Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1344,4,Mod(673,1344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1344.673");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1344.c (of order \(2\), degree \(1\), 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: | \(79.2985670477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.14024243776.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 17x^{4} - 164x^{3} + 299x^{2} + 2466x + 13042 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 17x^{4} - 164x^{3} + 299x^{2} + 2466x + 13042 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{5} - 54\nu^{4} + 553\nu^{3} - 453\nu^{2} - 157\nu - 47393 ) / 39375 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{5} + 236\nu^{4} - 427\nu^{3} + 2952\nu^{2} - 31287\nu + 6037 ) / 13125 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 28\nu^{4} + 54\nu^{3} - 529\nu^{2} - 4726\nu - 1699 ) / 1875 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -31\nu^{5} - 194\nu^{4} + 1183\nu^{3} + 4692\nu^{2} + 123\nu - 88273 ) / 13125 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + 34\nu^{4} + 162\nu^{3} + 563\nu^{2} - 1678\nu - 17197 ) / 1875 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 7\beta_{3} + 7\beta_{2} + 18\beta _1 + 28 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{5} - 7\beta_{4} - 5\beta_{3} + 2\beta_{2} + 153\beta _1 + 205 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 143\beta_{5} - 207\beta_{4} - 101\beta_{3} + 51\beta_{2} + 774\beta _1 + 736 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -129\beta_{5} - 785\beta_{4} - 813\beta_{3} + 889\beta_{2} + 9558\beta _1 + 3896 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(449\) | \(577\) | \(1093\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 673.1 |
|
0 | − | 3.00000i | 0 | − | 13.9325i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 673.2 | 0 | − | 3.00000i | 0 | 2.89754i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.3 | 0 | − | 3.00000i | 0 | 7.03493i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.4 | 0 | 3.00000i | 0 | − | 7.03493i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.5 | 0 | 3.00000i | 0 | − | 2.89754i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.6 | 0 | 3.00000i | 0 | 13.9325i | 0 | 7.00000 | 0 | −9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1344.4.c.c | yes | 6 |
| 4.b | odd | 2 | 1 | 1344.4.c.b | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 1344.4.c.c | yes | 6 |
| 8.d | odd | 2 | 1 | 1344.4.c.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1344.4.c.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1344.4.c.b | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 1344.4.c.c | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1344.4.c.c | yes | 6 | 8.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):
|
\( T_{5}^{6} + 252T_{5}^{4} + 11652T_{5}^{2} + 80656 \)
|
|
\( T_{23}^{3} + 98T_{23}^{2} - 4266T_{23} - 455616 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} + 252 T^{4} + \cdots + 80656 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 2361571216 \)
|
| $13$ |
\( T^{6} + 6884 T^{4} + \cdots + 107495424 \)
|
| $17$ |
\( (T^{3} + 36 T^{2} + \cdots + 340452)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 1405192126464 \)
|
| $23$ |
\( (T^{3} + 98 T^{2} + \cdots - 455616)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 173862579004416 \)
|
| $31$ |
\( (T^{3} - 52 T^{2} + \cdots + 6626656)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 134116781574400 \)
|
| $41$ |
\( (T^{3} + 108 T^{2} + \cdots - 856452)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 277856627817024 \)
|
| $47$ |
\( (T^{3} + 204 T^{2} + \cdots - 78456384)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 222264684199936 \)
|
| $61$ |
\( T^{6} + \cdots + 352990747650304 \)
|
| $67$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} + 582 T^{2} + \cdots - 89664720)^{2} \)
|
| $73$ |
\( (T^{3} + 138 T^{2} + \cdots - 401113080)^{2} \)
|
| $79$ |
\( (T^{3} - 676 T^{2} + \cdots + 347261568)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!56 \)
|
| $89$ |
\( (T^{3} - 1260 T^{2} + \cdots + 3313380)^{2} \)
|
| $97$ |
\( (T^{3} + 1170 T^{2} + \cdots - 273572696)^{2} \)
|
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