Properties

Label 162.4.c.g
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 29T + 841 \) Copy content Toggle raw display
$11$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$13$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$17$ \( (T - 72)^{2} \) Copy content Toggle raw display
$19$ \( (T + 106)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 174T + 30276 \) Copy content Toggle raw display
$29$ \( T^{2} + 210T + 44100 \) Copy content Toggle raw display
$31$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 218T + 47524 \) Copy content Toggle raw display
$47$ \( T^{2} - 474T + 224676 \) Copy content Toggle raw display
$53$ \( (T + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$61$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$67$ \( T^{2} - 142T + 20164 \) Copy content Toggle raw display
$71$ \( (T + 360)^{2} \) Copy content Toggle raw display
$73$ \( (T + 1159)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$83$ \( T^{2} - 735T + 540225 \) Copy content Toggle raw display
$89$ \( (T - 954)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 191T + 36481 \) Copy content Toggle raw display
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