# Properties

 Label 162.4.g.a Level $162$ Weight $4$ Character orbit 162.g Analytic conductor $9.558$ Analytic rank $0$ Dimension $234$ 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: $$234$$ Relative dimension: $$13$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$234 q + 36 q^{6}+O(q^{10})$$ 234 * q + 36 * q^6 $$\operatorname{Tr}(f)(q) =$$ $$234 q + 36 q^{6} - 90 q^{13} - 162 q^{18} - 144 q^{20} - 405 q^{21} - 756 q^{23} - 846 q^{25} + 702 q^{26} + 702 q^{27} + 504 q^{28} + 540 q^{29} + 1098 q^{30} + 2214 q^{31} + 684 q^{33} - 1242 q^{35} - 576 q^{36} - 72 q^{38} - 927 q^{41} - 774 q^{42} + 900 q^{43} - 3843 q^{45} + 2088 q^{46} + 297 q^{47} + 144 q^{48} + 810 q^{51} + 720 q^{52} + 1431 q^{53} + 2970 q^{55} + 1485 q^{57} - 126 q^{58} - 1179 q^{59} - 2259 q^{63} + 3627 q^{65} + 4680 q^{66} - 8046 q^{67} + 2304 q^{68} - 594 q^{69} + 1530 q^{70} + 720 q^{71} + 864 q^{72} - 3204 q^{73} - 3384 q^{74} - 9918 q^{75} - 144 q^{76} - 9792 q^{77} - 7524 q^{78} + 4527 q^{79} - 1440 q^{80} + 5832 q^{81} - 5904 q^{82} - 9621 q^{83} - 1224 q^{84} + 4059 q^{85} - 3600 q^{86} - 117 q^{87} - 576 q^{88} + 531 q^{89} + 1440 q^{90} - 4473 q^{91} + 1872 q^{92} + 135 q^{93} + 666 q^{94} + 8037 q^{95} + 1728 q^{96} - 16560 q^{97} + 8136 q^{98} + 18567 q^{99}+O(q^{100})$$ 234 * q + 36 * q^6 - 90 * q^13 - 162 * q^18 - 144 * q^20 - 405 * q^21 - 756 * q^23 - 846 * q^25 + 702 * q^26 + 702 * q^27 + 504 * q^28 + 540 * q^29 + 1098 * q^30 + 2214 * q^31 + 684 * q^33 - 1242 * q^35 - 576 * q^36 - 72 * q^38 - 927 * q^41 - 774 * q^42 + 900 * q^43 - 3843 * q^45 + 2088 * q^46 + 297 * q^47 + 144 * q^48 + 810 * q^51 + 720 * q^52 + 1431 * q^53 + 2970 * q^55 + 1485 * q^57 - 126 * q^58 - 1179 * q^59 - 2259 * q^63 + 3627 * q^65 + 4680 * q^66 - 8046 * q^67 + 2304 * q^68 - 594 * q^69 + 1530 * q^70 + 720 * q^71 + 864 * q^72 - 3204 * q^73 - 3384 * q^74 - 9918 * q^75 - 144 * q^76 - 9792 * q^77 - 7524 * q^78 + 4527 * q^79 - 1440 * q^80 + 5832 * q^81 - 5904 * q^82 - 9621 * q^83 - 1224 * q^84 + 4059 * q^85 - 3600 * q^86 - 117 * q^87 - 576 * q^88 + 531 * q^89 + 1440 * q^90 - 4473 * q^91 + 1872 * q^92 + 135 * q^93 + 666 * q^94 + 8037 * q^95 + 1728 * q^96 - 16560 * q^97 + 8136 * q^98 + 18567 * q^99

