Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [473,2,Mod(1,473)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, 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("473.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 473 = 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 473.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: | \(3.77692401561\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 17x^{9} + 15x^{8} + 102x^{7} - 77x^{6} - 255x^{5} + 150x^{4} + 248x^{3} - 59x^{2} - 93x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 17x^{9} + 15x^{8} + 102x^{7} - 77x^{6} - 255x^{5} + 150x^{4} + 248x^{3} - 59x^{2} - 93x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{10} - \nu^{9} - 17 \nu^{8} + 15 \nu^{7} + 99 \nu^{6} - 77 \nu^{5} - 222 \nu^{4} + 150 \nu^{3} + \cdots - 24 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4 \nu^{10} - 7 \nu^{9} - 65 \nu^{8} + 111 \nu^{7} + 363 \nu^{6} - 605 \nu^{5} - 789 \nu^{4} + 1266 \nu^{3} + \cdots - 204 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 7 \nu^{10} + 10 \nu^{9} + 116 \nu^{8} - 156 \nu^{7} - 669 \nu^{6} + 836 \nu^{5} + 1545 \nu^{4} + \cdots + 375 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7 \nu^{10} - 10 \nu^{9} - 116 \nu^{8} + 156 \nu^{7} + 669 \nu^{6} - 836 \nu^{5} - 1545 \nu^{4} + \cdots - 375 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 7 \nu^{10} + 10 \nu^{9} + 116 \nu^{8} - 156 \nu^{7} - 669 \nu^{6} + 836 \nu^{5} + 1554 \nu^{4} + \cdots + 429 ) / 9 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 19 \nu^{10} - 22 \nu^{9} - 311 \nu^{8} + 336 \nu^{7} + 1758 \nu^{6} - 1769 \nu^{5} - 3930 \nu^{4} + \cdots - 798 ) / 18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10 \nu^{10} - 13 \nu^{9} - 167 \nu^{8} + 201 \nu^{7} + 966 \nu^{6} - 1076 \nu^{5} - 2220 \nu^{4} + \cdots - 528 ) / 9 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 11 \nu^{10} + 17 \nu^{9} + 181 \nu^{8} - 258 \nu^{7} - 1032 \nu^{6} + 1342 \nu^{5} + 2343 \nu^{4} + \cdots + 570 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} - \beta_{7} + 8\beta_{6} + 8\beta_{5} + \beta_{3} + \beta_{2} + 29\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{7} + 10\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 45\beta_{2} + 2\beta _1 + 86 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{10} - 11 \beta_{9} - 12 \beta_{7} + 55 \beta_{6} + 55 \beta_{5} + \beta_{4} + 11 \beta_{3} + \cdots + 7 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{9} + 2 \beta_{8} + 78 \beta_{7} + 77 \beta_{6} - 12 \beta_{5} - 11 \beta_{4} - 14 \beta_{3} + \cdots + 520 \)
|
| \(\nu^{9}\) | \(=\) |
\( 17 \beta_{10} - 89 \beta_{9} + 2 \beta_{8} - 104 \beta_{7} + 365 \beta_{6} + 364 \beta_{5} + 14 \beta_{4} + \cdots + 98 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2 \beta_{10} - 18 \beta_{9} + 36 \beta_{8} + 557 \beta_{7} + 547 \beta_{6} - 100 \beta_{5} - 89 \beta_{4} + \cdots + 3226 \)
|
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 |
|
