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Properties

Label 89.8.g.a
Level $89$
Weight $8$
Character orbit 89.g
Analytic conductor $27.802$
Analytic rank $0$
Dimension $1020$
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] = [89,8,Mod(5,89)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([35]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("89.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 89 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 89.g (of order \(44\), degree \(20\), 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: \(27.8022672681\)
Analytic rank: \(0\)
Dimension: \(1020\)
Relative dimension: \(51\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1020 q - 2 q^{2} - 20 q^{3} - 6290 q^{4} - 22 q^{5} + 1852 q^{6} - 20 q^{7} + 3566 q^{8} - 23760 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1020 q - 2 q^{2} - 20 q^{3} - 6290 q^{4} - 22 q^{5} + 1852 q^{6} - 20 q^{7} + 3566 q^{8} - 23760 q^{9} - 22 q^{10} - 16722 q^{11} + 9052 q^{12} + 17226 q^{13} + 330112 q^{14} - 30060 q^{15} - 431778 q^{16} - 22 q^{17} - 22 q^{18} - 30652 q^{19} - 22 q^{20} - 22 q^{21} - 94514 q^{22} + 24136 q^{23} + 411814 q^{24} + 1861464 q^{25} + 47658 q^{26} + 908368 q^{27} - 530464 q^{28} - 155786 q^{29} - 1242616 q^{30} - 389634 q^{31} + 524782 q^{32} - 1066272 q^{33} - 2353684 q^{35} - 22 q^{36} - 1208780 q^{37} + 5485538 q^{38} - 302534 q^{39} - 7387512 q^{40} - 1564810 q^{41} - 22 q^{42} + 7216038 q^{43} - 2800402 q^{44} + 1792854 q^{45} + 67740 q^{46} - 22 q^{47} + 8660570 q^{48} - 22 q^{49} + 3077262 q^{50} + 1242076 q^{51} + 3299954 q^{52} - 20324238 q^{53} - 6661598 q^{54} - 20949668 q^{56} + 5236566 q^{57} + 13918306 q^{58} + 8583504 q^{59} - 8076230 q^{60} + 20148938 q^{61} - 1481048 q^{62} - 4903098 q^{63} + 25377694 q^{64} + 4012556 q^{65} + 22739768 q^{66} + 3698626 q^{67} + 2794 q^{68} - 58001086 q^{69} + 9788184 q^{70} - 22 q^{71} - 77404580 q^{72} - 2571136 q^{73} + 57383624 q^{74} + 85156694 q^{75} + 31912792 q^{76} - 33688902 q^{77} - 31908470 q^{78} - 21117074 q^{79} - 103088150 q^{80} - 53099914 q^{81} + 6653288 q^{82} - 33257458 q^{83} + 5775594 q^{84} + 128319980 q^{85} + 68836812 q^{86} - 40740986 q^{87} + 138377800 q^{88} + 18924138 q^{89} - 2319948 q^{90} + 39299622 q^{91} - 3694622 q^{92} - 64978988 q^{93} - 45366486 q^{94} + 117515586 q^{95} - 295610016 q^{96} - 19530672 q^{97} - 39727292 q^{98} - 73644252 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −14.1435 + 16.3225i 13.7257 − 10.2749i −48.1683 − 335.018i −265.744 + 413.506i −26.4172 + 369.361i −97.8765 − 449.931i 3823.94 + 2457.50i −533.329 + 1816.35i −2990.89 − 10186.0i
5.2 −13.7423 + 15.8594i −12.7052 + 9.51100i −44.4549 − 309.191i 78.9674 − 122.876i 23.7594 − 332.200i 42.8133 + 196.810i 3254.82 + 2091.74i −545.186 + 1856.73i 863.545 + 2940.96i
5.3 −13.4182 + 15.4855i 41.8102 − 31.2987i −41.5344 − 288.878i 160.004 − 248.971i −76.3436 + 1067.42i −370.026 − 1700.98i 2824.33 + 1815.09i 152.333 − 518.797i 1708.46 + 5818.49i
5.4 −13.2899 + 15.3374i −61.6665 + 46.1630i −40.3971 − 280.968i −102.275 + 159.142i 111.524 − 1559.30i −65.3820 − 300.556i 2660.89 + 1710.05i 1055.59 − 3595.00i −1081.61 − 3683.61i
