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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 $2$

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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.

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

Display 100 coefficients 10 coefficients

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:

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 \(T_{5}^{234} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database