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: |
\( x^{7} - 2x^{6} - 37x^{5} + 71x^{4} + 312x^{3} - 629x^{2} + 112x + 180 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{7} - 2x^{6} - 37x^{5} + 71x^{4} + 312x^{3} - 629x^{2} + 112x + 180 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 38\nu^{4} - 29\nu^{3} - 349\nu^{2} + 200\nu + 212 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 37\nu^{4} + 31\nu^{3} + 319\nu^{2} - 249\nu - 90 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 38\nu^{4} - 33\nu^{3} - 341\nu^{2} + 284\nu + 112 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 38\nu^{4} + 33\nu^{3} + 345\nu^{2} - 284\nu - 156 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{6} - 7\nu^{5} - 342\nu^{4} + 225\nu^{3} + 3099\nu^{2} - 1968\nu - 1524 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 2\beta_{2} + 21\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 26\beta_{5} + 28\beta_{4} + 4\beta_{3} + 4\beta_{2} + 7\beta _1 + 214 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{6} + 57\beta_{5} + 39\beta_{4} + 72\beta_{2} + 462\beta _1 - 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} + 638\beta_{5} + 725\beta_{4} + 152\beta_{3} + 158\beta_{2} + 319\beta _1 + 4577 \)
|
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 |
|
−4.75832 | 0 | 14.6416 | 3.92582 | 0 | −7.00000 | −31.6031 | 0 | −18.6803 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.75958 | 0 | 6.13441 | −3.12160 | 0 | −7.00000 | 7.01383 | 0 | 11.7359 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.422009 | 0 | −7.82191 | −0.947423 | 0 | −7.00000 | 6.67699 | 0 | 0.399821 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.10401 | 0 | −6.78116 | −14.0332 | 0 | −7.00000 | −16.3186 | 0 | −15.4928 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.18720 | 0 | −6.59056 | 20.2258 | 0 | −7.00000 | −17.3219 | 0 | 24.0121 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.61105 | 0 | 5.03969 | 11.1551 | 0 | −7.00000 | −10.6898 | 0 | 40.2815 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.03765 | 0 | 17.3779 | −6.20452 | 0 | −7.00000 | 47.2425 | 0 | −31.2562 | ||||||||||||||||||||||||||||||||||||||||
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.g | 7 | |
| 3.b | odd | 2 | 1 | 483.4.a.d | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.4.a.d | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 1449.4.a.g | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2T_{2}^{6} - 37T_{2}^{5} + 71T_{2}^{4} + 312T_{2}^{3} - 629T_{2}^{2} + 112T_{2} + 180 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots + 180 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots - 228084 \)
|
| $7$ |
\( (T + 7)^{7} \)
|
| $11$ |
\( T^{7} + \cdots - 5071866368 \)
|
| $13$ |
\( T^{7} - 17 T^{6} + \cdots - 323309364 \)
|
| $17$ |
\( T^{7} + \cdots - 111964950432 \)
|
| $19$ |
\( T^{7} + \cdots + 135809928192 \)
|
| $23$ |
\( (T + 23)^{7} \)
|
| $29$ |
\( T^{7} + \cdots + 319608557445276 \)
|
| $31$ |
\( T^{7} + \cdots + 273519093450528 \)
|
| $37$ |
\( T^{7} + \cdots - 13996222497396 \)
|
| $41$ |
\( T^{7} + \cdots + 42\!\cdots\!28 \)
|
| $43$ |
\( T^{7} + \cdots + 447690400483584 \)
|
| $47$ |
\( T^{7} + \cdots - 14\!\cdots\!24 \)
|
| $53$ |
\( T^{7} + \cdots - 16\!\cdots\!84 \)
|
| $59$ |
\( T^{7} + \cdots - 85\!\cdots\!44 \)
|
| $61$ |
\( T^{7} + \cdots - 39\!\cdots\!60 \)
|
| $67$ |
\( T^{7} + \cdots + 13\!\cdots\!88 \)
|
| $71$ |
\( T^{7} + \cdots - 17\!\cdots\!60 \)
|
| $73$ |
\( T^{7} + \cdots - 26\!\cdots\!76 \)
|
| $79$ |
\( T^{7} + \cdots - 71\!\cdots\!60 \)
|
| $83$ |
\( T^{7} + \cdots - 30\!\cdots\!12 \)
|
| $89$ |
\( T^{7} + \cdots - 21\!\cdots\!48 \)
|
| $97$ |
\( T^{7} + \cdots + 12\!\cdots\!64 \)
|
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