Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [103,8,Mod(1,103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("103.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 103 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 103.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: | \(32.1756576249\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
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 | −20.7827 | 33.4298 | 303.921 | −243.724 | −694.762 | −1694.08 | −3656.12 | −1069.45 | 5065.25 | ||||||||||||||||||
1.2 | −20.6574 | 46.4673 | 298.728 | 75.4363 | −959.894 | 112.969 | −3526.80 | −27.7873 | −1558.32 | ||||||||||||||||||
1.3 | −18.2028 | −40.9590 | 203.341 | −132.100 | 745.566 | −340.437 | −1371.41 | −509.364 | 2404.59 | ||||||||||||||||||
1.4 | −18.1848 | −92.4197 | 202.686 | −198.302 | 1680.63 | 538.310 | −1358.14 | 6354.40 | 3606.08 | ||||||||||||||||||
1.5 | −17.3474 | 87.7597 | 172.931 | 467.264 | −1522.40 | −1002.57 | −779.432 | 5514.76 | −8105.79 | ||||||||||||||||||
1.6 | −13.8714 | 24.6737 | 64.4145 | −100.756 | −342.258 | 1781.02 | 882.017 | −1578.21 | 1397.62 | ||||||||||||||||||
1.7 | −13.3268 | −73.6815 | 49.6030 | 444.430 | 981.937 | 751.237 | 1044.78 | 3241.96 | −5922.82 | ||||||||||||||||||
1.8 | −12.8065 | 16.6711 | 36.0070 | −184.175 | −213.499 | −819.277 | 1178.11 | −1909.07 | 2358.64 | ||||||||||||||||||
1.9 | −11.2327 | −47.2601 | −1.82614 | 33.9435 | 530.860 | 1327.58 | 1458.30 | 46.5205 | −381.278 | ||||||||||||||||||
1.10 | −8.28916 | 10.1290 | −59.2899 | 336.004 | −83.9610 | −1536.90 | 1552.48 | −2084.40 | −2785.19 | ||||||||||||||||||
1.11 | −7.05130 | −68.8783 | −78.2791 | 300.510 | 485.682 | −499.751 | 1454.54 | 2557.23 | −2118.98 | ||||||||||||||||||
1.12 | −6.62833 | 67.4256 | −84.0652 | 484.648 | −446.919 | 1346.24 | 1405.64 | 2359.21 | −3212.40 | ||||||||||||||||||
1.13 | −5.95705 | 72.0720 | −92.5135 | 16.6373 | −429.337 | 408.502 | 1313.61 | 3007.37 | −99.1094 | ||||||||||||||||||
1.14 | −4.47080 | 5.61726 | −108.012 | −478.616 | −25.1136 | 370.899 | 1055.16 | −2155.45 | 2139.80 | ||||||||||||||||||
1.15 | 0.682485 | 33.7116 | −127.534 | −330.528 | 23.0076 | −1374.53 | −174.398 | −1050.53 | −225.580 | ||||||||||||||||||
1.16 | 2.60267 | 76.1686 | −121.226 | −469.972 | 198.242 | 1258.56 | −648.654 | 3614.66 | −1223.18 | ||||||||||||||||||
1.17 | 3.05039 | −16.1277 | −118.695 | 221.532 | −49.1956 | 531.922 | −752.515 | −1926.90 | 675.757 | ||||||||||||||||||
1.18 | 5.04695 | −31.1930 | −102.528 | 523.347 | −157.430 | 1709.09 | −1163.46 | −1213.99 | 2641.31 | ||||||||||||||||||
1.19 | 5.56903 | −39.5094 | −96.9859 | −140.033 | −220.029 | −1518.37 | −1252.95 | −626.003 | −779.849 | ||||||||||||||||||
1.20 | 6.13722 | −29.4700 | −90.3346 | −100.733 | −180.863 | −488.558 | −1339.97 | −1318.52 | −618.221 | ||||||||||||||||||
See all 32 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(103\) | \(1\) |
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 | 103.8.a.b | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
103.8.a.b | ✓ | 32 | 1.a | even | 1 | 1 | trivial |