5.5 −13.2110 + 15.2463i 57.6955 − 43.1903i −39.7030 − 276.140i 7.59442 − 11.8172i −103.722 + 1450.23i 215.946 + 992.687i 2562.32 + 1646.70i 847.218 − 2885.36i 79.8380 + 271.903i
5.6 −13.0623 + 15.0747i −31.6453 + 23.6894i −38.4066 − 267.124i 176.239 − 274.233i 56.2503 − 786.482i 106.743 + 490.689i 2380.63 + 1529.94i −175.910 + 599.096i 1831.90 + 6238.87i
5.7 −10.7836 + 12.4449i 7.31019 − 5.47234i −20.3739 − 141.703i −217.635 + 338.647i −10.7272 + 149.986i −21.3591 − 98.1861i 210.019 + 134.971i −592.657 + 2018.40i −1867.54 − 6360.27i
5.8 −10.3677 + 11.9650i 16.4671 − 12.3271i −17.4549 − 121.401i −94.8847 + 147.643i −23.2324 + 324.832i 324.164 + 1490.16i −71.2603 − 45.7962i −496.942 + 1692.43i −782.813 − 2666.02i
5.9 −10.1483 + 11.7118i 42.4884 − 31.8064i −15.9613 − 111.013i 51.9915 − 80.9003i −58.6760 + 820.397i −134.210 − 616.952i −206.572 − 132.756i 177.466 − 604.393i 419.861 + 1429.92i
5.10 −9.82630 + 11.3402i −49.3333 + 36.9305i −13.8266 − 96.1663i −135.185 + 210.351i 65.9669 − 922.337i 299.279 + 1375.76i −389.358 − 250.225i 453.769 − 1545.39i −1057.05 − 3599.99i
5.11 −9.73252 + 11.2319i −24.8772 + 18.6228i −13.2179 − 91.9326i −50.5022 + 78.5830i 32.9475 − 460.666i −313.824 − 1442.62i −439.119 − 282.205i −344.084 + 1171.84i −391.124 − 1332.05i
5.12 −9.33993 + 10.7789i −70.0365 + 52.4287i −10.7331 − 74.6502i 232.726 − 362.129i 89.0150 − 1244.59i −188.436 − 866.224i −630.899 − 405.454i 1540.20 − 5245.43i 1729.69 + 5890.78i
5.13 −9.32058 + 10.7565i 10.3979 − 7.78374i −10.6133 − 73.8170i 235.841 − 366.975i −13.1881 + 184.394i 93.6158 + 430.345i −639.671 − 411.092i −568.620 + 1936.54i 1749.21 + 5957.25i
5.14 −8.18975 + 9.45147i 62.4472 − 46.7474i −4.04207 − 28.1132i −178.890 + 278.358i −69.5951 + 973.067i −86.4878 − 397.578i −1047.85 − 673.410i 1098.18 − 3740.07i −1165.83 − 3970.45i
5.15 −6.60251 + 7.61970i −40.8459 + 30.5769i 3.74959 + 26.0790i 114.524 − 178.203i 36.6989 − 513.117i −26.3803 − 121.268i −1309.14 − 841.332i 117.295 − 399.469i 601.707 + 2049.23i
5.16 −5.77906 + 6.66940i 70.2268 − 52.5711i 7.13305 + 49.6114i 275.970 − 429.418i −55.2276 + 772.182i 90.7948 + 417.377i −1322.37 − 849.834i 1551.93 − 5285.40i 1269.11 + 4322.19i
5.17 −5.07519 + 5.85708i −13.7243 + 10.2739i 9.66845 + 67.2455i −122.747 + 190.998i 9.47850 − 132.527i −56.1737 − 258.226i −1277.46 − 820.973i −533.345 + 1816.41i −495.727 − 1688.29i
5.18 −4.35401 + 5.02480i −61.4560 + 46.0054i 11.9251 + 82.9411i −282.740 + 439.953i 36.4124 − 509.112i −94.9930 − 436.676i −1184.63 − 761.313i 1044.20 − 3556.20i −979.617 − 3336.27i
5.19 −4.19463 + 4.84086i 32.5303 − 24.3519i 12.3773 + 86.0859i 21.8869 − 34.0566i −18.5685 + 259.622i 289.984 + 1333.04i −1158.38 − 744.447i −150.942 + 514.060i 73.0561 + 248.806i
5.20 −4.01891 + 4.63807i −44.9244 + 33.6300i 12.8563 + 89.4172i 105.456 − 164.092i 24.5689 − 343.518i 232.611 + 1069.29i −1127.23 − 724.427i 271.077 − 923.203i 337.254 + 1148.58i
See next 80 embeddings (of 1020 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.51
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.g even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 89.8.g.a ✓ 1020
89.g even 44 1 inner 89.8.g.a ✓ 1020
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
89.8.g.a ✓ 1020 1.a even 1 1 trivial
89.8.g.a ✓ 1020 89.g even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(89, [\chi])\).