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Properties

Label 162.4.a.f
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: \(x^{2} - x - 26\)
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.

\(f(q)\) \(=\) \( q -2 q^{2} + 4 q^{4} + ( -5 - \beta ) q^{5} + ( 9 - \beta ) q^{7} -8 q^{8} +O(q^{10})\) \( q -2 q^{2} + 4 q^{4} + ( -5 - \beta ) q^{5} + ( 9 - \beta ) q^{7} -8 q^{8} + ( 10 + 2 \beta ) q^{10} + ( -11 + 2 \beta ) q^{11} + ( 31 + \beta ) q^{13} + ( -18 + 2 \beta ) q^{14} + 16 q^{16} + ( 2 + \beta ) q^{17} + ( 64 - 5 \beta ) q^{19} + ( -20 - 4 \beta ) q^{20} + ( 22 - 4 \beta ) q^{22} + ( 35 + \beta ) q^{23} + ( 136 + 9 \beta ) q^{25} + ( -62 - 2 \beta ) q^{26} + ( 36 - 4 \beta ) q^{28} + ( 115 - 7 \beta ) q^{29} + ( 107 + 3 \beta ) q^{31} -32 q^{32} + ( -4 - 2 \beta ) q^{34} + ( 191 - 5 \beta ) q^{35} + ( 138 + 14 \beta ) q^{37} + ( -128 + 10 \beta ) q^{38} + ( 40 + 8 \beta ) q^{40} + ( 235 + 2 \beta ) q^{41} + ( -55 - 24 \beta ) q^{43} + ( -44 + 8 \beta ) q^{44} + ( -70 - 2 \beta ) q^{46} + ( 249 + 15 \beta ) q^{47} + ( -26 - 19 \beta ) q^{49} + ( -272 - 18 \beta ) q^{50} + ( 124 + 4 \beta ) q^{52} + ( 82 + 14 \beta ) q^{53} + ( -417 + 3 \beta ) q^{55} + ( -72 + 8 \beta ) q^{56} + ( -230 + 14 \beta ) q^{58} + ( 83 - 2 \beta ) q^{59} + ( -517 + 15 \beta ) q^{61} + ( -214 - 6 \beta ) q^{62} + 64 q^{64} + ( -391 - 35 \beta ) q^{65} + ( -577 + 12 \beta ) q^{67} + ( 8 + 4 \beta ) q^{68} + ( -382 + 10 \beta ) q^{70} + ( -172 - 32 \beta ) q^{71} + ( -166 - 21 \beta ) q^{73} + ( -276 - 28 \beta ) q^{74} + ( 256 - 20 \beta ) q^{76} + ( -571 + 31 \beta ) q^{77} + ( 181 + 13 \beta ) q^{79} + ( -80 - 16 \beta ) q^{80} + ( -470 - 4 \beta ) q^{82} + ( 621 + 21 \beta ) q^{83} + ( -246 - 6 \beta ) q^{85} + ( 110 + 48 \beta ) q^{86} + ( 88 - 16 \beta ) q^{88} + ( -226 + 40 \beta ) q^{89} + ( 43 - 21 \beta ) q^{91} + ( 140 + 4 \beta ) q^{92} + ( -498 - 30 \beta ) q^{94} + ( 860 - 44 \beta ) q^{95} + ( -53 + 22 \beta ) q^{97} + ( 52 + 38 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 9q^{5} + 19q^{7} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 9q^{5} + 19q^{7} - 16q^{8} + 18q^{10} - 24q^{11} + 61q^{13} - 38q^{14} + 32q^{16} + 3q^{17} + 133q^{19} - 36q^{20} + 48q^{22} + 69q^{23} + 263q^{25} - 122q^{26} + 76q^{28} + 237q^{29} + 211q^{31} - 64q^{32} - 6q^{34} + 387q^{35} + 262q^{37} - 266q^{38} + 72q^{40} + 468q^{41} - 86q^{43} - 96q^{44} - 138q^{46} + 483q^{47} - 33q^{49} - 526q^{50} + 244q^{52} + 150q^{53} - 837q^{55} - 152q^{56} - 474q^{58} + 168q^{59} - 1049q^{61} - 422q^{62} + 128q^{64} - 747q^{65} - 1166q^{67} + 12q^{68} - 774q^{70} - 312q^{71} - 311q^{73} - 524q^{74} + 532q^{76} - 1173q^{77} + 349q^{79} - 144q^{80} - 936q^{82} + 1221q^{83} - 486q^{85} + 172q^{86} + 192q^{88} - 492q^{89} + 107q^{91} + 276q^{92} - 966q^{94} + 1764q^{95} - 128q^{97} + 66q^{98} + O(q^{100}) \)

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
5.62348
−4.62348
−2.00000 0 4.00000 −19.8704 0 −5.87043 −8.00000 0 39.7409
1.2 −2.00000 0 4.00000 10.8704 0 24.8704 −8.00000 0 −21.7409
\(n\): 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.f 2
3.b odd 2 1 162.4.a.g 2
4.b odd 2 1 1296.4.a.l 2
9.c even 3 2 18.4.c.b ✓ 4
9.d odd 6 2 54.4.c.b 4
12.b even 2 1 1296.4.a.r 2
36.f odd 6 2 144.4.i.b 4
36.h even 6 2 432.4.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.b ✓ 4 9.c even 3 2
54.4.c.b 4 9.d odd 6 2
144.4.i.b 4 36.f odd 6 2
162.4.a.f 2 1.a even 1 1 trivial
162.4.a.g 2 3.b odd 2 1
432.4.i.b 4 36.h even 6 2
1296.4.a.l 2 4.b odd 2 1
1296.4.a.r 2 12.b even 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
$5$ \( 1 + 9 T + 34 T^{2} + 1125 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 19 T + 540 T^{2} - 6517 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 24 T + 1861 T^{2} + 31944 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 61 T + 5088 T^{2} - 134017 T^{3} + 4826809 T^{4} \)
$17$ \( 1 - 3 T + 9592 T^{2} - 14739 T^{3} + 24137569 T^{4} \)
$19$ \( 1 - 133 T + 12234 T^{2} - 912247 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 69 T + 25288 T^{2} - 839523 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 237 T + 51244 T^{2} - 5780193 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 211 T + 68586 T^{2} - 6285901 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 262 T + 72162 T^{2} - 13271086 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 468 T + 191653 T^{2} - 32255028 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 86 T + 24783 T^{2} + 6837602 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 483 T + 212812 T^{2} - 50146509 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 150 T + 257074 T^{2} - 22331550 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 168 T + 416869 T^{2} - 34503672 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 1049 T + 675906 T^{2} + 238103069 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 1166 T + 907395 T^{2} + 350689658 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 312 T + 498238 T^{2} + 111668232 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 311 T + 698028 T^{2} + 120984287 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 349 T + 976602 T^{2} - 172070611 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 1221 T + 1412098 T^{2} - 698151927 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 492 T + 1092454 T^{2} + 346844748 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 128 T + 1715097 T^{2} + 116822144 T^{3} + 832972004929 T^{4} \)
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