Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,22,Mod(4,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.4");
S:= CuspForms(chi, 22);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 22 \) |
Character orbit: | \([\chi]\) | \(=\) | 9.c (of order \(3\), 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: | \(25.1529609858\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −1351.65 | − | 2341.13i | 67379.3 | + | 76944.0i | −2.60534e6 | + | 4.51259e6i | −1.80542e6 | + | 3.12707e6i | 8.90624e7 | − | 2.61745e8i | 5.45249e8 | + | 9.44400e8i | 8.41682e9 | −1.38040e9 | + | 1.03689e10i | 9.76117e9 | ||||
4.2 | −1284.67 | − | 2225.12i | 28682.3 | − | 98171.7i | −2.25218e6 | + | 3.90090e6i | 1.41666e7 | − | 2.45373e7i | −2.55291e8 | + | 6.22970e7i | −3.23415e8 | − | 5.60171e8i | 6.18497e9 | −8.81501e9 | − | 5.63157e9i | −7.27978e10 | ||||
4.3 | −1137.04 | − | 1969.41i | −99416.7 | − | 24014.1i | −1.53715e6 | + | 2.66242e6i | −3.36439e6 | + | 5.82729e6i | 6.57472e7 | + | 2.23098e8i | 1.17458e8 | + | 2.03444e8i | 2.22212e9 | 9.30700e9 | + | 4.77480e9i | 1.53018e10 | ||||
4.4 | −959.466 | − | 1661.84i | −29737.5 | + | 97857.2i | −792574. | + | 1.37278e6i | −5.04427e6 | + | 8.73693e6i | 1.91156e8 | − | 4.44715e7i | −7.19629e8 | − | 1.24643e9i | −9.82502e8 | −8.69171e9 | − | 5.82006e9i | 1.93592e10 | ||||
4.5 | −887.730 | − | 1537.59i | 83627.0 | − | 58880.2i | −527555. | + | 913751.i | −1.80258e7 | + | 3.12216e7i | −1.64772e8 | − | 7.63146e7i | 1.15609e8 | + | 2.00240e8i | −1.85011e9 | 3.52659e9 | − | 9.84795e9i | 6.40082e10 | ||||
4.6 | −659.150 | − | 1141.68i | 100993. | + | 16150.4i | 179618. | − | 311108.i | 1.41824e7 | − | 2.45646e7i | −4.81307e7 | − | 1.25947e8i | −1.55043e8 | − | 2.68542e8i | −3.23826e9 | 9.93868e9 | + | 3.26215e9i | −3.73932e10 | ||||
4.7 | −626.415 | − | 1084.98i | −63755.9 | + | 79972.1i | 263783. | − | 456886.i | 1.51867e7 | − | 2.63042e7i | 1.26706e8 | + | 1.90783e7i | 5.82984e8 | + | 1.00976e9i | −3.28833e9 | −2.33073e9 | − | 1.01974e10i | −3.80528e10 | ||||
4.8 | −415.012 | − | 718.822i | −31036.5 | − | 97453.0i | 704106. | − | 1.21955e6i | 468737. | − | 811876.i | −5.71709e7 | + | 6.27539e7i | 2.46849e7 | + | 4.27554e7i | −2.90954e9 | −8.53382e9 | + | 6.04920e9i | −7.78126e8 | ||||
4.9 | −128.319 | − | 222.255i | 58395.5 | + | 83966.1i | 1.01564e6 | − | 1.75915e6i | −7.73413e6 | + | 1.33959e7i | 1.11687e7 | − | 2.37532e7i | 9.68611e7 | + | 1.67768e8i | −1.05952e9 | −3.64028e9 | + | 9.80650e9i | 3.96975e9 | ||||
4.10 | −5.94232 | − | 10.2924i | −100608. | + | 18394.3i | 1.04851e6 | − | 1.81606e6i | −1.81237e7 | + | 3.13912e7i | 787168. | + | 926195.i | 1.21750e8 | + | 2.10877e8i | −4.98461e7 | 9.78365e9 | − | 3.70124e9i | 4.30788e8 | ||||
4.11 | 156.290 | + | 270.702i | −101203. | − | 14776.5i | 999723. | − | 1.73157e6i | 1.75767e7 | − | 3.04438e7i | −1.18169e7 | − | 2.97052e7i | −6.23471e8 | − | 1.07988e9i | 1.28051e9 | 1.00237e10 | + | 2.99085e9i | 1.09883e10 | ||||
