Properties

Label 162.4.c.b
Level $162$
Weight $4$
Character orbit 162.c
Analytic conductor $9.558$
Analytic rank $0$
Dimension $2$
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.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -29 + 29 \zeta_{6} ) q^{7} + 8 q^{8} +O(q^{10})\) \( q + ( -2 + 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -29 + 29 \zeta_{6} ) q^{7} + 8 q^{8} + 6 q^{10} + ( 57 - 57 \zeta_{6} ) q^{11} -20 \zeta_{6} q^{13} -58 \zeta_{6} q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} -72 q^{17} -106 q^{19} + ( -12 + 12 \zeta_{6} ) q^{20} + 114 \zeta_{6} q^{22} -174 \zeta_{6} q^{23} + ( 116 - 116 \zeta_{6} ) q^{25} + 40 q^{26} + 116 q^{28} + ( 210 - 210 \zeta_{6} ) q^{29} -47 \zeta_{6} q^{31} -32 \zeta_{6} q^{32} + ( 144 - 144 \zeta_{6} ) q^{34} + 87 q^{35} + 2 q^{37} + ( 212 - 212 \zeta_{6} ) q^{38} -24 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( -218 + 218 \zeta_{6} ) q^{43} -228 q^{44} + 348 q^{46} + ( -474 + 474 \zeta_{6} ) q^{47} -498 \zeta_{6} q^{49} + 232 \zeta_{6} q^{50} + ( -80 + 80 \zeta_{6} ) q^{52} + 81 q^{53} -171 q^{55} + ( -232 + 232 \zeta_{6} ) q^{56} + 420 \zeta_{6} q^{58} -84 \zeta_{6} q^{59} + ( -56 + 56 \zeta_{6} ) q^{61} + 94 q^{62} + 64 q^{64} + ( -60 + 60 \zeta_{6} ) q^{65} + 142 \zeta_{6} q^{67} + 288 \zeta_{6} q^{68} + ( -174 + 174 \zeta_{6} ) q^{70} + 360 q^{71} -1159 q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + 424 \zeta_{6} q^{76} + 1653 \zeta_{6} q^{77} + ( 160 - 160 \zeta_{6} ) q^{79} + 48 q^{80} -12 q^{82} + ( -735 + 735 \zeta_{6} ) q^{83} + 216 \zeta_{6} q^{85} -436 \zeta_{6} q^{86} + ( 456 - 456 \zeta_{6} ) q^{88} -954 q^{89} + 580 q^{91} + ( -696 + 696 \zeta_{6} ) q^{92} -948 \zeta_{6} q^{94} + 318 \zeta_{6} q^{95} + ( -191 + 191 \zeta_{6} ) q^{97} + 996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} - 3q^{5} - 29q^{7} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} - 3q^{5} - 29q^{7} + 16q^{8} + 12q^{10} + 57q^{11} - 20q^{13} - 58q^{14} - 16q^{16} - 144q^{17} - 212q^{19} - 12q^{20} + 114q^{22} - 174q^{23} + 116q^{25} + 80q^{26} + 232q^{28} + 210q^{29} - 47q^{31} - 32q^{32} + 144q^{34} + 174q^{35} + 4q^{37} + 212q^{38} - 24q^{40} + 6q^{41} - 218q^{43} - 456q^{44} + 696q^{46} - 474q^{47} - 498q^{49} + 232q^{50} - 80q^{52} + 162q^{53} - 342q^{55} - 232q^{56} + 420q^{58} - 84q^{59} - 56q^{61} + 188q^{62} + 128q^{64} - 60q^{65} + 142q^{67} + 288q^{68} - 174q^{70} + 720q^{71} - 2318q^{73} - 4q^{74} + 424q^{76} + 1653q^{77} + 160q^{79} + 96q^{80} - 24q^{82} - 735q^{83} + 216q^{85} - 436q^{86} + 456q^{88} - 1908q^{89} + 1160q^{91} - 696q^{92} - 948q^{94} + 318q^{95} - 191q^{97} + 1992q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −1.50000 2.59808i 0 −14.5000 + 25.1147i 8.00000 0 6.00000
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.50000 + 2.59808i 0 −14.5000 25.1147i 8.00000 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.c.b 2
3.b odd 2 1 162.4.c.g 2
9.c even 3 1 54.4.a.c yes 1
9.c even 3 1 inner 162.4.c.b 2
9.d odd 6 1 54.4.a.b 1
9.d odd 6 1 162.4.c.g 2
36.f odd 6 1 432.4.a.j 1
36.h even 6 1 432.4.a.e 1
45.h odd 6 1 1350.4.a.o 1
45.j even 6 1 1350.4.a.a 1
45.k odd 12 2 1350.4.c.b 2
45.l even 12 2 1350.4.c.s 2
72.j odd 6 1 1728.4.a.v 1
72.l even 6 1 1728.4.a.u 1
72.n even 6 1 1728.4.a.l 1
72.p odd 6 1 1728.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 9.d odd 6 1
54.4.a.c yes 1 9.c even 3 1
162.4.c.b 2 1.a even 1 1 trivial
162.4.c.b 2 9.c even 3 1 inner
162.4.c.g 2 3.b odd 2 1
162.4.c.g 2 9.d odd 6 1
432.4.a.e 1 36.h even 6 1
432.4.a.j 1 36.f odd 6 1
1350.4.a.a 1 45.j even 6 1
1350.4.a.o 1 45.h odd 6 1
1350.4.c.b 2 45.k odd 12 2
1350.4.c.s 2 45.l even 12 2
1728.4.a.k 1 72.p odd 6 1
1728.4.a.l 1 72.n even 6 1
1728.4.a.u 1 72.l even 6 1
1728.4.a.v 1 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).