Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,5,Mod(3,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.3");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 98.h (of order \(42\), degree \(12\), 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: | \(10.1302563822\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −1.03334 | + | 2.63291i | −13.7798 | − | 1.03265i | −5.86441 | − | 5.44138i | 4.87396 | − | 7.14878i | 16.9581 | − | 35.2139i | −23.3515 | − | 43.0779i | 20.3866 | − | 9.81767i | 108.722 | + | 16.3872i | 13.7856 | + | 20.2198i |
3.2 | −1.03334 | + | 2.63291i | −9.18673 | − | 0.688451i | −5.86441 | − | 5.44138i | −15.1525 | + | 22.2246i | 11.3057 | − | 23.4764i | −28.8684 | + | 39.5931i | 20.3866 | − | 9.81767i | 3.82683 | + | 0.576802i | −42.8576 | − | 62.8606i |
3.3 | −1.03334 | + | 2.63291i | −9.09005 | − | 0.681205i | −5.86441 | − | 5.44138i | 10.9258 | − | 16.0251i | 11.1867 | − | 23.2294i | 40.8166 | + | 27.1109i | 20.3866 | − | 9.81767i | 2.06973 | + | 0.311961i | 30.9027 | + | 45.3259i |
3.4 | −1.03334 | + | 2.63291i | −0.341292 | − | 0.0255763i | −5.86441 | − | 5.44138i | −7.38538 | + | 10.8324i | 0.420011 | − | 0.872161i | 15.9874 | − | 46.3185i | 20.3866 | − | 9.81767i | −79.9795 | − | 12.0550i | −20.8890 | − | 30.6385i |
3.5 | −1.03334 | + | 2.63291i | 2.64055 | + | 0.197882i | −5.86441 | − | 5.44138i | 23.2214 | − | 34.0595i | −3.24960 | + | 6.74786i | −48.1538 | + | 9.06681i | 20.3866 | − | 9.81767i | −73.1619 | − | 11.0274i | 65.6800 | + | 96.3349i |
3.6 | −1.03334 | + | 2.63291i | 6.56959 | + | 0.492323i | −5.86441 | − | 5.44138i | −19.9016 | + | 29.1903i | −8.08486 | + | 16.7884i | 46.9335 | + | 14.0801i | 20.3866 | − | 9.81767i | −37.1782 | − | 5.60371i | −56.2903 | − | 82.5627i |
3.7 | −1.03334 | + | 2.63291i | 8.86092 | + | 0.664034i | −5.86441 | − | 5.44138i | 13.8478 | − | 20.3110i | −10.9047 | + | 22.6438i | 2.81979 | + | 48.9188i | 20.3866 | − | 9.81767i | −2.02032 | − | 0.304515i | 39.1675 | + | 57.4481i |
3.8 | −1.03334 | + | 2.63291i | 9.41284 | + | 0.705395i | −5.86441 | − | 5.44138i | −10.9686 | + | 16.0880i | −11.5839 | + | 24.0542i | −48.4695 | − | 7.19094i | 20.3866 | − | 9.81767i | 8.00874 | + | 1.20712i | −31.0238 | − | 45.5036i |
3.9 | −1.03334 | + | 2.63291i | 16.4510 | + | 1.23283i | −5.86441 | − | 5.44138i | 8.38583 | − | 12.2998i | −20.2455 | + | 42.0401i | 6.27978 | − | 48.5959i | 20.3866 | − | 9.81767i | 189.021 | + | 28.4904i | 23.7187 | + | 34.7890i |
3.10 | 1.03334 | − | 2.63291i | −14.1719 | − | 1.06204i | −5.86441 | − | 5.44138i | −9.39032 | + | 13.7731i | −17.4407 | + | 36.2160i | −46.9328 | − | 14.0822i | −20.3866 | + | 9.81767i | 119.621 | + | 18.0299i | 26.5598 | + | 38.9561i |
