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Properties

Label 54.8.e.b
Level $54$
Weight $8$
Character orbit 54.e
Analytic conductor $16.869$
Analytic rank $0$
Dimension $66$
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] = [54,8,Mod(7,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.e (of order \(9\), degree \(6\), 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: \(16.8687913761\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9} - 3000 q^{10} - 2523 q^{11} - 5376 q^{12} - 9924 q^{13} + 13416 q^{14} - 15084 q^{15} + 29478 q^{17} + 22848 q^{18} - 61731 q^{19} - 27264 q^{20} - 36372 q^{21} + 20184 q^{22} + 255234 q^{23} + 558999 q^{25} - 316368 q^{26} - 51597 q^{27} + 263424 q^{28} + 132153 q^{29} + 610632 q^{30} + 776898 q^{31} + 645795 q^{33} + 387840 q^{34} - 321081 q^{35} - 314688 q^{36} - 759795 q^{37} + 174504 q^{38} - 84012 q^{39} - 109056 q^{40} + 349287 q^{41} + 313752 q^{42} - 1043622 q^{43} + 255552 q^{44} + 845496 q^{45} - 524472 q^{46} + 1280709 q^{47} - 872448 q^{48} + 1011003 q^{49} - 1537704 q^{50} - 7812153 q^{51} - 635136 q^{52} - 2951976 q^{53} - 1256832 q^{54} + 2539854 q^{55} + 858624 q^{56} - 3973923 q^{57} - 47496 q^{58} + 639426 q^{59} - 94464 q^{60} + 1639350 q^{61} + 2859936 q^{62} + 6955635 q^{63} - 8650752 q^{64} - 4119567 q^{65} + 712512 q^{66} - 6408441 q^{67} - 1634496 q^{68} - 1007298 q^{69} + 2380872 q^{70} + 3531612 q^{71} + 6623232 q^{72} - 6666213 q^{73} + 7873368 q^{74} + 16535910 q^{75} - 1396032 q^{76} - 8256489 q^{77} - 4117104 q^{78} - 22540695 q^{79} - 3072000 q^{80} - 13134636 q^{81} + 3798096 q^{82} + 8663238 q^{83} + 8570304 q^{84} - 15143175 q^{85} + 8348976 q^{86} + 45908613 q^{87} + 11028480 q^{88} + 21718137 q^{89} - 27904392 q^{90} - 8606796 q^{91} - 8332800 q^{92} - 20412249 q^{93} - 10358784 q^{94} - 21603363 q^{95} - 5505024 q^{96} - 17397492 q^{97} + 44171448 q^{98} + 103927383 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 7.51754 − 2.73616i −45.8155 + 9.37779i 49.0268 − 41.1384i 28.4674 + 161.447i −318.761 + 195.856i −626.525 − 525.717i 256.000 − 443.405i 2011.11 − 859.295i 655.749 + 1135.79i
7.2 7.51754 − 2.73616i −45.6144 − 10.3117i 49.0268 − 41.1384i −8.01649 − 45.4638i −371.122 + 47.2898i 1037.92 + 870.920i 256.000 − 443.405i 1974.34 + 940.721i −184.661 − 319.841i
7.3 7.51754 − 2.73616i −31.2009 + 34.8354i 49.0268 − 41.1384i −25.2309 − 143.092i −139.239 + 347.247i 591.389 + 496.234i 256.000 − 443.405i −240.006 − 2173.79i −581.196 − 1006.66i
7.4 7.51754 − 2.73616i −30.6695 − 35.3041i 49.0268 − 41.1384i −78.9512 − 447.755i −327.157 − 181.484i −442.034 − 370.911i 256.000 − 443.405i −305.764 + 2165.52i −1818.65 − 3149.99i
7.5 7.51754 − 2.73616i −24.5414 − 39.8086i 49.0268 − 41.1384i 50.0623 + 283.917i −293.413 − 232.113i −793.355 − 665.704i 256.000 − 443.405i −982.442 + 1953.91i 1153.19 + 1997.38i
7.6 7.51754 − 2.73616i 6.43466 − 46.3206i 49.0268 − 41.1384i 60.2827 + 341.880i −78.3677 − 365.823i 1314.87 + 1103.31i 256.000 − 443.405i −2104.19 − 596.114i 1388.62 + 2405.15i
