Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
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: | \(75.5407791319\) |
Analytic rank: | \(1\) |
Dimension: | \(30\) |
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 | −11.3107 | −9.00000 | 95.9310 | 17.2447 | 101.796 | −21.0168 | −723.101 | 81.0000 | −195.049 | ||||||||||||||||||
1.2 | −10.5389 | −9.00000 | 79.0692 | −79.1689 | 94.8504 | 161.660 | −496.059 | 81.0000 | 834.356 | ||||||||||||||||||
1.3 | −9.78552 | −9.00000 | 63.7563 | 53.0529 | 88.0696 | −2.88714 | −310.752 | 81.0000 | −519.150 | ||||||||||||||||||
1.4 | −9.06695 | −9.00000 | 50.2095 | −0.166632 | 81.6025 | −235.849 | −165.105 | 81.0000 | 1.51084 | ||||||||||||||||||
1.5 | −8.01065 | −9.00000 | 32.1705 | −50.4239 | 72.0959 | 130.088 | −1.36618 | 81.0000 | 403.929 | ||||||||||||||||||
1.6 | −7.62164 | −9.00000 | 26.0894 | −4.86745 | 68.5948 | 215.306 | 45.0481 | 81.0000 | 37.0980 | ||||||||||||||||||
1.7 | −7.13980 | −9.00000 | 18.9767 | 54.3556 | 64.2582 | −91.4021 | 92.9836 | 81.0000 | −388.088 | ||||||||||||||||||
1.8 | −6.95119 | −9.00000 | 16.3190 | −101.673 | 62.5607 | −73.4089 | 109.002 | 81.0000 | 706.746 | ||||||||||||||||||
1.9 | −6.49004 | −9.00000 | 10.1206 | 53.7142 | 58.4103 | −12.0983 | 141.998 | 81.0000 | −348.607 | ||||||||||||||||||
1.10 | −5.25286 | −9.00000 | −4.40749 | 14.5253 | 47.2757 | 156.395 | 191.243 | 81.0000 | −76.2993 | ||||||||||||||||||
1.11 | −4.96468 | −9.00000 | −7.35194 | −22.5622 | 44.6821 | −186.074 | 195.370 | 81.0000 | 112.014 | ||||||||||||||||||
1.12 | −4.55243 | −9.00000 | −11.2754 | 65.2180 | 40.9718 | 201.026 | 197.008 | 81.0000 | −296.900 | ||||||||||||||||||
1.13 | −1.89378 | −9.00000 | −28.4136 | 13.1753 | 17.0440 | −20.4742 | 114.410 | 81.0000 | −24.9510 | ||||||||||||||||||
1.14 | −1.04956 | −9.00000 | −30.8984 | 80.7800 | 9.44602 | 79.1448 | 66.0156 | 81.0000 | −84.7833 | ||||||||||||||||||
1.15 | 0.00358760 | −9.00000 | −32.0000 | −98.0149 | −0.0322884 | −183.236 | −0.229607 | 81.0000 | −0.351639 | ||||||||||||||||||
1.16 | 0.614444 | −9.00000 | −31.6225 | −56.4863 | −5.52999 | −140.573 | −39.0924 | 81.0000 | −34.7077 | ||||||||||||||||||
1.17 | 0.892048 | −9.00000 | −31.2043 | −60.6723 | −8.02843 | 242.261 | −56.3812 | 81.0000 | −54.1226 | ||||||||||||||||||
1.18 | 1.56244 | −9.00000 | −29.5588 | 49.5691 | −14.0620 | −144.113 | −96.1822 | 81.0000 | 77.4490 | ||||||||||||||||||
1.19 | 2.78050 | −9.00000 | −24.2688 | 24.4329 | −25.0245 | −110.716 | −156.455 | 81.0000 | 67.9358 | ||||||||||||||||||
1.20 | 3.44137 | −9.00000 | −20.1570 | −70.8289 | −30.9724 | −20.9998 | −179.492 | 81.0000 | −243.749 | ||||||||||||||||||
See all 30 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 | 471.6.a.b | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.6.a.b | ✓ | 30 | 1.a | even | 1 | 1 | trivial |