Properties

Label 89.10.f.a
Level $89$
Weight $10$
Character orbit 89.f
Analytic conductor $45.838$
Analytic rank $0$
Dimension $660$
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] = [89,10,Mod(11,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([21]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.11");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 89.f (of order \(22\), degree \(10\), 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: \(45.8381894186\)
Analytic rank: \(0\)
Dimension: \(660\)
Relative dimension: \(66\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 660 q - 9 q^{2} - 825 q^{3} - 16393 q^{4} - 5127 q^{5} - 11 q^{6} - 11 q^{7} + 1015 q^{8} + 489535 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 660 q - 9 q^{2} - 825 q^{3} - 16393 q^{4} - 5127 q^{5} - 11 q^{6} - 11 q^{7} + 1015 q^{8} + 489535 q^{9} - 225841 q^{10} - 224944 q^{11} - 11 q^{13} + 1940664 q^{14} - 11 q^{15} - 5125641 q^{16} + 1457606 q^{17} + 2864796 q^{18} - 11 q^{19} - 5955661 q^{20} + 856653 q^{21} + 13531907 q^{22} - 11 q^{23} + 5621 q^{24} - 31973709 q^{25} - 4206763 q^{26} - 17737896 q^{27} - 11 q^{28} + 10200729 q^{29} - 81004858 q^{30} - 11 q^{31} - 44372422 q^{32} - 19516475 q^{33} + 35594328 q^{34} - 35970209 q^{35} + 71890930 q^{36} - 87132859 q^{38} + 42308203 q^{39} - 128175356 q^{40} + 60926635 q^{41} + 121333817 q^{42} + 296517661 q^{43} + 115664209 q^{44} + 48133749 q^{45} + 311125309 q^{46} - 46792325 q^{47} - 141127426 q^{48} + 164252871 q^{49} + 103838041 q^{50} - 427287971 q^{51} + 850769769 q^{53} - 329099771 q^{54} + 793594654 q^{55} - 2036185316 q^{56} - 154197519 q^{57} + 188920501 q^{58} + 405753711 q^{59} + 21484364 q^{60} + 344711653 q^{61} - 1167226764 q^{62} - 1320295229 q^{63} - 134364169 q^{64} + 1450391239 q^{65} + 863803105 q^{66} + 646170445 q^{67} + 1285178775 q^{68} - 221829309 q^{69} + 854155808 q^{70} - 1113621013 q^{71} - 2392386072 q^{72} - 314298299 q^{73} - 479039451 q^{74} - 3075936347 q^{75} - 6170784400 q^{76} - 2037231963 q^{78} - 79886655 q^{79} + 6302566534 q^{80} + 281226707 q^{81} + 5439310019 q^{82} + 1306521271 q^{83} + 1958614914 q^{84} + 4403895759 q^{85} - 3738814827 q^{86} - 4959185007 q^{87} + 2745817562 q^{88} + 3415767517 q^{89} - 10039266258 q^{90} + 1585238913 q^{91} + 8669558172 q^{92} + 1795690413 q^{93} - 8679076027 q^{94} + 10576695701 q^{95} + 12941524981 q^{96} - 2702894895 q^{97} + 1786224002 q^{98} - 9157402144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −29.3718 33.8968i 100.152 14.3997i −213.429 + 1484.43i −1324.01 + 850.889i −3429.76 2971.90i −2456.23 3821.96i 37267.6 23950.5i −9062.56 + 2661.01i 67731.0 + 19887.6i
11.2 −27.8962 32.1940i −111.443 + 16.0230i −185.387 + 1289.39i −383.758 + 246.626i 3624.67 + 3140.80i 3200.31 + 4979.78i 28334.0 18209.2i −6722.97 + 1974.04i 18645.3 + 5474.75i
11.3 −27.4737 31.7064i −251.974 + 36.2284i −177.623 + 1235.40i 988.921 635.541i 8071.35 + 6993.87i −4048.34 6299.35i 25979.7 16696.1i 43292.9 12711.9i −47320.1 13894.4i
11.4 −26.8447 30.9805i 276.126 39.7009i −166.285 + 1156.54i 514.520 330.662i −8642.48 7488.75i −609.321 948.122i 22637.3 14548.1i 55783.7 16379.6i −24056.2 7063.54i