7.7 7.51754 − 2.73616i 8.34949 + 46.0140i 49.0268 − 41.1384i −47.1028 − 267.133i 188.669 + 323.066i −885.713 − 743.201i 256.000 − 443.405i −2047.57 + 768.386i −1085.02 − 1879.30i
7.8 7.51754 − 2.73616i 33.7094 − 32.4142i 49.0268 − 41.1384i 13.6066 + 77.1671i 164.721 − 335.909i −711.029 − 596.624i 256.000 − 443.405i 85.6427 − 2185.32i 313.430 + 542.877i
7.9 7.51754 − 2.73616i 35.4628 − 30.4859i 49.0268 − 41.1384i −84.4758 − 479.086i 183.179 − 326.211i 666.476 + 559.239i 256.000 − 443.405i 328.222 − 2162.23i −1945.91 − 3370.41i
7.10 7.51754 − 2.73616i 37.5285 + 27.9037i 49.0268 − 41.1384i −21.4637 − 121.726i 358.471 + 107.083i 674.452 + 565.932i 256.000 − 443.405i 629.771 + 2094.36i −494.417 − 856.356i
7.11 7.51754 − 2.73616i 43.9672 + 15.9339i 49.0268 − 41.1384i 72.2267 + 409.618i 374.123 − 0.517653i −152.227 − 127.734i 256.000 − 443.405i 1679.22 + 1401.13i 1663.75 + 2881.70i
13.1 −1.38919 − 7.87846i −45.0005 − 12.7261i −60.1403 + 21.8893i −56.5716 − 47.4692i −37.7478 + 372.214i 1413.94 + 514.632i 256.000 + 443.405i 1863.09 + 1145.36i −295.396 + 511.640i
13.2 −1.38919 − 7.87846i −44.6459 − 13.9191i −60.1403 + 21.8893i 200.637 + 168.354i −47.6395 + 371.077i −437.811 − 159.350i 256.000 + 443.405i 1799.52 + 1242.86i 1047.65 − 1814.59i
13.3 −1.38919 − 7.87846i −43.4440 + 17.3094i −60.1403 + 21.8893i −179.661 − 150.754i 196.723 + 318.226i −1557.21 − 566.780i 256.000 + 443.405i 1587.77 − 1503.98i −938.125 + 1624.88i
13.4 −1.38919 − 7.87846i −23.2566 + 40.5725i −60.1403 + 21.8893i 297.214 + 249.392i 351.957 + 126.863i 210.527 + 76.6255i 256.000 + 443.405i −1105.26 − 1887.16i 1551.94 − 2688.04i
13.5 −1.38919 − 7.87846i −7.96623 − 46.0819i −60.1403 + 21.8893i −406.101 − 340.759i −351.988 + 126.778i −407.930 − 148.474i 256.000 + 443.405i −2060.08 + 734.197i −2120.51 + 3672.83i
13.6 −1.38919 − 7.87846i −2.46752 − 46.7002i −60.1403 + 21.8893i 138.474 + 116.193i −364.498 + 84.3155i 95.8895 + 34.9009i 256.000 + 443.405i −2174.82 + 230.467i 723.058 − 1252.37i
13.7 −1.38919 − 7.87846i 10.3801 + 45.5988i −60.1403 + 21.8893i −119.330 − 100.129i 344.829 − 145.124i 96.2629 + 35.0368i 256.000 + 443.405i −1971.51 + 946.640i −623.095 + 1079.23i
13.8 −1.38919 − 7.87846i 33.5262 − 32.6035i −60.1403 + 21.8893i −0.605177 − 0.507804i −303.440 − 218.843i 1422.78 + 517.850i 256.000 + 443.405i 61.0178 − 2186.15i −3.16001 + 5.47330i
13.9 −1.38919 − 7.87846i 38.4590 − 26.6064i −60.1403 + 21.8893i 337.131 + 282.886i −263.044 − 266.037i −1612.97 − 587.074i 256.000 + 443.405i 771.197 − 2046.52i 1760.37 − 3049.05i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.8.e.b ✓ 66
27.e even 9 1 inner 54.8.e.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.e.b ✓ 66 1.a even 1 1 trivial
54.8.e.b ✓ 66 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{66} - 213 T_{5}^{65} - 256815 T_{5}^{64} + 11205726 T_{5}^{63} + 31529071476 T_{5}^{62} + \cdots + 21\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display