# Properties

 Label 162.6.c.e Level 162 Weight 6 Character orbit 162.c Analytic conductor 25.982 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$25.9821788097$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -4 + 4 \zeta_{6} ) q^{2} -16 \zeta_{6} q^{4} + 66 \zeta_{6} q^{5} + ( -176 + 176 \zeta_{6} ) q^{7} + 64 q^{8} +O(q^{10})$$ $$q + ( -4 + 4 \zeta_{6} ) q^{2} -16 \zeta_{6} q^{4} + 66 \zeta_{6} q^{5} + ( -176 + 176 \zeta_{6} ) q^{7} + 64 q^{8} -264 q^{10} + ( 60 - 60 \zeta_{6} ) q^{11} + 658 \zeta_{6} q^{13} -704 \zeta_{6} q^{14} + ( -256 + 256 \zeta_{6} ) q^{16} -414 q^{17} + 956 q^{19} + ( 1056 - 1056 \zeta_{6} ) q^{20} + 240 \zeta_{6} q^{22} -600 \zeta_{6} q^{23} + ( -1231 + 1231 \zeta_{6} ) q^{25} -2632 q^{26} + 2816 q^{28} + ( -5574 + 5574 \zeta_{6} ) q^{29} + 3592 \zeta_{6} q^{31} -1024 \zeta_{6} q^{32} + ( 1656 - 1656 \zeta_{6} ) q^{34} -11616 q^{35} -8458 q^{37} + ( -3824 + 3824 \zeta_{6} ) q^{38} + 4224 \zeta_{6} q^{40} -19194 \zeta_{6} q^{41} + ( -13316 + 13316 \zeta_{6} ) q^{43} -960 q^{44} + 2400 q^{46} + ( 19680 - 19680 \zeta_{6} ) q^{47} -14169 \zeta_{6} q^{49} -4924 \zeta_{6} q^{50} + ( 10528 - 10528 \zeta_{6} ) q^{52} -31266 q^{53} + 3960 q^{55} + ( -11264 + 11264 \zeta_{6} ) q^{56} -22296 \zeta_{6} q^{58} -26340 \zeta_{6} q^{59} + ( 31090 - 31090 \zeta_{6} ) q^{61} -14368 q^{62} + 4096 q^{64} + ( -43428 + 43428 \zeta_{6} ) q^{65} + 16804 \zeta_{6} q^{67} + 6624 \zeta_{6} q^{68} + ( 46464 - 46464 \zeta_{6} ) q^{70} + 6120 q^{71} -25558 q^{73} + ( 33832 - 33832 \zeta_{6} ) q^{74} -15296 \zeta_{6} q^{76} + 10560 \zeta_{6} q^{77} + ( -74408 + 74408 \zeta_{6} ) q^{79} -16896 q^{80} + 76776 q^{82} + ( 6468 - 6468 \zeta_{6} ) q^{83} -27324 \zeta_{6} q^{85} -53264 \zeta_{6} q^{86} + ( 3840 - 3840 \zeta_{6} ) q^{88} -32742 q^{89} -115808 q^{91} + ( -9600 + 9600 \zeta_{6} ) q^{92} + 78720 \zeta_{6} q^{94} + 63096 \zeta_{6} q^{95} + ( -166082 + 166082 \zeta_{6} ) q^{97} + 56676 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} - 16q^{4} + 66q^{5} - 176q^{7} + 128q^{8} + O(q^{10})$$ $$2q - 4q^{2} - 16q^{4} + 66q^{5} - 176q^{7} + 128q^{8} - 528q^{10} + 60q^{11} + 658q^{13} - 704q^{14} - 256q^{16} - 828q^{17} + 1912q^{19} + 1056q^{20} + 240q^{22} - 600q^{23} - 1231q^{25} - 5264q^{26} + 5632q^{28} - 5574q^{29} + 3592q^{31} - 1024q^{32} + 1656q^{34} - 23232q^{35} - 16916q^{37} - 3824q^{38} + 4224q^{40} - 19194q^{41} - 13316q^{43} - 1920q^{44} + 4800q^{46} + 19680q^{47} - 14169q^{49} - 4924q^{50} + 10528q^{52} - 62532q^{53} + 7920q^{55} - 11264q^{56} - 22296q^{58} - 26340q^{59} + 31090q^{61} - 28736q^{62} + 8192q^{64} - 43428q^{65} + 16804q^{67} + 6624q^{68} + 46464q^{70} + 12240q^{71} - 51116q^{73} + 33832q^{74} - 15296q^{76} + 10560q^{77} - 74408q^{79} - 33792q^{80} + 153552q^{82} + 6468q^{83} - 27324q^{85} - 53264q^{86} + 3840q^{88} - 65484q^{89} - 231616q^{91} - 9600q^{92} + 78720q^{94} + 63096q^{95} - 166082q^{97} + 113352q^{98} + O(q^{100})$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−2.00000 + 3.46410i 0 −8.00000 13.8564i 33.0000 + 57.1577i 0 −88.0000 + 152.420i 64.0000 0 −264.000
109.1 −2.00000 3.46410i 0 −8.00000 + 13.8564i 33.0000 57.1577i 0 −88.0000 152.420i 64.0000 0 −264.000
 $$n$$: 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
