Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,8,Mod(11,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.k (of order \(4\), degree \(2\), 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: | \(14.9944812232\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(54\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −11.3125 | + | 0.167474i | 4.08471 | + | 46.5866i | 127.944 | − | 3.78909i | −280.037 | + | 280.037i | −54.0103 | − | 526.326i | −532.864 | −1446.73 | + | 64.2912i | −2153.63 | + | 380.586i | 3121.01 | − | 3214.81i | ||
11.2 | −11.1336 | + | 2.01052i | 1.90114 | + | 46.7267i | 119.916 | − | 44.7688i | 277.365 | − | 277.365i | −115.112 | − | 516.416i | 418.647 | −1245.09 | + | 739.532i | −2179.77 | + | 177.668i | −2530.44 | + | 3645.73i | ||
11.3 | −11.0402 | − | 2.47286i | −21.9431 | − | 41.2977i | 115.770 | + | 54.6016i | 47.3788 | − | 47.3788i | 140.132 | + | 510.195i | −321.346 | −1143.10 | − | 889.093i | −1224.00 | + | 1812.40i | −640.231 | + | 405.908i | ||
11.4 | −10.9905 | − | 2.68503i | 42.7767 | − | 18.8985i | 113.581 | + | 59.0195i | 312.645 | − | 312.645i | −520.879 | + | 92.8472i | 1303.49 | −1089.84 | − | 953.621i | 1472.69 | − | 1616.83i | −4275.58 | + | 2596.66i | ||
11.5 | −10.8871 | + | 3.07743i | 22.4120 | − | 41.0451i | 109.059 | − | 67.0087i | −78.5897 | + | 78.5897i | −117.689 | + | 515.835i | −861.367 | −981.123 | + | 1065.15i | −1182.40 | − | 1839.81i | 613.761 | − | 1097.47i | ||
11.6 | −10.7935 | − | 3.39113i | −42.3573 | + | 19.8208i | 105.000 | + | 73.2046i | −65.9576 | + | 65.9576i | 524.399 | − | 70.2967i | 652.374 | −885.079 | − | 1146.21i | 1401.28 | − | 1679.11i | 935.586 | − | 488.244i | ||
11.7 | −10.5420 | + | 4.10672i | −46.5860 | − | 4.09257i | 94.2696 | − | 86.5866i | 254.666 | − | 254.666i | 507.918 | − | 148.172i | −665.648 | −638.208 | + | 1299.94i | 2153.50 | + | 381.312i | −1638.86 | + | 3730.54i | ||
11.8 | −10.3588 | + | 4.54910i | 45.2039 | + | 11.9836i | 86.6113 | − | 94.2469i | −93.8822 | + | 93.8822i | −522.775 | + | 81.5006i | 809.729 | −468.455 | + | 1370.29i | 1899.79 | + | 1083.41i | 545.432 | − | 1399.59i | ||
11.9 | −10.3435 | − | 4.58397i | 45.5791 | + | 10.4665i | 85.9744 | + | 94.8283i | 25.0262 | − | 25.0262i | −423.468 | − | 317.193i | −1569.84 | −454.583 | − | 1374.96i | 1967.91 | + | 954.104i | −373.577 | + | 144.138i | ||
11.10 | −9.75422 | + | 5.73194i | −37.4127 | − | 28.0587i | 62.2898 | − | 111.821i | −287.248 | + | 287.248i | 525.763 | + | 59.2434i | 1269.27 | 33.3635 | + | 1447.77i | 612.421 | + | 2099.50i | 1155.40 | − | 4448.37i | ||
11.11 | −9.27322 | − | 6.48131i | 27.2746 | − | 37.9881i | 43.9852 | + | 120.205i | −379.549 | + | 379.549i | −499.136 | + | 175.497i | 864.981 | 371.203 | − | 1399.77i | −699.194 | − | 2072.22i | 5979.61 | − | 1059.67i | ||
11.12 | −8.00843 | + | 7.99156i | −34.6593 | + | 31.3963i | 0.269890 | − | 128.000i | −153.366 | + | 153.366i | 26.6610 | − | 528.418i | −1061.45 | 1020.76 | + | 1027.23i | 215.539 | − | 2176.35i | 2.58700 | − | 2453.85i | ||
11.13 | −7.71273 | − | 8.27730i | −22.7540 | + | 40.8565i | −9.02745 | + | 127.681i | 277.521 | − | 277.521i | 513.677 | − | 126.773i | −1218.23 | 1126.48 | − | 910.049i | −1151.51 | − | 1859.30i | −4437.56 | − | 156.679i | ||
11.14 | −7.70062 | − | 8.28857i | 25.6221 | + | 39.1217i | −9.40086 | + | 127.654i | −45.8713 | + | 45.8713i | 126.957 | − | 513.632i | 943.511 | 1130.46 | − | 905.098i | −874.018 | + | 2004.76i | 733.445 | + | 26.9701i | ||
11.15 | −7.46649 | + | 8.50009i | 0.872512 | − | 46.7572i | −16.5031 | − | 126.932i | 294.772 | − | 294.772i | 390.926 | + | 356.529i | 694.446 | 1202.15 | + | 807.456i | −2185.48 | − | 81.5925i | 304.678 | + | 4706.51i | ||
11.16 | −7.22422 | − | 8.70693i | −28.4787 | − | 37.0940i | −23.6212 | + | 125.802i | 118.895 | − | 118.895i | −117.238 | + | 515.937i | 553.079 | 1265.99 | − | 703.151i | −564.924 | + | 2112.78i | −1894.14 | − | 176.286i | ||
11.17 | −6.86971 | + | 8.98928i | 34.0729 | + | 32.0318i | −33.6142 | − | 123.507i | 201.123 | − | 201.123i | −522.014 | + | 86.2416i | −1025.82 | 1341.16 | + | 546.292i | 134.926 | + | 2182.83i | 426.294 | + | 3189.61i | ||
11.18 | −6.30926 | − | 9.39113i | −45.6917 | − | 9.96321i | −48.3865 | + | 118.502i | −316.866 | + | 316.866i | 194.715 | + | 491.957i | −1511.21 | 1418.15 | − | 293.257i | 1988.47 | + | 910.472i | 4974.91 | + | 976.537i | ||
11.19 | −5.66532 | + | 9.79307i | −17.1135 | + | 43.5216i | −63.8083 | − | 110.962i | 27.9730 | − | 27.9730i | −329.256 | − | 414.157i | 1719.05 | 1448.15 | + | 3.75424i | −1601.26 | − | 1489.61i | 115.466 | + | 432.418i | ||
11.20 | −5.44512 | + | 9.91719i | 40.1781 | − | 23.9317i | −68.7013 | − | 108.001i | −101.752 | + | 101.752i | 18.5603 | + | 528.764i | −33.7333 | 1445.15 | − | 93.2472i | 1041.55 | − | 1923.05i | −455.042 | − | 1563.15i | ||
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 48.8.k.a | ✓ | 108 |
3.b | odd | 2 | 1 | inner | 48.8.k.a | ✓ | 108 |
16.f | odd | 4 | 1 | inner | 48.8.k.a | ✓ | 108 |
48.k | even | 4 | 1 | inner | 48.8.k.a | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.8.k.a | ✓ | 108 | 1.a | even | 1 | 1 | trivial |
48.8.k.a | ✓ | 108 | 3.b | odd | 2 | 1 | inner |
48.8.k.a | ✓ | 108 | 16.f | odd | 4 | 1 | inner |
48.8.k.a | ✓ | 108 | 48.k | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(48, [\chi])\).