Properties

Label 162.4.c.f
Level 162
Weight 4
Character orbit 162.c
Analytic conductor 9.558
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.55830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -6 \zeta_{6} q^{5} + ( 16 - 16 \zeta_{6} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} -4 \zeta_{6} q^{4} -6 \zeta_{6} q^{5} + ( 16 - 16 \zeta_{6} ) q^{7} -8 q^{8} -12 q^{10} + ( -12 + 12 \zeta_{6} ) q^{11} -38 \zeta_{6} q^{13} -32 \zeta_{6} q^{14} + ( -16 + 16 \zeta_{6} ) q^{16} -126 q^{17} + 20 q^{19} + ( -24 + 24 \zeta_{6} ) q^{20} + 24 \zeta_{6} q^{22} -168 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} -76 q^{26} -64 q^{28} + ( -30 + 30 \zeta_{6} ) q^{29} + 88 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( -252 + 252 \zeta_{6} ) q^{34} -96 q^{35} + 254 q^{37} + ( 40 - 40 \zeta_{6} ) q^{38} + 48 \zeta_{6} q^{40} -42 \zeta_{6} q^{41} + ( 52 - 52 \zeta_{6} ) q^{43} + 48 q^{44} -336 q^{46} + ( 96 - 96 \zeta_{6} ) q^{47} + 87 \zeta_{6} q^{49} -178 \zeta_{6} q^{50} + ( -152 + 152 \zeta_{6} ) q^{52} + 198 q^{53} + 72 q^{55} + ( -128 + 128 \zeta_{6} ) q^{56} + 60 \zeta_{6} q^{58} + 660 \zeta_{6} q^{59} + ( 538 - 538 \zeta_{6} ) q^{61} + 176 q^{62} + 64 q^{64} + ( -228 + 228 \zeta_{6} ) q^{65} -884 \zeta_{6} q^{67} + 504 \zeta_{6} q^{68} + ( -192 + 192 \zeta_{6} ) q^{70} + 792 q^{71} + 218 q^{73} + ( 508 - 508 \zeta_{6} ) q^{74} -80 \zeta_{6} q^{76} + 192 \zeta_{6} q^{77} + ( 520 - 520 \zeta_{6} ) q^{79} + 96 q^{80} -84 q^{82} + ( 492 - 492 \zeta_{6} ) q^{83} + 756 \zeta_{6} q^{85} -104 \zeta_{6} q^{86} + ( 96 - 96 \zeta_{6} ) q^{88} + 810 q^{89} -608 q^{91} + ( -672 + 672 \zeta_{6} ) q^{92} -192 \zeta_{6} q^{94} -120 \zeta_{6} q^{95} + ( -1154 + 1154 \zeta_{6} ) q^{97} + 174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 6q^{5} + 16q^{7} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 6q^{5} + 16q^{7} - 16q^{8} - 24q^{10} - 12q^{11} - 38q^{13} - 32q^{14} - 16q^{16} - 252q^{17} + 40q^{19} - 24q^{20} + 24q^{22} - 168q^{23} + 89q^{25} - 152q^{26} - 128q^{28} - 30q^{29} + 88q^{31} + 32q^{32} - 252q^{34} - 192q^{35} + 508q^{37} + 40q^{38} + 48q^{40} - 42q^{41} + 52q^{43} + 96q^{44} - 672q^{46} + 96q^{47} + 87q^{49} - 178q^{50} - 152q^{52} + 396q^{53} + 144q^{55} - 128q^{56} + 60q^{58} + 660q^{59} + 538q^{61} + 352q^{62} + 128q^{64} - 228q^{65} - 884q^{67} + 504q^{68} - 192q^{70} + 1584q^{71} + 436q^{73} + 508q^{74} - 80q^{76} + 192q^{77} + 520q^{79} + 192q^{80} - 168q^{82} + 492q^{83} + 756q^{85} - 104q^{86} + 96q^{88} + 1620q^{89} - 1216q^{91} - 672q^{92} - 192q^{94} - 120q^{95} - 1154q^{97} + 348q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 1.73205i 0 −2.00000 3.46410i −3.00000 5.19615i 0 8.00000 13.8564i −8.00000 0 −12.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.00000 + 5.19615i 0 8.00000 + 13.8564i −8.00000 0 −12.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.4.c.f 2
3.b odd 2 1 162.4.c.c 2
9.c even 3 1 6.4.a.a 1
9.c even 3 1 inner 162.4.c.f 2
9.d odd 6 1 18.4.a.a 1
9.d odd 6 1 162.4.c.c 2
36.f odd 6 1 48.4.a.c 1
36.h even 6 1 144.4.a.c 1
45.h odd 6 1 450.4.a.h 1
45.j even 6 1 150.4.a.i 1
45.k odd 12 2 150.4.c.d 2
45.l even 12 2 450.4.c.e 2
63.g even 3 1 294.4.e.h 2
63.h even 3 1 294.4.e.h 2
63.i even 6 1 882.4.g.f 2
