Properties

Label 162.4.c.h
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} - 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 12 \zeta_{6} q^{5} + ( - 7 \zeta_{6} + 7) q^{7} - 8 q^{8} + 24 q^{10} + ( - 60 \zeta_{6} + 60) q^{11} + 79 \zeta_{6} q^{13} - 14 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 108 q^{17} + 11 q^{19} + ( - 48 \zeta_{6} + 48) q^{20} - 120 \zeta_{6} q^{22} - 132 \zeta_{6} q^{23} + (19 \zeta_{6} - 19) q^{25} + 158 q^{26} - 28 q^{28} + ( - 96 \zeta_{6} + 96) q^{29} - 20 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( - 216 \zeta_{6} + 216) q^{34} + 84 q^{35} - 169 q^{37} + ( - 22 \zeta_{6} + 22) q^{38} - 96 \zeta_{6} q^{40} + 192 \zeta_{6} q^{41} + (488 \zeta_{6} - 488) q^{43} - 240 q^{44} - 264 q^{46} + ( - 204 \zeta_{6} + 204) q^{47} + 294 \zeta_{6} q^{49} + 38 \zeta_{6} q^{50} + ( - 316 \zeta_{6} + 316) q^{52} - 360 q^{53} + 720 q^{55} + (56 \zeta_{6} - 56) q^{56} - 192 \zeta_{6} q^{58} + 156 \zeta_{6} q^{59} + (83 \zeta_{6} - 83) q^{61} - 40 q^{62} + 64 q^{64} + (948 \zeta_{6} - 948) q^{65} - 47 \zeta_{6} q^{67} - 432 \zeta_{6} q^{68} + ( - 168 \zeta_{6} + 168) q^{70} - 216 q^{71} - 511 q^{73} + (338 \zeta_{6} - 338) q^{74} - 44 \zeta_{6} q^{76} - 420 \zeta_{6} q^{77} + ( - 529 \zeta_{6} + 529) q^{79} - 192 q^{80} + 384 q^{82} + (1128 \zeta_{6} - 1128) q^{83} + 1296 \zeta_{6} q^{85} + 976 \zeta_{6} q^{86} + (480 \zeta_{6} - 480) q^{88} - 36 q^{89} + 553 q^{91} + (528 \zeta_{6} - 528) q^{92} - 408 \zeta_{6} q^{94} + 132 \zeta_{6} q^{95} + (605 \zeta_{6} - 605) q^{97} + 588 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} + 7 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 12 q^{5} + 7 q^{7} - 16 q^{8} + 48 q^{10} + 60 q^{11} + 79 q^{13} - 14 q^{14} - 16 q^{16} + 216 q^{17} + 22 q^{19} + 48 q^{20} - 120 q^{22} - 132 q^{23} - 19 q^{25} + 316 q^{26} - 56 q^{28} + 96 q^{29} - 20 q^{31} + 32 q^{32} + 216 q^{34} + 168 q^{35} - 338 q^{37} + 22 q^{38} - 96 q^{40} + 192 q^{41} - 488 q^{43} - 480 q^{44} - 528 q^{46} + 204 q^{47} + 294 q^{49} + 38 q^{50} + 316 q^{52} - 720 q^{53} + 1440 q^{55} - 56 q^{56} - 192 q^{58} + 156 q^{59} - 83 q^{61} - 80 q^{62} + 128 q^{64} - 948 q^{65} - 47 q^{67} - 432 q^{68} + 168 q^{70} - 432 q^{71} - 1022 q^{73} - 338 q^{74} - 44 q^{76} - 420 q^{77} + 529 q^{79} - 384 q^{80} + 768 q^{82} - 1128 q^{83} + 1296 q^{85} + 976 q^{86} - 480 q^{88} - 72 q^{89} + 1106 q^{91} - 528 q^{92} - 408 q^{94} + 132 q^{95} - 605 q^{97} + 1176 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 6.00000 + 10.3923i 0 3.50000 6.06218i −8.00000 0 24.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 6.00000 10.3923i 0 3.50000 + 6.06218i −8.00000 0 24.0000
\(n\): 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.h 2
3.b odd 2 1 162.4.c.a 2
9.c even 3 1 54.4.a.a 1
9.c even 3 1 inner 162.4.c.h 2
9.d odd 6 1 54.4.a.d yes 1
9.d odd 6 1 162.4.c.a 2
36.f odd 6 1 432.4.a.b 1
36.h even 6 1 432.4.a.m 1
45.h odd 6 1 1350.4.a.h 1
45.j even 6 1 1350.4.a.v 1
45.k odd 12 2 1350.4.c.a 2
45.l even 12 2 1350.4.c.t 2
72.j odd 6 1 1728.4.a.e 1
72.l even 6 1 1728.4.a.f 1
72.n even 6 1 1728.4.a.ba 1
72.p odd 6 1 1728.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.a 1 9.c even 3 1
54.4.a.d yes 1 9.d odd 6 1
162.4.c.a 2 3.b odd 2 1
162.4.c.a 2 9.d odd 6 1
162.4.c.h 2 1.a even 1 1 trivial
162.4.c.h 2 9.c even 3 1 inner
432.4.a.b 1 36.f odd 6 1
432.4.a.m 1 36.h even 6 1
1350.4.a.h 1 45.h odd 6 1
1350.4.a.v 1 45.j even 6 1
1350.4.c.a 2 45.k odd 12 2
1350.4.c.t 2 45.l even 12 2
1728.4.a.e 1 72.j odd 6 1
1728.4.a.f 1 72.l even 6 1
1728.4.a.ba 1 72.n even 6 1
1728.4.a.bb 1 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12T_{5} + 144 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 60T + 3600 \) Copy content Toggle raw display
$13$ \( T^{2} - 79T + 6241 \) Copy content Toggle raw display
$17$ \( (T - 108)^{2} \) Copy content Toggle raw display
$19$ \( (T - 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 132T + 17424 \) Copy content Toggle raw display
$29$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$31$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$37$ \( (T + 169)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 192T + 36864 \) Copy content Toggle raw display
$43$ \( T^{2} + 488T + 238144 \) Copy content Toggle raw display
$47$ \( T^{2} - 204T + 41616 \) Copy content Toggle raw display
$53$ \( (T + 360)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 156T + 24336 \) Copy content Toggle raw display
$61$ \( T^{2} + 83T + 6889 \) Copy content Toggle raw display
$67$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$71$ \( (T + 216)^{2} \) Copy content Toggle raw display
$73$ \( (T + 511)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 529T + 279841 \) Copy content Toggle raw display
$83$ \( T^{2} + 1128 T + 1272384 \) Copy content Toggle raw display
$89$ \( (T + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 605T + 366025 \) Copy content Toggle raw display
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