# Properties

 Label 216.3.p.b Level $216$ Weight $3$ Character orbit 216.p Analytic conductor $5.886$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 72) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 5q^{2} + 7q^{4} - 46q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 5q^{2} + 7q^{4} - 46q^{8} - 12q^{10} + 16q^{11} - 6q^{14} + 31q^{16} + 4q^{17} - 76q^{19} + 12q^{20} + 35q^{22} + 118q^{25} + 72q^{26} - 36q^{28} + 5q^{32} + 5q^{34} + 108q^{35} + 169q^{38} - 6q^{40} - 20q^{41} - 16q^{43} - 362q^{44} - 96q^{46} + 166q^{49} - 73q^{50} - 24q^{52} - 186q^{56} + 36q^{58} + 64q^{59} - 384q^{62} - 518q^{64} + 102q^{65} - 64q^{67} + 295q^{68} - 6q^{70} - 292q^{73} - 318q^{74} + 197q^{76} + 720q^{80} + 386q^{82} - 554q^{83} + 295q^{86} + 59q^{88} + 688q^{89} - 204q^{91} + 378q^{92} - 66q^{94} + 92q^{97} + 614q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.99426 + 0.151428i 0 3.95414 0.603974i 1.70411 0.983869i 0 −8.69613 5.02071i −7.79412 + 1.80325i 0 −3.24945 + 2.22014i
19.2 −1.94382 + 0.470721i 0 3.55684 1.82999i 5.15803 2.97799i 0 4.09037 + 2.36158i −6.05243 + 5.23145i 0 −8.62446 + 8.21666i
19.3 −1.89833 0.629552i 0 3.20733 + 2.39020i −5.84790 + 3.37629i 0 −3.50808 2.02539i −4.58382 6.55657i 0 13.2268 2.72775i
19.4 −1.75466 0.959768i 0 2.15769 + 3.36814i 6.07342 3.50649i 0 8.07247 + 4.66064i −0.553388 7.98084i 0 −14.0222 + 0.323638i
19.5 −1.67744 + 1.08913i 0 1.62758 3.65390i −5.42020 + 3.12935i 0 5.96345 + 3.44300i 1.24943 + 7.90183i 0 5.68375 11.1526i
19.6 −1.35259 1.47326i 0 −0.340985 + 3.98544i −0.0166003 + 0.00958419i 0 −4.07208 2.35102i 6.33280 4.88832i 0 0.0365734 + 0.0114930i
19.7 −0.827034 1.82099i 0 −2.63203 + 3.01205i −3.84571 + 2.22032i 0 0.704321 + 0.406640i 7.66169 + 2.30184i 0 7.22371 + 5.16672i
19.8 −0.104500 + 1.99727i 0 −3.97816 0.417428i 5.42020 3.12935i 0 −5.96345 3.44300i 1.24943 7.90183i 0 5.68375 + 11.1526i
19.9 −0.0392136 1.99962i 0 −3.99692 + 0.156824i 1.50948 0.871501i 0 −7.93804 4.58303i 0.470322 + 7.98616i 0 −1.80186 2.98421i
19.10 0.263796 1.98253i 0 −3.86082 1.04596i −4.40783 + 2.54486i 0 10.9609 + 6.32830i −3.09212 + 7.37826i 0 3.88249 + 9.40996i
19.11 0.460815 1.94619i 0 −3.57530 1.79367i 8.07964 4.66478i 0 4.91220 + 2.83606i −5.13836 + 6.13166i 0 −5.35532 17.8741i
19.12 0.564251 + 1.91875i 0 −3.36324 + 2.16532i −5.15803 + 2.97799i 0 −4.09037 2.36158i −6.05243 5.23145i 0 −8.62446 8.21666i
19.13 0.865989 + 1.80279i 0 −2.50013 + 3.12240i −1.70411 + 0.983869i 0 8.69613 + 5.02071i −7.79412 1.80325i 0 −3.24945 2.22014i
19.14 1.45504 1.37217i 0 0.234289 3.99313i −8.07964 + 4.66478i 0 −4.91220 2.83606i −5.13836 6.13166i 0 −5.35532 + 17.8741i
19.15 1.49437 + 1.32923i 0 0.466308 + 3.97273i 5.84790 3.37629i 0 3.50808 + 2.02539i −4.58382 + 6.55657i 0 13.2268 + 2.72775i
