Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1323,4,Mod(1,1323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1323.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1323 = 3^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1323.a (trivial) |
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: | yes |
| Analytic conductor: | \(78.0595269376\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.346909504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 10\nu^{2} + 27\nu + 8 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{5} - 6\nu^{4} - 24\nu^{3} + 30\nu^{2} + 51\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 16\nu^{3} - 22\nu^{2} - 57\nu + 16 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 6\nu^{4} + 36\nu^{3} - 30\nu^{2} - 135\nu - 46 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} + 2\nu^{4} - 68\nu^{3} - 22\nu^{2} + 201\nu + 76 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} - 6\beta _1 + 2 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 4\beta _1 + 25 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{4} + 9\beta_{2} - 42\beta _1 + 62 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 6\beta_{3} + 10\beta_{2} - 44\beta _1 + 173 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4\beta_{5} - 27\beta_{4} - 4\beta_{3} + 83\beta_{2} - 330\beta _1 + 662 ) / 12 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.77246 | 0 | 14.7764 | −1.85905 | 0 | 0 | −32.3399 | 0 | 8.87225 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.78484 | 0 | −0.244684 | 20.8311 | 0 | 0 | 22.9601 | 0 | −58.0112 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.57109 | 0 | −5.53167 | −6.97204 | 0 | 0 | 21.2595 | 0 | 10.9537 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.57109 | 0 | −5.53167 | −6.97204 | 0 | 0 | −21.2595 | 0 | −10.9537 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.78484 | 0 | −0.244684 | 20.8311 | 0 | 0 | −22.9601 | 0 | 58.0112 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.77246 | 0 | 14.7764 | −1.85905 | 0 | 0 | 32.3399 | 0 | −8.87225 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1323.4.a.bc | yes | 6 |
| 3.b | odd | 2 | 1 | 1323.4.a.bb | ✓ | 6 | |
| 7.b | odd | 2 | 1 | 1323.4.a.bb | ✓ | 6 | |
| 21.c | even | 2 | 1 | inner | 1323.4.a.bc | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1323.4.a.bb | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 1323.4.a.bb | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 1323.4.a.bc | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1323.4.a.bc | yes | 6 | 21.c | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):
|
\( T_{2}^{6} - 33T_{2}^{4} + 252T_{2}^{2} - 436 \)
|
|
\( T_{5}^{3} - 12T_{5}^{2} - 171T_{5} - 270 \)
|
|
\( T_{13}^{6} - 8505T_{13}^{4} + 17399448T_{13}^{2} - 5371563600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 33 T^{4} + \cdots - 436 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - 12 T^{2} + \cdots - 270)^{2} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 5580 T^{4} + \cdots - 879150400 \)
|
| $13$ |
\( T^{6} + \cdots - 5371563600 \)
|
| $17$ |
\( (T^{3} - 21 T^{2} + \cdots - 91719)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 232895741184 \)
|
| $23$ |
\( T^{6} + \cdots - 103079719936 \)
|
| $29$ |
\( T^{6} + \cdots - 149059723600 \)
|
| $31$ |
\( T^{6} + \cdots - 2588939818704 \)
|
| $37$ |
\( (T^{3} + 156 T^{2} + \cdots - 706050)^{2} \)
|
| $41$ |
\( (T^{3} - 180 T^{2} + \cdots + 10490310)^{2} \)
|
| $43$ |
\( (T^{3} - 327 T^{2} + \cdots + 15007815)^{2} \)
|
| $47$ |
\( (T^{3} - 906 T^{2} + \cdots + 63550440)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 12\!\cdots\!56 \)
|
| $59$ |
\( (T^{3} + 3 T^{2} + \cdots - 60900795)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 20403087817536 \)
|
| $67$ |
\( (T^{3} - 21 T^{2} + \cdots - 4249360)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 27\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} - 978 T^{2} + \cdots + 324650908)^{2} \)
|
| $83$ |
\( (T^{3} - 1446 T^{2} + \cdots + 66493008)^{2} \)
|
| $89$ |
\( (T^{3} - 759 T^{2} + \cdots + 79092180)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
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