Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [87,6,Mod(7,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("87.7");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 87.g (of order \(7\), degree \(6\), 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: | \(13.9533923237\) |
Analytic rank: | \(0\) |
Dimension: | \(78\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | −2.26086 | − | 9.90549i | 8.10872 | + | 3.90495i | −64.1762 | + | 30.9056i | 3.70316 | + | 16.2246i | 20.3478 | − | 89.1494i | 99.3933 | + | 47.8653i | 248.515 | + | 311.628i | 50.5027 | + | 63.3284i | 152.340 | − | 73.3632i |
7.2 | −1.93840 | − | 8.49270i | 8.10872 | + | 3.90495i | −39.5375 | + | 19.0403i | −7.82896 | − | 34.3009i | 17.4456 | − | 76.4343i | −90.1162 | − | 43.3977i | 64.5416 | + | 80.9326i | 50.5027 | + | 63.3284i | −276.131 | + | 132.978i |
7.3 | −1.60675 | − | 7.03963i | 8.10872 | + | 3.90495i | −18.1438 | + | 8.73759i | 13.2265 | + | 57.9493i | 14.4608 | − | 63.3567i | −17.0594 | − | 8.21538i | −53.4025 | − | 66.9646i | 50.5027 | + | 63.3284i | 386.690 | − | 186.220i |
7.4 | −0.991659 | − | 4.34474i | 8.10872 | + | 3.90495i | 10.9376 | − | 5.26727i | 11.7683 | + | 51.5605i | 8.92493 | − | 39.1027i | −152.983 | − | 73.6729i | −122.645 | − | 153.793i | 50.5027 | + | 63.3284i | 212.347 | − | 102.261i |
7.5 | −0.852092 | − | 3.73326i | 8.10872 | + | 3.90495i | 15.6198 | − | 7.52212i | −11.0422 | − | 48.3789i | 7.66883 | − | 33.5993i | 218.824 | + | 105.380i | −117.792 | − | 147.706i | 50.5027 | + | 63.3284i | −171.202 | + | 82.4466i |
7.6 | −0.682925 | − | 2.99209i | 8.10872 | + | 3.90495i | 20.3448 | − | 9.79754i | −17.6030 | − | 77.1239i | 6.14632 | − | 26.9288i | −99.2164 | − | 47.7801i | −104.441 | − | 130.965i | 50.5027 | + | 63.3284i | −218.740 | + | 105.340i |
7.7 | 0.0330315 | + | 0.144721i | 8.10872 | + | 3.90495i | 28.8112 | − | 13.8747i | 15.6670 | + | 68.6416i | −0.297284 | + | 1.30249i | 67.3517 | + | 32.4349i | 5.92131 | + | 7.42508i | 50.5027 | + | 63.3284i | −9.41635 | + | 4.53467i |
7.8 | 0.621155 | + | 2.72146i | 8.10872 | + | 3.90495i | 21.8105 | − | 10.5034i | −4.60003 | − | 20.1541i | −5.59040 | + | 24.4931i | 28.2944 | + | 13.6259i | 97.8263 | + | 122.670i | 50.5027 | + | 63.3284i | 51.9911 | − | 25.0376i |
7.9 | 1.38807 | + | 6.08151i | 8.10872 | + | 3.90495i | −6.22706 | + | 2.99879i | 14.1607 | + | 62.0419i | −12.4926 | + | 54.7336i | −214.780 | − | 103.433i | 97.5760 | + | 122.356i | 50.5027 | + | 63.3284i | −357.653 | + | 172.236i |
7.10 | 1.72031 | + | 7.53718i | 8.10872 | + | 3.90495i | −25.0187 | + | 12.0484i | −9.36503 | − | 41.0309i | −15.4828 | + | 67.8347i | 139.550 | + | 67.2038i | 20.3961 | + | 25.5759i | 50.5027 | + | 63.3284i | 293.146 | − | 141.172i |
7.11 | 1.73808 | + | 7.61502i | 8.10872 | + | 3.90495i | −26.1367 | + | 12.5868i | −21.6674 | − | 94.9309i | −15.6427 | + | 68.5352i | −141.588 | − | 68.1854i | 14.5636 | + | 18.2622i | 50.5027 | + | 63.3284i | 685.241 | − | 329.995i |
