Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [71,9,Mod(7,71)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(71, base_ring=CyclotomicField(70))
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("71.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 71 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 71.h (of order \(70\), degree \(24\), 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: | \(1128\) |
Relative dimension: | \(47\) over \(\Q(\zeta_{70})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{70}]$ |
$q$-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 | −29.8117 | − | 8.22750i | −6.23568 | − | 14.5891i | 601.281 | + | 359.249i | 300.369 | + | 924.441i | 65.8643 | + | 486.230i | 831.034 | + | 37.3218i | −9498.26 | − | 9934.40i | 4360.10 | − | 4560.31i | −1348.67 | − | 30030.4i |
7.2 | −27.9049 | − | 7.70125i | 30.2488 | + | 70.7707i | 499.609 | + | 298.502i | −201.133 | − | 619.023i | −299.067 | − | 2207.80i | −1735.55 | − | 77.9435i | −6521.40 | − | 6820.85i | 440.568 | − | 460.798i | 845.329 | + | 18822.7i |
7.3 | −27.5284 | − | 7.59736i | −32.1065 | − | 75.1170i | 480.332 | + | 286.985i | −39.5634 | − | 121.764i | 313.152 | + | 2311.78i | −2612.63 | − | 117.333i | −5990.27 | − | 6265.33i | −77.6695 | + | 81.2359i | 164.036 | + | 3652.54i |
7.4 | −26.3814 | − | 7.28081i | −53.2959 | − | 124.692i | 423.207 | + | 252.855i | −71.1650 | − | 219.023i | 498.164 | + | 3677.59i | 2315.48 | + | 103.988i | −4482.15 | − | 4687.97i | −8173.58 | + | 8548.90i | 282.767 | + | 6296.29i |
7.5 | −25.5518 | − | 7.05184i | 27.9094 | + | 65.2972i | 383.402 | + | 229.072i | −57.6539 | − | 177.440i | −252.668 | − | 1865.27i | 3990.36 | + | 179.207i | −3491.80 | − | 3652.14i | 1049.27 | − | 1097.45i | 221.877 | + | 4940.48i |
7.6 | −24.9077 | − | 6.87410i | 55.6598 | + | 130.223i | 353.379 | + | 211.134i | 176.810 | + | 544.167i | −491.197 | − | 3626.16i | 593.892 | + | 26.6717i | −2779.31 | − | 2906.93i | −9325.84 | + | 9754.06i | −663.293 | − | 14769.4i |
7.7 | −24.6515 | − | 6.80337i | −30.6649 | − | 71.7441i | 341.646 | + | 204.124i | −287.767 | − | 885.657i | 267.833 | + | 1977.22i | 1192.92 | + | 53.5740i | −2509.15 | − | 2624.37i | 327.181 | − | 342.204i | 1068.43 | + | 23790.5i |
7.8 | −20.7036 | − | 5.71382i | −0.572781 | − | 1.34009i | 176.227 | + | 105.291i | 86.7721 | + | 267.057i | 4.20159 | + | 31.0174i | −743.577 | − | 33.3941i | 752.725 | + | 787.288i | 4532.59 | − | 4740.72i | −270.575 | − | 6024.83i |
7.9 | −20.3971 | − | 5.62924i | 29.4906 | + | 68.9966i | 164.590 | + | 98.3379i | 294.687 | + | 906.955i | −213.123 | − | 1573.34i | −3342.08 | − | 150.093i | 939.804 | + | 982.958i | 643.219 | − | 672.755i | −905.301 | − | 20158.1i |
7.10 | −20.0821 | − | 5.54230i | −60.7634 | − | 142.163i | 152.809 | + | 91.2992i | 335.668 | + | 1033.08i | 432.345 | + | 3191.70i | −3094.13 | − | 138.958i | 1122.86 | + | 1174.42i | −11984.1 | + | 12534.4i | −1015.27 | − | 22606.8i |
