# Properties

 Label 147.2.e.a Level $147$ Weight $2$ Character orbit 147.e Analytic conductor $1.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -2 q^{6} -\zeta_{6} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} -2 q^{6} -\zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{10} + ( 2 - 2 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{12} - q^{13} -2 q^{15} + 4 \zeta_{6} q^{16} + ( -2 + 2 \zeta_{6} ) q^{18} + \zeta_{6} q^{19} + 4 q^{20} -4 q^{22} + ( 1 - \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} - q^{27} + 4 q^{29} + 4 \zeta_{6} q^{30} + ( 9 - 9 \zeta_{6} ) q^{31} + ( 8 - 8 \zeta_{6} ) q^{32} -2 \zeta_{6} q^{33} + 2 q^{36} -3 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + ( -1 + \zeta_{6} ) q^{39} + 10 q^{41} + 5 q^{43} + 4 \zeta_{6} q^{44} + ( -2 + 2 \zeta_{6} ) q^{45} -6 \zeta_{6} q^{47} + 4 q^{48} -2 q^{50} + ( 2 - 2 \zeta_{6} ) q^{52} + ( -12 + 12 \zeta_{6} ) q^{53} + 2 \zeta_{6} q^{54} -4 q^{55} + q^{57} -8 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + ( 4 - 4 \zeta_{6} ) q^{60} + 10 \zeta_{6} q^{61} -18 q^{62} -8 q^{64} + 2 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{66} + ( 5 - 5 \zeta_{6} ) q^{67} -6 q^{71} + ( -3 + 3 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} -\zeta_{6} q^{75} -2 q^{76} + 2 q^{78} + \zeta_{6} q^{79} + ( 8 - 8 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -20 \zeta_{6} q^{82} -6 q^{83} -10 \zeta_{6} q^{86} + ( 4 - 4 \zeta_{6} ) q^{87} + 16 \zeta_{6} q^{89} + 4 q^{90} -9 \zeta_{6} q^{93} + ( -12 + 12 \zeta_{6} ) q^{94} + ( 2 - 2 \zeta_{6} ) q^{95} -8 \zeta_{6} q^{96} + 6 q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + q^{3} - 2q^{4} - 2q^{5} - 4q^{6} - q^{9} + O(q^{10})$$ $$2q - 2q^{2} + q^{3} - 2q^{4} - 2q^{5} - 4q^{6} - q^{9} - 4q^{10} + 2q^{11} + 2q^{12} - 2q^{13} - 4q^{15} + 4q^{16} - 2q^{18} + q^{19} + 8q^{20} - 8q^{22} + q^{25} + 2q^{26} - 2q^{27} + 8q^{29} + 4q^{30} + 9q^{31} + 8q^{32} - 2q^{33} + 4q^{36} - 3q^{37} + 2q^{38} - q^{39} + 20q^{41} + 10q^{43} + 4q^{44} - 2q^{45} - 6q^{47} + 8q^{48} - 4q^{50} + 2q^{52} - 12q^{53} + 2q^{54} - 8q^{55} + 2q^{57} - 8q^{58} - 12q^{59} + 4q^{60} + 10q^{61} - 36q^{62} - 16q^{64} + 2q^{65} - 4q^{66} + 5q^{67} - 12q^{71} - 3q^{73} - 6q^{74} - q^{75} - 4q^{76} + 4q^{78} + q^{79} + 8q^{80} - q^{81} - 20q^{82} - 12q^{83} - 10q^{86} + 4q^{87} + 16q^{89} + 8q^{90} - 9q^{93} - 12q^{94} + 2q^{95} - 8q^{96} + 12q^{97} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0.500000 0.866025i −1.00000 + 1.73205i −1.00000 1.73205i −2.00000 0 0 −0.500000 0.866025i −2.00000 + 3.46410i
79.1 −1.00000 + 1.73205i 0.500000 + 0.866025i −1.00000 1.73205i −1.00000 + 1.73205i −2.00000 0 0 −0.500000 + 0.866025i −2.00000 3.46410i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.e.a 2