7.12 | 1.77168 | + | 7.76224i | 8.10872 | + | 3.90495i | −28.2825 | + | 13.6201i | 19.5439 | + | 85.6275i | −15.9451 | + | 69.8601i | 133.220 | + | 64.1553i | 3.02217 | + | 3.78968i | 50.5027 | + | 63.3284i | −630.035 | + | 303.409i |
7.13 | 2.43957 | + | 10.6884i | 8.10872 | + | 3.90495i | −79.4602 | + | 38.2660i | 1.75680 | + | 7.69705i | −21.9561 | + | 96.1959i | −66.8029 | − | 32.1706i | −384.116 | − | 481.666i | 50.5027 | + | 63.3284i | −77.9836 | + | 37.5549i |
16.1 | −8.99529 | − | 4.33190i | −5.61141 | + | 7.03648i | 42.1982 | + | 52.9148i | 44.5360 | + | 21.4474i | 80.9576 | − | 38.9871i | 87.7619 | − | 110.050i | −79.2699 | − | 347.304i | −18.0242 | − | 78.9692i | −307.706 | − | 385.851i |
16.2 | −7.92947 | − | 3.81863i | −5.61141 | + | 7.03648i | 28.3429 | + | 35.5409i | −91.4078 | − | 44.0197i | 71.3652 | − | 34.3677i | 111.730 | − | 140.105i | −26.3573 | − | 115.479i | −18.0242 | − | 78.9692i | 556.721 | + | 698.106i |
16.3 | −6.29108 | − | 3.02962i | −5.61141 | + | 7.03648i | 10.4473 | + | 13.1006i | 74.0869 | + | 35.6784i | 56.6197 | − | 27.2666i | −44.0006 | + | 55.1750i | 23.6853 | + | 103.772i | −18.0242 | − | 78.9692i | −357.994 | − | 448.911i |
16.4 | −5.63420 | − | 2.71329i | −5.61141 | + | 7.03648i | 4.43057 | + | 5.55576i | −10.4355 | − | 5.02547i | 50.7078 | − | 24.4196i | −39.0334 | + | 48.9464i | 34.6407 | + | 151.771i | −18.0242 | − | 78.9692i | 45.1601 | + | 56.6289i |
16.5 | −1.55088 | − | 0.746863i | −5.61141 | + | 7.03648i | −18.1043 | − | 22.7020i | −73.4400 | − | 35.3668i | 13.9579 | − | 6.72176i | −101.424 | + | 127.182i | 23.3793 | + | 102.431i | −18.0242 | − | 78.9692i | 87.4822 | + | 109.699i |
16.6 | −0.807834 | − | 0.389033i | −5.61141 | + | 7.03648i | −19.4504 | − | 24.3901i | 43.7893 | + | 21.0878i | 7.27051 | − | 3.50129i | 16.5251 | − | 20.7218i | 12.6088 | + | 55.2427i | −18.0242 | − | 78.9692i | −27.1707 | − | 34.0709i |
16.7 | −0.507638 | − | 0.244466i | −5.61141 | + | 7.03648i | −19.7537 | − | 24.7704i | −38.1099 | − | 18.3528i | 4.56874 | − | 2.20019i | 66.6920 | − | 83.6292i | 7.98428 | + | 34.9814i | −18.0242 | − | 78.9692i | 14.8594 | + | 18.6331i |
See all 78 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.d | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 87.6.g.b | ✓ | 78 |
29.d | even | 7 | 1 | inner | 87.6.g.b | ✓ | 78 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
87.6.g.b | ✓ | 78 | 1.a | even | 1 | 1 | trivial |
87.6.g.b | ✓ | 78 | 29.d | even | 7 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{78} - 10 T_{2}^{77} + 359 T_{2}^{76} - 3561 T_{2}^{75} + 81951 T_{2}^{74} - 789996 T_{2}^{73} + 14705941 T_{2}^{72} - 126989374 T_{2}^{71} + 2167074599 T_{2}^{70} - 16945371146 T_{2}^{69} + \cdots + 30\!\cdots\!44 \)
acting on \(S_{6}^{\mathrm{new}}(87, [\chi])\).