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Properties

Label 162.4.a.a
Level 162
Weight 4
Character orbit 162.a
Self dual yes
Analytic conductor 9.558
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.55830942093\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} + 4q^{4} + 9q^{5} - 31q^{7} - 8q^{8} + O(q^{10}) \) \( q - 2q^{2} + 4q^{4} + 9q^{5} - 31q^{7} - 8q^{8} - 18q^{10} + 15q^{11} - 37q^{13} + 62q^{14} + 16q^{16} + 42q^{17} - 28q^{19} + 36q^{20} - 30q^{22} - 195q^{23} - 44q^{25} + 74q^{26} - 124q^{28} - 111q^{29} - 205q^{31} - 32q^{32} - 84q^{34} - 279q^{35} - 166q^{37} + 56q^{38} - 72q^{40} + 261q^{41} - 43q^{43} + 60q^{44} + 390q^{46} - 177q^{47} + 618q^{49} + 88q^{50} - 148q^{52} - 114q^{53} + 135q^{55} + 248q^{56} + 222q^{58} - 159q^{59} + 191q^{61} + 410q^{62} + 64q^{64} - 333q^{65} - 421q^{67} + 168q^{68} + 558q^{70} - 156q^{71} + 182q^{73} + 332q^{74} - 112q^{76} - 465q^{77} + 1133q^{79} + 144q^{80} - 522q^{82} + 1083q^{83} + 378q^{85} + 86q^{86} - 120q^{88} + 1050q^{89} + 1147q^{91} - 780q^{92} + 354q^{94} - 252q^{95} - 901q^{97} - 1236q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−2.00000 0 4.00000 9.00000 0 −31.0000 −8.00000 0 −18.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.a.a 1
3.b odd 2 1 162.4.a.d 1
4.b odd 2 1 1296.4.a.g 1
9.c even 3 2 54.4.c.a 2
9.d odd 6 2 18.4.c.a ✓ 2
12.b even 2 1 1296.4.a.b 1
36.f odd 6 2 432.4.i.a 2
36.h even 6 2 144.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a ✓ 2 9.d odd 6 2
54.4.c.a 2 9.c even 3 2
144.4.i.a 2 36.h even 6 2
162.4.a.a 1 1.a even 1 1 trivial
162.4.a.d 1 3.b odd 2 1
432.4.i.a 2 36.f odd 6 2
1296.4.a.b 1 12.b even 2 1
1296.4.a.g 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ 1
$5$ \( 1 - 9 T + 125 T^{2} \)
$7$ \( 1 + 31 T + 343 T^{2} \)
$11$ \( 1 - 15 T + 1331 T^{2} \)
$13$ \( 1 + 37 T + 2197 T^{2} \)
$17$ \( 1 - 42 T + 4913 T^{2} \)
$19$ \( 1 + 28 T + 6859 T^{2} \)
$23$ \( 1 + 195 T + 12167 T^{2} \)
$29$ \( 1 + 111 T + 24389 T^{2} \)
$31$ \( 1 + 205 T + 29791 T^{2} \)
$37$ \( 1 + 166 T + 50653 T^{2} \)
$41$ \( 1 - 261 T + 68921 T^{2} \)
$43$ \( 1 + 43 T + 79507 T^{2} \)
$47$ \( 1 + 177 T + 103823 T^{2} \)
$53$ \( 1 + 114 T + 148877 T^{2} \)
$59$ \( 1 + 159 T + 205379 T^{2} \)
$61$ \( 1 - 191 T + 226981 T^{2} \)
$67$ \( 1 + 421 T + 300763 T^{2} \)
$71$ \( 1 + 156 T + 357911 T^{2} \)
$73$ \( 1 - 182 T + 389017 T^{2} \)
$79$ \( 1 - 1133 T + 493039 T^{2} \)
$83$ \( 1 - 1083 T + 571787 T^{2} \)
$89$ \( 1 - 1050 T + 704969 T^{2} \)
$97$ \( 1 + 901 T + 912673 T^{2} \)
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