Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,4,Mod(1,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("471.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 471.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: | \(27.7898996127\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −5.40228 | −3.00000 | 21.1846 | 6.01793 | 16.2068 | −1.25342 | −71.2270 | 9.00000 | −32.5106 | ||||||||||||||||||
1.2 | −5.06198 | −3.00000 | 17.6236 | 12.0952 | 15.1859 | −27.2034 | −48.7145 | 9.00000 | −61.2257 | ||||||||||||||||||
1.3 | −4.42661 | −3.00000 | 11.5949 | −9.85596 | 13.2798 | −17.0380 | −15.9132 | 9.00000 | 43.6285 | ||||||||||||||||||
1.4 | −3.69988 | −3.00000 | 5.68908 | 8.40381 | 11.0996 | 21.0620 | 8.55011 | 9.00000 | −31.0931 | ||||||||||||||||||
1.5 | −3.62090 | −3.00000 | 5.11092 | −9.00865 | 10.8627 | 5.02048 | 10.4611 | 9.00000 | 32.6194 | ||||||||||||||||||
1.6 | −3.40822 | −3.00000 | 3.61596 | −8.48893 | 10.2247 | 10.4594 | 14.9418 | 9.00000 | 28.9321 | ||||||||||||||||||
1.7 | −2.31678 | −3.00000 | −2.63252 | −1.90474 | 6.95035 | −14.6237 | 24.6332 | 9.00000 | 4.41286 | ||||||||||||||||||
1.8 | −1.56491 | −3.00000 | −5.55104 | 8.64342 | 4.69474 | 19.5992 | 21.2062 | 9.00000 | −13.5262 | ||||||||||||||||||
1.9 | −1.38179 | −3.00000 | −6.09066 | 20.1959 | 4.14537 | −9.60234 | 19.4703 | 9.00000 | −27.9065 | ||||||||||||||||||
1.10 | −1.22585 | −3.00000 | −6.49728 | −2.10116 | 3.67756 | 35.0092 | 17.7716 | 9.00000 | 2.57572 | ||||||||||||||||||
1.11 | 0.196646 | −3.00000 | −7.96133 | −2.23494 | −0.589937 | −24.0345 | −3.13873 | 9.00000 | −0.439492 | ||||||||||||||||||
1.12 | 1.29561 | −3.00000 | −6.32139 | −17.0439 | −3.88684 | 13.0789 | −18.5550 | 9.00000 | −22.0823 | ||||||||||||||||||
1.13 | 1.36089 | −3.00000 | −6.14798 | 17.6223 | −4.08267 | 31.9947 | −19.2538 | 9.00000 | 23.9820 | ||||||||||||||||||
1.14 | 1.51003 | −3.00000 | −5.71981 | 14.7843 | −4.53009 | −36.4705 | −20.7173 | 9.00000 | 22.3248 | ||||||||||||||||||
1.15 | 2.29403 | −3.00000 | −2.73741 | −14.4094 | −6.88210 | −20.5162 | −24.6320 | 9.00000 | −33.0556 | ||||||||||||||||||
1.16 | 2.43477 | −3.00000 | −2.07188 | −12.7413 | −7.30432 | −29.4801 | −24.5227 | 9.00000 | −31.0221 | ||||||||||||||||||
1.17 | 2.99489 | −3.00000 | 0.969360 | 5.29125 | −8.98467 | 8.08775 | −21.0560 | 9.00000 | 15.8467 | ||||||||||||||||||
1.18 | 4.33495 | −3.00000 | 10.7918 | −21.4542 | −13.0048 | −11.7470 | 12.1021 | 9.00000 | −93.0027 | ||||||||||||||||||
1.19 | 4.50022 | −3.00000 | 12.2520 | 17.0216 | −13.5007 | 15.6710 | 19.1349 | 9.00000 | 76.6007 | ||||||||||||||||||
1.20 | 5.00169 | −3.00000 | 17.0169 | 16.1971 | −15.0051 | −23.8262 | 45.0996 | 9.00000 | 81.0129 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(157\) | \(1\) |
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 | 471.4.a.c | ✓ | 22 |
3.b | odd | 2 | 1 | 1413.4.a.e | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.4.a.c | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
1413.4.a.e | 22 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 4 T_{2}^{21} - 125 T_{2}^{20} + 491 T_{2}^{19} + 6583 T_{2}^{18} - 25317 T_{2}^{17} + \cdots + 752167488 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(471))\).