Properties

Label 162.4.a.h
Level $162$
Weight $4$
Character orbit 162.a
Self dual yes
Analytic conductor $9.558$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + (\beta + 6) q^{5} + (2 \beta + 8) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + (\beta + 6) q^{5} + (2 \beta + 8) q^{7} + 8 q^{8} + (2 \beta + 12) q^{10} + ( - 8 \beta + 18) q^{11} + ( - 14 \beta + 5) q^{13} + (4 \beta + 16) q^{14} + 16 q^{16} + (11 \beta + 60) q^{17} + (22 \beta - 4) q^{19} + (4 \beta + 24) q^{20} + ( - 16 \beta + 36) q^{22} + ( - 4 \beta + 90) q^{23} + (12 \beta - 62) q^{25} + ( - 28 \beta + 10) q^{26} + (8 \beta + 32) q^{28} + (7 \beta + 162) q^{29} + ( - 36 \beta - 124) q^{31} + 32 q^{32} + (22 \beta + 120) q^{34} + (20 \beta + 102) q^{35} + (2 \beta - 217) q^{37} + (44 \beta - 8) q^{38} + (8 \beta + 48) q^{40} + ( - 44 \beta + 96) q^{41} + ( - 6 \beta - 304) q^{43} + ( - 32 \beta + 72) q^{44} + ( - 8 \beta + 180) q^{46} + ( - 36 \beta - 192) q^{47} + (32 \beta - 171) q^{49} + (24 \beta - 124) q^{50} + ( - 56 \beta + 20) q^{52} + (76 \beta - 204) q^{53} + ( - 30 \beta - 108) q^{55} + (16 \beta + 64) q^{56} + (14 \beta + 324) q^{58} + (32 \beta - 504) q^{59} + ( - 18 \beta + 371) q^{61} + ( - 72 \beta - 248) q^{62} + 64 q^{64} + ( - 79 \beta - 348) q^{65} + (138 \beta - 52) q^{67} + (44 \beta + 240) q^{68} + (40 \beta + 204) q^{70} + (8 \beta - 570) q^{71} + ( - 96 \beta + 425) q^{73} + (4 \beta - 434) q^{74} + (88 \beta - 16) q^{76} + ( - 28 \beta - 288) q^{77} + ( - 50 \beta - 220) q^{79} + (16 \beta + 96) q^{80} + ( - 88 \beta + 192) q^{82} + ( - 60 \beta + 132) q^{83} + (126 \beta + 657) q^{85} + ( - 12 \beta - 608) q^{86} + ( - 64 \beta + 144) q^{88} + (89 \beta + 384) q^{89} + ( - 102 \beta - 716) q^{91} + ( - 16 \beta + 360) q^{92} + ( - 72 \beta - 384) q^{94} + (128 \beta + 570) q^{95} + ( - 56 \beta - 382) q^{97} + (64 \beta - 342) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−1.73205
1.73205
2.00000 0 4.00000 0.803848 0 −2.39230 8.00000 0 1.60770
1.2 2.00000 0 4.00000 11.1962 0 18.3923 8.00000 0 22.3923
\(n\): 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.h yes 2
3.b odd 2 1 162.4.a.e 2
4.b odd 2 1 1296.4.a.s 2
9.c even 3 2 162.4.c.i 4
9.d odd 6 2 162.4.c.j 4
12.b even 2 1 1296.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.e 2 3.b odd 2 1
162.4.a.h yes 2 1.a even 1 1 trivial
162.4.c.i 4 9.c even 3 2
162.4.c.j 4 9.d odd 6 2
1296.4.a.j 2 12.b even 2 1
1296.4.a.s 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 16T - 44 \) Copy content Toggle raw display
$11$ \( T^{2} - 36T - 1404 \) Copy content Toggle raw display
$13$ \( T^{2} - 10T - 5267 \) Copy content Toggle raw display
$17$ \( T^{2} - 120T + 333 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 13052 \) Copy content Toggle raw display
$23$ \( T^{2} - 180T + 7668 \) Copy content Toggle raw display
$29$ \( T^{2} - 324T + 24921 \) Copy content Toggle raw display
$31$ \( T^{2} + 248T - 19616 \) Copy content Toggle raw display
$37$ \( T^{2} + 434T + 46981 \) Copy content Toggle raw display
$41$ \( T^{2} - 192T - 43056 \) Copy content Toggle raw display
$43$ \( T^{2} + 608T + 91444 \) Copy content Toggle raw display
$47$ \( T^{2} + 384T + 1872 \) Copy content Toggle raw display
$53$ \( T^{2} + 408T - 114336 \) Copy content Toggle raw display
$59$ \( T^{2} + 1008 T + 226368 \) Copy content Toggle raw display
$61$ \( T^{2} - 742T + 128893 \) Copy content Toggle raw display
$67$ \( T^{2} + 104T - 511484 \) Copy content Toggle raw display
$71$ \( T^{2} + 1140 T + 323172 \) Copy content Toggle raw display
$73$ \( T^{2} - 850T - 68207 \) Copy content Toggle raw display
$79$ \( T^{2} + 440T - 19100 \) Copy content Toggle raw display
$83$ \( T^{2} - 264T - 79776 \) Copy content Toggle raw display
$89$ \( T^{2} - 768T - 66411 \) Copy content Toggle raw display
$97$ \( T^{2} + 764T + 61252 \) Copy content Toggle raw display
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