Properties

Label 162.4.a.g
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: \(\Q(\sqrt{105}) \)
Defining polynomial: \( x^{2} - x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(-1 + 3\sqrt{105})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - \beta + 4) q^{5} + (\beta + 10) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + ( - \beta + 4) q^{5} + (\beta + 10) q^{7} + 8 q^{8} + ( - 2 \beta + 8) q^{10} + (2 \beta + 13) q^{11} + ( - \beta + 30) q^{13} + (2 \beta + 20) q^{14} + 16 q^{16} + (\beta - 1) q^{17} + (5 \beta + 69) q^{19} + ( - 4 \beta + 16) q^{20} + (4 \beta + 26) q^{22} + (\beta - 34) q^{23} + ( - 9 \beta + 127) q^{25} + ( - 2 \beta + 60) q^{26} + (4 \beta + 40) q^{28} + ( - 7 \beta - 122) q^{29} + ( - 3 \beta + 104) q^{31} + 32 q^{32} + (2 \beta - 2) q^{34} + ( - 5 \beta - 196) q^{35} + ( - 14 \beta + 124) q^{37} + (10 \beta + 138) q^{38} + ( - 8 \beta + 32) q^{40} + (2 \beta - 233) q^{41} + (24 \beta - 31) q^{43} + (8 \beta + 52) q^{44} + (2 \beta - 68) q^{46} + (15 \beta - 234) q^{47} + (19 \beta - 7) q^{49} + ( - 18 \beta + 254) q^{50} + ( - 4 \beta + 120) q^{52} + (14 \beta - 68) q^{53} + ( - 3 \beta - 420) q^{55} + (8 \beta + 80) q^{56} + ( - 14 \beta - 244) q^{58} + ( - 2 \beta - 85) q^{59} + ( - 15 \beta - 532) q^{61} + ( - 6 \beta + 208) q^{62} + 64 q^{64} + ( - 35 \beta + 356) q^{65} + ( - 12 \beta - 589) q^{67} + (4 \beta - 4) q^{68} + ( - 10 \beta - 392) q^{70} + ( - 32 \beta + 140) q^{71} + (21 \beta - 145) q^{73} + ( - 28 \beta + 248) q^{74} + (20 \beta + 276) q^{76} + (31 \beta + 602) q^{77} + ( - 13 \beta + 168) q^{79} + ( - 16 \beta + 64) q^{80} + (4 \beta - 466) q^{82} + (21 \beta - 600) q^{83} + (6 \beta - 240) q^{85} + (48 \beta - 62) q^{86} + (16 \beta + 104) q^{88} + (40 \beta + 266) q^{89} + (21 \beta + 64) q^{91} + (4 \beta - 136) q^{92} + (30 \beta - 468) q^{94} + ( - 44 \beta - 904) q^{95} + ( - 22 \beta - 75) q^{97} + (38 \beta - 14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26} + 76 q^{28} - 237 q^{29} + 211 q^{31} + 64 q^{32} - 6 q^{34} - 387 q^{35} + 262 q^{37} + 266 q^{38} + 72 q^{40} - 468 q^{41} - 86 q^{43} + 96 q^{44} - 138 q^{46} - 483 q^{47} - 33 q^{49} + 526 q^{50} + 244 q^{52} - 150 q^{53} - 837 q^{55} + 152 q^{56} - 474 q^{58} - 168 q^{59} - 1049 q^{61} + 422 q^{62} + 128 q^{64} + 747 q^{65} - 1166 q^{67} - 12 q^{68} - 774 q^{70} + 312 q^{71} - 311 q^{73} + 524 q^{74} + 532 q^{76} + 1173 q^{77} + 349 q^{79} + 144 q^{80} - 936 q^{82} - 1221 q^{83} - 486 q^{85} - 172 q^{86} + 192 q^{88} + 492 q^{89} + 107 q^{91} - 276 q^{92} - 966 q^{94} - 1764 q^{95} - 128 q^{97} - 66 q^{98}+O(q^{100}) \) 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
5.62348
−4.62348
2.00000 0 4.00000 −10.8704 0 24.8704 8.00000 0 −21.7409
1.2 2.00000 0 4.00000 19.8704 0 −5.87043 8.00000 0 39.7409
\(n\): 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.g 2
3.b odd 2 1 162.4.a.f 2
4.b odd 2 1 1296.4.a.r 2
9.c even 3 2 54.4.c.b 4
9.d odd 6 2 18.4.c.b 4
12.b even 2 1 1296.4.a.l 2
36.f odd 6 2 432.4.i.b 4
36.h even 6 2 144.4.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.b 4 9.d odd 6 2
54.4.c.b 4 9.c even 3 2
144.4.i.b 4 36.h even 6 2
162.4.a.f 2 3.b odd 2 1
162.4.a.g 2 1.a even 1 1 trivial
432.4.i.b 4 36.f odd 6 2
1296.4.a.l 2 12.b even 2 1
1296.4.a.r 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 9T_{5} - 216 \) 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
$5$ \( T^{2} - 9T - 216 \) Copy content Toggle raw display
$7$ \( T^{2} - 19T - 146 \) Copy content Toggle raw display
$11$ \( T^{2} - 24T - 801 \) Copy content Toggle raw display
$13$ \( T^{2} - 61T + 694 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 234 \) Copy content Toggle raw display
$19$ \( T^{2} - 133T - 1484 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 954 \) Copy content Toggle raw display
$29$ \( T^{2} + 237T + 2466 \) Copy content Toggle raw display
$31$ \( T^{2} - 211T + 9004 \) Copy content Toggle raw display
$37$ \( T^{2} - 262T - 29144 \) Copy content Toggle raw display
$41$ \( T^{2} + 468T + 53811 \) Copy content Toggle raw display
$43$ \( T^{2} + 86T - 134231 \) Copy content Toggle raw display
$47$ \( T^{2} + 483T + 5166 \) Copy content Toggle raw display
$53$ \( T^{2} + 150T - 40680 \) Copy content Toggle raw display
$59$ \( T^{2} + 168T + 6111 \) Copy content Toggle raw display
$61$ \( T^{2} + 1049 T + 221944 \) Copy content Toggle raw display
$67$ \( T^{2} + 1166 T + 305869 \) Copy content Toggle raw display
$71$ \( T^{2} - 312T - 217584 \) Copy content Toggle raw display
$73$ \( T^{2} + 311T - 80006 \) Copy content Toggle raw display
$79$ \( T^{2} - 349T - 9476 \) Copy content Toggle raw display
$83$ \( T^{2} + 1221 T + 268524 \) Copy content Toggle raw display
$89$ \( T^{2} - 492T - 317484 \) Copy content Toggle raw display
$97$ \( T^{2} + 128T - 110249 \) Copy content Toggle raw display
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