# 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.

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

## 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 $$T_{5}^{24} - 12 T_{5}^{23} - 360 T_{5}^{22} + 6699 T_{5}^{21} + 31221 T_{5}^{20} - 1138725 T_{5}^{19} + 32801526 T_{5}^{18} - 187511760 T_{5}^{17} + 1422469026 T_{5}^{16} + 40147364502 T_{5}^{15} + \cdots + 37\!\cdots\!01$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.