Properties

Label 216.3.x.a
Level $216$
Weight $3$
Character orbit 216.x
Analytic conductor $5.886$
Analytic rank $0$
Dimension $420$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(420\)
Relative dimension: \(70\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 420q - 6q^{2} - 6q^{4} - 6q^{6} - 12q^{7} - 9q^{8} - 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 420q - 6q^{2} - 6q^{4} - 6q^{6} - 12q^{7} - 9q^{8} - 12q^{9} - 3q^{10} - 51q^{12} + 39q^{14} - 12q^{15} - 6q^{16} - 18q^{17} - 27q^{18} + 57q^{20} - 6q^{22} - 12q^{23} + 126q^{24} - 12q^{25} - 12q^{28} + 87q^{30} - 12q^{31} + 84q^{32} - 36q^{33} - 18q^{34} - 36q^{36} - 108q^{38} - 12q^{39} + 69q^{40} - 84q^{41} + 114q^{42} - 657q^{44} - 3q^{46} - 12q^{47} - 453q^{48} - 12q^{49} + 153q^{50} + 21q^{52} - 90q^{54} - 24q^{55} + 99q^{56} - 66q^{57} + 129q^{58} + 210q^{60} - 900q^{62} + 468q^{63} - 3q^{64} - 12q^{65} - 855q^{66} + 279q^{68} + 153q^{70} - 18q^{71} - 156q^{72} - 6q^{73} + 423q^{74} - 54q^{76} + 642q^{78} - 12q^{79} - 12q^{81} - 12q^{82} + 192q^{84} + 606q^{86} - 588q^{87} + 186q^{88} - 18q^{89} - 534q^{90} - 735q^{92} + 345q^{94} - 162q^{95} - 486q^{96} - 12q^{97} - 1548q^{98} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.99762 0.0976193i −1.40653 + 2.64984i 3.98094 + 0.390012i −7.28649 + 2.65207i 3.06839 5.15606i −0.503077 2.85309i −7.91432 1.16771i −5.04332 7.45419i 14.8145 4.58651i
5.2 −1.98021 + 0.280686i −1.96788 + 2.26438i 3.84243 1.11163i 8.66971 3.15552i 3.26123 5.03630i −0.680238 3.85782i −7.29678 + 3.27978i −1.25487 8.91209i −16.2821 + 8.68204i
5.3 −1.97743 0.299639i 1.51508 + 2.58931i 3.82043 + 1.18503i 0.829293 0.301838i −2.22009 5.57415i 1.72439 + 9.77949i −7.19954 3.48806i −4.40908 + 7.84602i −1.73031 + 0.348374i
5.4 −1.97718 + 0.301233i 1.67983 2.48559i 3.81852 1.19119i −4.31690 + 1.57122i −2.57260 + 5.42049i 1.15077 + 6.52637i −7.19109 + 3.50546i −3.35631 8.35076i 8.06201 4.40700i
5.5 −1.94771 0.454324i −0.0785589 2.99897i 3.58718 + 1.76979i 0.792146 0.288318i −1.20949 + 5.87683i −1.04591 5.93165i −6.18274 5.07678i −8.98766 + 0.471192i −1.67386 + 0.201669i
5.6 −1.93633 + 0.500639i 2.99956 0.0513619i 3.49872 1.93880i −3.68720 + 1.34203i −5.78241 + 1.60115i −0.898934 5.09811i −5.80403 + 5.50574i 8.99472 0.308126i 6.46775 4.44457i
5.7 −1.91201 0.586697i −2.76561 1.16249i 3.31157 + 2.24354i 3.88145 1.41273i 4.60586 + 3.84527i 0.551431 + 3.12732i −5.01548 6.23257i 6.29724 + 6.42999i −8.25022 + 0.423924i
5.8 −1.88656 + 0.664009i −2.86047 0.904260i 3.11818 2.50538i −1.90855 + 0.694657i 5.99688 0.193444i −1.88770 10.7057i −4.21904 + 6.79704i 7.36463 + 5.17322i 3.13934 2.57781i
