Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,8,Mod(2,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.2");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.f (of order \(10\), degree \(4\), minimal) |
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: | no |
Analytic conductor: | \(10.3087058410\) |
Analytic rank: | \(0\) |
Dimension: | \(104\) |
Relative dimension: | \(26\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −6.60852 | + | 20.3389i | −0.462116 | + | 46.7631i | −266.445 | − | 193.584i | 133.323 | − | 43.3192i | −948.057 | − | 318.434i | 864.199 | − | 1189.47i | 3483.52 | − | 2530.93i | −2186.57 | − | 43.2200i | 2997.91i | ||
2.2 | −6.54483 | + | 20.1429i | 34.8194 | − | 31.2188i | −259.349 | − | 188.428i | −311.568 | + | 101.235i | 400.951 | + | 905.686i | 7.55299 | − | 10.3958i | 3299.65 | − | 2397.34i | 237.777 | − | 2174.04i | − | 6938.46i | |
2.3 | −5.89289 | + | 18.1365i | −33.9797 | − | 32.1307i | −190.651 | − | 138.516i | 349.084 | − | 113.424i | 782.976 | − | 426.928i | −252.753 | + | 347.884i | 1660.92 | − | 1206.73i | 122.233 | + | 2183.58i | 6999.55i | ||
2.4 | −5.37068 | + | 16.5293i | −46.7045 | + | 2.38546i | −140.818 | − | 102.310i | −418.675 | + | 136.036i | 211.405 | − | 784.802i | 46.5852 | − | 64.1191i | 647.639 | − | 470.537i | 2175.62 | − | 222.823i | − | 7651.00i | |
2.5 | −5.00631 | + | 15.4078i | 43.8799 | + | 16.1726i | −108.784 | − | 79.0360i | 190.716 | − | 61.9674i | −468.861 | + | 595.129i | −557.986 | + | 768.002i | 84.7229 | − | 61.5548i | 1663.89 | + | 1419.30i | 3248.75i | ||
2.6 | −4.09750 | + | 12.6108i | −17.5014 | + | 43.3670i | −38.6886 | − | 28.1089i | 4.24286 | − | 1.37859i | −475.181 | − | 398.404i | −778.147 | + | 1071.03i | −860.103 | + | 624.902i | −1574.40 | − | 1517.97i | 59.1546i | ||
2.7 | −3.82493 | + | 11.7719i | 1.38201 | − | 46.7449i | −20.3938 | − | 14.8170i | −22.0489 | + | 7.16411i | 544.992 | + | 195.065i | 49.3778 | − | 67.9627i | −1029.34 | + | 747.857i | −2183.18 | − | 129.204i | − | 286.960i | |
2.8 | −2.99455 | + | 9.21628i | 22.8616 | + | 40.7964i | 27.5817 | + | 20.0393i | −419.491 | + | 136.301i | −444.451 | + | 88.5320i | 451.568 | − | 621.530i | −1270.78 | + | 923.277i | −1141.69 | + | 1865.34i | − | 4274.31i | |
2.9 | −2.85991 | + | 8.80190i | 44.0288 | − | 15.7627i | 34.2599 | + | 24.8913i | 118.636 | − | 38.5472i | 12.8232 | + | 432.617i | 768.087 | − | 1057.18i | −1275.45 | + | 926.669i | 1690.07 | − | 1388.03i | 1154.47i | ||
2.10 | −2.74972 | + | 8.46278i | −41.5222 | + | 21.5152i | 39.4965 | + | 28.6959i | 404.353 | − | 131.382i | −67.9040 | − | 410.554i | 811.138 | − | 1116.44i | −1272.91 | + | 924.821i | 1261.19 | − | 1786.72i | 3783.21i | ||