11.5 −26.5347 30.6227i 57.2379 8.22957i −160.793 + 1118.34i 1789.65 1150.14i −1770.80 1534.41i −680.903 1059.51i 21060.6 13534.8i −15677.2 + 4603.26i −82708.0 24285.3i
11.6 −24.4588 28.2269i 162.887 23.4196i −125.662 + 874.001i 299.561 192.516i −4645.08 4024.98i 3266.28 + 5082.43i 11656.6 7491.26i 7098.00 2084.16i −12761.0 3746.97i
11.7 −24.0656 27.7732i −110.529 + 15.8917i −119.332 + 829.971i 1590.10 1021.89i 3101.32 + 2687.31i 2917.13 + 4539.15i 10094.1 6487.05i −6921.55 + 2032.35i −66647.9 19569.6i
11.8 −23.8701 27.5475i −96.7895 + 13.9162i −116.221 + 808.333i −236.144 + 151.761i 2693.73 + 2334.13i −6039.03 9396.92i 9341.70 6003.54i −9711.15 + 2851.45i 9817.41 + 2882.65i
11.9 −23.6242 27.2637i −245.101 + 35.2402i −112.345 + 781.376i −2304.70 + 1481.14i 6751.07 + 5849.84i 2623.57 + 4082.35i 8418.97 5410.54i 39946.8 11729.4i 94828.0 + 27844.0i
11.10 −23.1036 26.6629i −44.4319 + 6.38834i −104.272 + 725.230i −1564.09 + 1005.18i 1196.87 + 1037.09i −1347.71 2097.08i 6549.91 4209.37i −16952.3 + 4977.65i 62937.2 + 18480.0i
11.11 −22.5505 26.0247i 122.825 17.6596i −95.8928 + 666.949i −945.067 + 607.358i −3229.35 2798.25i 5568.18 + 8664.25i 4687.40 3012.41i −4111.55 + 1207.26i 37118.0 + 10898.8i
11.12 −22.2001 25.6203i 121.444 17.4610i −90.6890 + 630.756i 992.362 637.752i −3143.42 2723.79i −5968.88 9287.77i 3571.77 2295.44i −4441.94 + 1304.27i −38369.9 11266.4i
11.13 −20.5229 23.6847i 180.388 25.9359i −66.9096 + 465.366i −1796.71 + 1154.67i −4316.36 3740.15i −2806.51 4367.02i −1103.30 + 709.049i 12981.4 3811.68i 64221.6 + 18857.2i
11.14 −18.9979 21.9247i −176.954 + 25.4421i −46.9086 + 326.256i −667.049 + 428.686i 3919.55 + 3396.31i −1569.31 2441.89i −4451.24 + 2860.64i 11779.6 3458.79i 22071.3 + 6480.72i
11.15 −18.6156 21.4835i −248.677 + 35.7544i −42.1371 + 293.070i 754.308 484.765i 5397.41 + 4676.88i 5204.71 + 8098.69i −5163.46 + 3318.35i 41676.3 12237.3i −24456.4 7181.04i
11.16 −17.6929 20.4187i −78.9795 + 11.3555i −31.0189 + 215.741i 256.751 165.004i 1629.24 + 1411.74i 3112.86 + 4843.70i −6683.18 + 4295.02i −12776.9 + 3751.63i −7911.82 2323.12i
11.17 −15.6407 18.0504i 57.7859 8.30835i −8.31807 + 57.8534i 915.893 588.609i −1053.78 913.107i −1849.17 2877.36i −9113.01 + 5856.58i −15615.5 + 4585.13i −24949.8 7325.93i
11.18 −15.6053 18.0095i −169.737 + 24.4045i −7.95116 + 55.3015i 2179.61 1400.75i 3088.32 + 2676.05i −1004.03 1562.30i −9144.08 + 5876.54i 9329.46 2739.38i −59240.4 17394.5i
11.19 −15.4169 17.7921i 217.920 31.3321i −6.01132 + 41.8096i 614.570 394.960i −3917.12 3394.20i −2069.29 3219.88i −9303.62 + 5979.08i 27621.6 8110.45i −16501.9 4845.41i
11.20 −13.7990 15.9249i 202.658 29.1378i 9.67508 67.2917i 2267.64 1457.33i −3260.50 2825.24i 6234.30 + 9700.76i −10281.2 + 6607.30i 21335.5 6264.68i −54499.1 16002.4i
See next 80 embeddings (of 660 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.f even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.10.f.a 660
89.f even 22 1 inner 89.10.f.a 660
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.10.f.a 660 1.a even 1 1 trivial
89.10.f.a 660 89.f even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(89, [\chi])\).