Properties

Label 162.3.d.b
Level $162$
Weight $3$
Character orbit 162.d
Analytic conductor $4.414$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.41418028264\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{5} + ( 4 - 4 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{5} + ( 4 - 4 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 6 q^{10} + 12 \beta_{1} q^{11} -8 \beta_{2} q^{13} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} + 9 \beta_{3} q^{17} -16 q^{19} + 6 \beta_{1} q^{20} + 24 \beta_{2} q^{22} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{23} + ( -7 + 7 \beta_{2} ) q^{25} -8 \beta_{3} q^{26} + 8 q^{28} + 3 \beta_{1} q^{29} -44 \beta_{2} q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -18 + 18 \beta_{2} ) q^{34} -12 \beta_{3} q^{35} -34 q^{37} -16 \beta_{1} q^{38} + 12 \beta_{2} q^{40} + ( -33 \beta_{1} + 33 \beta_{3} ) q^{41} + ( 40 - 40 \beta_{2} ) q^{43} + 24 \beta_{3} q^{44} + 24 q^{46} -60 \beta_{1} q^{47} + 33 \beta_{2} q^{49} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{50} + ( 16 - 16 \beta_{2} ) q^{52} -27 \beta_{3} q^{53} + 72 q^{55} + 8 \beta_{1} q^{56} + 6 \beta_{2} q^{58} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{59} + ( -50 + 50 \beta_{2} ) q^{61} -44 \beta_{3} q^{62} -8 q^{64} -24 \beta_{1} q^{65} -8 \beta_{2} q^{67} + ( -18 \beta_{1} + 18 \beta_{3} ) q^{68} + ( 24 - 24 \beta_{2} ) q^{70} + 36 \beta_{3} q^{71} -16 q^{73} -34 \beta_{1} q^{74} -32 \beta_{2} q^{76} + ( 48 \beta_{1} - 48 \beta_{3} ) q^{77} + ( 76 - 76 \beta_{2} ) q^{79} + 12 \beta_{3} q^{80} -66 q^{82} + 84 \beta_{1} q^{83} + 54 \beta_{2} q^{85} + ( 40 \beta_{1} - 40 \beta_{3} ) q^{86} + ( -48 + 48 \beta_{2} ) q^{88} -9 \beta_{3} q^{89} -32 q^{91} + 24 \beta_{1} q^{92} -120 \beta_{2} q^{94} + ( -48 \beta_{1} + 48 \beta_{3} ) q^{95} + ( -176 + 176 \beta_{2} ) q^{97} + 33 \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + 8q^{7} + O(q^{10}) \) \( 4q + 4q^{4} + 8q^{7} + 24q^{10} - 16q^{13} - 8q^{16} - 64q^{19} + 48q^{22} - 14q^{25} + 32q^{28} - 88q^{31} - 36q^{34} - 136q^{37} + 24q^{40} + 80q^{43} + 96q^{46} + 66q^{49} + 32q^{52} + 288q^{55} + 12q^{58} - 100q^{61} - 32q^{64} - 16q^{67} + 48q^{70} - 64q^{73} - 64q^{76} + 152q^{79} - 264q^{82} + 108q^{85} - 96q^{88} - 128q^{91} - 240q^{94} - 352q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −3.67423 2.12132i 0 2.00000 + 3.46410i 2.82843i 0 6.00000
53.2 1.22474 0.707107i 0 1.00000 1.73205i 3.67423 + 2.12132i 0 2.00000 + 3.46410i 2.82843i 0 6.00000
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.67423 + 2.12132i 0 2.00000 3.46410i 2.82843i 0 6.00000
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.67423 2.12132i 0 2.00000 3.46410i 2.82843i 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.3.d.b 4
3.b odd 2 1 inner 162.3.d.b 4
4.b odd 2 1 1296.3.q.f 4
9.c even 3 1 18.3.b.a 2
9.c even 3 1 inner 162.3.d.b 4
9.d odd 6 1 18.3.b.a 2
9.d odd 6 1 inner 162.3.d.b 4
12.b even 2 1 1296.3.q.f 4
36.f odd 6 1 144.3.e.b 2
36.f odd 6 1 1296.3.q.f 4
36.h even 6 1 144.3.e.b 2
36.h even 6 1 1296.3.q.f 4