2.11 | −0.997925 | + | 3.07130i | −46.7172 | − | 2.12148i | 95.1172 | + | 69.1067i | −23.9618 | + | 7.78566i | 53.1360 | − | 141.365i | −324.042 | + | 446.006i | −641.580 | + | 466.135i | 2178.00 | + | 198.219i | − | 81.3633i | |
2.12 | −0.334381 | + | 1.02912i | −25.6322 | − | 39.1151i | 102.607 | + | 74.5483i | −221.683 | + | 72.0293i | 48.8250 | − | 13.2993i | 142.147 | − | 195.649i | −223.083 | + | 162.079i | −872.979 | + | 2005.21i | − | 252.224i | |
2.13 | −0.119980 | + | 0.369259i | 44.9099 | − | 13.0422i | 103.432 | + | 75.1479i | −475.452 | + | 154.484i | −0.572341 | + | 18.1482i | −851.641 | + | 1172.18i | −80.3650 | + | 58.3886i | 1846.80 | − | 1171.44i | − | 194.100i | |
2.14 | 0.119980 | − | 0.369259i | 26.2818 | − | 38.6816i | 103.432 | + | 75.1479i | 475.452 | − | 154.484i | −11.1303 | − | 14.3458i | −851.641 | + | 1172.18i | 80.3650 | − | 58.3886i | −805.539 | − | 2033.24i | − | 194.100i | |
2.15 | 0.334381 | − | 1.02912i | 29.2799 | + | 36.4649i | 102.607 | + | 74.5483i | 221.683 | − | 72.0293i | 47.3173 | − | 17.9393i | 142.147 | − | 195.649i | 223.083 | − | 162.079i | −472.379 | + | 2135.38i | − | 252.224i | |
2.16 | 0.997925 | − | 3.07130i | −12.4188 | + | 45.0863i | 95.1172 | + | 69.1067i | 23.9618 | − | 7.78566i | 126.080 | + | 83.1345i | −324.042 | + | 446.006i | 641.580 | − | 466.135i | −1878.55 | − | 1119.83i | − | 81.3633i | |
2.17 | 2.74972 | − | 8.46278i | −33.2933 | + | 32.8414i | 39.4965 | + | 28.6959i | −404.353 | + | 131.382i | 186.382 | + | 372.058i | 811.138 | − | 1116.44i | 1272.91 | − | 924.821i | 29.8835 | − | 2186.80i | 3783.21i | ||
2.18 | 2.85991 | − | 8.80190i | 28.5969 | − | 37.0030i | 34.2599 | + | 24.8913i | −118.636 | + | 38.5472i | −243.912 | − | 357.532i | 768.087 | − | 1057.18i | 1275.45 | − | 926.669i | −551.438 | − | 2116.34i | 1154.47i | ||
2.19 | 2.99455 | − | 9.21628i | −31.7351 | − | 34.3495i | 27.5817 | + | 20.0393i | 419.491 | − | 136.301i | −411.607 | + | 189.618i | 451.568 | − | 621.530i | 1270.78 | − | 923.277i | −172.770 | + | 2180.16i | − | 4274.31i | |
2.20 | 3.82493 | − | 11.7719i | 44.8842 | + | 13.1306i | −20.3938 | − | 14.8170i | 22.0489 | − | 7.16411i | 326.251 | − | 478.149i | 49.3778 | − | 67.9627i | 1029.34 | − | 747.857i | 1842.17 | + | 1178.71i | − | 286.960i | |
See next 80 embeddings (of 104 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
33.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 33.8.f.a | ✓ | 104 |
3.b | odd | 2 | 1 | inner | 33.8.f.a | ✓ | 104 |
11.d | odd | 10 | 1 | inner | 33.8.f.a | ✓ | 104 |
33.f | even | 10 | 1 | inner | 33.8.f.a | ✓ | 104 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.8.f.a | ✓ | 104 | 1.a | even | 1 | 1 | trivial |
33.8.f.a | ✓ | 104 | 3.b | odd | 2 | 1 | inner |
33.8.f.a | ✓ | 104 | 11.d | odd | 10 | 1 | inner |
33.8.f.a | ✓ | 104 | 33.f | even | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(33, [\chi])\).