−2.58731 | −2.11846 | 4.69419 | −0.422094 | 5.48113 | 4.63658 | −6.97072 | 1.48789 | 1.09209 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.53952 | 2.75256 | 4.44918 | 2.43343 | −6.99018 | 0.182933 | −6.21975 | 4.57656 | −6.17976 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.67713 | −1.90897 | 0.812772 | 2.49122 | 3.20159 | 3.46916 | 1.99114 | 0.644160 | −4.17810 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.36651 | 3.02245 | −0.132656 | −2.95221 | −4.13020 | 4.38627 | 2.91429 | 6.13521 | 4.03422 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.05891 | −0.172440 | −0.878703 | −2.97856 | 0.182599 | −2.08395 | 3.04830 | −2.97026 | 3.15403 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.331559 | 3.17650 | −1.89007 | 3.05541 | 1.05319 | 0.592473 | −1.28979 | 7.09012 | 1.01305 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.383643 | −2.66301 | −1.85282 | −2.11027 | −1.02165 | −0.738563 | −1.47811 | 4.09164 | −0.809591 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.804900 | 1.27167 | −1.35214 | 1.45736 | 1.02356 | 3.48877 | −2.69813 | −1.38286 | 1.17303 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.98512 | −0.521173 | 1.94070 | 4.27081 | −1.03459 | 2.44087 | −0.117710 | −2.72838 | 8.47808 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.20294 | 2.30856 | 2.85295 | 0.366391 | 5.08562 | −2.89896 | 1.87901 | 2.32944 | 0.807138 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.52123 | 0.852332 | 4.35658 | −2.61150 | 2.14892 | 3.52442 | 5.94148 | −2.27353 | −6.58418 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(43\) | \(-1\) |
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 | 473.2.a.g | ✓ | 11 |
| 3.b | odd | 2 | 1 | 4257.2.a.t | 11 | ||
| 4.b | odd | 2 | 1 | 7568.2.a.br | 11 | ||
| 11.b | odd | 2 | 1 | 5203.2.a.k | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 473.2.a.g | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 4257.2.a.t | 11 | 3.b | odd | 2 | 1 | ||
| 5203.2.a.k | 11 | 11.b | odd | 2 | 1 | ||
| 7568.2.a.br | 11 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(473))\):
|
\( T_{2}^{11} + T_{2}^{10} - 17 T_{2}^{9} - 15 T_{2}^{8} + 102 T_{2}^{7} + 77 T_{2}^{6} - 255 T_{2}^{5} + \cdots + 18 \)
|
|
\( T_{3}^{11} - 6 T_{3}^{10} - 7 T_{3}^{9} + 94 T_{3}^{8} - 53 T_{3}^{7} - 483 T_{3}^{6} + 524 T_{3}^{5} + \cdots + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + T^{10} + \cdots + 18 \)
|
| $3$ |
\( T^{11} - 6 T^{10} + \cdots + 64 \)
|
| $5$ |
\( T^{11} - 3 T^{10} + \cdots + 864 \)
|
| $7$ |
\( T^{11} - 17 T^{10} + \cdots + 1024 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} - 11 T^{10} + \cdots + 384 \)
|
| $17$ |
\( T^{11} + 5 T^{10} + \cdots + 2304 \)
|
| $19$ |
\( T^{11} - 5 T^{10} + \cdots + 512 \)
|
| $23$ |
\( T^{11} - 11 T^{10} + \cdots + 309216 \)
|
| $29$ |
\( T^{11} + 16 T^{10} + \cdots + 6635124 \)
|
| $31$ |
\( T^{11} - 33 T^{10} + \cdots - 64672 \)
|
| $37$ |
\( T^{11} - 8 T^{10} + \cdots - 1443264 \)
|
| $41$ |
\( T^{11} - 2 T^{10} + \cdots - 3252096 \)
|
| $43$ |
\( (T - 1)^{11} \)
|
| $47$ |
\( T^{11} - 15 T^{10} + \cdots - 2502144 \)
|
| $53$ |
\( T^{11} + 5 T^{10} + \cdots + 6087516 \)
|
| $59$ |
\( T^{11} + \cdots + 5642666784 \)
|
| $61$ |
\( T^{11} + 12 T^{10} + \cdots - 2638072 \)
|
| $67$ |
\( T^{11} - 32 T^{10} + \cdots - 29750608 \)
|
| $71$ |
\( T^{11} - 6 T^{10} + \cdots - 32099328 \)
|
| $73$ |
\( T^{11} + \cdots + 7837916288 \)
|
| $79$ |
\( T^{11} + \cdots + 922355712 \)
|
| $83$ |
\( T^{11} + \cdots + 465463392 \)
|
| $89$ |
\( T^{11} + \cdots - 7968616608 \)
|
| $97$ |
\( T^{11} + \cdots + 1086402906 \)
|
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