Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8003,2,Mod(1,8003)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, 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("8003.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8003 = 53 \cdot 151 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8003.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: | \(63.9042767376\) |
Analytic rank: | \(1\) |
Dimension: | \(147\) |
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 | −2.76716 | −2.20267 | 5.65717 | 0.836026 | 6.09513 | −1.25752 | −10.1200 | 1.85174 | −2.31342 | ||||||||||||||||||
1.2 | −2.74141 | 2.30375 | 5.51534 | −1.31886 | −6.31554 | 1.25952 | −9.63701 | 2.30728 | 3.61554 | ||||||||||||||||||
1.3 | −2.73133 | −0.575375 | 5.46014 | −2.64139 | 1.57154 | −4.90025 | −9.45076 | −2.66894 | 7.21448 | ||||||||||||||||||
1.4 | −2.69892 | 1.36364 | 5.28418 | 3.25523 | −3.68037 | 0.105698 | −8.86375 | −1.14047 | −8.78561 | ||||||||||||||||||
1.5 | −2.69176 | −2.97285 | 5.24555 | 2.64194 | 8.00219 | −3.02993 | −8.73623 | 5.83784 | −7.11145 | ||||||||||||||||||
1.6 | −2.68460 | −1.10941 | 5.20709 | 3.69885 | 2.97832 | 2.40865 | −8.60976 | −1.76921 | −9.92993 | ||||||||||||||||||
1.7 | −2.63479 | 0.411921 | 4.94210 | −2.62173 | −1.08532 | 0.446803 | −7.75181 | −2.83032 | 6.90770 | ||||||||||||||||||
1.8 | −2.57093 | 1.68353 | 4.60966 | 0.0447393 | −4.32822 | −3.24015 | −6.70923 | −0.165740 | −0.115021 | ||||||||||||||||||
1.9 | −2.46705 | 3.03255 | 4.08634 | −1.27620 | −7.48147 | 0.505683 | −5.14712 | 6.19639 | 3.14846 | ||||||||||||||||||
1.10 | −2.46471 | −1.16892 | 4.07480 | −2.61176 | 2.88105 | −0.765646 | −5.11379 | −1.63363 | 6.43723 | ||||||||||||||||||
1.11 | −2.45879 | −0.163481 | 4.04563 | 2.51114 | 0.401964 | −0.639591 | −5.02975 | −2.97327 | −6.17435 | ||||||||||||||||||
1.12 | −2.44772 | −0.651188 | 3.99131 | −1.37302 | 1.59392 | 0.683234 | −4.87416 | −2.57595 | 3.36077 | ||||||||||||||||||
1.13 | −2.41146 | 1.80060 | 3.81513 | 0.537438 | −4.34207 | 4.37842 | −4.37711 | 0.242159 | −1.29601 | ||||||||||||||||||
1.14 | −2.40138 | 1.08484 | 3.76663 | −1.87801 | −2.60511 | −2.74065 | −4.24236 | −1.82313 | 4.50981 | ||||||||||||||||||
1.15 | −2.37028 | −0.677898 | 3.61822 | 2.58676 | 1.60681 | −3.88263 | −3.83563 | −2.54045 | −6.13133 | ||||||||||||||||||
1.16 | −2.33017 | −2.83003 | 3.42967 | −2.62223 | 6.59443 | 3.30505 | −3.33137 | 5.00905 | 6.11023 | ||||||||||||||||||
1.17 | −2.31195 | 2.01743 | 3.34512 | −2.72339 | −4.66419 | −3.92860 | −3.10985 | 1.07001 | 6.29634 | ||||||||||||||||||
1.18 | −2.30251 | −2.68436 | 3.30157 | −0.273715 | 6.18077 | 3.09928 | −2.99689 | 4.20577 | 0.630233 | ||||||||||||||||||
1.19 | −2.28070 | −3.18519 | 3.20158 | −4.19393 | 7.26446 | −3.02488 | −2.74045 | 7.14546 | 9.56509 | ||||||||||||||||||
1.20 | −2.26111 | −1.67317 | 3.11264 | 0.0442572 | 3.78322 | 2.69750 | −2.51581 | −0.200518 | −0.100071 | ||||||||||||||||||
See next 80 embeddings (of 147 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(53\) | \(-1\) |
\(151\) | \(-1\) |
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 | 8003.2.a.a | ✓ | 147 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8003.2.a.a | ✓ | 147 | 1.a | even | 1 | 1 | trivial |