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Properties

Label 79.9.d.a
Level $79$
Weight $9$
Character orbit 79.d
Analytic conductor $32.183$
Analytic rank $0$
Dimension $104$
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] = [79,9,Mod(24,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(6))
 
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("79.24");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 79.d (of order \(6\), 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: \(32.1829101948\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q + 2 q^{2} - 84 q^{3} - 6970 q^{4} - q^{5} - 3 q^{6} - 2190 q^{7} - 9488 q^{8} + 102844 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q + 2 q^{2} - 84 q^{3} - 6970 q^{4} - q^{5} - 3 q^{6} - 2190 q^{7} - 9488 q^{8} + 102844 q^{9} + 72676 q^{10} + 6857 q^{11} + 33657 q^{13} - 788662 q^{16} - 9632 q^{18} - 226852 q^{19} - 219402 q^{20} - 1277350 q^{21} + 1006000 q^{22} + 127586 q^{23} + 1364772 q^{24} - 4283835 q^{25} + 944269 q^{26} + 911979 q^{28} - 2019099 q^{29} + 2356671 q^{30} + 1188207 q^{31} + 153591 q^{32} + 7980111 q^{34} - 943968 q^{35} + 14864875 q^{36} - 9064860 q^{37} + 7808566 q^{38} - 19696677 q^{39} - 7113169 q^{40} - 7814845 q^{42} - 7737366 q^{43} + 7731459 q^{44} + 3113873 q^{45} + 60808472 q^{46} + 8350800 q^{47} - 16645308 q^{48} + 25296310 q^{49} - 14955846 q^{50} + 16214026 q^{51} - 8014188 q^{52} + 25500174 q^{53} + 51886776 q^{54} - 20267 q^{55} + 28740777 q^{56} - 4124757 q^{59} - 37655733 q^{60} + 62152546 q^{62} - 58843464 q^{63} + 126504224 q^{64} - 13768202 q^{65} - 63847536 q^{66} + 102166380 q^{67} + 273753135 q^{68} - 176013324 q^{70} - 94449421 q^{72} + 75645012 q^{73} - 159249108 q^{74} + 60857715 q^{75} - 61609776 q^{76} - 44619834 q^{77} - 195605926 q^{79} - 126767414 q^{80} - 382160972 q^{81} - 36258447 q^{82} + 2233106 q^{83} + 262310811 q^{84} - 13982544 q^{85} - 88175784 q^{86} - 525770194 q^{87} + 25733396 q^{88} + 20473934 q^{89} + 246245830 q^{90} + 53888898 q^{92} - 272190233 q^{95} + 386146828 q^{97} - 365283121 q^{98} - 400375519 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −15.3691 + 26.6201i 56.4183 + 32.5731i −344.420 − 596.553i 437.940 + 758.534i −1734.20 + 1001.24i 740.469 − 427.510i 13304.7 −1158.48 − 2006.55i −26923.0
24.2 −15.0188 + 26.0134i 29.0890 + 16.7946i −323.130 − 559.678i −506.009 − 876.433i −873.766 + 504.469i −3245.93 + 1874.04i 11722.5 −2716.39 − 4704.92i 30398.6
24.3 −14.9097 + 25.8244i −93.0405 − 53.7170i −316.598 − 548.364i −3.80900 − 6.59738i 2774.41 − 1601.81i −1116.75 + 644.758i 11247.8 2490.52 + 4313.72i 227.164
24.4 −14.1175 + 24.4522i 124.967 + 72.1498i −270.607 − 468.706i −164.063 − 284.165i −3528.44 + 2037.15i 360.604 − 208.195i 8053.03 7130.68 + 12350.7i 9264.61
