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Properties

Label 162.4.c.h
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( 7 - 7 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( 7 - 7 \zeta_{6} ) q^{7} -8 q^{8} + 24 q^{10} + ( 60 - 60 \zeta_{6} ) q^{11} + 79 \zeta_{6} q^{13} -14 \zeta_{6} q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 108 q^{17} + 11 q^{19} + ( 48 - 48 \zeta_{6} ) q^{20} -120 \zeta_{6} q^{22} -132 \zeta_{6} q^{23} + ( -19 + 19 \zeta_{6} ) q^{25} + 158 q^{26} -28 q^{28} + ( 96 - 96 \zeta_{6} ) q^{29} -20 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( 216 - 216 \zeta_{6} ) q^{34} + 84 q^{35} -169 q^{37} + ( 22 - 22 \zeta_{6} ) q^{38} -96 \zeta_{6} q^{40} + 192 \zeta_{6} q^{41} + ( -488 + 488 \zeta_{6} ) q^{43} -240 q^{44} -264 q^{46} + ( 204 - 204 \zeta_{6} ) q^{47} + 294 \zeta_{6} q^{49} + 38 \zeta_{6} q^{50} + ( 316 - 316 \zeta_{6} ) q^{52} -360 q^{53} + 720 q^{55} + ( -56 + 56 \zeta_{6} ) q^{56} -192 \zeta_{6} q^{58} + 156 \zeta_{6} q^{59} + ( -83 + 83 \zeta_{6} ) q^{61} -40 q^{62} + 64 q^{64} + ( -948 + 948 \zeta_{6} ) q^{65} -47 \zeta_{6} q^{67} -432 \zeta_{6} q^{68} + ( 168 - 168 \zeta_{6} ) q^{70} -216 q^{71} -511 q^{73} + ( -338 + 338 \zeta_{6} ) q^{74} -44 \zeta_{6} q^{76} -420 \zeta_{6} q^{77} + ( 529 - 529 \zeta_{6} ) q^{79} -192 q^{80} + 384 q^{82} + ( -1128 + 1128 \zeta_{6} ) q^{83} + 1296 \zeta_{6} q^{85} + 976 \zeta_{6} q^{86} + ( -480 + 480 \zeta_{6} ) q^{88} -36 q^{89} + 553 q^{91} + ( -528 + 528 \zeta_{6} ) q^{92} -408 \zeta_{6} q^{94} + 132 \zeta_{6} q^{95} + ( -605 + 605 \zeta_{6} ) q^{97} + 588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} + 12q^{5} + 7q^{7} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} + 12q^{5} + 7q^{7} - 16q^{8} + 48q^{10} + 60q^{11} + 79q^{13} - 14q^{14} - 16q^{16} + 216q^{17} + 22q^{19} + 48q^{20} - 120q^{22} - 132q^{23} - 19q^{25} + 316q^{26} - 56q^{28} + 96q^{29} - 20q^{31} + 32q^{32} + 216q^{34} + 168q^{35} - 338q^{37} + 22q^{38} - 96q^{40} + 192q^{41} - 488q^{43} - 480q^{44} - 528q^{46} + 204q^{47} + 294q^{49} + 38q^{50} + 316q^{52} - 720q^{53} + 1440q^{55} - 56q^{56} - 192q^{58} + 156q^{59} - 83q^{61} - 80q^{62} + 128q^{64} - 948q^{65} - 47q^{67} - 432q^{68} + 168q^{70} - 432q^{71} - 1022q^{73} - 338q^{74} - 44q^{76} - 420q^{77} + 529q^{79} - 384q^{80} + 768q^{82} - 1128q^{83} + 1296q^{85} + 976q^{86} - 480q^{88} - 72q^{89} + 1106q^{91} - 528q^{92} - 408q^{94} + 132q^{95} - 605q^{97} + 1176q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 6.00000 + 10.3923i 0 3.50000 − 6.06218i −8.00000 0 24.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 6.00000 − 10.3923i 0 3.50000 + 6.06218i −8.00000 0 24.0000
\(n\): 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.h 2
3.b odd 2 1 162.4.c.a 2
9.c even 3 1 54.4.a.a ✓ 1
9.c even 3 1 inner 162.4.c.h 2
9.d odd 6 1 54.4.a.d yes 1
9.d odd 6 1 162.4.c.a 2
36.f odd 6 1 432.4.a.b 1
36.h even 6 1 432.4.a.m 1
45.h odd 6 1 1350.4.a.h 1
45.j even 6 1 1350.4.a.v 1
45.k odd 12 2 1350.4.c.a 2
45.l even 12 2 1350.4.c.t 2
72.j odd 6 1 1728.4.a.e 1
72.l even 6 1 1728.4.a.f 1
72.n even 6 1 1728.4.a.ba 1
72.p odd 6 1 1728.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.a ✓ 1 9.c even 3 1
54.4.a.d yes 1 9.d odd 6 1
162.4.c.a 2 3.b odd 2 1
162.4.c.a 2 9.d odd 6 1
162.4.c.h 2 1.a even 1 1 trivial
162.4.c.h 2 9.c even 3 1 inner
432.4.a.b 1 36.f odd 6 1
432.4.a.m 1 36.h even 6 1
1350.4.a.h 1 45.h odd 6 1
1350.4.a.v 1 45.j even 6 1
1350.4.c.a 2 45.k odd 12 2
1350.4.c.t 2 45.l even 12 2
1728.4.a.e 1 72.j odd 6 1
1728.4.a.f 1 72.l even 6 1
1728.4.a.ba 1 72.n even 6 1
1728.4.a.bb 1 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12 T_{5} + 144 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).