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 162.6.c.e 2
3.b odd 2 1 162.6.c.h 2
9.c even 3 1 6.6.a.a 1
9.c even 3 1 inner 162.6.c.e 2
9.d odd 6 1 18.6.a.b 1
9.d odd 6 1 162.6.c.h 2
36.f odd 6 1 48.6.a.c 1
36.h even 6 1 144.6.a.j 1
45.h odd 6 1 450.6.a.m 1
45.j even 6 1 150.6.a.d 1
45.k odd 12 2 150.6.c.b 2
45.l even 12 2 450.6.c.j 2
63.g even 3 1 294.6.e.g 2
63.h even 3 1 294.6.e.g 2
63.k odd 6 1 294.6.e.a 2
63.l odd 6 1 294.6.a.m 1
63.o even 6 1 882.6.a.a 1
63.t odd 6 1 294.6.e.a 2
72.j odd 6 1 576.6.a.j 1
72.l even 6 1 576.6.a.i 1
72.n even 6 1 192.6.a.o 1
72.p odd 6 1 192.6.a.g 1
99.h odd 6 1 726.6.a.a 1
117.t even 6 1 1014.6.a.c 1
144.v odd 12 2 768.6.d.p 2
144.x even 12 2 768.6.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 9.c even 3 1
18.6.a.b 1 9.d odd 6 1
48.6.a.c 1 36.f odd 6 1
144.6.a.j 1 36.h even 6 1
150.6.a.d 1 45.j even 6 1
150.6.c.b 2 45.k odd 12 2
162.6.c.e 2 1.a even 1 1 trivial
162.6.c.e 2 9.c even 3 1 inner
162.6.c.h 2 3.b odd 2 1
162.6.c.h 2 9.d odd 6 1
192.6.a.g 1 72.p odd 6 1
192.6.a.o 1 72.n even 6 1
294.6.a.m 1 63.l odd 6 1
294.6.e.a 2 63.k odd 6 1
294.6.e.a 2 63.t odd 6 1
294.6.e.g 2 63.g even 3 1
294.6.e.g 2 63.h even 3 1
450.6.a.m 1 45.h odd 6 1
450.6.c.j 2 45.l even 12 2
576.6.a.i 1 72.l even 6 1
576.6.a.j 1 72.j odd 6 1
726.6.a.a 1 99.h odd 6 1
768.6.d.c 2 144.x even 12 2
768.6.d.p 2 144.v odd 12 2
882.6.a.a 1 63.o even 6 1
1014.6.a.c 1 117.t even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 66 T_{5} + 4356$$ acting on $$S_{6}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T + 16 T^{2}$$
$3$ 1
$5$ $$1 - 66 T + 1231 T^{2} - 206250 T^{3} + 9765625 T^{4}$$
$7$ $$1 + 176 T + 14169 T^{2} + 2958032 T^{3} + 282475249 T^{4}$$
$11$ $$1 - 60 T - 157451 T^{2} - 9663060 T^{3} + 25937424601 T^{4}$$
$13$ $$1 - 658 T + 61671 T^{2} - 244310794 T^{3} + 137858491849 T^{4}$$
$17$ $$( 1 + 414 T + 1419857 T^{2} )^{2}$$
$19$ $$( 1 - 956 T + 2476099 T^{2} )^{2}$$
$23$ $$1 + 600 T - 6076343 T^{2} + 3861805800 T^{3} + 41426511213649 T^{4}$$
$29$ $$1 + 5574 T + 10558327 T^{2} + 114329144526 T^{3} + 420707233300201 T^{4}$$
$31$ $$1 - 3592 T - 15726687 T^{2} - 102835910392 T^{3} + 819628286980801 T^{4}$$
$37$ $$( 1 + 8458 T + 69343957 T^{2} )^{2}$$
$41$ $$1 + 19194 T + 252553435 T^{2} + 2223743921994 T^{3} + 13422659310152401 T^{4}$$
$43$ $$1 + 13316 T + 30307413 T^{2} + 1957564426988 T^{3} + 21611482313284249 T^{4}$$
$47$ $$1 - 19680 T + 157957393 T^{2} - 4513509737760 T^{3} + 52599132235830049 T^{4}$$
$53$ $$( 1 + 31266 T + 418195493 T^{2} )^{2}$$
$59$ $$1 + 26340 T - 21128699 T^{2} + 18831106035660 T^{3} + 511116753300641401 T^{4}$$
$61$ $$1 - 31090 T + 121991799 T^{2} - 26258498998090 T^{3} + 713342911662882601 T^{4}$$
$67$ $$1 - 16804 T - 1067750691 T^{2} - 22687502298028 T^{3} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 6120 T + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 + 25558 T + 2073071593 T^{2} )^{2}$$
$79$ $$1 + 74408 T + 2459494065 T^{2} + 228957612536792 T^{3} + 9468276082626847201 T^{4}$$
$83$ $$1 - 6468 T - 3897205619 T^{2} - 25477714878924 T^{3} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 32742 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 166082 T + 18995890467 T^{2} + 1426202644563074 T^{3} + 73742412689492826049 T^{4}$$