Properties

Label 9.22.c.a
Level $9$
Weight $22$
Character orbit 9.c
Analytic conductor $25.153$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 1023 q^{2} + 128841 q^{3} - 19922945 q^{4} + 32234853 q^{5} + 187115031 q^{6} - 189623959 q^{7} - 1296271662 q^{8} + 10117230687 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 1023 q^{2} + 128841 q^{3} - 19922945 q^{4} + 32234853 q^{5} + 187115031 q^{6} - 189623959 q^{7} - 1296271662 q^{8} + 10117230687 q^{9} + 4194300 q^{10} + 146068576386 q^{11} - 287594221404 q^{12} - 177565977277 q^{13} + 1549677244440 q^{14} - 5069511951597 q^{15} - 18691699769345 q^{16} - 18615603749598 q^{17} + 35716321080900 q^{18} - 9768733723954 q^{19} + 76202257650204 q^{20} - 25195769607435 q^{21} + 86758343554047 q^{22} + 356460494884095 q^{23} + 410684147823429 q^{24} - 13\!\cdots\!29 q^{25}+ \cdots + 16\!\cdots\!41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.20
Significant digits:
Format:

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 \(S_{22}^{\mathrm{new}}(9, [\chi])\).