Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [71,9,Mod(14,71)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(71, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("71.14");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 71 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 71.e (of order \(10\), degree \(4\), 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: | \(28.9238813143\) |
Analytic rank: | \(0\) |
Dimension: | \(188\) |
Relative dimension: | \(47\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 | −9.55562 | + | 29.4092i | −36.4701 | + | 112.243i | −566.481 | − | 411.572i | −865.123 | + | 628.549i | −2952.49 | − | 2145.11i | 1604.44 | + | 521.316i | 11112.7 | − | 8073.87i | −5960.56 | − | 4330.60i | −10218.3 | − | 31448.7i |
14.2 | −9.48670 | + | 29.1971i | −18.9267 | + | 58.2505i | −555.363 | − | 403.495i | 474.158 | − | 344.496i | −1521.19 | − | 1105.21i | 399.230 | + | 129.718i | 10691.3 | − | 7767.66i | 2273.07 | + | 1651.48i | 5560.07 | + | 17112.2i |
14.3 | −9.23578 | + | 28.4248i | 34.7597 | − | 106.979i | −515.562 | − | 374.578i | 556.251 | − | 404.140i | 2719.84 | + | 1976.08i | 816.157 | + | 265.185i | 9218.94 | − | 6697.95i | −4928.39 | − | 3580.69i | 6350.19 | + | 19543.9i |
14.4 | −8.72538 | + | 26.8540i | 13.7218 | − | 42.2315i | −437.895 | − | 318.149i | −120.642 | + | 87.6517i | 1014.36 | + | 736.972i | −2163.49 | − | 702.961i | 6516.48 | − | 4734.50i | 3712.75 | + | 2697.47i | −1301.15 | − | 4004.52i |
14.5 | −8.44070 | + | 25.9778i | 2.81292 | − | 8.65727i | −396.492 | − | 288.068i | −624.581 | + | 453.784i | 201.154 | + | 146.147i | −3601.98 | − | 1170.35i | 5172.95 | − | 3758.37i | 5240.92 | + | 3807.75i | −6516.42 | − | 20055.5i |
14.6 | −8.09329 | + | 24.9086i | 31.8997 | − | 98.1772i | −347.828 | − | 252.712i | −564.927 | + | 410.444i | 2187.28 | + | 1589.15i | 2666.06 | + | 866.254i | 3685.51 | − | 2677.68i | −3313.21 | − | 2407.19i | −5651.45 | − | 17393.4i |
14.7 | −7.67748 | + | 23.6288i | −14.9010 | + | 45.8605i | −292.270 | − | 212.347i | 443.724 | − | 322.384i | −969.228 | − | 704.186i | 1661.54 | + | 539.868i | 2115.83 | − | 1537.24i | 3426.81 | + | 2489.73i | 4210.89 | + | 12959.8i |
14.8 | −7.48640 | + | 23.0408i | −43.7983 | + | 134.797i | −267.723 | − | 194.512i | 378.287 | − | 274.842i | −2777.94 | − | 2018.29i | −2111.84 | − | 686.179i | 1468.48 | − | 1066.91i | −10944.1 | − | 7951.33i | 3500.55 | + | 10773.6i |
14.9 | −6.53857 | + | 20.1236i | 1.61600 | − | 4.97354i | −155.100 | − | 112.687i | −471.892 | + | 342.850i | 89.5195 | + | 65.0397i | 2413.84 | + | 784.305i | −1100.46 | + | 799.529i | 5285.84 | + | 3840.38i | −3813.89 | − | 11737.9i |
14.10 | −6.45743 | + | 19.8739i | −29.1194 | + | 89.6202i | −146.166 | − | 106.196i | −428.254 | + | 311.145i | −1593.07 | − | 1157.43i | 2120.69 | + | 689.055i | −1273.50 | + | 925.251i | −1875.89 | − | 1362.91i | −3418.25 | − | 10520.3i |