63.j odd 6 1 882.4.g.i 2
63.k odd 6 1 294.4.e.g 2
63.l odd 6 1 294.4.a.e 1
63.n odd 6 1 882.4.g.i 2
63.o even 6 1 882.4.a.n 1
63.s even 6 1 882.4.g.f 2
63.t odd 6 1 294.4.e.g 2
72.j odd 6 1 576.4.a.q 1
72.l even 6 1 576.4.a.r 1
72.n even 6 1 192.4.a.i 1
72.p odd 6 1 192.4.a.c 1
99.g even 6 1 2178.4.a.e 1
99.h odd 6 1 726.4.a.f 1
117.t even 6 1 1014.4.a.g 1
117.y odd 12 2 1014.4.b.d 2
144.v odd 12 2 768.4.d.c 2
144.x even 12 2 768.4.d.n 2
153.h even 6 1 1734.4.a.d 1
171.o odd 6 1 2166.4.a.i 1
180.p odd 6 1 1200.4.a.b 1
180.x even 12 2 1200.4.f.j 2
252.bi even 6 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 9.c even 3 1
18.4.a.a 1 9.d odd 6 1
48.4.a.c 1 36.f odd 6 1
144.4.a.c 1 36.h even 6 1
150.4.a.i 1 45.j even 6 1
150.4.c.d 2 45.k odd 12 2
162.4.c.c 2 3.b odd 2 1
162.4.c.c 2 9.d odd 6 1
162.4.c.f 2 1.a even 1 1 trivial
162.4.c.f 2 9.c even 3 1 inner
192.4.a.c 1 72.p odd 6 1
192.4.a.i 1 72.n even 6 1
294.4.a.e 1 63.l odd 6 1
294.4.e.g 2 63.k odd 6 1
294.4.e.g 2 63.t odd 6 1
294.4.e.h 2 63.g even 3 1
294.4.e.h 2 63.h even 3 1
450.4.a.h 1 45.h odd 6 1
450.4.c.e 2 45.l even 12 2
576.4.a.q 1 72.j odd 6 1
576.4.a.r 1 72.l even 6 1
726.4.a.f 1 99.h odd 6 1
768.4.d.c 2 144.v odd 12 2
768.4.d.n 2 144.x even 12 2
882.4.a.n 1 63.o even 6 1
882.4.g.f 2 63.i even 6 1
882.4.g.f 2 63.s even 6 1
882.4.g.i 2 63.j odd 6 1
882.4.g.i 2 63.n odd 6 1
1014.4.a.g 1 117.t even 6 1
1014.4.b.d 2 117.y odd 12 2
1200.4.a.b 1 180.p odd 6 1
1200.4.f.j 2 180.x even 12 2
1734.4.a.d 1 153.h even 6 1
2166.4.a.i 1 171.o odd 6 1
2178.4.a.e 1 99.g even 6 1
2352.4.a.e 1 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6 T_{5} + 36 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ 1
$5$ \( 1 + 6 T - 89 T^{2} + 750 T^{3} + 15625 T^{4} \)
$7$ \( 1 - 16 T - 87 T^{2} - 5488 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 12 T - 1187 T^{2} + 15972 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 38 T - 753 T^{2} + 83486 T^{3} + 4826809 T^{4} \)
$17$ \( ( 1 + 126 T + 4913 T^{2} )^{2} \)
$19$ \( ( 1 - 20 T + 6859 T^{2} )^{2} \)
$23$ \( 1 + 168 T + 16057 T^{2} + 2044056 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 30 T - 23489 T^{2} + 731670 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 88 T - 22047 T^{2} - 2621608 T^{3} + 887503681 T^{4} \)
$37$ \( ( 1 - 254 T + 50653 T^{2} )^{2} \)
$41$ \( 1 + 42 T - 67157 T^{2} + 2894682 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 52 T - 76803 T^{2} - 4134364 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 96 T - 94607 T^{2} - 9967008 T^{3} + 10779215329 T^{4} \)
$53$ \( ( 1 - 198 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 - 538 T + 62463 T^{2} - 122115778 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 884 T + 480693 T^{2} + 265874492 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 - 792 T + 357911 T^{2} )^{2} \)
$73$ \( ( 1 - 218 T + 389017 T^{2} )^{2} \)
$79$ \( 1 - 520 T - 222639 T^{2} - 256380280 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 492 T - 329723 T^{2} - 281319204 T^{3} + 326940373369 T^{4} \)
$89$ \( ( 1 - 810 T + 704969 T^{2} )^{2} \)
$97$ \( 1 + 1154 T + 419043 T^{2} + 1053224642 T^{3} + 832972004929 T^{4} \)
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