Properties

Label 162.2.c.d
Level $162$
Weight $2$
Character orbit 162.c
Analytic conductor $1.294$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - q^{8} + 3 q^{10} + \zeta_{6} q^{13} - 4 \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 3 q^{17} - 4 q^{19} + ( - 3 \zeta_{6} + 3) q^{20} + (4 \zeta_{6} - 4) q^{25} + q^{26} - 4 q^{28} + (9 \zeta_{6} - 9) q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{34} + 12 q^{35} - q^{37} + (4 \zeta_{6} - 4) q^{38} - 3 \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} + ( - 12 \zeta_{6} + 12) q^{47} - 9 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} - 6 q^{53} + (4 \zeta_{6} - 4) q^{56} + 9 \zeta_{6} q^{58} + ( - \zeta_{6} + 1) q^{61} + 4 q^{62} + q^{64} + (3 \zeta_{6} - 3) q^{65} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + ( - 12 \zeta_{6} + 12) q^{70} - 12 q^{71} + 11 q^{73} + (\zeta_{6} - 1) q^{74} + 4 \zeta_{6} q^{76} + ( - 16 \zeta_{6} + 16) q^{79} - 3 q^{80} - 6 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} - 9 \zeta_{6} q^{85} + 8 \zeta_{6} q^{86} - 3 q^{89} + 4 q^{91} - 12 \zeta_{6} q^{94} - 12 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} - 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 3 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 3 q^{5} + 4 q^{7} - 2 q^{8} + 6 q^{10} + q^{13} - 4 q^{14} - q^{16} - 6 q^{17} - 8 q^{19} + 3 q^{20} - 4 q^{25} + 2 q^{26} - 8 q^{28} - 9 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} + 24 q^{35} - 2 q^{37} - 4 q^{38} - 3 q^{40} - 6 q^{41} - 8 q^{43} + 12 q^{47} - 9 q^{49} + 4 q^{50} + q^{52} - 12 q^{53} - 4 q^{56} + 9 q^{58} + q^{61} + 8 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 3 q^{68} + 12 q^{70} - 24 q^{71} + 22 q^{73} - q^{74} + 4 q^{76} + 16 q^{79} - 6 q^{80} - 12 q^{82} + 12 q^{83} - 9 q^{85} + 8 q^{86} - 6 q^{89} + 8 q^{91} - 12 q^{94} - 12 q^{95} - 2 q^{97} - 18 q^{98}+O(q^{100}) \) 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
0.500000 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 2.00000 3.46410i −1.00000 0 3.00000
109.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 2.00000 + 3.46410i −1.00000 0 3.00000
\(n\): 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 162.2.c.d 2
3.b odd 2 1 162.2.c.a 2
4.b odd 2 1 1296.2.i.n 2
9.c even 3 1 162.2.a.a 1
9.c even 3 1 inner 162.2.c.d 2
9.d odd 6 1 162.2.a.d yes 1
9.d odd 6 1 162.2.c.a 2
12.b even 2 1 1296.2.i.b 2
36.f odd 6 1 1296.2.a.c 1
36.f odd 6 1 1296.2.i.n 2
36.h even 6 1 1296.2.a.l 1
36.h even 6 1 1296.2.i.b 2
45.h odd 6 1 4050.2.a.r 1
45.j even 6 1 4050.2.a.bh 1
45.k odd 12 2 4050.2.c.g 2
45.l even 12 2 4050.2.c.n 2
63.l odd 6 1 7938.2.a.n 1
63.o even 6 1 7938.2.a.s 1
72.j odd 6 1 5184.2.a.c 1
72.l even 6 1 5184.2.a.h 1
72.n even 6 1 5184.2.a.y 1
72.p odd 6 1 5184.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 9.c even 3 1
162.2.a.d yes 1 9.d odd 6 1
162.2.c.a 2 3.b odd 2 1
162.2.c.a 2 9.d odd 6 1
162.2.c.d 2 1.a even 1 1 trivial
162.2.c.d 2 9.c even 3 1 inner
1296.2.a.c 1 36.f odd 6 1
1296.2.a.l 1 36.h even 6 1
1296.2.i.b 2 12.b even 2 1
1296.2.i.b 2 36.h even 6 1
1296.2.i.n 2 4.b odd 2 1
1296.2.i.n 2 36.f odd 6 1
4050.2.a.r 1 45.h odd 6 1
4050.2.a.bh 1 45.j even 6 1
4050.2.c.g 2 45.k odd 12 2
4050.2.c.n 2 45.l even 12 2
5184.2.a.c 1 72.j odd 6 1
5184.2.a.h 1 72.l even 6 1
5184.2.a.y 1 72.n even 6 1
5184.2.a.bd 1 72.p odd 6 1
7938.2.a.n 1 63.l odd 6 1
7938.2.a.s 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(162, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$89$ \( (T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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