Properties

Label 162.2.c.c
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} -3 q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} -3 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 q^{26} - q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} + \zeta_{6} q^{32} -3 q^{35} + 2 q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} -3 q^{44} + 6 q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + 9 q^{53} -9 q^{55} + ( -1 + \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} -12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} -5 q^{62} + q^{64} + ( 12 - 12 \zeta_{6} ) q^{65} -14 \zeta_{6} q^{67} + ( -3 + 3 \zeta_{6} ) q^{70} -7 q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{76} -3 \zeta_{6} q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} + 3 q^{80} + 6 q^{82} + ( 3 - 3 \zeta_{6} ) q^{83} -10 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} -18 q^{89} + 4 q^{91} + ( 6 - 6 \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{94} -6 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{97} + 6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 3q^{5} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 3q^{5} + q^{7} - 2q^{8} - 6q^{10} + 3q^{11} + 4q^{13} - q^{14} - q^{16} + 4q^{19} - 3q^{20} - 3q^{22} + 6q^{23} - 4q^{25} + 8q^{26} - 2q^{28} - 6q^{29} - 5q^{31} + q^{32} - 6q^{35} + 4q^{37} + 2q^{38} + 3q^{40} + 6q^{41} + 10q^{43} - 6q^{44} + 12q^{46} - 6q^{47} + 6q^{49} + 4q^{50} + 4q^{52} + 18q^{53} - 18q^{55} - q^{56} + 6q^{58} - 12q^{59} - 8q^{61} - 10q^{62} + 2q^{64} + 12q^{65} - 14q^{67} - 3q^{70} - 14q^{73} + 2q^{74} - 2q^{76} - 3q^{77} - 8q^{79} + 6q^{80} + 12q^{82} + 3q^{83} - 10q^{86} - 3q^{88} - 36q^{89} + 8q^{91} + 6q^{92} + 6q^{94} - 6q^{95} + q^{97} + 12q^{98} + O(q^{100}) \)

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
0.500000 0.866025i 0 −0.500000 0.866025i −1.50000 2.59808i 0 0.500000 0.866025i −1.00000 0 −3.00000
109.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.50000 + 2.59808i 0 0.500000 + 0.866025i −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.c 2
3.b odd 2 1 162.2.c.b 2
4.b odd 2 1 1296.2.i.c 2
9.c even 3 1 54.2.a.a 1
9.c even 3 1 inner 162.2.c.c 2
9.d odd 6 1 54.2.a.b yes 1
9.d odd 6 1 162.2.c.b 2
12.b even 2 1 1296.2.i.o 2
36.f odd 6 1 432.2.a.g 1
36.f odd 6 1 1296.2.i.c 2
36.h even 6 1 432.2.a.b 1
36.h even 6 1 1296.2.i.o 2
45.h odd 6 1 1350.2.a.h 1
45.j even 6 1 1350.2.a.r 1
45.k odd 12 2 1350.2.c.b 2
45.l even 12 2 1350.2.c.k 2
63.l odd 6 1 2646.2.a.a 1
63.o even 6 1 2646.2.a.bd 1
72.j odd 6 1 1728.2.a.y 1
72.l even 6 1 1728.2.a.z 1
72.n even 6 1 1728.2.a.c 1
72.p odd 6 1 1728.2.a.d 1
99.g even 6 1 6534.2.a.b 1
99.h odd 6 1 6534.2.a.bc 1
117.n odd 6 1 9126.2.a.r 1
117.t even 6 1 9126.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 9.c even 3 1
54.2.a.b yes 1 9.d odd 6 1
162.2.c.b 2 3.b odd 2 1
162.2.c.b 2 9.d odd 6 1
162.2.c.c 2 1.a even 1 1 trivial
162.2.c.c 2 9.c even 3 1 inner
432.2.a.b 1 36.h even 6 1
432.2.a.g 1 36.f odd 6 1
1296.2.i.c 2 4.b odd 2 1
1296.2.i.c 2 36.f odd 6 1
1296.2.i.o 2 12.b even 2 1
1296.2.i.o 2 36.h even 6 1
1350.2.a.h 1 45.h odd 6 1
1350.2.a.r 1 45.j even 6 1
1350.2.c.b 2 45.k odd 12 2
1350.2.c.k 2 45.l even 12 2
1728.2.a.c 1 72.n even 6 1
1728.2.a.d 1 72.p odd 6 1
1728.2.a.y 1 72.j odd 6 1
1728.2.a.z 1 72.l even 6 1
2646.2.a.a 1 63.l odd 6 1
2646.2.a.bd 1 63.o even 6 1
6534.2.a.b 1 99.g even 6 1
6534.2.a.bc 1 99.h odd 6 1
9126.2.a.r 1 117.n odd 6 1
9126.2.a.u 1 117.t even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{7}^{2} - T_{7} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 9 + 3 T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 9 - 3 T + T^{2} \)
$13$ \( 16 - 4 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 100 - 10 T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( ( -9 + T )^{2} \)
$59$ \( 144 + 12 T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 196 + 14 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 7 + T )^{2} \)
$79$ \( 64 + 8 T + T^{2} \)
$83$ \( 9 - 3 T + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 1 - T + T^{2} \)
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