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: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 722x^{3} + 11881x^{2} + 54936x + 127008 \)
|
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 722x^{3} + 11881x^{2} + 54936x + 127008 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5869\nu^{5} + 25724\nu^{4} - 85196\nu^{3} - 2684972\nu^{2} - 64680643\nu - 172672164 ) / 142698276 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -111\nu^{5} + 583\nu^{4} - 12321\nu^{3} - 40071\nu^{2} - 55944\nu - 7048478 ) / 1132526 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19855\nu^{5} - 99182\nu^{4} + 1637642\nu^{3} + 7733918\nu^{2} + 357126139\nu + 918082116 ) / 142698276 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1721\nu^{5} + 29445\nu^{4} - 191031\nu^{3} - 621281\nu^{2} - 867384\nu + 78246822 ) / 1132526 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -233885\nu^{5} + 988654\nu^{4} + 653126\nu^{3} - 206179030\nu^{2} - 2162653733\nu - 5879361852 ) / 47566092 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 9\beta_{3} + 150\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 118\beta_{3} - 118\beta_{2} + 763\beta _1 - 763 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 111\beta_{4} - 1721\beta_{2} - 18380 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 472\beta_{5} + 472\beta_{4} - 16851\beta_{3} - 16851\beta_{2} - 139347\beta _1 - 139347 ) / 2 \)
|
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 | − | 15.6195i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 673.2 | 0 | − | 3.00000i | 0 | 7.86469i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.3 | 0 | − | 3.00000i | 0 | 11.7548i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.4 | 0 | 3.00000i | 0 | − | 11.7548i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.5 | 0 | 3.00000i | 0 | − | 7.86469i | 0 | 7.00000 | 0 | −9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 673.6 | 0 | 3.00000i | 0 | 15.6195i | 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.d | yes | 6 |
| 4.b | odd | 2 | 1 | 1344.4.c.a | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 1344.4.c.d | yes | 6 |
| 8.d | odd | 2 | 1 | 1344.4.c.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1344.4.c.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1344.4.c.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 1344.4.c.d | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1344.4.c.d | 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 \(S_{4}^{\mathrm{new}}(1344, [\chi])\):
|
\( T_{5}^{6} + 444T_{5}^{4} + 57348T_{5}^{2} + 2085136 \)
|
|
\( T_{23}^{3} + 30T_{23}^{2} - 21186T_{23} + 470448 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( T^{6} + 444 T^{4} + \cdots + 2085136 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( T^{6} + 6396 T^{4} + \cdots + 906973456 \)
|
| $13$ |
\( T^{6} + 5028 T^{4} + \cdots + 369869824 \)
|
| $17$ |
\( (T^{3} + 8 T^{2} + \cdots - 66092)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 71892624384 \)
|
| $23$ |
\( (T^{3} + 30 T^{2} + \cdots + 470448)^{2} \)
|
| $29$ |
\( T^{6} + 6456 T^{4} + \cdots + 97140736 \)
|
| $31$ |
\( (T^{3} + 276 T^{2} + \cdots - 913888)^{2} \)
|
| $37$ |
\( T^{6} + 9816 T^{4} + \cdots + 488233216 \)
|
| $41$ |
\( (T^{3} + 136 T^{2} + \cdots - 9550804)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 4829129310784 \)
|
| $47$ |
\( (T^{3} + 788 T^{2} + \cdots + 13844416)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 924071477319936 \)
|
| $59$ |
\( T^{6} + \cdots + 4716055035904 \)
|
| $61$ |
\( T^{6} + \cdots + 268125124864 \)
|
| $67$ |
\( T^{6} + \cdots + 15\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} + 1274 T^{2} + \cdots - 4238528)^{2} \)
|
| $73$ |
\( (T^{3} - 222 T^{2} + \cdots + 226444392)^{2} \)
|
| $79$ |
\( (T^{3} + 1764 T^{2} + \cdots + 157706752)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 964272413085696 \)
|
| $89$ |
\( (T^{3} - 8 T^{2} + \cdots + 690068)^{2} \)
|
| $97$ |
\( (T^{3} - 774 T^{2} + \cdots - 107008376)^{2} \)
|
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