Properties

Label 72.8.n.a
Level $72$
Weight $8$
Character orbit 72.n
Analytic conductor $22.492$
Analytic rank $0$
Dimension $164$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,8,Mod(13,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.13");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.n (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: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 164 q - q^{2} - q^{4} - 217 q^{6} - 2 q^{7} - 2602 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q - q^{2} - q^{4} - 217 q^{6} - 2 q^{7} - 2602 q^{8} - 4 q^{9} - 260 q^{10} + 14444 q^{12} + 14750 q^{14} - 4378 q^{15} - q^{16} - 8 q^{17} - 17786 q^{18} - 76372 q^{20} - 257 q^{22} + 146002 q^{23} - 22421 q^{24} + 1156248 q^{25} + 19424 q^{26} + 32764 q^{28} + 152578 q^{30} - 2 q^{31} - 431261 q^{32} + 94638 q^{33} + 195093 q^{34} - 323541 q^{36} - 870761 q^{38} - 146350 q^{39} - 321626 q^{40} + 300602 q^{41} + 3851402 q^{42} - 1783398 q^{44} - 1081992 q^{46} - 2076462 q^{47} + 4480743 q^{48} - 8235432 q^{49} + 578333 q^{50} - 1154556 q^{52} + 6341533 q^{54} - 312508 q^{55} - 361846 q^{56} + 351916 q^{57} - 1295024 q^{58} + 13532922 q^{60} - 63104 q^{62} + 11485202 q^{63} + 2821274 q^{64} - 312502 q^{65} - 945990 q^{66} - 4367667 q^{68} - 979666 q^{70} - 20043024 q^{71} + 15838855 q^{72} - 8 q^{73} - 10609486 q^{74} - 3634515 q^{76} - 13229042 q^{78} - 2 q^{79} - 34658016 q^{80} - 335456 q^{81} + 1782378 q^{82} + 6111092 q^{84} + 22805525 q^{86} + 2067066 q^{87} - 5528213 q^{88} + 7614440 q^{89} - 48494288 q^{90} - 29403282 q^{92} - 6809706 q^{94} - 27279752 q^{95} - 43951874 q^{96} - 2 q^{97} + 95826882 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −11.3038 + 0.473330i −23.1239 40.6483i 127.552 10.7009i 156.431 + 90.3155i 280.628 + 448.536i 183.338 + 317.552i −1436.76 + 181.334i −1117.57 + 1879.89i −1811.01 946.865i
13.2 −11.2976 + 0.602606i −46.7246 1.95340i 127.274 13.6161i −29.6157 17.0986i 529.055 6.08770i −438.175 758.942i −1429.69 + 230.525i 2179.37 + 182.543i 344.891 + 175.328i
13.3 −11.2966 0.622684i 19.3085 + 42.5932i 127.225 + 14.0684i −458.263 264.578i −191.598 493.180i −430.048 744.865i −1428.44 238.145i −1441.36 + 1644.82i 5012.04 + 3274.18i
13.4 −11.2910 0.716509i 41.0005 22.4936i 126.973 + 16.1802i −73.9107 42.6724i −479.053 + 224.598i 208.656 + 361.403i −1422.06 273.668i 1175.08 1844.50i 803.951 + 534.772i
13.5 −11.1884 + 1.67919i −34.9028 + 31.1255i 122.361 37.5749i −271.359 156.669i 338.240 406.853i 704.673 + 1220.53i −1305.92 + 625.870i 249.406 2172.73i 3299.15 + 1297.21i
13.6 −11.1359 + 1.99766i 18.1379 + 43.1047i 120.019 44.4916i 205.108 + 118.419i −288.092 443.778i 841.491 + 1457.51i −1247.64 + 735.213i −1529.03 + 1563.66i −2520.63 908.972i
