# Properties

 Label 162.4.e.b Level 162 Weight 4 Character orbit 162.e Analytic conductor 9.558 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 162.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.55830942093$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$5$$ over $$\Q(\zeta_{9})$$ Twist minimal: no (minimal twist has level 54) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 12q^{5} + 33q^{7} - 120q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 12q^{5} + 33q^{7} - 120q^{8} - 30q^{10} + 39q^{11} - 60q^{13} + 66q^{14} - 102q^{17} - 171q^{19} - 96q^{20} + 78q^{22} - 48q^{23} - 432q^{25} + 468q^{26} + 336q^{28} + 381q^{29} - 801q^{31} - 222q^{34} - 624q^{35} - 555q^{37} - 606q^{38} + 96q^{40} - 1401q^{41} + 648q^{43} - 132q^{44} - 348q^{46} - 540q^{47} + 15q^{49} + 828q^{50} - 240q^{52} + 1794q^{53} + 3906q^{55} + 264q^{56} - 444q^{58} + 1500q^{59} - 378q^{61} - 744q^{62} - 960q^{64} - 3666q^{65} + 3087q^{67} + 24q^{68} + 2118q^{70} - 120q^{71} - 2604q^{73} + 1974q^{74} - 1212q^{76} + 6504q^{77} - 2625q^{79} + 480q^{80} + 3408q^{82} + 5211q^{83} - 1395q^{85} + 1296q^{86} - 912q^{88} - 2604q^{89} - 3399q^{91} - 372q^{92} + 4500q^{94} - 7545q^{95} + 8940q^{97} - 4002q^{98} + O(q^{100})$$

## 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 −1.87939 + 0.684040i 0 3.06418 2.57115i −3.05422 17.3214i 0 −19.9833 16.7680i −4.00000 + 6.92820i 0 17.5886 + 30.4643i
19.2 −1.87939 + 0.684040i 0 3.06418 2.57115i −2.39299 13.5713i 0 20.6081 + 17.2922i −4.00000 + 6.92820i 0 13.7807 + 23.8688i
19.3 −1.87939 + 0.684040i 0 3.06418 2.57115i −0.127307 0.721992i 0 1.18538 + 0.994655i −4.00000 + 6.92820i 0 0.733130 + 1.26982i
19.4 −1.87939 + 0.684040i 0 3.06418 2.57115i 2.36712 + 13.4246i 0 −10.7136 8.98975i −4.00000 + 6.92820i 0 −13.6317 23.6108i
19.5 −1.87939 + 0.684040i 0 3.06418 2.57115i 3.01146 + 17.0788i 0 16.7015 + 14.0142i −4.00000 + 6.92820i 0 −17.3423 30.0378i
37.1 1.53209 1.28558i 0 0.694593 3.93923i −13.2521 + 4.82337i 0 3.00195 + 17.0249i −4.00000 6.92820i 0 −14.1026 + 24.4264i
37.2 1.53209 1.28558i 0 0.694593 3.93923i −11.4993 + 4.18541i 0 3.15689 + 17.9036i −4.00000 6.92820i 0 −12.2373 + 21.1956i
37.3 1.53209 1.28558i 0 0.694593 3.93923i −3.40278 + 1.23851i 0 −3.51598 19.9401i −4.00000 6.92820i 0 −3.62116 + 6.27204i
37.4 1.53209 1.28558i 0 0.694593 3.93923i 7.46345 2.71647i 0 −2.66046 15.0882i −4.00000 6.92820i 0 7.94244 13.7567i
37.5 1.53209 1.28558i 0 0.694593 3.93923i 17.2977 6.29584i 0 6.03855 + 34.2463i −4.00000 6.92820i 0 18.4078 31.8832i
73.1 0.347296 1.96962i 0 −3.75877 1.36808i −7.73816 + 6.49309i 0 7.54967 2.74785i −4.00000 + 6.92820i 0 10.1015 + 17.4962i
73.2 0.347296 1.96962i 0 −3.75877 1.36808i −6.77242 + 5.68273i 0 30.7336 11.1861i −4.00000 + 6.92820i 0 8.84076 + 15.3127i
73.3 0.347296 1.96962i 0 −3.75877 1.36808i −0.657259 + 0.551505i 0 −20.5288 + 7.47187i −4.00000 + 6.92820i 0 0.857990 + 1.48608i
73.4 0.347296 1.96962i 0 −3.75877 1.36808i 9.84993 8.26508i 0 −26.8516 + 9.77320i −4.00000 + 6.92820i 0 −12.8582 22.2710i
73.5 0.347296 1.96962i 0 −3.75877 1.36808i 14.9069 12.5084i 0 11.7781 4.28689i −4.00000 + 6.92820i 0 −19.4596 33.7050i
91.1 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −7.73816 6.49309i 0 7.54967 + 2.74785i −4.00000 6.92820i 0 10.1015 17.4962i
91.2 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −6.77242 5.68273i 0 30.7336 + 11.1861i −4.00000 6.92820i 0 8.84076 15.3127i
91.3 0.347296 + 1.96962i 0 −3.75877 + 1.36808i −0.657259 0.551505i 0 −20.5288 7.47187i −4.00000 6.92820i 0 0.857990 1.48608i
91.4 0.347296 + 1.96962i 0 −3.75877 + 1.36808i 9.84993 + 8.26508i 0 −26.8516 9.77320i −4.00000 6.92820i 0 −12.8582 + 22.2710i
91.5 0.347296 + 1.96962i 0 −3.75877 + 1.36808i 14.9069 + 12.5084i 0 11.7781 + 4.28689i −4.00000 6.92820i 0 −19.4596 + 33.7050i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 162.4.e.b 30
3.b odd 2 1 54.4.e.b 30
27.e even 9 1 inner 162.4.e.b 30
27.e even 9 1 1458.4.a.j 15
27.f odd 18 1 54.4.e.b 30
27.f odd 18 1 1458.4.a.i 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.b 30 3.b odd 2 1
54.4.e.b 30 27.f odd 18 1
162.4.e.b 30 1.a even 1 1 trivial
162.4.e.b 30 27.e even 9 1 inner
1458.4.a.i 15 27.f odd 18 1
1458.4.a.j 15 27.e even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{30} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database