# Properties

 Label 162.4.g.b Level $162$ Weight $4$ Character orbit 162.g Analytic conductor $9.558$ Analytic rank $0$ Dimension $252$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 162.g (of order $$27$$, degree $$18$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.55830942093$$ Analytic rank: $$0$$ Dimension: $$252$$ Relative dimension: $$14$$ over $$\Q(\zeta_{27})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$252q - 36q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$252q - 36q^{6} + 90q^{13} - 252q^{18} - 144q^{20} + 189q^{21} + 1512q^{23} + 846q^{25} + 702q^{26} + 702q^{27} - 504q^{28} + 540q^{29} - 342q^{30} - 2214q^{31} - 1548q^{33} - 1242q^{35} - 576q^{36} - 882q^{38} - 927q^{41} + 774q^{42} - 900q^{43} + 2817q^{45} - 2088q^{46} + 297q^{47} - 144q^{48} + 2151q^{51} - 720q^{52} + 1431q^{53} - 2970q^{55} + 729q^{57} + 126q^{58} + 2628q^{59} + 261q^{63} + 3627q^{65} + 4680q^{66} + 2538q^{67} - 936q^{68} + 4716q^{69} - 3150q^{70} + 720q^{71} - 1440q^{72} + 3204q^{73} - 2088q^{74} + 918q^{75} + 1764q^{76} - 9792q^{77} - 3132q^{78} + 3897q^{79} - 1440q^{80} - 17352q^{81} + 5904q^{82} - 1035q^{83} - 1224q^{84} + 2421q^{85} - 3600q^{86} - 10827q^{87} + 1224q^{88} - 4086q^{89} + 4473q^{91} + 1872q^{92} + 8955q^{93} - 6174q^{94} + 8037q^{95} + 1152q^{96} + 6192q^{97} + 8136q^{98} + 2187q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.19432 1.60425i −4.99896 1.41789i −1.14721 + 3.83196i 13.9580 9.18030i 3.69570 + 9.71297i −16.3937 + 17.3763i 7.51754 2.73616i 22.9792 + 14.1759i −31.3977 11.4278i
7.2 −1.19432 1.60425i −4.83160 + 1.91197i −1.14721 + 3.83196i 1.51049 0.993463i 8.83774 + 5.46758i 8.41336 8.91764i 7.51754 2.73616i 19.6887 18.4758i −3.39776 1.23668i
7.3 −1.19432 1.60425i −4.55952 2.49214i −1.14721 + 3.83196i −10.2500 + 6.74151i 1.44751 + 10.2910i −13.2497 + 14.0439i 7.51754 2.73616i 14.5785 + 22.7259i 23.0567 + 8.39197i
7.4 −1.19432 1.60425i −3.47518 3.86305i −1.14721 + 3.83196i 5.82437 3.83075i −2.04682 + 10.1887i 21.8767 23.1880i 7.51754 2.73616i −2.84630 + 26.8496i −13.1016 4.76860i
7.5 −1.19432 1.60425i −3.14794 + 4.13406i −1.14721 + 3.83196i −12.1611 + 7.99847i 10.3917 + 0.112698i −13.0673 + 13.8505i 7.51754 2.73616i −7.18091 26.0276i 27.3557 + 9.95667i
7.6 −1.19432 1.60425i −0.821241 5.13084i −1.14721 + 3.83196i 1.88361 1.23887i −7.25032 + 7.44533i −2.53643 + 2.68846i 7.51754 2.73616i −25.6511 + 8.42732i −4.23709 1.54217i
7.7 −1.19432 1.60425i −0.639106 + 5.15670i −1.14721 + 3.83196i 4.47381 2.94247i 9.03591 5.13345i 8.78882 9.31560i 7.51754 2.73616i −26.1831 6.59135i −10.0636 3.66285i
7.8 −1.19432 1.60425i 0.771405 5.13857i −1.14721 + 3.83196i −17.0853 + 11.2372i −9.16484 + 4.89956i 7.54659 7.99892i 7.51754 2.73616i −25.8099 7.92784i 38.4326 + 13.9883i
7.9 −1.19432 1.60425i 2.06068 + 4.77007i −1.14721 + 3.83196i 6.94326 4.56666i 5.19127 9.00282i −24.6415 + 26.1185i 7.51754 2.73616i −18.5072 + 19.6592i −15.6185 5.68467i
7.10 −1.19432 1.60425i 2.83197 + 4.35660i −1.14721 + 3.83196i −15.4172 + 10.1401i 3.60678 9.74634i 13.2031 13.9945i 7.51754 2.73616i −10.9599 + 24.6755i 34.6802 + 12.6226i
7.11 −1.19432 1.60425i 3.30686 4.00807i −1.14721 + 3.83196i 18.1458 11.9347i −10.3794 0.518112i −8.00040 + 8.47993i 7.51754 2.73616i −5.12931 26.5083i −40.8181 14.8566i
7.12 −1.19432 1.60425i 3.85407 3.48513i −1.14721 + 3.83196i −1.04141 + 0.684943i −10.1940 2.02053i 3.33832 3.53842i 7.51754 2.73616i 2.70774 26.8639i 2.34259 + 0.852632i
7.13 −1.19432 1.60425i 4.68723 + 2.24275i −1.14721 + 3.83196i 16.0370 10.5477i −2.00012 10.1980i 18.3946 19.4971i 7.51754 2.73616i 16.9402 + 21.0245i −36.0744 13.1300i
7.14 −1.19432 1.60425i 5.17909 0.420697i −1.14721 + 3.83196i −4.05716 + 2.66844i −6.86038 7.80610i −14.1577 + 15.0063i 7.51754 2.73616i 26.6460 4.35766i 9.12637 + 3.32173i
13.1 −1.78727 0.897598i −5.19032 + 0.246219i 2.38863 + 3.20849i 4.68865 + 15.6612i 9.49748 + 4.21876i −0.254938 0.591012i −1.38919 7.87846i 26.8788 2.55591i 5.67759 32.1992i
13.2 −1.78727 0.897598i −5.16958 0.524798i 2.38863 + 3.20849i −4.83465 16.1489i 8.76836 + 5.57816i −8.83771 20.4881i −1.38919 7.87846i 26.4492 + 5.42597i −5.85438 + 33.2019i
13.3 −1.78727 0.897598i −3.98094 3.33948i 2.38863 + 3.20849i −3.33686 11.1459i 4.11748 + 9.54182i 11.7915 + 27.3357i −1.38919 7.87846i 4.69572 + 26.5885i −4.04068 + 22.9158i
13.4 −1.78727 0.897598i −3.27564 + 4.03363i 2.38863 + 3.20849i −0.224791 0.750853i 9.47502 4.26895i −4.80994 11.1507i −1.38919 7.87846i −5.54032 26.4255i −0.272204 + 1.54374i
13.5 −1.78727 0.897598i −3.15164 4.13124i 2.38863 + 3.20849i 1.14961 + 3.83998i 1.92461 + 10.2125i −0.280834 0.651047i −1.38919 7.87846i −7.13437 + 26.0404i 1.39209 7.89495i
13.6 −1.78727 0.897598i −0.479474 + 5.17398i 2.38863 + 3.20849i −5.85860 19.5691i 5.50111 8.81691i 9.36681 + 21.7147i −1.38919 7.87846i −26.5402 4.96158i −7.09430 + 40.2338i
See next 80 embeddings (of 252 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.g.b 252
81.g even 27 1 inner 162.4.g.b 252

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.g.b 252 1.a even 1 1 trivial
162.4.g.b 252 81.g even 27 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{252} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database