## 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 −5.09650 1.01278i −1.14721 + 3.83196i 8.61968 5.66925i −4.46208 9.38562i −8.82631 + 9.35534i −7.51754 + 2.73616i 24.9485 + 10.3233i 19.3895 + 7.05721i
7.2 1.19432 + 1.60425i −5.05924 + 1.18494i −1.14721 + 3.83196i −13.6695 + 8.99057i −7.94328 6.70107i −2.17193 + 2.30211i −7.51754 + 2.73616i 24.1918 11.9898i −30.7488 11.1917i
7.3 1.19432 + 1.60425i −4.09632 + 3.19690i −1.14721 + 3.83196i 9.47743 6.23340i −10.0209 2.75340i 22.4337 23.7784i −7.51754 + 2.73616i 6.55972 26.1910i 21.3190 + 7.75947i
7.4 1.19432 + 1.60425i −3.95175 3.37396i −1.14721 + 3.83196i −2.77429 + 1.82468i 0.693022 10.3692i 8.88110 9.41341i −7.51754 + 2.73616i 4.23273 + 26.6662i −6.24061 2.27140i
7.5 1.19432 + 1.60425i −1.73958 4.89631i −1.14721 + 3.83196i 11.9812 7.88019i 5.77728 8.63846i −7.16944 + 7.59916i −7.51754 + 2.73616i −20.9477 + 17.0351i 26.9512 + 9.80942i
7.6 1.19432 + 1.60425i −0.706260 + 5.14793i −1.14721 + 3.83196i 0.363184 0.238870i −9.10205 + 5.01525i −1.02179 + 1.08304i −7.51754 + 2.73616i −26.0024 7.27155i 0.816963 + 0.297350i
7.7 1.19432 + 1.60425i −0.319078 + 5.18635i −1.14721 + 3.83196i −6.04374 + 3.97503i −8.70126 + 5.68226i −7.98413 + 8.46269i −7.51754 + 2.73616i −26.7964 3.30970i −13.5951 4.94820i
7.8 1.19432 + 1.60425i −0.182976 5.19293i −1.14721 + 3.83196i −11.2321 + 7.38746i 8.11221 6.49554i 21.4630 22.7494i −7.51754 + 2.73616i −26.9330 + 1.90037i −25.2660 9.19607i
7.9 1.19432 + 1.60425i 1.72596 4.90113i −1.14721 + 3.83196i −5.18053 + 3.40729i 9.92396 3.08463i −19.8455 + 21.0350i −7.51754 + 2.73616i −21.0421 16.9183i −11.6533 4.24147i
7.10 1.19432 + 1.60425i 3.86822 + 3.46942i −1.14721 + 3.83196i 6.35824 4.18188i −0.945916 + 10.3492i 19.8464 21.0360i −7.51754 + 2.73616i 2.92628 + 26.8410i 14.3025 + 5.20569i
7.11 1.19432 + 1.60425i 4.33355 + 2.86711i −1.14721 + 3.83196i 12.9644 8.52679i 0.576086 + 10.3763i −19.6582 + 20.8365i −7.51754 + 2.73616i 10.5594 + 24.8495i 29.1626 + 10.6143i
7.12 1.19432 + 1.60425i 4.45599 2.67286i −1.14721 + 3.83196i 10.0588 6.61580i 9.60979 + 3.95626i 6.90035 7.31394i −7.51754 + 2.73616i 12.7116 23.8205i 22.6268 + 8.23549i
7.13 1.19432 + 1.60425i 4.90776 + 1.70702i −1.14721 + 3.83196i −12.1586 + 7.99685i 3.12293 + 9.91198i −2.36202 + 2.50359i −7.51754 + 2.73616i 21.1721 + 16.7553i −27.3501 9.95464i
13.1 1.78727 + 0.897598i −5.19043 + 0.243791i 2.38863 + 3.20849i 0.946763 + 3.16241i −9.49550 4.22320i −13.1374 30.4558i 1.38919 + 7.87846i 26.8811 2.53076i −1.14646 + 6.50187i
13.2 1.78727 + 0.897598i −5.01248 + 1.36931i 2.38863 + 3.20849i −4.36579 14.5828i −10.1877 2.05188i 6.73427 + 15.6118i 1.38919 + 7.87846i 23.2500 13.7273i 5.28663 29.9820i
13.3 1.78727 + 0.897598i −4.27995 2.94653i 2.38863 + 3.20849i −2.62306 8.76162i −5.00461 9.10790i 0.612588 + 1.42014i 1.38919 + 7.87846i 9.63595 + 25.2220i 3.17632 18.0138i
13.4 1.78727 + 0.897598i −3.53329 + 3.80997i 2.38863 + 3.20849i 3.44623 + 11.5112i −9.73474 + 3.63796i 4.85587 + 11.2572i 1.38919 + 7.87846i −2.03179 26.9234i −4.17312 + 23.6669i
13.5 1.78727 + 0.897598i −2.81153 4.36982i 2.38863 + 3.20849i 5.60704 + 18.7288i −1.10260 10.3336i −5.60229 12.9876i 1.38919 + 7.87846i −11.1906 + 24.5717i −6.78969 + 38.5062i
13.6 1.78727 + 0.897598i −1.70142 + 4.90970i 2.38863 + 3.20849i −3.38775 11.3159i −7.44783 + 7.24774i −10.6530 24.6965i 1.38919 + 7.87846i −21.2103 16.7069i 4.10230 23.2653i
13.7 1.78727 + 0.897598i −1.30843 5.02872i 2.38863 + 3.20849i 0.879965 + 2.93929i 2.17526 10.1621i 10.8789 + 25.2202i 1.38919 + 7.87846i −23.5760 + 13.1595i −1.06557 + 6.04315i
See next 80 embeddings (of 234 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 157.13 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.a 234
81.g even 27 1 inner 162.4.g.a 234

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.g.a 234 1.a even 1 1 trivial
162.4.g.a 234 81.g even 27 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{234} + 423 T_{5}^{232} - 591 T_{5}^{231} + 72126 T_{5}^{230} + 703053 T_{5}^{229} - 2853324 T_{5}^{228} - 795696507 T_{5}^{227} + 17289328635 T_{5}^{226} - 500223424476 T_{5}^{225} + \cdots + 30\!\cdots\!29$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.