Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,6,Mod(5,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([11, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.5");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.g (of order \(22\), degree \(10\), 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: | \(11.0664835671\) |
Analytic rank: | \(0\) |
Dimension: | \(380\) |
Relative dimension: | \(38\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −3.01088 | + | 10.2541i | 2.12177 | + | 15.4434i | −69.1616 | − | 44.4475i | −4.96747 | + | 34.5495i | −164.747 | − | 24.7413i | −54.7844 | + | 25.0192i | 405.553 | − | 351.413i | −233.996 | + | 65.5346i | −339.319 | − | 154.962i |
5.2 | −2.97006 | + | 10.1151i | −14.6309 | − | 5.37922i | −66.5741 | − | 42.7846i | 8.40244 | − | 58.4402i | 97.8662 | − | 132.017i | −141.246 | + | 64.5047i | 375.549 | − | 325.415i | 185.128 | + | 157.406i | 566.174 | + | 258.563i |
5.3 | −2.79794 | + | 9.52891i | −3.60982 | − | 15.1647i | −56.0515 | − | 36.0221i | −6.21680 | + | 43.2388i | 154.603 | + | 8.03237i | 210.991 | − | 96.3564i | 259.904 | − | 225.208i | −216.938 | + | 109.484i | −394.624 | − | 180.219i |
5.4 | −2.60390 | + | 8.86808i | 14.0479 | + | 6.75686i | −44.9425 | − | 28.8828i | 14.8404 | − | 103.218i | −96.4999 | + | 106.984i | 66.5201 | − | 30.3787i | 149.641 | − | 129.665i | 151.690 | + | 189.840i | 876.699 | + | 400.375i |
5.5 | −2.59429 | + | 8.83533i | 15.4604 | − | 1.99434i | −44.4125 | − | 28.5422i | −6.70991 | + | 46.6684i | −22.4879 | + | 141.771i | −7.07279 | + | 3.23003i | 144.705 | − | 125.387i | 235.045 | − | 61.6665i | −394.923 | − | 180.355i |
5.6 | −2.54608 | + | 8.67115i | 8.15133 | − | 13.2874i | −41.7861 | − | 26.8543i | 0.168773 | − | 1.17384i | 94.4634 | + | 104.512i | −160.739 | + | 73.4069i | 120.693 | − | 104.581i | −110.112 | − | 216.620i | 9.74886 | + | 4.45216i |
5.7 | −2.40375 | + | 8.18642i | −12.7578 | + | 8.95766i | −34.3194 | − | 22.0557i | 4.22700 | − | 29.3994i | −42.6647 | − | 125.972i | 170.161 | − | 77.7098i | 56.7143 | − | 49.1432i | 82.5208 | − | 228.559i | 230.515 | + | 105.273i |
5.8 | −1.97556 | + | 6.72815i | −11.2628 | + | 10.7772i | −14.4450 | − | 9.28323i | −7.37109 | + | 51.2671i | −50.2603 | − | 97.0690i | −115.006 | + | 52.5213i | −78.5868 | + | 68.0959i | 10.7027 | − | 242.764i | −330.370 | − | 150.875i |
5.9 | −1.91071 | + | 6.50727i | −13.3402 | − | 8.06476i | −11.7737 | − | 7.56648i | −10.6354 | + | 73.9705i | 77.9687 | − | 71.3986i | −54.2822 | + | 24.7899i | −92.2823 | + | 79.9631i | 112.919 | + | 215.170i | −461.025 | − | 210.543i |
5.10 | −1.75424 | + | 5.97440i | −0.0984677 | − | 15.5881i | −5.69594 | − | 3.66056i | 11.1842 | − | 77.7880i | 93.3025 | + | 26.7571i | −8.66991 | + | 3.95942i | −118.723 | + | 102.874i | −242.981 | + | 3.06986i | 445.117 | + | 203.278i |