45.h odd 6 1 450.3.d.f 2
45.j even 6 1 450.3.d.f 2
45.k odd 12 2 450.3.b.b 4
45.l even 12 2 450.3.b.b 4
63.g even 3 1 882.3.s.b 4
63.h even 3 1 882.3.s.b 4
63.i even 6 1 882.3.s.d 4
63.j odd 6 1 882.3.s.b 4
63.k odd 6 1 882.3.s.d 4
63.l odd 6 1 882.3.b.a 2
63.n odd 6 1 882.3.s.b 4
63.o even 6 1 882.3.b.a 2
63.s even 6 1 882.3.s.d 4
63.t odd 6 1 882.3.s.d 4
72.j odd 6 1 576.3.e.c 2
72.l even 6 1 576.3.e.f 2
72.n even 6 1 576.3.e.c 2
72.p odd 6 1 576.3.e.f 2
99.g even 6 1 2178.3.c.d 2
99.h odd 6 1 2178.3.c.d 2
117.n odd 6 1 3042.3.c.e 2
117.t even 6 1 3042.3.c.e 2
117.y odd 12 2 3042.3.d.a 4
117.z even 12 2 3042.3.d.a 4
144.u even 12 2 2304.3.h.c 4
144.v odd 12 2 2304.3.h.c 4
144.w odd 12 2 2304.3.h.f 4
144.x even 12 2 2304.3.h.f 4
180.n even 6 1 3600.3.l.d 2
180.p odd 6 1 3600.3.l.d 2
180.v odd 12 2 3600.3.c.b 4
180.x even 12 2 3600.3.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 9.c even 3 1
18.3.b.a 2 9.d odd 6 1
144.3.e.b 2 36.f odd 6 1
144.3.e.b 2 36.h even 6 1
162.3.d.b 4 1.a even 1 1 trivial
162.3.d.b 4 3.b odd 2 1 inner
162.3.d.b 4 9.c even 3 1 inner
162.3.d.b 4 9.d odd 6 1 inner
450.3.b.b 4 45.k odd 12 2
450.3.b.b 4 45.l even 12 2
450.3.d.f 2 45.h odd 6 1
450.3.d.f 2 45.j even 6 1
576.3.e.c 2 72.j odd 6 1
576.3.e.c 2 72.n even 6 1
576.3.e.f 2 72.l even 6 1
576.3.e.f 2 72.p odd 6 1
882.3.b.a 2 63.l odd 6 1
882.3.b.a 2 63.o even 6 1
882.3.s.b 4 63.g even 3 1
882.3.s.b 4 63.h even 3 1
882.3.s.b 4 63.j odd 6 1
882.3.s.b 4 63.n odd 6 1
882.3.s.d 4 63.i even 6 1
882.3.s.d 4 63.k odd 6 1
882.3.s.d 4 63.s even 6 1
882.3.s.d 4 63.t odd 6 1
1296.3.q.f 4 4.b odd 2 1
1296.3.q.f 4 12.b even 2 1
1296.3.q.f 4 36.f odd 6 1
1296.3.q.f 4 36.h even 6 1
2178.3.c.d 2 99.g even 6 1
2178.3.c.d 2 99.h odd 6 1
2304.3.h.c 4 144.u even 12 2
2304.3.h.c 4 144.v odd 12 2
2304.3.h.f 4 144.w odd 12 2
2304.3.h.f 4 144.x even 12 2
3042.3.c.e 2 117.n odd 6 1
3042.3.c.e 2 117.t even 6 1
3042.3.d.a 4 117.y odd 12 2
3042.3.d.a 4 117.z even 12 2
3600.3.c.b 4 180.v odd 12 2
3600.3.c.b 4 180.x even 12 2
3600.3.l.d 2 180.n even 6 1
3600.3.l.d 2 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 18 T_{5}^{2} + 324 \) acting on \(S_{3}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 - 18 T^{2} + T^{4} \)
$7$ \( ( 16 - 4 T + T^{2} )^{2} \)
$11$ \( 82944 - 288 T^{2} + T^{4} \)
$13$ \( ( 64 + 8 T + T^{2} )^{2} \)
$17$ \( ( 162 + T^{2} )^{2} \)
$19$ \( ( 16 + T )^{4} \)
$23$ \( 82944 - 288 T^{2} + T^{4} \)
$29$ \( 324 - 18 T^{2} + T^{4} \)
$31$ \( ( 1936 + 44 T + T^{2} )^{2} \)
$37$ \( ( 34 + T )^{4} \)
$41$ \( 4743684 - 2178 T^{2} + T^{4} \)
$43$ \( ( 1600 - 40 T + T^{2} )^{2} \)
$47$ \( 51840000 - 7200 T^{2} + T^{4} \)
$53$ \( ( 1458 + T^{2} )^{2} \)
$59$ \( 1327104 - 1152 T^{2} + T^{4} \)
$61$ \( ( 2500 + 50 T + T^{2} )^{2} \)
$67$ \( ( 64 + 8 T + T^{2} )^{2} \)
$71$ \( ( 2592 + T^{2} )^{2} \)
$73$ \( ( 16 + T )^{4} \)
$79$ \( ( 5776 - 76 T + T^{2} )^{2} \)
$83$ \( 199148544 - 14112 T^{2} + T^{4} \)
$89$ \( ( 162 + T^{2} )^{2} \)
$97$ \( ( 30976 + 176 T + T^{2} )^{2} \)
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