Properties

Label 162.4.c.d
Level $162$
Weight $4$
Character orbit 162.c
Analytic conductor $9.558$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: 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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (8 \zeta_{6} - 8) q^{7} + 8 q^{8} - 42 q^{10} + ( - 36 \zeta_{6} + 36) q^{11} + 49 \zeta_{6} q^{13} - 16 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 21 q^{17} - 112 q^{19} + ( - 84 \zeta_{6} + 84) q^{20} + 72 \zeta_{6} q^{22} + 180 \zeta_{6} q^{23} + (316 \zeta_{6} - 316) q^{25} - 98 q^{26} + 32 q^{28} + (135 \zeta_{6} - 135) q^{29} - 308 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 42 \zeta_{6} + 42) q^{34} - 168 q^{35} - q^{37} + ( - 224 \zeta_{6} + 224) q^{38} + 168 \zeta_{6} q^{40} - 42 \zeta_{6} q^{41} + (20 \zeta_{6} - 20) q^{43} - 144 q^{44} - 360 q^{46} + ( - 84 \zeta_{6} + 84) q^{47} + 279 \zeta_{6} q^{49} - 632 \zeta_{6} q^{50} + ( - 196 \zeta_{6} + 196) q^{52} + 174 q^{53} + 756 q^{55} + (64 \zeta_{6} - 64) q^{56} - 270 \zeta_{6} q^{58} + 504 \zeta_{6} q^{59} + ( - 385 \zeta_{6} + 385) q^{61} + 616 q^{62} + 64 q^{64} + (1029 \zeta_{6} - 1029) q^{65} - 272 \zeta_{6} q^{67} + 84 \zeta_{6} q^{68} + ( - 336 \zeta_{6} + 336) q^{70} + 888 q^{71} + 371 q^{73} + ( - 2 \zeta_{6} + 2) q^{74} + 448 \zeta_{6} q^{76} + 288 \zeta_{6} q^{77} + ( - 652 \zeta_{6} + 652) q^{79} - 336 q^{80} + 84 q^{82} + ( - 84 \zeta_{6} + 84) q^{83} - 441 \zeta_{6} q^{85} - 40 \zeta_{6} q^{86} + ( - 288 \zeta_{6} + 288) q^{88} - 21 q^{89} - 392 q^{91} + ( - 720 \zeta_{6} + 720) q^{92} + 168 \zeta_{6} q^{94} - 2352 \zeta_{6} q^{95} + ( - 1246 \zeta_{6} + 1246) q^{97} - 558 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 21 q^{5} - 8 q^{7} + 16 q^{8} - 84 q^{10} + 36 q^{11} + 49 q^{13} - 16 q^{14} - 16 q^{16} - 42 q^{17} - 224 q^{19} + 84 q^{20} + 72 q^{22} + 180 q^{23} - 316 q^{25} - 196 q^{26} + 64 q^{28} - 135 q^{29} - 308 q^{31} - 32 q^{32} + 42 q^{34} - 336 q^{35} - 2 q^{37} + 224 q^{38} + 168 q^{40} - 42 q^{41} - 20 q^{43} - 288 q^{44} - 720 q^{46} + 84 q^{47} + 279 q^{49} - 632 q^{50} + 196 q^{52} + 348 q^{53} + 1512 q^{55} - 64 q^{56} - 270 q^{58} + 504 q^{59} + 385 q^{61} + 1232 q^{62} + 128 q^{64} - 1029 q^{65} - 272 q^{67} + 84 q^{68} + 336 q^{70} + 1776 q^{71} + 742 q^{73} + 2 q^{74} + 448 q^{76} + 288 q^{77} + 652 q^{79} - 672 q^{80} + 168 q^{82} + 84 q^{83} - 441 q^{85} - 40 q^{86} + 288 q^{88} - 42 q^{89} - 784 q^{91} + 720 q^{92} + 168 q^{94} - 2352 q^{95} + 1246 q^{97} - 1116 q^{98}+O(q^{100}) \) 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 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.d 2
3.b odd 2 1 162.4.c.e 2
9.c even 3 1 162.4.a.c yes 1
9.c even 3 1 inner 162.4.c.d 2
9.d odd 6 1 162.4.a.b 1
9.d odd 6 1 162.4.c.e 2
36.f odd 6 1 1296.4.a.a 1
36.h even 6 1 1296.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 9.d odd 6 1
162.4.a.c yes 1 9.c even 3 1
162.4.c.d 2 1.a even 1 1 trivial
162.4.c.d 2 9.c even 3 1 inner
162.4.c.e 2 3.b odd 2 1
162.4.c.e 2 9.d odd 6 1
1296.4.a.a 1 36.f odd 6 1
1296.4.a.h 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 21T_{5} + 441 \) 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
$5$ \( T^{2} - 21T + 441 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$11$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$13$ \( T^{2} - 49T + 2401 \) Copy content Toggle raw display
$17$ \( (T + 21)^{2} \) Copy content Toggle raw display
$19$ \( (T + 112)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 180T + 32400 \) Copy content Toggle raw display
$29$ \( T^{2} + 135T + 18225 \) Copy content Toggle raw display
$31$ \( T^{2} + 308T + 94864 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$43$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$47$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$53$ \( (T - 174)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 504T + 254016 \) Copy content Toggle raw display
$61$ \( T^{2} - 385T + 148225 \) Copy content Toggle raw display
$67$ \( T^{2} + 272T + 73984 \) Copy content Toggle raw display
$71$ \( (T - 888)^{2} \) Copy content Toggle raw display
$73$ \( (T - 371)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 652T + 425104 \) Copy content Toggle raw display
$83$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$89$ \( (T + 21)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 1246 T + 1552516 \) Copy content Toggle raw display
show more
show less