3.11 | 1.03334 | − | 2.63291i | −13.2432 | − | 0.992441i | −5.86441 | − | 5.44138i | 25.8416 | − | 37.9027i | −16.2977 | + | 33.8426i | 23.3218 | − | 43.0940i | −20.3866 | + | 9.81767i | 94.3023 | + | 14.2138i | −73.0912 | − | 107.205i |
3.12 | 1.03334 | − | 2.63291i | −11.5689 | − | 0.866972i | −5.86441 | − | 5.44138i | −9.61510 | + | 14.1028i | −14.2373 | + | 29.5641i | 43.1303 | + | 23.2546i | −20.3866 | + | 9.81767i | 52.9935 | + | 7.98749i | 27.1956 | + | 39.8886i |
3.13 | 1.03334 | − | 2.63291i | −3.49449 | − | 0.261876i | −5.86441 | − | 5.44138i | 9.11024 | − | 13.3623i | −4.30049 | + | 8.93006i | −34.1673 | + | 35.1226i | −20.3866 | + | 9.81767i | −67.9524 | − | 10.2422i | −25.7676 | − | 37.7942i |
3.14 | 1.03334 | − | 2.63291i | −1.48521 | − | 0.111301i | −5.86441 | − | 5.44138i | −10.4504 | + | 15.3280i | −1.82778 | + | 3.79542i | 33.1945 | − | 36.0434i | −20.3866 | + | 9.81767i | −77.9018 | − | 11.7418i | 29.5583 | + | 43.3540i |
3.15 | 1.03334 | − | 2.63291i | 7.10780 | + | 0.532656i | −5.86441 | − | 5.44138i | −17.6874 | + | 25.9427i | 8.74721 | − | 18.1638i | −35.5979 | + | 33.6717i | −20.3866 | + | 9.81767i | −29.8583 | − | 4.50041i | 50.0276 | + | 73.3770i |
3.16 | 1.03334 | − | 2.63291i | 8.32236 | + | 0.623675i | −5.86441 | − | 5.44138i | 9.25164 | − | 13.5697i | 10.2419 | − | 21.2676i | −16.2717 | − | 46.2194i | −20.3866 | + | 9.81767i | −11.2225 | − | 1.69152i | −26.1676 | − | 38.3808i |
3.17 | 1.03334 | − | 2.63291i | 12.2032 | + | 0.914503i | −5.86441 | − | 5.44138i | −15.7636 | + | 23.1209i | 15.0179 | − | 31.1849i | 45.1496 | + | 19.0398i | −20.3866 | + | 9.81767i | 67.9864 | + | 10.2473i | 44.5861 | + | 65.3959i |
3.18 | 1.03334 | − | 2.63291i | 15.1566 | + | 1.13583i | −5.86441 | − | 5.44138i | 24.6502 | − | 36.1553i | 18.6525 | − | 38.7324i | 32.5086 | + | 36.6632i | −20.3866 | + | 9.81767i | 148.338 | + | 22.3584i | −69.7214 | − | 102.263i |
5.1 | −2.79684 | + | 0.421555i | −9.54857 | + | 14.0052i | 7.64458 | − | 2.35804i | −30.6644 | − | 2.29798i | 20.8018 | − | 43.1954i | −42.9807 | + | 23.5299i | −20.3866 | + | 9.81767i | −75.3773 | − | 192.058i | 86.7321 | − | 6.49967i |
5.2 | −2.79684 | + | 0.421555i | −7.75332 | + | 11.3720i | 7.64458 | − | 2.35804i | 20.2326 | + | 1.51622i | 16.8908 | − | 35.0742i | 21.8955 | − | 43.8359i | −20.3866 | + | 9.81767i | −39.6166 | − | 100.941i | −57.2263 | + | 4.28852i |
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
49.h | odd | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 98.5.h.a | ✓ | 216 |
49.h | odd | 42 | 1 | inner | 98.5.h.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
98.5.h.a | ✓ | 216 | 1.a | even | 1 | 1 | trivial |
98.5.h.a | ✓ | 216 | 49.h | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(98, [\chi])\).