Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [447,2,Mod(1,447)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(447, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("447.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 447 = 3 \cdot 149 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 447.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: | \(3.56931297035\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 4x^{8} - 6x^{7} + 37x^{6} - 3x^{5} - 101x^{4} + 49x^{3} + 72x^{2} - 21x - 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4x^{8} - 6x^{7} + 37x^{6} - 3x^{5} - 101x^{4} + 49x^{3} + 72x^{2} - 21x - 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 10\nu^{6} + 17\nu^{5} + 31\nu^{4} - 38\nu^{3} - 29\nu^{2} + 9\nu + 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{8} + 2\nu^{7} + 10\nu^{6} - 16\nu^{5} - 33\nu^{4} + 33\nu^{3} + 38\nu^{2} - 7\nu - 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{8} + 4\nu^{7} + 21\nu^{6} - 35\nu^{5} - 71\nu^{4} + 82\nu^{3} + 80\nu^{2} - 27\nu - 20 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{8} + 4\nu^{7} + 21\nu^{6} - 35\nu^{5} - 71\nu^{4} + 83\nu^{3} + 79\nu^{2} - 31\nu - 18 \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{8} + 7\nu^{7} + 29\nu^{6} - 61\nu^{5} - 86\nu^{4} + 143\nu^{3} + 75\nu^{2} - 49\nu - 17 \)
|
| \(\beta_{8}\) | \(=\) |
\( 3\nu^{8} - 7\nu^{7} - 29\nu^{6} + 61\nu^{5} + 87\nu^{4} - 145\nu^{3} - 80\nu^{2} + 56\nu + 20 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 2\beta_{6} - 2\beta_{5} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{8} + 2\beta_{7} + 9\beta_{6} - 9\beta_{5} + \beta_{4} + \beta_{3} + 10\beta_{2} + 20\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{8} + 11\beta_{7} + 21\beta_{6} - 20\beta_{5} + \beta_{4} + 3\beta_{3} + 45\beta_{2} + 14\beta _1 + 71 \)
|
| \(\nu^{7}\) | \(=\) |
\( 24\beta_{8} + 25\beta_{7} + 68\beta_{6} - 67\beta_{5} + 11\beta_{4} + 16\beta_{3} + 79\beta_{2} + 113\beta _1 + 69 \)
|
| \(\nu^{8}\) | \(=\) |
\( 93 \beta_{8} + 95 \beta_{7} + 169 \beta_{6} - 157 \beta_{5} + 15 \beta_{4} + 46 \beta_{3} + 288 \beta_{2} + \cdots + 380 \)
|
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 |
|
−2.09964 | 1.00000 | 2.40847 | 1.32093 | −2.09964 | −4.08420 | −0.857648 | 1.00000 | −2.77347 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.96824 | 1.00000 | 1.87398 | 3.76140 | −1.96824 | 4.20310 | 0.248040 | 1.00000 | −7.40335 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.720171 | 1.00000 | −1.48135 | −3.36155 | −0.720171 | 0.451528 | 2.50717 | 1.00000 | 2.42089 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.359705 | 1.00000 | −1.87061 | 2.42650 | −0.359705 | 2.04650 | 1.39228 | 1.00000 | −0.872823 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.621098 | 1.00000 | −1.61424 | 0.389054 | 0.621098 | 0.149999 | −2.24480 | 1.00000 | 0.241641 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.78117 | 1.00000 | 1.17257 | 1.18684 | 1.78117 | 2.59967 | −1.47379 | 1.00000 | 2.11396 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.82168 | 1.00000 | 1.31853 | 4.39502 | 1.82168 | −3.34291 | −1.24142 | 1.00000 | 8.00633 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.27383 | 1.00000 | 3.17032 | 0.319256 | 2.27383 | 0.431913 | 2.66112 | 1.00000 | 0.725935 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.64997 | 1.00000 | 5.02232 | −2.43743 | 2.64997 | −1.45560 | 8.00904 | 1.00000 | −6.45912 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(149\) | \(1\) |
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 | 447.2.a.c | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1341.2.a.d | 9 | ||
| 4.b | odd | 2 | 1 | 7152.2.a.z | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 447.2.a.c | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1341.2.a.d | 9 | 3.b | odd | 2 | 1 | ||
| 7152.2.a.z | 9 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - 4T_{2}^{8} - 6T_{2}^{7} + 37T_{2}^{6} - 3T_{2}^{5} - 101T_{2}^{4} + 49T_{2}^{3} + 72T_{2}^{2} - 21T_{2} - 13 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(447))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 4 T^{8} + \cdots - 13 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( T^{9} - 8 T^{8} + \cdots - 64 \)
|
| $7$ |
\( T^{9} - T^{8} + \cdots + 13 \)
|
| $11$ |
\( T^{9} - 13 T^{8} + \cdots + 17707 \)
|
| $13$ |
\( T^{9} + 2 T^{8} + \cdots - 106688 \)
|
| $17$ |
\( T^{9} - 10 T^{8} + \cdots + 3136 \)
|
| $19$ |
\( T^{9} + 11 T^{8} + \cdots + 245 \)
|
| $23$ |
\( T^{9} - 11 T^{8} + \cdots - 145829 \)
|
| $29$ |
\( T^{9} - 164 T^{7} + \cdots - 246080 \)
|
| $31$ |
\( T^{9} + 13 T^{8} + \cdots + 2219947 \)
|
| $37$ |
\( T^{9} - T^{8} + \cdots + 383 \)
|
| $41$ |
\( T^{9} - 17 T^{8} + \cdots - 1478759 \)
|
| $43$ |
\( T^{9} + 8 T^{8} + \cdots + 124736 \)
|
| $47$ |
\( T^{9} - 10 T^{8} + \cdots + 7744 \)
|
| $53$ |
\( T^{9} - 2 T^{8} + \cdots - 499904 \)
|
| $59$ |
\( T^{9} - 25 T^{8} + \cdots + 87095 \)
|
| $61$ |
\( T^{9} + 13 T^{8} + \cdots - 1131067 \)
|
| $67$ |
\( T^{9} - T^{8} + \cdots - 8795843 \)
|
| $71$ |
\( T^{9} - 13 T^{8} + \cdots - 3679753 \)
|
| $73$ |
\( T^{9} - 5 T^{8} + \cdots - 3095969 \)
|
| $79$ |
\( T^{9} + 22 T^{8} + \cdots + 1078720 \)
|
| $83$ |
\( T^{9} - T^{8} + \cdots - 50251 \)
|
| $89$ |
\( T^{9} - 23 T^{8} + \cdots - 65699195 \)
|
| $97$ |
\( T^{9} - 16 T^{8} + \cdots - 217472192 \)
|
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