4.12 | 342.083 | + | 592.505i | 63482.3 | − | 80189.5i | 814534. | − | 1.41081e6i | 9.58081e6 | − | 1.65944e7i | 6.92289e7 | + | 1.01821e7i | 7.21094e8 | + | 1.24897e9i | 2.54935e9 | −2.40035e9 | − | 1.01812e10i | 1.31097e10 | ||||
4.13 | 370.915 | + | 642.444i | 94939.8 | − | 38036.5i | 773419. | − | 1.33960e6i | −2.26405e6 | + | 3.92145e6i | 5.96510e7 | + | 4.68852e7i | −5.05165e8 | − | 8.74972e8i | 2.70323e9 | 7.56680e9 | − | 7.22237e9i | −3.35909e9 | ||||
4.14 | 543.657 | + | 941.641i | −36859.5 | + | 95403.0i | 457451. | − | 792329.i | 748351. | − | 1.29618e6i | −1.09874e8 | + | 1.71580e7i | 3.81975e7 | + | 6.61600e7i | 3.27505e9 | −7.74310e9 | − | 7.03302e9i | 1.62738e9 | ||||
4.15 | 823.990 | + | 1427.19i | −21016.3 | − | 100093.i | −309344. | + | 535799.i | −1.17845e7 | + | 2.04114e7i | 1.25535e8 | − | 1.12470e8i | −3.12041e8 | − | 5.40471e8i | 2.43648e9 | −9.57698e9 | + | 4.20718e9i | −3.88413e10 | ||||
4.16 | 926.101 | + | 1604.05i | 63319.1 | + | 80318.4i | −666752. | + | 1.15485e6i | 1.45987e7 | − | 2.52856e7i | −7.01953e7 | + | 1.75950e8i | −1.00025e8 | − | 1.73248e8i | 1.41443e9 | −2.44174e9 | + | 1.01714e10i | 5.40794e10 | ||||
4.17 | 991.417 | + | 1717.18i | −93811.7 | − | 40739.6i | −917239. | + | 1.58870e6i | 5.12169e6 | − | 8.87103e6i | −2.30492e7 | − | 2.01482e8i | 4.75645e8 | + | 8.23841e8i | 5.20838e8 | 7.14093e9 | + | 7.64370e9i | 2.03109e10 | ||||
4.18 | 1133.55 | + | 1963.36i | 99710.4 | + | 22763.9i | −1.52129e6 | + | 2.63495e6i | −1.61745e7 | + | 2.80151e7i | 6.83328e7 | + | 2.21572e8i | 3.77107e8 | + | 6.53168e8i | −2.14339e9 | 9.42397e9 | + | 4.53959e9i | −7.33384e10 | ||||
4.19 | 1290.74 | + | 2235.62i | −69485.1 | + | 75047.8i | −2.28343e6 | + | 3.95502e6i | −6.95783e6 | + | 1.20513e7i | −2.57466e8 | − | 5.84754e7i | −3.00160e8 | − | 5.19892e8i | −6.37551e9 | −8.03999e8 | − | 1.04294e10i | −3.59230e10 | ||||
4.20 | 1388.16 | + | 2404.36i | 50822.3 | − | 88755.0i | −2.80539e6 | + | 4.85908e6i | 1.57654e7 | − | 2.73064e7i | 2.83948e8 | − | 1.01067e6i | −2.72501e8 | − | 4.71986e8i | −9.75494e9 | −5.29453e9 | − | 9.02147e9i | 8.75393e10 | ||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 9.22.c.a | ✓ | 40 |
3.b | odd | 2 | 1 | 27.22.c.a | 40 | ||
9.c | even | 3 | 1 | inner | 9.22.c.a | ✓ | 40 |
9.c | even | 3 | 1 | 81.22.a.c | 20 | ||
9.d | odd | 6 | 1 | 27.22.c.a | 40 | ||
9.d | odd | 6 | 1 | 81.22.a.d | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9.22.c.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
9.22.c.a | ✓ | 40 | 9.c | even | 3 | 1 | inner |
27.22.c.a | 40 | 3.b | odd | 2 | 1 | ||
27.22.c.a | 40 | 9.d | odd | 6 | 1 | ||
81.22.a.c | 20 | 9.c | even | 3 | 1 | ||
81.22.a.d | 20 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(9, [\chi])\).