Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1339,4,Mod(1,1339)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1339, 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("1339.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1339 = 13 \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1339.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: | \(79.0035574977\) |
Analytic rank: | \(0\) |
Dimension: | \(81\) |
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 | −5.49150 | 2.90028 | 22.1566 | 5.78569 | −15.9269 | −13.7516 | −77.7408 | −18.5884 | −31.7721 | ||||||||||||||||||
1.2 | −5.48493 | 5.86488 | 22.0845 | 15.9853 | −32.1685 | 10.3210 | −77.2526 | 7.39676 | −87.6785 | ||||||||||||||||||
1.3 | −5.16274 | −8.76065 | 18.6539 | 10.7511 | 45.2289 | −33.6084 | −55.0031 | 49.7489 | −55.5053 | ||||||||||||||||||
1.4 | −5.11659 | −4.25777 | 18.1795 | −12.5358 | 21.7853 | −16.5271 | −52.0845 | −8.87137 | 64.1405 | ||||||||||||||||||
1.5 | −5.01493 | −4.02441 | 17.1495 | 5.04403 | 20.1821 | 31.0445 | −45.8841 | −10.8041 | −25.2955 | ||||||||||||||||||
1.6 | −4.98675 | 3.24508 | 16.8677 | 1.77219 | −16.1824 | −7.78574 | −44.2211 | −16.4694 | −8.83745 | ||||||||||||||||||
1.7 | −4.81639 | −10.0473 | 15.1976 | 17.0888 | 48.3918 | 23.6945 | −34.6664 | 73.9488 | −82.3063 | ||||||||||||||||||
1.8 | −4.56725 | 2.74545 | 12.8598 | −6.65464 | −12.5392 | −13.0337 | −22.1958 | −19.4625 | 30.3934 | ||||||||||||||||||
1.9 | −4.38698 | 6.27446 | 11.2456 | −5.24908 | −27.5259 | 27.3853 | −14.2382 | 12.3689 | 23.0276 | ||||||||||||||||||
1.10 | −4.25477 | −2.33039 | 10.1031 | −10.1776 | 9.91529 | 6.77364 | −8.94819 | −21.5693 | 43.3035 | ||||||||||||||||||
1.11 | −4.23979 | 10.0611 | 9.97578 | 0.270687 | −42.6568 | 4.36470 | −8.37689 | 74.2252 | −1.14765 | ||||||||||||||||||
1.12 | −4.22150 | 8.06806 | 9.82110 | −19.4222 | −34.0594 | 7.23234 | −7.68778 | 38.0936 | 81.9908 | ||||||||||||||||||
1.13 | −4.14109 | 2.03460 | 9.14866 | 21.3331 | −8.42547 | −34.9102 | −4.75673 | −22.8604 | −88.3426 | ||||||||||||||||||
1.14 | −3.96488 | −5.53216 | 7.72029 | 11.3747 | 21.9343 | 20.9003 | 1.10900 | 3.60475 | −45.0995 | ||||||||||||||||||
1.15 | −3.86408 | −8.10434 | 6.93114 | −6.45482 | 31.3159 | 13.3220 | 4.13016 | 38.6804 | 24.9420 | ||||||||||||||||||
1.16 | −3.54273 | 3.14722 | 4.55090 | 19.9871 | −11.1497 | 31.0178 | 12.2192 | −17.0950 | −70.8088 | ||||||||||||||||||
1.17 | −3.45822 | 9.42402 | 3.95926 | 17.1535 | −32.5903 | −5.97598 | 13.9738 | 61.8121 | −59.3204 | ||||||||||||||||||
1.18 | −3.45037 | −1.33681 | 3.90505 | 9.92993 | 4.61250 | 2.13656 | 14.1291 | −25.2129 | −34.2619 | ||||||||||||||||||
1.19 | −3.27427 | −0.503529 | 2.72086 | 6.03692 | 1.64869 | −19.6821 | 17.2853 | −26.7465 | −19.7665 | ||||||||||||||||||
1.20 | −3.18679 | −6.03709 | 2.15566 | −21.7124 | 19.2390 | −5.02844 | 18.6247 | 9.44643 | 69.1929 | ||||||||||||||||||
See all 81 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(13\) | \(-1\) |
\(103\) | \(-1\) |
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 | 1339.4.a.d | ✓ | 81 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1339.4.a.d | ✓ | 81 | 1.a | even | 1 | 1 | trivial |