | 8.32034i | 0 | −7.65080 | 0 | 29.0613 | 0 | −42.2280 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 289.8 | 0 | 10.1579i | 0 | 8.43433 | 0 | −12.6526 | 0 | −76.1822 | 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 | 464.4.e.b | 8 | |
| 4.b | odd | 2 | 1 | 58.4.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 522.4.d.c | 8 | ||
| 29.b | even | 2 | 1 | inner | 464.4.e.b | 8 | |
| 116.d | odd | 2 | 1 | 58.4.b.a | ✓ | 8 | |
| 116.e | even | 4 | 1 | 1682.4.a.h | 4 | ||
| 116.e | even | 4 | 1 | 1682.4.a.i | 4 | ||
| 348.b | even | 2 | 1 | 522.4.d.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.4.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 58.4.b.a | ✓ | 8 | 116.d | odd | 2 | 1 | |
| 464.4.e.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 464.4.e.b | 8 | 29.b | even | 2 | 1 | inner | |
| 522.4.d.c | 8 | 12.b | even | 2 | 1 | ||
| 522.4.d.c | 8 | 348.b | even | 2 | 1 | ||
| 1682.4.a.h | 4 | 116.e | even | 4 | 1 | ||
| 1682.4.a.i | 4 | 116.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 199T_{3}^{6} + 11895T_{3}^{4} + 218821T_{3}^{2} + 1196836 \)
acting on \(S_{4}^{\mathrm{new}}(464, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 199 T^{6} + \cdots + 1196836 \)
|
| $5$ |
\( (T^{4} + 11 T^{3} + \cdots + 4726)^{2} \)
|
| $7$ |
\( (T^{4} + 6 T^{3} + \cdots + 113472)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 272440153764 \)
|
| $13$ |
\( (T^{4} - 31 T^{3} + \cdots + 3936938)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 11395989504 \)
|
| $19$ |
\( T^{8} + \cdots + 787819809424384 \)
|
| $23$ |
\( (T^{4} + 138 T^{3} + \cdots + 78758208)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 35\!\cdots\!41 \)
|
| $31$ |
\( T^{8} + \cdots + 236683394092324 \)
|
| $37$ |
\( T^{8} + \cdots + 11\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots + 56\!\cdots\!76 \)
|
| $43$ |
\( T^{8} + \cdots + 15\!\cdots\!24 \)
|
| $47$ |
\( T^{8} + \cdots + 15\!\cdots\!24 \)
|
| $53$ |
\( (T^{4} + 513 T^{3} + \cdots + 9621413154)^{2} \)
|
| $59$ |
\( (T^{4} - 554 T^{3} + \cdots + 14879171872)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 32\!\cdots\!64 \)
|
| $67$ |
\( (T^{4} - 356 T^{3} + \cdots - 2663354368)^{2} \)
|
| $71$ |
\( (T^{4} + 1824 T^{3} + \cdots - 217774980096)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $79$ |
\( T^{8} + \cdots + 24\!\cdots\!56 \)
|
| $83$ |
\( (T^{4} + 1066 T^{3} + \cdots + 30856265504)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 88\!\cdots\!24 \)
|
| $97$ |
\( T^{8} + \cdots + 87\!\cdots\!16 \)
|
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