19.16 1.58502 1.21972i 0 1.02458 3.86655i 4.40783 2.54486i 0 −10.9609 6.32830i −3.09212 7.37826i 0 3.88249 9.40996i
19.17 1.70852 + 1.03970i 0 1.83805 + 3.55269i −6.07342 + 3.50649i 0 −8.07247 4.66064i −0.553388 + 7.98084i 0 −14.0222 0.323638i
19.18 1.75132 0.965848i 0 2.13428 3.38303i −1.50948 + 0.871501i 0 7.93804 + 4.58303i 0.470322 7.98616i 0 −1.80186 + 2.98421i
19.19 1.95218 + 0.434750i 0 3.62198 + 1.69742i 0.0166003 0.00958419i 0 4.07208 + 2.35102i 6.33280 + 4.88832i 0 0.0365734 0.0114930i
19.20 1.99054 0.194264i 0 3.92452 0.773382i 3.84571 2.22032i 0 −0.704321 0.406640i 7.66169 2.30184i 0 7.22371 5.16672i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.c even 3 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.p.b 40
3.b odd 2 1 72.3.p.b 40
4.b odd 2 1 864.3.t.b 40
8.b even 2 1 864.3.t.b 40
8.d odd 2 1 inner 216.3.p.b 40
9.c even 3 1 inner 216.3.p.b 40
9.c even 3 1 648.3.b.e 20
9.d odd 6 1 72.3.p.b 40
9.d odd 6 1 648.3.b.f 20
12.b even 2 1 288.3.t.b 40
24.f even 2 1 72.3.p.b 40
24.h odd 2 1 288.3.t.b 40
36.f odd 6 1 864.3.t.b 40
36.f odd 6 1 2592.3.b.f 20
36.h even 6 1 288.3.t.b 40
36.h even 6 1 2592.3.b.e 20
72.j odd 6 1 288.3.t.b 40
72.j odd 6 1 2592.3.b.e 20
72.l even 6 1 72.3.p.b 40
72.l even 6 1 648.3.b.f 20
72.n even 6 1 864.3.t.b 40
72.n even 6 1 2592.3.b.f 20
72.p odd 6 1 inner 216.3.p.b 40
72.p odd 6 1 648.3.b.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.p.b 40 3.b odd 2 1
72.3.p.b 40 9.d odd 6 1
72.3.p.b 40 24.f even 2 1
72.3.p.b 40 72.l even 6 1
216.3.p.b 40 1.a even 1 1 trivial
216.3.p.b 40 8.d odd 2 1 inner
216.3.p.b 40 9.c even 3 1 inner
216.3.p.b 40 72.p odd 6 1 inner
288.3.t.b 40 12.b even 2 1
288.3.t.b 40 24.h odd 2 1
288.3.t.b 40 36.h even 6 1
288.3.t.b 40 72.j odd 6 1
648.3.b.e 20 9.c even 3 1
648.3.b.e 20 72.p odd 6 1
648.3.b.f 20 9.d odd 6 1
648.3.b.f 20 72.l even 6 1
864.3.t.b 40 4.b odd 2 1
864.3.t.b 40 8.b even 2 1
864.3.t.b 40 36.f odd 6 1
864.3.t.b 40 72.n even 6 1
2592.3.b.e 20 36.h even 6 1
2592.3.b.e 20 72.j odd 6 1
2592.3.b.f 20 36.f odd 6 1
2592.3.b.f 20 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$23\!\cdots\!89$$$$T_{5}^{28} -$$$$10\!\cdots\!87$$$$T_{5}^{26} +$$$$37\!\cdots\!45$$$$T_{5}^{24} -$$$$11\!\cdots\!58$$$$T_{5}^{22} +$$$$26\!\cdots\!49$$$$T_{5}^{20} -$$$$50\!\cdots\!43$$$$T_{5}^{18} +$$$$74\!\cdots\!08$$$$T_{5}^{16} -$$$$81\!\cdots\!17$$$$T_{5}^{14} +$$$$63\!\cdots\!73$$$$T_{5}^{12} -$$$$30\!\cdots\!76$$$$T_{5}^{10} +$$$$10\!\cdots\!00$$$$T_{5}^{8} -$$$$20\!\cdots\!96$$$$T_{5}^{6} +$$$$26\!\cdots\!24$$$$T_{5}^{4} -$$$$97\!\cdots\!60$$$$T_{5}^{2} +$$$$35\!\cdots\!00$$">$$T_{5}^{40} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.