Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,7,Mod(7,69)]
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([0, 19]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.7");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.f (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: | \(15.8737317698\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −10.0393 | + | 11.5859i | −13.1138 | + | 8.42776i | −24.3387 | − | 169.280i | −28.9992 | + | 13.2435i | 34.0100 | − | 236.544i | −177.630 | − | 604.953i | 1380.21 | + | 887.009i | 100.946 | − | 221.041i | 137.692 | − | 468.937i |
7.2 | −9.70976 | + | 11.2057i | 13.1138 | − | 8.42776i | −22.1792 | − | 154.260i | 28.9455 | − | 13.2190i | −32.8937 | + | 228.781i | 60.2734 | + | 205.272i | 1145.64 | + | 736.258i | 100.946 | − | 221.041i | −132.927 | + | 452.706i |
7.3 | −7.55085 | + | 8.71415i | −13.1138 | + | 8.42776i | −9.81285 | − | 68.2498i | −14.6117 | + | 6.67294i | 25.5800 | − | 177.913i | 9.29392 | + | 31.6522i | 48.0315 | + | 30.8680i | 100.946 | − | 221.041i | 52.1818 | − | 177.715i |
7.4 | −7.46772 | + | 8.61821i | 13.1138 | − | 8.42776i | −9.39853 | − | 65.3682i | −12.8791 | + | 5.88169i | −25.2984 | + | 175.954i | −31.2375 | − | 106.385i | 19.5745 | + | 12.5798i | 100.946 | − | 221.041i | 45.4879 | − | 154.918i |
7.5 | −7.37873 | + | 8.51551i | −13.1138 | + | 8.42776i | −8.96008 | − | 62.3187i | 186.661 | − | 85.2451i | 24.9969 | − | 173.857i | 65.3761 | + | 222.650i | −9.86246 | − | 6.33822i | 100.946 | − | 221.041i | −651.414 | + | 2218.51i |
7.6 | −5.37670 | + | 6.20505i | −13.1138 | + | 8.42776i | −0.485518 | − | 3.37685i | −45.2350 | + | 20.6582i | 18.2146 | − | 126.686i | −13.8057 | − | 47.0180i | −418.489 | − | 268.946i | 100.946 | − | 221.041i | 115.031 | − | 391.758i |
7.7 | −4.49904 | + | 5.19217i | 13.1138 | − | 8.42776i | 2.39089 | + | 16.6290i | 160.721 | − | 73.3986i | −15.2414 | + | 106.006i | −169.038 | − | 575.691i | −466.992 | − | 300.117i | 100.946 | − | 221.041i | −341.990 | + | 1164.71i |
7.8 | −4.02561 | + | 4.64580i | 13.1138 | − | 8.42776i | 3.73022 | + | 25.9443i | −145.258 | + | 66.3370i | −13.6375 | + | 94.8511i | 94.3193 | + | 321.222i | −466.519 | − | 299.813i | 100.946 | − | 221.041i | 276.562 | − | 941.884i |
7.9 | −3.87564 | + | 4.47273i | 13.1138 | − | 8.42776i | 4.12344 | + | 28.6791i | 175.461 | − | 80.1304i | −13.1295 | + | 91.3177i | 148.103 | + | 504.394i | −462.896 | − | 297.485i | 100.946 | − | 221.041i | −321.623 | + | 1095.35i |
7.10 | −2.99454 | + | 3.45588i | −13.1138 | + | 8.42776i | 6.13230 | + | 42.6511i | −203.315 | + | 92.8510i | 10.1446 | − | 70.5571i | −106.030 | − | 361.104i | −411.960 | − | 264.751i | 100.946 | − | 221.041i | 287.953 | − | 980.679i |
7.11 | −0.826670 | + | 0.954028i | −13.1138 | + | 8.42776i | 8.88136 | + | 61.7712i | 46.0457 | − | 21.0284i | 2.80051 | − | 19.4779i | 134.637 | + | 458.531i | −134.239 | − | 86.2703i | 100.946 | − | 221.041i | −18.0029 | + | 61.3124i |
7.12 | 0.0241383 | − | 0.0278571i | 13.1138 | − | 8.42776i | 9.10796 | + | 63.3472i | −89.8556 | + | 41.0357i | 0.0817732 | − | 0.568745i | −160.665 | − | 547.174i | 3.96908 | + | 2.55077i | 100.946 | − | 221.041i | −1.02583 | + | 3.49365i |
7.13 | 0.203656 | − | 0.235032i | −13.1138 | + | 8.42776i | 9.09439 | + | 63.2528i | 171.608 | − | 78.3705i | −0.689926 | + | 4.79854i | −84.9865 | − | 289.438i | 33.4624 | + | 21.5050i | 100.946 | − | 221.041i | 16.5294 | − | 56.2939i |
7.14 | 0.439258 | − | 0.506931i | −13.1138 | + | 8.42776i | 9.04412 | + | 62.9032i | −167.785 | + | 76.6249i | −1.48807 | + | 10.3498i | 132.687 | + | 451.891i | 71.9745 | + | 46.2552i | 100.946 | − | 221.041i | −34.8575 | + | 118.714i |
7.15 | 0.991787 | − | 1.14458i | 13.1138 | − | 8.42776i | 8.78172 | + | 61.0782i | −45.4301 | + | 20.7472i | 3.35987 | − | 23.3684i | 28.4560 | + | 96.9124i | 160.160 | + | 102.928i | 100.946 | − | 221.041i | −21.3100 | + | 72.5754i |
7.16 | 3.76160 | − | 4.34111i | 13.1138 | − | 8.42776i | 4.41249 | + | 30.6896i | 172.538 | − | 78.7957i | 12.7431 | − | 88.6305i | 43.8846 | + | 149.457i | 459.089 | + | 295.039i | 100.946 | − | 221.041i | 306.959 | − | 1045.41i |
7.17 | 4.04842 | − | 4.67213i | −13.1138 | + | 8.42776i | 3.66909 | + | 25.5191i | −6.23883 | + | 2.84918i | −13.7148 | + | 95.3887i | −83.3001 | − | 283.694i | 466.929 | + | 300.077i | 100.946 | − | 221.041i | −11.9457 | + | 40.6833i |
7.18 | 5.80964 | − | 6.70469i | 13.1138 | − | 8.42776i | −2.09271 | − | 14.5551i | −138.264 | + | 63.1430i | 19.6813 | − | 136.886i | 130.352 | + | 443.939i | 367.902 | + | 236.436i | 100.946 | − | 221.041i | −379.910 | + | 1293.85i |
7.19 | 6.08888 | − | 7.02694i | −13.1138 | + | 8.42776i | −3.19530 | − | 22.2238i | −77.8070 | + | 35.5333i | −20.6273 | + | 143.466i | −42.4272 | − | 144.494i | 324.984 | + | 208.854i | 100.946 | − | 221.041i | −224.067 | + | 763.103i |
7.20 | 6.34640 | − | 7.32413i | 13.1138 | − | 8.42776i | −4.25801 | − | 29.6151i | 79.9255 | − | 36.5007i | 21.4997 | − | 149.533i | −34.3232 | − | 116.894i | 277.849 | + | 178.563i | 100.946 | − | 221.041i | 239.903 | − | 817.033i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.d | odd | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.7.f.a | ✓ | 240 |
23.d | odd | 22 | 1 | inner | 69.7.f.a | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.7.f.a | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
69.7.f.a | ✓ | 240 | 23.d | odd | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).