Properties

Label 162.4.c.e
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: 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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -21 \zeta_{6} q^{5} + ( -8 + 8 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -21 \zeta_{6} q^{5} + ( -8 + 8 \zeta_{6} ) q^{7} -8 q^{8} -42 q^{10} + ( -36 + 36 \zeta_{6} ) q^{11} + 49 \zeta_{6} q^{13} + 16 \zeta_{6} q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} + 21 q^{17} -112 q^{19} + ( -84 + 84 \zeta_{6} ) q^{20} + 72 \zeta_{6} q^{22} -180 \zeta_{6} q^{23} + ( -316 + 316 \zeta_{6} ) q^{25} + 98 q^{26} + 32 q^{28} + ( 135 - 135 \zeta_{6} ) q^{29} -308 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( 42 - 42 \zeta_{6} ) q^{34} + 168 q^{35} - q^{37} + ( -224 + 224 \zeta_{6} ) q^{38} + 168 \zeta_{6} q^{40} + 42 \zeta_{6} q^{41} + ( -20 + 20 \zeta_{6} ) q^{43} + 144 q^{44} -360 q^{46} + ( -84 + 84 \zeta_{6} ) q^{47} + 279 \zeta_{6} q^{49} + 632 \zeta_{6} q^{50} + ( 196 - 196 \zeta_{6} ) q^{52} -174 q^{53} + 756 q^{55} + ( 64 - 64 \zeta_{6} ) q^{56} -270 \zeta_{6} q^{58} -504 \zeta_{6} q^{59} + ( 385 - 385 \zeta_{6} ) q^{61} -616 q^{62} + 64 q^{64} + ( 1029 - 1029 \zeta_{6} ) q^{65} -272 \zeta_{6} q^{67} -84 \zeta_{6} q^{68} + ( 336 - 336 \zeta_{6} ) q^{70} -888 q^{71} + 371 q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 448 \zeta_{6} q^{76} -288 \zeta_{6} q^{77} + ( 652 - 652 \zeta_{6} ) q^{79} + 336 q^{80} + 84 q^{82} + ( -84 + 84 \zeta_{6} ) q^{83} -441 \zeta_{6} q^{85} + 40 \zeta_{6} q^{86} + ( 288 - 288 \zeta_{6} ) q^{88} + 21 q^{89} -392 q^{91} + ( -720 + 720 \zeta_{6} ) q^{92} + 168 \zeta_{6} q^{94} + 2352 \zeta_{6} q^{95} + ( 1246 - 1246 \zeta_{6} ) q^{97} + 558 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 21q^{5} - 8q^{7} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 21q^{5} - 8q^{7} - 16q^{8} - 84q^{10} - 36q^{11} + 49q^{13} + 16q^{14} - 16q^{16} + 42q^{17} - 224q^{19} - 84q^{20} + 72q^{22} - 180q^{23} - 316q^{25} + 196q^{26} + 64q^{28} + 135q^{29} - 308q^{31} + 32q^{32} + 42q^{34} + 336q^{35} - 2q^{37} - 224q^{38} + 168q^{40} + 42q^{41} - 20q^{43} + 288q^{44} - 720q^{46} - 84q^{47} + 279q^{49} + 632q^{50} + 196q^{52} - 348q^{53} + 1512q^{55} + 64q^{56} - 270q^{58} - 504q^{59} + 385q^{61} - 1232q^{62} + 128q^{64} + 1029q^{65} - 272q^{67} - 84q^{68} + 336q^{70} - 1776q^{71} + 742q^{73} - 2q^{74} + 448q^{76} - 288q^{77} + 652q^{79} + 672q^{80} + 168q^{82} - 84q^{83} - 441q^{85} + 40q^{86} + 288q^{88} + 42q^{89} - 784q^{91} - 720q^{92} + 168q^{94} + 2352q^{95} + 1246q^{97} + 1116q^{98} + O(q^{100}) \)

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 −10.5000 18.1865i 0 −4.00000 + 6.92820i −8.00000 0 −42.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.5000 + 18.1865i 0 −4.00000 6.92820i −8.00000 0 −42.0000
\(n\): 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.e 2
3.b odd 2 1 162.4.c.d 2
9.c even 3 1 162.4.a.b 1
9.c even 3 1 inner 162.4.c.e 2
9.d odd 6 1 162.4.a.c yes 1
9.d odd 6 1 162.4.c.d 2
36.f odd 6 1 1296.4.a.h 1
36.h even 6 1 1296.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 9.c even 3 1
162.4.a.c yes 1 9.d odd 6 1
162.4.c.d 2 3.b odd 2 1
162.4.c.d 2 9.d odd 6 1
162.4.c.e 2 1.a even 1 1 trivial
162.4.c.e 2 9.c even 3 1 inner
1296.4.a.a 1 36.h even 6 1
1296.4.a.h 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 21 T_{5} + 441 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ 1
$5$ \( 1 + 21 T + 316 T^{2} + 2625 T^{3} + 15625 T^{4} \)
$7$ \( 1 + 8 T - 279 T^{2} + 2744 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 36 T - 35 T^{2} + 47916 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 - 21 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 + 112 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 180 T + 20233 T^{2} + 2190060 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 135 T - 6164 T^{2} - 3292515 T^{3} + 594823321 T^{4} \)
$31$ \( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \)
$37$ \( ( 1 + T + 50653 T^{2} )^{2} \)
$41$ \( 1 - 42 T - 67157 T^{2} - 2894682 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 20 T - 79107 T^{2} + 1590140 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 84 T - 96767 T^{2} + 8721132 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 + 174 T + 148877 T^{2} )^{2} \)
$59$ \( 1 + 504 T + 48637 T^{2} + 103511016 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 385 T - 78756 T^{2} - 87387685 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 272 T - 226779 T^{2} + 81807536 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 888 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 371 T + 389017 T^{2} )^{2} \)
$79$ \( 1 - 652 T - 67935 T^{2} - 321461428 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 84 T - 564731 T^{2} + 48030108 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 - 21 T + 704969 T^{2} )^{2} \)
$97$ \( 1 - 1246 T + 639843 T^{2} - 1137190558 T^{3} + 832972004929 T^{4} \)
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