5.11 | −1.54386 | + | 5.25792i | 2.93920 | + | 15.3089i | 1.65792 | + | 1.06548i | 4.93387 | − | 34.3158i | −85.0305 | − | 8.18070i | −31.8114 | + | 14.5278i | −140.687 | + | 121.906i | −225.722 | + | 89.9917i | 172.813 | + | 78.9209i |
5.12 | −1.50479 | + | 5.12484i | 10.2377 | + | 11.7554i | 2.92056 | + | 1.87693i | −14.4825 | + | 100.728i | −75.6499 | + | 34.7774i | 170.076 | − | 77.6713i | −143.185 | + | 124.071i | −33.3771 | + | 240.697i | −494.421 | − | 225.795i |
5.13 | −1.29764 | + | 4.41935i | 13.8639 | − | 7.12680i | 9.07333 | + | 5.83108i | 3.05854 | − | 21.2726i | 13.5055 | + | 70.5176i | 139.466 | − | 63.6921i | −148.933 | + | 129.051i | 141.417 | − | 197.611i | 90.0424 | + | 41.1210i |
5.14 | −1.13703 | + | 3.87238i | −14.7589 | − | 5.01748i | 13.2177 | + | 8.49447i | 6.88235 | − | 47.8678i | 36.2109 | − | 51.4470i | −29.0361 | + | 13.2603i | −145.526 | + | 126.099i | 192.650 | + | 148.105i | 177.537 | + | 81.0783i |
5.15 | −0.903285 | + | 3.07631i | 14.5683 | + | 5.54656i | 18.2724 | + | 11.7429i | 0.440615 | − | 3.06454i | −30.2222 | + | 39.8065i | −221.259 | + | 101.046i | −130.168 | + | 112.791i | 181.471 | + | 161.608i | 9.02947 | + | 4.12362i |
5.16 | −0.617954 | + | 2.10456i | 7.38321 | − | 13.7291i | 22.8728 | + | 14.6995i | −12.1594 | + | 84.5703i | 24.3312 | + | 24.0223i | −68.9240 | + | 31.4765i | −98.1155 | + | 85.0176i | −133.976 | − | 202.730i | −170.469 | − | 77.8506i |
5.17 | −0.327846 | + | 1.11654i | −6.80137 | − | 14.0265i | 25.7809 | + | 16.5684i | −0.383188 | + | 2.66513i | 17.8909 | − | 2.99549i | 83.3649 | − | 38.0715i | −55.0938 | + | 47.7391i | −150.483 | + | 190.798i | −2.85010 | − | 1.30160i |
5.18 | −0.235789 | + | 0.803025i | −13.0330 | + | 8.55220i | 26.3309 | + | 16.9218i | 11.5441 | − | 80.2911i | −3.79458 | − | 12.4824i | −106.024 | + | 48.4193i | −40.0374 | + | 34.6926i | 96.7196 | − | 222.922i | 61.7537 | + | 28.2020i |
5.19 | −0.190214 | + | 0.647810i | −15.2666 | + | 3.15135i | 26.5366 | + | 17.0541i | −6.91235 | + | 48.0764i | 0.862446 | − | 10.4893i | 156.619 | − | 71.5256i | −32.4235 | + | 28.0951i | 223.138 | − | 96.2208i | −29.8296 | − | 13.6227i |
5.20 | 0.190214 | − | 0.647810i | −0.946612 | + | 15.5597i | 26.5366 | + | 17.0541i | 6.91235 | − | 48.0764i | 9.89966 | + | 3.57290i | 156.619 | − | 71.5256i | 32.4235 | − | 28.0951i | −241.208 | − | 29.4580i | −29.8296 | − | 13.6227i |
See next 80 embeddings (of 380 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
23.d | odd | 22 | 1 | inner |
69.g | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.6.g.a | ✓ | 380 |
3.b | odd | 2 | 1 | inner | 69.6.g.a | ✓ | 380 |
23.d | odd | 22 | 1 | inner | 69.6.g.a | ✓ | 380 |
69.g | even | 22 | 1 | inner | 69.6.g.a | ✓ | 380 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.6.g.a | ✓ | 380 | 1.a | even | 1 | 1 | trivial |
69.6.g.a | ✓ | 380 | 3.b | odd | 2 | 1 | inner |
69.6.g.a | ✓ | 380 | 23.d | odd | 22 | 1 | inner |
69.6.g.a | ✓ | 380 | 69.g | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(69, [\chi])\).