7.11 | −18.9677 | − | 5.23475i | −22.9381 | − | 53.6663i | 112.608 | + | 67.2802i | 235.044 | + | 723.392i | 154.153 | + | 1138.00i | 4231.00 | + | 190.014i | 1697.34 | + | 1775.28i | 2180.15 | − | 2280.26i | −671.471 | − | 14951.5i |
7.12 | −17.0539 | − | 4.70658i | 1.56231 | + | 3.65520i | 48.9215 | + | 29.2292i | −355.084 | − | 1092.84i | −9.43996 | − | 69.6886i | 2393.30 | + | 107.483i | 2433.10 | + | 2544.82i | 4523.14 | − | 4730.84i | 912.050 | + | 20308.4i |
7.13 | −16.2933 | − | 4.49666i | −10.8948 | − | 25.4898i | 25.4877 | + | 15.2282i | −10.5522 | − | 32.4764i | 62.8940 | + | 464.302i | −1827.73 | − | 82.0833i | 2643.43 | + | 2764.81i | 4003.03 | − | 4186.84i | 25.8949 | + | 576.596i |
7.14 | −15.9670 | − | 4.40661i | 51.3872 | + | 120.226i | 15.7637 | + | 9.41834i | −267.295 | − | 822.651i | −290.709 | − | 2146.10i | −3970.61 | − | 178.320i | 2720.16 | + | 2845.06i | −7279.68 | + | 7613.94i | 642.803 | + | 14313.1i |
7.15 | −14.3352 | − | 3.95627i | −43.7785 | − | 102.425i | −29.9160 | − | 17.8740i | 78.9573 | + | 243.005i | 222.354 | + | 1641.48i | 1308.82 | + | 58.7790i | 2989.02 | + | 3126.27i | −4040.23 | + | 4225.75i | −170.475 | − | 3795.92i |
7.16 | −13.6164 | − | 3.75789i | 55.4877 | + | 129.820i | −48.4779 | − | 28.9642i | −100.486 | − | 309.264i | −267.694 | − | 1976.20i | 2624.71 | + | 117.876i | 3050.21 | + | 3190.27i | −9240.25 | + | 9664.54i | 206.078 | + | 4588.69i |
7.17 | −12.8627 | − | 3.54987i | −41.6820 | − | 97.5198i | −66.9162 | − | 39.9806i | −275.639 | − | 848.330i | 189.959 | + | 1402.33i | −4784.63 | − | 214.878i | 3079.43 | + | 3220.83i | −3238.67 | + | 3387.38i | 533.993 | + | 11890.3i |
7.18 | −11.4259 | − | 3.15335i | 24.4083 | + | 57.1060i | −99.1551 | − | 59.2424i | −100.071 | − | 307.987i | −98.8116 | − | 729.457i | 342.125 | + | 15.3648i | 3043.07 | + | 3182.81i | 1868.73 | − | 1954.53i | 172.212 | + | 3834.60i |
7.19 | −7.16801 | − | 1.97824i | 39.8175 | + | 93.1578i | −172.296 | − | 102.942i | 178.026 | + | 547.908i | −101.124 | − | 746.525i | −656.248 | − | 29.4721i | 2346.89 | + | 2454.65i | −2558.87 | + | 2676.37i | −192.197 | − | 4279.59i |
7.20 | −6.32703 | − | 1.74615i | −53.6449 | − | 125.509i | −182.781 | − | 109.206i | −184.357 | − | 567.392i | 120.256 | + | 887.768i | 1980.49 | + | 88.9441i | 2126.94 | + | 2224.61i | −8340.55 | + | 8723.53i | 175.680 | + | 3911.82i |
See next 80 embeddings (of 1128 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
71.h | odd | 70 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 71.9.h.a | ✓ | 1128 |
71.h | odd | 70 | 1 | inner | 71.9.h.a | ✓ | 1128 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
71.9.h.a | ✓ | 1128 | 1.a | even | 1 | 1 | trivial |
71.9.h.a | ✓ | 1128 | 71.h | odd | 70 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(71, [\chi])\).