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: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 34x^{5} + 7x^{4} + 295x^{3} + 84x^{2} - 524x - 288 \)
|
| 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
\( x^{7} - x^{6} - 34x^{5} + 7x^{4} + 295x^{3} + 84x^{2} - 524x - 288 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} - 4\nu^{5} - 112\nu^{4} + 149\nu^{3} + 926\nu^{2} - 1360\nu - 824 ) / 136 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} + \nu^{5} + 164\nu^{4} + 69\nu^{3} - 1175\nu^{2} - 646\nu + 784 ) / 136 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 4\nu^{5} + 112\nu^{4} - 81\nu^{3} - 1062\nu^{2} + 340\nu + 1504 ) / 68 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\nu^{6} - 26\nu^{5} - 320\nu^{4} + 501\nu^{3} + 2092\nu^{2} - 1972\nu - 1616 ) / 136 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 2\beta_{2} + 17\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 5\beta_{5} - 2\beta_{4} + 3\beta_{3} + 25\beta_{2} + 35\beta _1 + 181 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{6} + 36\beta_{5} - 16\beta_{4} + 60\beta_{3} + 77\beta_{2} + 363\beta _1 + 382 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32\beta_{6} + 185\beta_{5} - 96\beta_{4} + 138\beta_{3} + 628\beta_{2} + 1091\beta _1 + 3958 \)
|
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 |
|
−4.21883 | 0 | 9.79849 | 19.6249 | 0 | 7.00000 | −7.58752 | 0 | −82.7940 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.29766 | 0 | −2.72078 | 7.00327 | 0 | 7.00000 | 24.6327 | 0 | −16.0911 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.603389 | 0 | −7.63592 | −14.1843 | 0 | 7.00000 | 9.43454 | 0 | 8.55866 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.61094 | 0 | −5.40488 | 4.04614 | 0 | 7.00000 | −21.5944 | 0 | 6.51808 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.37660 | 0 | −2.35176 | 18.5873 | 0 | 7.00000 | −24.6020 | 0 | 44.1746 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.01159 | 0 | 8.09283 | −9.29066 | 0 | 7.00000 | 0.372383 | 0 | −37.2703 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.12074 | 0 | 18.2220 | 15.2134 | 0 | 7.00000 | 52.3444 | 0 | 77.9040 | ||||||||||||||||||||||||||||||||||||||||
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.h | 7 | |
| 3.b | odd | 2 | 1 | 483.4.a.c | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.4.a.c | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 1449.4.a.h | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 6T_{2}^{6} - 19T_{2}^{5} + 143T_{2}^{4} - 2T_{2}^{3} - 677T_{2}^{2} + 388T_{2} + 460 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 6 T^{6} + \cdots + 460 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 41 T^{6} + \cdots - 20722732 \)
|
| $7$ |
\( (T - 7)^{7} \)
|
| $11$ |
\( T^{7} + \cdots - 224017353856 \)
|
| $13$ |
\( T^{7} + 87 T^{6} + \cdots + 868269144 \)
|
| $17$ |
\( T^{7} + \cdots - 12051705641600 \)
|
| $19$ |
\( T^{7} + \cdots + 7302931600672 \)
|
| $23$ |
\( (T + 23)^{7} \)
|
| $29$ |
\( T^{7} + \cdots + 10\!\cdots\!40 \)
|
| $31$ |
\( T^{7} + \cdots - 623763925110016 \)
|
| $37$ |
\( T^{7} + \cdots + 701389545056212 \)
|
| $41$ |
\( T^{7} + \cdots + 237579020750652 \)
|
| $43$ |
\( T^{7} + \cdots + 25\!\cdots\!40 \)
|
| $47$ |
\( T^{7} + \cdots + 14\!\cdots\!40 \)
|
| $53$ |
\( T^{7} + \cdots + 15\!\cdots\!28 \)
|
| $59$ |
\( T^{7} + \cdots - 27\!\cdots\!96 \)
|
| $61$ |
\( T^{7} + \cdots + 18\!\cdots\!36 \)
|
| $67$ |
\( T^{7} + \cdots - 21\!\cdots\!16 \)
|
| $71$ |
\( T^{7} + \cdots + 76\!\cdots\!08 \)
|
| $73$ |
\( T^{7} + \cdots - 20\!\cdots\!04 \)
|
| $79$ |
\( T^{7} + \cdots + 82\!\cdots\!60 \)
|
| $83$ |
\( T^{7} + \cdots - 75\!\cdots\!56 \)
|
| $89$ |
\( T^{7} + \cdots - 43\!\cdots\!92 \)
|
| $97$ |
\( T^{7} + \cdots - 83\!\cdots\!52 \)
|
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