Properties

Label 162.3.d.c
Level $162$
Weight $3$
Character orbit 162.d
Analytic conductor $4.414$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.41418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + 2 \beta_{3}) q^{5} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + 2 \beta_{3}) q^{5} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} - 2 \beta_{3}) q^{8} + ( - \beta_{4} + 3) q^{10} + (4 \beta_{6} - 2 \beta_{5}) q^{11} + ( - \beta_{2} + 16 \beta_1) q^{13} + (4 \beta_{7} - 4 \beta_{6}) q^{14} + (4 \beta_1 - 4) q^{16} + ( - 5 \beta_{7} - 5 \beta_{5} + 5 \beta_{3}) q^{17} + ( - 2 \beta_{4} + 14) q^{19} + ( - 2 \beta_{6} + 4 \beta_{5}) q^{20} + 4 \beta_{2} q^{22} + ( - 4 \beta_{7} + 4 \beta_{6} - 2 \beta_{3}) q^{23} + ( - 3 \beta_{4} + 3 \beta_{2} + 7 \beta_1 - 7) q^{25} + ( - 2 \beta_{7} + 17 \beta_{5} - 17 \beta_{3}) q^{26} + ( - 4 \beta_{4} - 4) q^{28} + (\beta_{6} - 23 \beta_{5}) q^{29} - 8 \beta_1 q^{31} - 4 \beta_{3} q^{32} + (5 \beta_{4} - 5 \beta_{2} - 15 \beta_1 + 15) q^{34} + (4 \beta_{7} - 26 \beta_{5} + 26 \beta_{3}) q^{35} + (8 \beta_{4} - 19) q^{37} + ( - 4 \beta_{6} + 16 \beta_{5}) q^{38} + ( - 2 \beta_{2} + 6 \beta_1) q^{40} + ( - 8 \beta_{7} + 8 \beta_{6} - 7 \beta_{3}) q^{41} + (6 \beta_{4} - 6 \beta_{2} - 22 \beta_1 + 22) q^{43} + (8 \beta_{7} - 4 \beta_{5} + 4 \beta_{3}) q^{44} + 4 \beta_{4} q^{46} + 12 \beta_{5} q^{47} + ( - 8 \beta_{2} - 63 \beta_1) q^{49} + (6 \beta_{7} - 6 \beta_{6} - 4 \beta_{3}) q^{50} + (2 \beta_{4} - 2 \beta_{2} + 32 \beta_1 - 32) q^{52} + (8 \beta_{7} + 35 \beta_{5} - 35 \beta_{3}) q^{53} + (6 \beta_{4} - 54) q^{55} - 8 \beta_{6} q^{56} + (\beta_{2} - 45 \beta_1) q^{58} + ( - 4 \beta_{7} + 4 \beta_{6} + 52 \beta_{3}) q^{59} + ( - 13 \beta_1 + 13) q^{61} + ( - 8 \beta_{5} + 8 \beta_{3}) q^{62} - 8 q^{64} + ( - 19 \beta_{6} + 47 \beta_{5}) q^{65} + (6 \beta_{2} + 10 \beta_1) q^{67} + ( - 10 \beta_{7} + 10 \beta_{6} + 10 \beta_{3}) q^{68} + ( - 4 \beta_{4} + 4 \beta_{2} - 48 \beta_1 + 48) q^{70} + (16 \beta_{7} - 38 \beta_{5} + 38 \beta_{3}) q^{71} + (3 \beta_{4} + 56) q^{73} + (16 \beta_{6} - 27 \beta_{5}) q^{74} + ( - 4 \beta_{2} + 28 \beta_1) q^{76} + (8 \beta_{7} - 8 \beta_{6} - 104 \beta_{3}) q^{77} + (14 \beta_{4} - 14 \beta_{2} + 26 \beta_1 - 26) q^{79} + ( - 4 \beta_{7} + 8 \beta_{5} - 8 \beta_{3}) q^{80} + (8 \beta_{4} - 6) q^{82} + (12 \beta_{6} - 48 \beta_{5}) q^{83} + 45 \beta_1 q^{85} + ( - 12 \beta_{7} + 12 \beta_{6} + 16 \beta_{3}) q^{86} + ( - 8 \beta_{4} + 8 \beta_{2}) q^{88} + ( - 29 \beta_{7} + 34 \beta_{5} - 34 \beta_{3}) q^{89} + ( - 30 \beta_{4} + 22) q^{91} + (8 \beta_{6} - 4 \beta_{5}) q^{92} + 24 \beta_1 q^{94} + (20 \beta_{7} - 20 \beta_{6} + 58 \beta_{3}) q^{95} + ( - 16 \beta_{4} + 16 \beta_{2} + 8 \beta_1 - 8) q^{97} + ( - 16 \beta_{7} - 55 \beta_{5} + 55 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 8 q^{7} + 24 q^{10} + 64 q^{13} - 16 q^{16} + 112 q^{19} - 28 q^{25} - 32 q^{28} - 32 q^{31} + 60 q^{34} - 152 q^{37} + 24 q^{40} + 88 q^{43} - 252 q^{49} - 128 q^{52} - 432 q^{55} - 180 q^{58} + 52 q^{61} - 64 q^{64} + 40 q^{67} + 192 q^{70} + 448 q^{73} + 112 q^{76} - 104 q^{79} - 48 q^{82} + 180 q^{85} + 176 q^{91} + 96 q^{94} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 0.965926i
−1.22474 + 0.707107i 0 1.00000 1.73205i −5.01910 2.89778i 0 4.19615 + 7.26795i 2.82843i 0 8.19615
