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

Related objects

Downloads

Learn more about

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: \(x^{2} - x - 26\)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( 4 - \beta ) q^{5} + ( 10 + \beta ) q^{7} + 8 q^{8} +O(q^{10})\) \( q + 2 q^{2} + 4 q^{4} + ( 4 - \beta ) q^{5} + ( 10 + \beta ) q^{7} + 8 q^{8} + ( 8 - 2 \beta ) q^{10} + ( 13 + 2 \beta ) q^{11} + ( 30 - \beta ) q^{13} + ( 20 + 2 \beta ) q^{14} + 16 q^{16} + ( -1 + \beta ) q^{17} + ( 69 + 5 \beta ) q^{19} + ( 16 - 4 \beta ) q^{20} + ( 26 + 4 \beta ) q^{22} + ( -34 + \beta ) q^{23} + ( 127 - 9 \beta ) q^{25} + ( 60 - 2 \beta ) q^{26} + ( 40 + 4 \beta ) q^{28} + ( -122 - 7 \beta ) q^{29} + ( 104 - 3 \beta ) q^{31} + 32 q^{32} + ( -2 + 2 \beta ) q^{34} + ( -196 - 5 \beta ) q^{35} + ( 124 - 14 \beta ) q^{37} + ( 138 + 10 \beta ) q^{38} + ( 32 - 8 \beta ) q^{40} + ( -233 + 2 \beta ) q^{41} + ( -31 + 24 \beta ) q^{43} + ( 52 + 8 \beta ) q^{44} + ( -68 + 2 \beta ) q^{46} + ( -234 + 15 \beta ) q^{47} + ( -7 + 19 \beta ) q^{49} + ( 254 - 18 \beta ) q^{50} + ( 120 - 4 \beta ) q^{52} + ( -68 + 14 \beta ) q^{53} + ( -420 - 3 \beta ) q^{55} + ( 80 + 8 \beta ) q^{56} + ( -244 - 14 \beta ) q^{58} + ( -85 - 2 \beta ) q^{59} + ( -532 - 15 \beta ) q^{61} + ( 208 - 6 \beta ) q^{62} + 64 q^{64} + ( 356 - 35 \beta ) q^{65} + ( -589 - 12 \beta ) q^{67} + ( -4 + 4 \beta ) q^{68} + ( -392 - 10 \beta ) q^{70} + ( 140 - 32 \beta ) q^{71} + ( -145 + 21 \beta ) q^{73} + ( 248 - 28 \beta ) q^{74} + ( 276 + 20 \beta ) q^{76} + ( 602 + 31 \beta ) q^{77} + ( 168 - 13 \beta ) q^{79} + ( 64 - 16 \beta ) q^{80} + ( -466 + 4 \beta ) q^{82} + ( -600 + 21 \beta ) q^{83} + ( -240 + 6 \beta ) q^{85} + ( -62 + 48 \beta ) q^{86} + ( 104 + 16 \beta ) q^{88} + ( 266 + 40 \beta ) q^{89} + ( 64 + 21 \beta ) q^{91} + ( -136 + 4 \beta ) q^{92} + ( -468 + 30 \beta ) q^{94} + ( -904 - 44 \beta ) q^{95} + ( -75 - 22 \beta ) q^{97} + ( -14 + 38 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 9q^{5} + 19q^{7} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 9q^{5} + 19q^{7} + 16q^{8} + 18q^{10} + 24q^{11} + 61q^{13} + 38q^{14} + 32q^{16} - 3q^{17} + 133q^{19} + 36q^{20} + 48q^{22} - 69q^{23} + 263q^{25} + 122q^{26} + 76q^{28} - 237q^{29} + 211q^{31} + 64q^{32} - 6q^{34} - 387q^{35} + 262q^{37} + 266q^{38} + 72q^{40} - 468q^{41} - 86q^{43} + 96q^{44} - 138q^{46} - 483q^{47} - 33q^{49} + 526q^{50} + 244q^{52} - 150q^{53} - 837q^{55} + 152q^{56} - 474q^{58} - 168q^{59} - 1049q^{61} + 422q^{62} + 128q^{64} + 747q^{65} - 1166q^{67} - 12q^{68} - 774q^{70} + 312q^{71} - 311q^{73} + 524q^{74} + 532q^{76} + 1173q^{77} + 349q^{79} + 144q^{80} - 936q^{82} - 1221q^{83} - 486q^{85} - 172q^{86} + 192q^{88} + 492q^{89} + 107q^{91} - 276q^{92} - 966q^{94} - 1764q^{95} - 128q^{97} - 66q^{98} + O(q^{100}) \)

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
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 \( T_{5}^{2} - 9 T_{5} - 216 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )^{2} \)
$3$ 1
$5$ \( 1 - 9 T + 34 T^{2} - 1125 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4} \)
$11$ \( 1 - 24 T + 1861 T^{2} - 31944 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 3 T + 9592 T^{2} + 14739 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 69 T + 25288 T^{2} + 839523 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 237 T + 51244 T^{2} + 5780193 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 468 T + 191653 T^{2} + 32255028 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 483 T + 212812 T^{2} + 50146509 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 150 T + 257074 T^{2} + 22331550 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 168 T + 416869 T^{2} + 34503672 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 312 T + 498238 T^{2} - 111668232 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 1221 T + 1412098 T^{2} + 698151927 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 492 T + 1092454 T^{2} - 346844748 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4} \)
show more
show less