Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,4,Mod(289,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.289");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.q (of order \(3\), degree \(2\), 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: | \(36.8171918436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.31902363.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 17x^{4} + 85x^{2} + 108 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 312) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 17x^{4} + 85x^{2} + 108 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 11\nu^{3} + 25\nu + 6 ) / 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} - 12\nu^{4} - 23\nu^{3} - 108\nu^{2} - 73\nu - 138 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 12\nu^{4} - 23\nu^{3} + 108\nu^{2} - 73\nu + 138 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{4} + 2\nu^{3} + 39\nu^{2} - 47\nu + 102 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{4} - 13\nu^{2} - 34 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - 4\beta_{4} + 3\beta_{3} + 3\beta_{2} + 14\beta _1 - 7 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{3} + \beta_{2} - 45 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} - 3\beta_{3} - 3\beta_{2} - 10\beta _1 + 5 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 18\beta_{5} + 13\beta_{3} - 13\beta_{2} + 313 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -38\beta_{5} - 76\beta_{4} + 189\beta_{3} + 189\beta_{2} + 818\beta _1 - 409 ) / 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) | \(1\) |
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 | −1.50000 | − | 2.59808i | 0 | −16.6097 | 0 | −13.4641 | + | 23.3205i | 0 | −4.50000 | + | 7.79423i | 0 | ||||||||||||||||||||||||||||||
| 289.2 | 0 | −1.50000 | − | 2.59808i | 0 | 0.369140 | 0 | 4.19539 | − | 7.26663i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||||
| 289.3 | 0 | −1.50000 | − | 2.59808i | 0 | 4.24054 | 0 | −5.23127 | + | 9.06083i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||||
| 529.1 | 0 | −1.50000 | + | 2.59808i | 0 | −16.6097 | 0 | −13.4641 | − | 23.3205i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||||
| 529.2 | 0 | −1.50000 | + | 2.59808i | 0 | 0.369140 | 0 | 4.19539 | + | 7.26663i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||||
| 529.3 | 0 | −1.50000 | + | 2.59808i | 0 | 4.24054 | 0 | −5.23127 | − | 9.06083i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.4.q.g | 6 | |
| 4.b | odd | 2 | 1 | 312.4.q.b | ✓ | 6 | |
| 13.c | even | 3 | 1 | inner | 624.4.q.g | 6 | |
| 52.j | odd | 6 | 1 | 312.4.q.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.4.q.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 312.4.q.b | ✓ | 6 | 52.j | odd | 6 | 1 | |
| 624.4.q.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 624.4.q.g | 6 | 13.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{3} + 12T_{5}^{2} - 75T_{5} + 26 \)
|
|
\( T_{7}^{6} + 29T_{7}^{5} + 873T_{7}^{4} + 3800T_{7}^{3} + 69580T_{7}^{2} + 75648T_{7} + 5588496 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{3} \)
|
| $5$ |
\( (T^{3} + 12 T^{2} + \cdots + 26)^{2} \)
|
| $7$ |
\( T^{6} + 29 T^{5} + \cdots + 5588496 \)
|
| $11$ |
\( T^{6} - 58 T^{5} + \cdots + 209438784 \)
|
| $13$ |
\( T^{6} + \cdots + 10604499373 \)
|
| $17$ |
\( T^{6} + \cdots + 1498618944 \)
|
| $19$ |
\( T^{6} + \cdots + 16273594624 \)
|
| $23$ |
\( T^{6} + \cdots + 1242985536 \)
|
| $29$ |
\( T^{6} + \cdots + 967913598276 \)
|
| $31$ |
\( (T^{3} + 313 T^{2} + \cdots - 2735088)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 27344302555684 \)
|
| $41$ |
\( T^{6} + 86 T^{5} + \cdots + 62536464 \)
|
| $43$ |
\( T^{6} + \cdots + 28155715990416 \)
|
| $47$ |
\( (T^{3} - 314 T^{2} + \cdots - 9961176)^{2} \)
|
| $53$ |
\( (T^{3} - 270 T^{2} + \cdots + 8662464)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 41\!\cdots\!44 \)
|
| $61$ |
\( T^{6} + \cdots + 6740051937921 \)
|
| $67$ |
\( T^{6} + \cdots + 43\!\cdots\!24 \)
|
| $71$ |
\( T^{6} + \cdots + 42\!\cdots\!76 \)
|
| $73$ |
\( (T^{3} + 369 T^{2} + \cdots - 527382961)^{2} \)
|
| $79$ |
\( (T^{3} - 849 T^{2} + \cdots + 134237808)^{2} \)
|
| $83$ |
\( (T^{3} + 1140 T^{2} + \cdots - 368244224)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 80\!\cdots\!84 \)
|
| $97$ |
\( T^{6} + \cdots + 46\!\cdots\!36 \)
|
| show more | |
| show less |