5.9 −1.86507 + 0.722171i −1.09942 2.79129i 2.95694 2.69379i 6.30966 2.29653i 4.06627 + 4.41197i 1.48683 + 8.43221i −3.56950 + 7.15952i −6.58256 + 6.13758i −10.1094 + 8.83983i
5.10 −1.85293 0.752764i 2.85359 0.925745i 2.86669 + 2.78964i 6.33667 2.30636i −5.98437 0.432745i −0.404112 2.29183i −3.21184 7.32694i 7.28599 5.28340i −13.4775 0.496500i
5.11 −1.71853 1.02307i −2.36755 1.84247i 1.90667 + 3.51633i −7.66878 + 2.79121i 2.18374 + 5.58849i 0.943246 + 5.34941i 0.320773 7.99357i 2.21063 + 8.72428i 16.0346 + 3.04890i
5.12 −1.68433 + 1.07844i −2.97464 + 0.389273i 1.67392 3.63290i −3.75258 + 1.36583i 4.59046 3.86364i 2.21734 + 12.5752i 1.09844 + 7.92423i 8.69693 2.31589i 4.84761 6.34745i
5.13 −1.65199 + 1.12735i 0.0725877 + 2.99912i 1.45817 3.72475i 2.01543 0.733558i −3.50097 4.87270i 0.236981 + 1.34399i 1.79021 + 7.79713i −8.98946 + 0.435399i −2.50251 + 3.48393i
5.14 −1.64616 + 1.13585i 2.96853 + 0.433379i 1.41970 3.73958i 6.44507 2.34581i −5.37894 + 2.65839i 1.04940 + 5.95146i 1.91054 + 7.76852i 8.62436 + 2.57300i −7.94514 + 11.1822i
5.15 −1.62902 1.16029i −2.58836 + 1.51670i 1.30744 + 3.78029i 0.174835 0.0636348i 5.97632 + 0.532510i −0.481503 2.73074i 2.25640 7.67520i 4.39922 7.85155i −0.358646 0.0991975i
5.16 −1.59804 1.20262i 2.77706 + 1.13487i 1.10743 + 3.84364i −6.71804 + 2.44517i −3.07303 5.15330i −0.389417 2.20849i 2.85270 7.47409i 6.42414 + 6.30321i 13.6763 + 4.17175i
5.17 −1.54566 1.26923i 0.649225 + 2.92891i 0.778100 + 3.92359i 2.81183 1.02342i 2.71399 5.35110i −1.90676 10.8138i 3.77727 7.05211i −8.15701 + 3.80304i −5.64509 1.98701i
5.18 −1.42825 + 1.40003i 1.48437 2.60704i 0.0798093 3.99920i 4.33790 1.57887i 1.52990 + 5.80167i −1.85705 10.5318i 5.48503 + 5.82361i −4.59332 7.73960i −3.98515 + 8.32823i
5.19 −1.37398 + 1.45333i 1.48412 + 2.60718i −0.224349 3.99370i −5.29643 + 1.92774i −5.82825 1.42529i −0.799013 4.53143i 6.11243 + 5.16122i −4.59476 + 7.73875i 4.47555 10.3462i
5.20 −1.34054 1.48423i 1.66458 2.49583i −0.405897 + 3.97935i −1.96446 + 0.715007i −5.93583 + 0.875140i 2.11808 + 12.0122i 6.45041 4.73204i −3.45835 8.30902i 3.69468 + 1.95723i
See next 80 embeddings (of 420 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
27.f odd 18 1 inner
216.x odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.x.a 420
8.b even 2 1 inner 216.3.x.a 420
27.f odd 18 1 inner 216.3.x.a 420
216.x odd 18 1 inner 216.3.x.a 420
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.x.a 420 1.a even 1 1 trivial
216.3.x.a 420 8.b even 2 1 inner
216.3.x.a 420 27.f odd 18 1 inner
216.3.x.a 420 216.x odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(216, [\chi])\).