# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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