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: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 3 \beta_1) q^{5} + ( - 4 \beta_{2} + 4) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - 3 \beta_{3} + 3 \beta_1) q^{5} + ( - 4 \beta_{2} + 4) q^{7} + 2 \beta_{3} q^{8} + 6 q^{10} + 12 \beta_1 q^{11} - 8 \beta_{2} q^{13} + ( - 4 \beta_{3} + 4 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} + 9 \beta_{3} q^{17} - 16 q^{19} + 6 \beta_1 q^{20} + 24 \beta_{2} q^{22} + ( - 12 \beta_{3} + 12 \beta_1) q^{23} + (7 \beta_{2} - 7) q^{25} - 8 \beta_{3} q^{26} + 8 q^{28} + 3 \beta_1 q^{29} - 44 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} + (18 \beta_{2} - 18) q^{34} - 12 \beta_{3} q^{35} - 34 q^{37} - 16 \beta_1 q^{38} + 12 \beta_{2} q^{40} + (33 \beta_{3} - 33 \beta_1) q^{41} + ( - 40 \beta_{2} + 40) q^{43} + 24 \beta_{3} q^{44} + 24 q^{46} - 60 \beta_1 q^{47} + 33 \beta_{2} q^{49} + (7 \beta_{3} - 7 \beta_1) q^{50} + ( - 16 \beta_{2} + 16) q^{52} - 27 \beta_{3} q^{53} + 72 q^{55} + 8 \beta_1 q^{56} + 6 \beta_{2} q^{58} + (24 \beta_{3} - 24 \beta_1) q^{59} + (50 \beta_{2} - 50) q^{61} - 44 \beta_{3} q^{62} - 8 q^{64} - 24 \beta_1 q^{65} - 8 \beta_{2} q^{67} + (18 \beta_{3} - 18 \beta_1) q^{68} + ( - 24 \beta_{2} + 24) q^{70} + 36 \beta_{3} q^{71} - 16 q^{73} - 34 \beta_1 q^{74} - 32 \beta_{2} q^{76} + ( - 48 \beta_{3} + 48 \beta_1) q^{77} + ( - 76 \beta_{2} + 76) q^{79} + 12 \beta_{3} q^{80} - 66 q^{82} + 84 \beta_1 q^{83} + 54 \beta_{2} q^{85} + ( - 40 \beta_{3} + 40 \beta_1) q^{86} + (48 \beta_{2} - 48) q^{88} - 9 \beta_{3} q^{89} - 32 q^{91} + 24 \beta_1 q^{92} - 120 \beta_{2} q^{94} + (48 \beta_{3} - 48 \beta_1) q^{95} + (176 \beta_{2} - 176) q^{97} + 33 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 8 q^{7} + 24 q^{10} - 16 q^{13} - 8 q^{16} - 64 q^{19} + 48 q^{22} - 14 q^{25} + 32 q^{28} - 88 q^{31} - 36 q^{34} - 136 q^{37} + 24 q^{40} + 80 q^{43} + 96 q^{46} + 66 q^{49} + 32 q^{52} + 288 q^{55} + 12 q^{58} - 100 q^{61} - 32 q^{64} - 16 q^{67} + 48 q^{70} - 64 q^{73} - 64 q^{76} + 152 q^{79} - 264 q^{82} + 108 q^{85} - 96 q^{88} - 128 q^{91} - 240 q^{94} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \( T_{5}^{4} - 18T_{5}^{2} + 324 \) acting on \(S_{3}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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