13.7 −10.9054 + 3.01190i 34.0751 + 32.0294i 109.857 65.6921i 318.063 + 183.634i −468.074 246.664i −768.035 1330.28i −1000.18 + 1047.28i 135.231 + 2182.82i −4021.70 1044.63i
13.8 −10.8000 + 3.37058i 5.68683 46.4183i 105.278 72.8043i −392.053 226.352i 95.0392 + 520.484i −563.528 976.059i −891.609 + 1141.13i −2122.32 527.946i 4997.09 + 1123.14i
13.9 −10.6787 3.73693i 27.7884 37.6139i 100.071 + 79.8114i 390.655 + 225.545i −437.305 + 297.825i −517.710 896.700i −770.378 1226.24i −642.611 2090.46i −3328.86 3868.38i
13.10 −10.6617 3.78537i 45.3121 + 11.5679i 99.3419 + 80.7167i −121.295 70.0297i −439.313 294.856i 248.188 + 429.875i −753.607 1236.62i 1919.37 + 1048.33i 1028.12 + 1205.78i
13.11 −10.5554 4.07240i −14.9325 + 44.3173i 94.8311 + 85.9713i 159.215 + 91.9229i 338.096 406.973i −195.310 338.288i −650.867 1293.65i −1741.04 1323.54i −1306.23 1618.67i
13.12 −9.94286 + 5.39810i −42.4698 + 19.5784i 69.7211 107.345i 395.210 + 228.175i 316.586 423.921i −20.8306 36.0796i −113.767 + 1443.68i 1420.38 1662.98i −5161.23 135.326i
13.13 −9.91032 + 5.45762i 46.3951 5.87285i 68.4287 108.174i −56.1042 32.3918i −427.739 + 311.409i −152.197 263.613i −87.7794 + 1445.49i 2118.02 544.944i 732.793 + 14.8171i
13.14 −9.87518 5.52094i 5.24322 46.4705i 67.0384 + 109.041i −315.728 182.286i −308.339 + 429.957i 611.374 + 1058.93i −60.0094 1446.91i −2132.02 487.310i 2111.48 + 3543.22i
13.15 −9.70577 + 5.81361i 18.5086 42.9468i 60.4040 112.851i 223.068 + 128.788i 70.0355 + 524.434i 512.210 + 887.174i 69.8041 + 1446.47i −1501.86 1589.77i −2913.77 + 46.8387i
13.16 −9.44441 6.22921i −46.3527 + 6.19919i 50.3939 + 117.662i −277.107 159.988i 476.390 + 230.193i 41.7818 + 72.3683i 257.003 1425.17i 2110.14 574.698i 1620.52 + 3237.15i
13.17 −9.38659 6.31601i −26.9410 38.2254i 48.2160 + 118.572i −76.9557 44.4304i 11.4521 + 528.966i −589.934 1021.79i 296.316 1417.52i −735.363 + 2059.66i 441.728 + 903.103i
13.18 −9.24600 6.52008i −45.8311 9.30116i 42.9772 + 120.569i 410.301 + 236.888i 363.110 + 384.821i 806.324 + 1396.59i 388.755 1395.00i 2013.98 + 852.565i −2249.12 4865.46i
13.19 −8.85286 + 7.04464i −37.3311 28.1671i 28.7461 124.730i −339.926 196.256i 528.914 13.6246i 470.927 + 815.670i 624.195 + 1306.73i 600.225 + 2103.02i 4391.87 657.226i
13.20 −8.70945 + 7.22118i −18.9377 + 42.7594i 23.7091 125.785i −134.181 77.4696i −143.836 509.163i −357.842 619.801i 701.823 + 1266.73i −1469.72 1619.53i 1728.07 294.229i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.n.a 164
8.b even 2 1 inner 72.8.n.a 164
9.c even 3 1 inner 72.8.n.a 164
72.n even 6 1 inner 72.8.n.a 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.n.a 164 1.a even 1 1 trivial
72.8.n.a 164 8.b even 2 1 inner
72.8.n.a 164 9.c even 3 1 inner
72.8.n.a 164 72.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(72, [\chi])\).