24.5 −13.7620 + 23.8365i −78.4835 − 45.3125i −250.786 − 434.373i 0.360886 + 0.625074i 2160.18 − 1247.18i 3876.54 − 2238.12i 6759.11 825.938 + 1430.57i −19.8661
24.6 −13.3447 + 23.1137i −8.40275 − 4.85133i −228.163 − 395.191i −520.834 − 902.111i 224.265 − 129.479i 2375.07 − 1371.25i 5346.61 −3233.43 − 5600.46i 27801.5
24.7 −13.3133 + 23.0593i −30.9959 − 17.8955i −226.487 − 392.286i 301.995 + 523.070i 825.313 − 476.495i −1789.02 + 1032.89i 5244.72 −2640.00 − 4572.62i −16082.2
24.8 −12.7749 + 22.1268i 71.6387 + 41.3606i −198.397 − 343.634i −64.8593 − 112.340i −1830.36 + 1056.76i 1381.12 − 797.388i 3597.28 140.898 + 244.042i 3314.29
24.9 −10.6445 + 18.4369i −133.419 − 77.0297i −98.6128 − 170.802i −537.219 − 930.491i 2840.38 − 1639.89i −1470.77 + 849.149i −1251.25 8586.64 + 14872.5i 22873.8
24.10 −10.3541 + 17.9339i 108.428 + 62.6012i −86.4158 − 149.677i 412.942 + 715.236i −2245.36 + 1296.36i −3031.47 + 1750.22i −1722.27 4557.32 + 7893.50i −17102.6
24.11 −10.2271 + 17.7138i −134.612 − 77.7186i −81.1857 − 140.618i 558.024 + 966.526i 2753.38 − 1589.67i 293.734 − 169.587i −1915.09 8799.85 + 15241.8i −22827.8
24.12 −9.91079 + 17.1660i 32.4527 + 18.7366i −68.4477 − 118.555i −95.6981 − 165.754i −643.263 + 371.388i −2023.42 + 1168.22i −2360.84 −2578.38 − 4465.89i 3793.78
24.13 −9.72047 + 16.8364i −71.3495 − 41.1937i −60.9751 − 105.612i −33.8371 − 58.6076i 1387.10 − 800.844i −1660.37 + 958.614i −2606.05 113.338 + 196.306i 1315.65
24.14 −9.54467 + 16.5318i −4.74562 − 2.73988i −54.2014 − 93.8795i 493.596 + 854.933i 90.5907 − 52.3026i 2316.07 − 1337.18i −2817.53 −3265.49 − 5655.99i −18844.8
24.15 −8.79366 + 15.2311i 76.6186 + 44.2358i −26.6571 − 46.1714i −47.8459 − 82.8715i −1347.52 + 777.989i 2915.82 − 1683.45i −3564.70 633.111 + 1096.58i 1682.96
24.16 −8.30444 + 14.3837i −51.0078 − 29.4494i −9.92754 − 17.1950i −336.886 − 583.504i 847.183 − 489.121i 395.741 − 228.481i −3922.10 −1545.97 − 2677.70i 11190.6
24.17 −7.25689 + 12.5693i 131.859 + 76.1289i 22.6752 + 39.2747i −556.119 − 963.226i −1913.77 + 1104.92i −416.100 + 240.235i −4373.73 8310.71 + 14394.6i 16142.8
24.18 −4.84781 + 8.39666i −69.6948 − 40.2383i 80.9974 + 140.292i 178.891 + 309.848i 675.735 − 390.136i 1504.73 − 868.754i −4052.72 −42.2535 − 73.1852i −3468.92
24.19 −4.27662 + 7.40733i 65.1843 + 37.6342i 91.4210 + 158.346i −329.338 − 570.431i −557.538 + 321.894i −3363.41 + 1941.86i −3753.52 −447.836 − 775.674i 5633.82
24.20 −4.08487 + 7.07520i −101.942 − 58.8562i 94.6277 + 163.900i −161.684 − 280.046i 832.839 − 480.840i 2719.28 − 1569.98i −3637.62 3647.61 + 6317.84i 2641.84
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.9.d.a ✓ 104
79.d odd 6 1 inner 79.9.d.a ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.9.d.a ✓ 104 1.a even 1 1 trivial
79.9.d.a ✓ 104 79.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(79, [\chi])\).