Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1449,4,Mod(1,1449)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1449.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1449 = 3^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1449.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: | \(85.4937675983\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2x^{6} - 37x^{5} + 71x^{4} + 340x^{3} - 633x^{2} - 288x - 28 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 483) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2x^{6} - 37x^{5} + 71x^{4} + 340x^{3} - 633x^{2} - 288x - 28 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 34\nu^{4} + 33\nu^{3} + 261\nu^{2} - 320\nu - 28 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 36\nu^{4} - 97\nu^{3} - 323\nu^{2} + 742\nu + 232 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 36\nu^{4} + 105\nu^{3} + 315\nu^{2} - 894\nu - 120 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{6} + 9\nu^{5} + 182\nu^{4} - 325\nu^{3} - 1597\nu^{2} + 3012\nu + 724 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{2} + 19\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 4\beta_{5} + 2\beta_{4} + 3\beta_{3} + 25\beta_{2} + 5\beta _1 + 197 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} + 28\beta_{5} + 34\beta_{4} + \beta_{3} + 38\beta_{2} + 392\beta _1 - 54 \)
|
| \(\nu^{6}\) | \(=\) |
\( 33\beta_{6} + 131\beta_{5} + 69\beta_{4} + 111\beta_{3} + 594\beta_{2} + 255\beta _1 + 3900 \)
|
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 |
|
−4.91050 | 0 | 16.1130 | 15.9911 | 0 | 7.00000 | −39.8391 | 0 | −78.5242 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.24981 | 0 | 2.56124 | 5.04837 | 0 | 7.00000 | 17.6749 | 0 | −16.4062 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.68486 | 0 | −0.791514 | −10.1369 | 0 | 7.00000 | 23.6040 | 0 | 27.2161 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.152027 | 0 | −7.97689 | 3.54652 | 0 | 7.00000 | −2.42892 | 0 | 0.539166 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.243077 | 0 | −7.94091 | 14.8004 | 0 | 7.00000 | −3.87487 | 0 | 3.59763 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.81685 | 0 | 6.56836 | −16.0851 | 0 | 7.00000 | −5.46436 | 0 | −61.3944 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.63321 | 0 | 13.4667 | −2.16438 | 0 | 7.00000 | 25.3283 | 0 | −10.0280 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1449.4.a.f | 7 | |
| 3.b | odd | 2 | 1 | 483.4.a.e | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.4.a.e | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 1449.4.a.f | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 2T_{2}^{6} - 37T_{2}^{5} - 71T_{2}^{4} + 340T_{2}^{3} + 633T_{2}^{2} - 288T_{2} + 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 2 T^{6} + \cdots + 28 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots + 1495424 \)
|
| $7$ |
\( (T - 7)^{7} \)
|
| $11$ |
\( T^{7} + \cdots - 3671855200 \)
|
| $13$ |
\( T^{7} + \cdots - 74470948284 \)
|
| $17$ |
\( T^{7} + \cdots + 3016423566736 \)
|
| $19$ |
\( T^{7} + \cdots - 359743419648 \)
|
| $23$ |
\( (T - 23)^{7} \)
|
| $29$ |
\( T^{7} + \cdots - 233333051936380 \)
|
| $31$ |
\( T^{7} + \cdots + 21\!\cdots\!64 \)
|
| $37$ |
\( T^{7} + \cdots - 22625380981168 \)
|
| $41$ |
\( T^{7} + \cdots - 36\!\cdots\!80 \)
|
| $43$ |
\( T^{7} + \cdots + 31\!\cdots\!44 \)
|
| $47$ |
\( T^{7} + \cdots - 34\!\cdots\!28 \)
|
| $53$ |
\( T^{7} + \cdots + 69\!\cdots\!80 \)
|
| $59$ |
\( T^{7} + \cdots + 15\!\cdots\!64 \)
|
| $61$ |
\( T^{7} + \cdots + 50\!\cdots\!64 \)
|
| $67$ |
\( T^{7} + \cdots + 31\!\cdots\!80 \)
|
| $71$ |
\( T^{7} + \cdots + 87\!\cdots\!00 \)
|
| $73$ |
\( T^{7} + \cdots - 43\!\cdots\!44 \)
|
| $79$ |
\( T^{7} + \cdots + 17\!\cdots\!36 \)
|
| $83$ |
\( T^{7} + \cdots + 36\!\cdots\!80 \)
|
| $89$ |
\( T^{7} + \cdots - 56\!\cdots\!44 \)
|
| $97$ |
\( T^{7} + \cdots + 45\!\cdots\!08 \)
|
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