# Properties

 Label 162.4.e.a Level $162$ Weight $4$ Character orbit 162.e Analytic conductor $9.558$ Analytic rank $0$ Dimension $24$ 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: $$24$$ Relative dimension: $$4$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{5} - 33q^{7} + 96q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{5} - 33q^{7} + 96q^{8} + 30q^{10} + 12q^{11} + 60q^{13} + 66q^{14} - 102q^{17} + 171q^{19} - 96q^{20} - 24q^{22} + 708q^{23} + 864q^{25} + 468q^{26} - 336q^{28} + 381q^{29} + 909q^{31} - 48q^{34} - 624q^{35} + 555q^{37} - 66q^{38} - 96q^{40} - 618q^{41} - 1161q^{43} - 132q^{44} + 348q^{46} + 378q^{47} + 579q^{49} - 36q^{50} + 240q^{52} + 1794q^{53} - 3906q^{55} + 264q^{56} + 444q^{58} - 1038q^{59} + 324q^{61} - 744q^{62} - 768q^{64} - 5718q^{65} - 576q^{67} - 1056q^{68} - 1038q^{70} - 120q^{71} + 3036q^{73} + 1110q^{74} + 132q^{76} + 3804q^{77} - 2991q^{79} + 480q^{80} - 3408q^{82} - 513q^{83} - 2925q^{85} + 2322q^{86} + 480q^{88} - 1065q^{89} + 2859q^{91} - 1884q^{92} - 828q^{94} - 6357q^{95} - 2055q^{97} - 1356q^{98} + O(q^{100})$$

## 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 −1.31542 7.46011i 0 −2.90083 2.43409i 4.00000 6.92820i 0 −7.57519 13.1206i
19.2 1.87939 0.684040i 0 3.06418 2.57115i −1.18374 6.71331i 0 −25.7220 21.5833i 4.00000 6.92820i 0 −6.81687 11.8072i
19.3 1.87939 0.684040i 0 3.06418 2.57115i −0.203469 1.15393i 0 19.9691 + 16.7561i 4.00000 6.92820i 0 −1.17173 2.02949i
19.4 1.87939 0.684040i 0 3.06418 2.57115i 2.50669 + 14.2161i 0 0.855575 + 0.717913i 4.00000 6.92820i 0 14.4354 + 25.0029i
37.1 −1.53209 + 1.28558i 0 0.694593 3.93923i −18.7075 + 6.80898i 0 −3.26433 18.5129i 4.00000 + 6.92820i 0 19.9081 34.4819i
37.2 −1.53209 + 1.28558i 0 0.694593 3.93923i −3.45007 + 1.25572i 0 0.157049 + 0.890672i 4.00000 + 6.92820i 0 3.67149 6.35921i
37.3 −1.53209 + 1.28558i 0 0.694593 3.93923i −1.07676 + 0.391909i 0 0.470326 + 2.66735i 4.00000 + 6.92820i 0 1.14586 1.98470i
37.4 −1.53209 + 1.28558i 0 0.694593 3.93923i 19.8413 7.22164i 0 −3.38399 19.1916i 4.00000 + 6.92820i 0 −21.1147 + 36.5717i
73.1 −0.347296 + 1.96962i 0 −3.75877 1.36808i −5.41266 + 4.54176i 0 −4.71753 + 1.71704i 4.00000 6.92820i 0 −7.06572 12.2382i
73.2 −0.347296 + 1.96962i 0 −3.75877 1.36808i −3.46830 + 2.91025i 0 22.9979 8.37056i 4.00000 6.92820i 0 −4.52754 7.84193i
73.3 −0.347296 + 1.96962i 0 −3.75877 1.36808i 5.66862 4.75653i 0 −11.1935 + 4.07412i 4.00000 6.92820i 0 7.39985 + 12.8169i
73.4 −0.347296 + 1.96962i 0 −3.75877 1.36808i 12.8013 10.7416i 0 −9.76778 + 3.55518i 4.00000 6.92820i 0 16.7110 + 28.9442i
91.1 −0.347296 1.96962i 0 −3.75877 + 1.36808i −5.41266 4.54176i 0 −4.71753 1.71704i 4.00000 + 6.92820i 0 −7.06572 + 12.2382i
91.2 −0.347296 1.96962i 0 −3.75877 + 1.36808i −3.46830 2.91025i 0 22.9979 + 8.37056i 4.00000 + 6.92820i 0 −4.52754 + 7.84193i
91.3 −0.347296 1.96962i 0 −3.75877 + 1.36808i 5.66862 + 4.75653i 0 −11.1935 4.07412i 4.00000 + 6.92820i 0 7.39985 12.8169i
91.4 −0.347296 1.96962i 0 −3.75877 + 1.36808i 12.8013 + 10.7416i 0 −9.76778 3.55518i 4.00000 + 6.92820i 0 16.7110 28.9442i
127.1 −1.53209 1.28558i 0 0.694593 + 3.93923i −18.7075 6.80898i 0 −3.26433 + 18.5129i 4.00000 6.92820i 0 19.9081 + 34.4819i
127.2 −1.53209 1.28558i 0 0.694593 + 3.93923i −3.45007 1.25572i 0 0.157049 0.890672i 4.00000 6.92820i 0 3.67149 + 6.35921i
127.3 −1.53209 1.28558i 0 0.694593 + 3.93923i −1.07676 0.391909i 0 0.470326 2.66735i 4.00000 6.92820i 0 1.14586 + 1.98470i
127.4 −1.53209 1.28558i 0 0.694593 + 3.93923i 19.8413 + 7.22164i 0 −3.38399 + 19.1916i 4.00000 6.92820i 0 −21.1147 36.5717i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.4 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.a 24
3.b odd 2 1 54.4.e.a 24
27.e even 9 1 inner 162.4.e.a 24
27.e even 9 1 1458.4.a.e 12
27.f odd 18 1 54.4.e.a 24
27.f odd 18 1 1458.4.a.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.e.a 24 3.b odd 2 1
54.4.e.a 24 27.f odd 18 1
162.4.e.a 24 1.a even 1 1 trivial
162.4.e.a 24 27.e even 9 1 inner
1458.4.a.e 12 27.e even 9 1
1458.4.a.h 12 27.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{24} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database