3.b odd 2 1 441.2.e.e 2
4.b odd 2 1 2352.2.q.c 2
7.b odd 2 1 21.2.e.a 2
7.c even 3 1 147.2.a.b 1
7.c even 3 1 inner 147.2.e.a 2
7.d odd 6 1 21.2.e.a 2
7.d odd 6 1 147.2.a.c 1
21.c even 2 1 63.2.e.b 2
21.g even 6 1 63.2.e.b 2
21.g even 6 1 441.2.a.b 1
21.h odd 6 1 441.2.a.a 1
21.h odd 6 1 441.2.e.e 2
28.d even 2 1 336.2.q.f 2
28.f even 6 1 336.2.q.f 2
28.f even 6 1 2352.2.a.d 1
28.g odd 6 1 2352.2.a.w 1
28.g odd 6 1 2352.2.q.c 2
35.c odd 2 1 525.2.i.e 2
35.f even 4 2 525.2.r.e 4
35.i odd 6 1 525.2.i.e 2
35.i odd 6 1 3675.2.a.a 1
35.j even 6 1 3675.2.a.c 1
35.k even 12 2 525.2.r.e 4
56.e even 2 1 1344.2.q.c 2
56.h odd 2 1 1344.2.q.m 2
56.j odd 6 1 1344.2.q.m 2
56.j odd 6 1 9408.2.a.bg 1
56.k odd 6 1 9408.2.a.k 1
56.m even 6 1 1344.2.q.c 2
56.m even 6 1 9408.2.a.cv 1
56.p even 6 1 9408.2.a.bz 1
63.i even 6 1 567.2.g.f 2
63.k odd 6 1 567.2.h.f 2
63.l odd 6 1 567.2.g.a 2
63.l odd 6 1 567.2.h.f 2
63.o even 6 1 567.2.g.f 2
63.o even 6 1 567.2.h.a 2
63.s even 6 1 567.2.h.a 2
63.t odd 6 1 567.2.g.a 2
84.h odd 2 1 1008.2.s.d 2
84.j odd 6 1 1008.2.s.d 2
84.j odd 6 1 7056.2.a.bp 1
84.n even 6 1 7056.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 7.b odd 2 1
21.2.e.a 2 7.d odd 6 1
63.2.e.b 2 21.c even 2 1
63.2.e.b 2 21.g even 6 1
147.2.a.b 1 7.c even 3 1
147.2.a.c 1 7.d odd 6 1
147.2.e.a 2 1.a even 1 1 trivial
147.2.e.a 2 7.c even 3 1 inner
336.2.q.f 2 28.d even 2 1
336.2.q.f 2 28.f even 6 1
441.2.a.a 1 21.h odd 6 1
441.2.a.b 1 21.g even 6 1
441.2.e.e 2 3.b odd 2 1
441.2.e.e 2 21.h odd 6 1
525.2.i.e 2 35.c odd 2 1
525.2.i.e 2 35.i odd 6 1
525.2.r.e 4 35.f even 4 2
525.2.r.e 4 35.k even 12 2
567.2.g.a 2 63.l odd 6 1
567.2.g.a 2 63.t odd 6 1
567.2.g.f 2 63.i even 6 1
567.2.g.f 2 63.o even 6 1
567.2.h.a 2 63.o even 6 1
567.2.h.a 2 63.s even 6 1
567.2.h.f 2 63.k odd 6 1
567.2.h.f 2 63.l odd 6 1
1008.2.s.d 2 84.h odd 2 1
1008.2.s.d 2 84.j odd 6 1
1344.2.q.c 2 56.e even 2 1
1344.2.q.c 2 56.m even 6 1
1344.2.q.m 2 56.h odd 2 1
1344.2.q.m 2 56.j odd 6 1
2352.2.a.d 1 28.f even 6 1
2352.2.a.w 1 28.g odd 6 1
2352.2.q.c 2 4.b odd 2 1
2352.2.q.c 2 28.g odd 6 1
3675.2.a.a 1 35.i odd 6 1
3675.2.a.c 1 35.j even 6 1
7056.2.a.m 1 84.n even 6 1
7056.2.a.bp 1 84.j odd 6 1
9408.2.a.k 1 56.k odd 6 1
9408.2.a.bg 1 56.j odd 6 1
9408.2.a.bz 1 56.p even 6 1
9408.2.a.cv 1 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{2} + 2 T_{2} + 4$$ $$T_{5}^{2} + 2 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$81 - 9 T + T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$9 + 3 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$256 - 16 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$