53.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.34486 + 0.776457i 0 −6.19615 10.7321i 2.82843i 0 −2.19615
53.3 1.22474 0.707107i 0 1.00000 1.73205i −1.34486 0.776457i 0 −6.19615 10.7321i 2.82843i 0 −2.19615
53.4 1.22474 0.707107i 0 1.00000 1.73205i 5.01910 + 2.89778i 0 4.19615 + 7.26795i 2.82843i 0 8.19615
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −5.01910 + 2.89778i 0 4.19615 7.26795i 2.82843i 0 8.19615
107.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.34486 0.776457i 0 −6.19615 + 10.7321i 2.82843i 0 −2.19615
107.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.34486 + 0.776457i 0 −6.19615 + 10.7321i 2.82843i 0 −2.19615
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 5.01910 2.89778i 0 4.19615 7.26795i 2.82843i 0 8.19615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.d.c 8
3.b odd 2 1 inner 162.3.d.c 8
4.b odd 2 1 1296.3.q.o 8
9.c even 3 1 162.3.b.b 4
9.c even 3 1 inner 162.3.d.c 8
9.d odd 6 1 162.3.b.b 4
9.d odd 6 1 inner 162.3.d.c 8
12.b even 2 1 1296.3.q.o 8
36.f odd 6 1 1296.3.e.d 4
36.f odd 6 1 1296.3.q.o 8
36.h even 6 1 1296.3.e.d 4
36.h even 6 1 1296.3.q.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.3.b.b 4 9.c even 3 1
162.3.b.b 4 9.d odd 6 1
162.3.d.c 8 1.a even 1 1 trivial
162.3.d.c 8 3.b odd 2 1 inner
162.3.d.c 8 9.c even 3 1 inner
162.3.d.c 8 9.d odd 6 1 inner
1296.3.e.d 4 36.f odd 6 1
1296.3.e.d 4 36.h even 6 1
1296.3.q.o 8 4.b odd 2 1
1296.3.q.o 8 12.b even 2 1
1296.3.q.o 8 36.f odd 6 1
1296.3.q.o 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 36T_{5}^{6} + 1215T_{5}^{4} - 2916T_{5}^{2} + 6561 \) acting on \(S_{3}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 36 T^{6} + 1215 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + 120 T^{2} - 416 T + 10816)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 216 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 32 T^{3} + 795 T^{2} + \cdots + 52441)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 900 T^{2} + 50625)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28 T + 88)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 216 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 2052 T^{6} + \cdots + 996005996001 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 38 T - 1367)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 1764 T^{6} + \cdots + 512249392656 \) Copy content Toggle raw display
$43$ \( (T^{4} - 44 T^{3} + 2424 T^{2} + \cdots + 238144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7812 T^{2} + 4743684)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 994737284775936 \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + 1272 T^{2} + \cdots + 760384)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 10512 T^{2} + 2742336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 112 T + 2893)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 52 T^{3} + 7320 T^{2} + \cdots + 21307456)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 10944 T^{6} + \cdots + 6295362011136 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24228 T^{2} + \cdots + 112211649)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + 7104 T^{2} + \cdots + 46895104)^{2} \) Copy content Toggle raw display
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