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: | \(0\) |
Dimension: | \(179\) |
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 | −2.81186 | −1.67550 | 5.90655 | −2.52541 | 4.71126 | 3.13505 | −10.9847 | −0.192706 | 7.10109 | ||||||||||||||||||
1.2 | −2.76280 | 0.0815042 | 5.63307 | 2.29049 | −0.225180 | −4.47769 | −10.0374 | −2.99336 | −6.32817 | ||||||||||||||||||
1.3 | −2.76009 | 2.82654 | 5.61808 | 3.76331 | −7.80149 | −3.62929 | −9.98623 | 4.98931 | −10.3871 | ||||||||||||||||||
1.4 | −2.73492 | −2.66806 | 5.47977 | 3.59226 | 7.29692 | 3.26568 | −9.51690 | 4.11853 | −9.82454 | ||||||||||||||||||
1.5 | −2.72330 | 1.38182 | 5.41638 | 0.695892 | −3.76310 | 2.22178 | −9.30382 | −1.09059 | −1.89512 | ||||||||||||||||||
1.6 | −2.69327 | −0.453777 | 5.25372 | −2.96502 | 1.22215 | 4.09118 | −8.76316 | −2.79409 | 7.98562 | ||||||||||||||||||
1.7 | −2.68916 | 1.31955 | 5.23156 | 0.565625 | −3.54848 | −3.46548 | −8.69016 | −1.25878 | −1.52105 | ||||||||||||||||||
1.8 | −2.63717 | 2.57307 | 4.95467 | −2.90004 | −6.78562 | 1.48673 | −7.79198 | 3.62067 | 7.64791 | ||||||||||||||||||
1.9 | −2.62553 | −3.09813 | 4.89340 | −2.94807 | 8.13422 | 1.83284 | −7.59671 | 6.59839 | 7.74024 | ||||||||||||||||||
1.10 | −2.59548 | −2.23952 | 4.73650 | −0.915423 | 5.81261 | −1.92878 | −7.10252 | 2.01543 | 2.37596 | ||||||||||||||||||
1.11 | −2.59447 | −1.57672 | 4.73128 | 2.96648 | 4.09075 | 0.648960 | −7.08623 | −0.513957 | −7.69644 | ||||||||||||||||||
1.12 | −2.59279 | 0.176343 | 4.72258 | −4.20399 | −0.457220 | 0.364097 | −7.05909 | −2.96890 | 10.9001 | ||||||||||||||||||
1.13 | −2.58313 | −0.944028 | 4.67256 | 2.39597 | 2.43855 | 2.66460 | −6.90357 | −2.10881 | −6.18909 | ||||||||||||||||||
1.14 | −2.56401 | 2.69258 | 4.57412 | −0.262266 | −6.90379 | 3.89440 | −6.60007 | 4.24998 | 0.672451 | ||||||||||||||||||
1.15 | −2.52781 | 2.99330 | 4.38982 | 1.15266 | −7.56649 | −2.64816 | −6.04101 | 5.95984 | −2.91370 | ||||||||||||||||||
1.16 | −2.48629 | −1.57430 | 4.18165 | −3.66584 | 3.91418 | −5.15105 | −5.42422 | −0.521566 | 9.11435 | ||||||||||||||||||
1.17 | −2.47698 | 3.33984 | 4.13542 | −4.25899 | −8.27272 | 3.67754 | −5.28940 | 8.15455 | 10.5494 | ||||||||||||||||||
1.18 | −2.47596 | −1.58949 | 4.13039 | −0.715359 | 3.93552 | −0.683204 | −5.27477 | −0.473513 | 1.77120 | ||||||||||||||||||
1.19 | −2.36148 | 1.10160 | 3.57657 | 1.64504 | −2.60141 | 0.267733 | −3.72302 | −1.78647 | −3.88472 | ||||||||||||||||||
1.20 | −2.33610 | −3.21364 | 3.45737 | 1.58018 | 7.50740 | 0.0419320 | −3.40456 | 7.32750 | −3.69146 | ||||||||||||||||||
See next 80 embeddings (of 179 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.d | ✓ | 179 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8003.2.a.d | ✓ | 179 | 1.a | even | 1 | 1 | trivial |