Properties

Label 162.4.a.c
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

Related objects

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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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 −21.0000 0 8.00000 8.00000 0 −42.0000
\(n\): 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.c yes 1
3.b odd 2 1 162.4.a.b 1
4.b odd 2 1 1296.4.a.a 1
9.c even 3 2 162.4.c.d 2
9.d odd 6 2 162.4.c.e 2
12.b even 2 1 1296.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.b 1 3.b odd 2 1
162.4.a.c yes 1 1.a even 1 1 trivial
162.4.c.d 2 9.c even 3 2
162.4.c.e 2 9.d odd 6 2
1296.4.a.a 1 4.b odd 2 1
1296.4.a.h 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 21 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ 1
$5$ \( 1 + 21 T + 125 T^{2} \)
$7$ \( 1 - 8 T + 343 T^{2} \)
$11$ \( 1 + 36 T + 1331 T^{2} \)
$13$ \( 1 + 49 T + 2197 T^{2} \)
$17$ \( 1 + 21 T + 4913 T^{2} \)
$19$ \( 1 + 112 T + 6859 T^{2} \)
$23$ \( 1 + 180 T + 12167 T^{2} \)
$29$ \( 1 - 135 T + 24389 T^{2} \)
$31$ \( 1 - 308 T + 29791 T^{2} \)
$37$ \( 1 + T + 50653 T^{2} \)
$41$ \( 1 - 42 T + 68921 T^{2} \)
$43$ \( 1 - 20 T + 79507 T^{2} \)
$47$ \( 1 + 84 T + 103823 T^{2} \)
$53$ \( 1 - 174 T + 148877 T^{2} \)
$59$ \( 1 + 504 T + 205379 T^{2} \)
$61$ \( 1 + 385 T + 226981 T^{2} \)
$67$ \( 1 - 272 T + 300763 T^{2} \)
$71$ \( 1 - 888 T + 357911 T^{2} \)
$73$ \( 1 - 371 T + 389017 T^{2} \)
$79$ \( 1 + 652 T + 493039 T^{2} \)
$83$ \( 1 + 84 T + 571787 T^{2} \)
$89$ \( 1 + 21 T + 704969 T^{2} \)
$97$ \( 1 + 1246 T + 912673 T^{2} \)
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