14.11 | −5.93473 | + | 18.2652i | 43.1747 | − | 132.878i | −91.2892 | − | 66.3255i | 146.058 | − | 106.118i | 2170.82 | + | 1577.19i | −1719.22 | − | 558.608i | −2224.33 | + | 1616.07i | −10484.6 | − | 7617.48i | 1071.45 | + | 3297.57i |
14.12 | −5.83043 | + | 17.9442i | 6.58777 | − | 20.2751i | −80.8925 | − | 58.7719i | 675.661 | − | 490.897i | 325.410 | + | 236.425i | −3087.36 | − | 1003.14i | −2381.40 | + | 1730.19i | 4940.28 | + | 3589.32i | 4869.36 | + | 14986.4i |
14.13 | −5.26657 | + | 16.2088i | 19.4523 | − | 59.8680i | −27.8812 | − | 20.2569i | 856.389 | − | 622.203i | 867.943 | + | 630.598i | 3680.93 | + | 1196.01i | −3054.56 | + | 2219.27i | 2102.18 | + | 1527.32i | 5574.95 | + | 17157.9i |
14.14 | −4.97778 | + | 15.3200i | −31.0907 | + | 95.6873i | −2.81636 | − | 2.04620i | −559.299 | + | 406.354i | −1311.17 | − | 952.619i | −3234.47 | − | 1050.94i | −3290.82 | + | 2390.92i | −2881.46 | − | 2093.50i | −3441.29 | − | 10591.2i |
14.15 | −3.86234 | + | 11.8870i | −7.77024 | + | 23.9144i | 80.7241 | + | 58.6495i | −308.789 | + | 224.348i | −254.260 | − | 184.731i | −143.740 | − | 46.7041i | −3597.55 | + | 2613.78i | 4796.44 | + | 3484.82i | −1474.19 | − | 4537.10i |
14.16 | −3.73678 | + | 11.5006i | 33.6182 | − | 103.466i | 88.8075 | + | 64.5225i | −962.003 | + | 698.936i | 1064.30 | + | 773.260i | −1774.46 | − | 576.558i | −3578.35 | + | 2599.83i | −4267.09 | − | 3100.22i | −4443.41 | − | 13675.4i |
14.17 | −3.43660 | + | 10.5768i | −45.3188 | + | 139.477i | 107.051 | + | 77.7768i | 388.730 | − | 282.429i | −1319.47 | − | 958.652i | 4438.96 | + | 1442.30i | −3493.78 | + | 2538.38i | −12092.0 | − | 8785.38i | 1651.28 | + | 5082.10i |
14.18 | −2.83536 | + | 8.72635i | 26.1649 | − | 80.5273i | 138.998 | + | 100.988i | −68.6655 | + | 49.8884i | 628.523 | + | 456.648i | 2633.26 | + | 855.599i | −3175.68 | + | 2307.26i | −492.089 | − | 357.524i | −240.652 | − | 740.650i |
14.19 | −2.43649 | + | 7.49875i | 16.1739 | − | 49.7781i | 156.814 | + | 113.932i | −5.52655 | + | 4.01527i | 333.866 | + | 242.568i | −761.080 | − | 247.290i | −2869.40 | + | 2084.74i | 3091.69 | + | 2246.25i | −16.6441 | − | 51.2254i |
14.20 | −1.99278 | + | 6.13314i | −22.7009 | + | 69.8663i | 173.464 | + | 126.029i | 339.186 | − | 246.433i | −383.262 | − | 278.456i | 653.541 | + | 212.348i | −2454.22 | + | 1783.10i | 941.995 | + | 684.399i | 835.485 | + | 2571.36i |
See next 80 embeddings (of 188 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
71.e | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 71.9.e.a | ✓ | 188 |
71.e | odd | 10 | 1 | inner | 71.9.e.a | ✓ | 188 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
71.9.e.a | ✓ | 188 | 1.a | even | 1 | 1 | trivial |
71.9.e.a | ✓ | 188 | 71.e | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(71, [\chi])\).