# Properties

 Label 147.2.e.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + 3 q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + 2 \zeta_{6} q^{5} - q^{6} + 3 q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} -2 q^{13} -2 q^{15} + \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -4 \zeta_{6} q^{19} + 2 q^{20} -4 q^{22} + ( -3 + 3 \zeta_{6} ) q^{24} + ( 1 - \zeta_{6} ) q^{25} -2 \zeta_{6} q^{26} + q^{27} -2 q^{29} -2 \zeta_{6} q^{30} + ( 5 - 5 \zeta_{6} ) q^{32} -4 \zeta_{6} q^{33} + 6 q^{34} - q^{36} -6 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 2 - 2 \zeta_{6} ) q^{39} + 6 \zeta_{6} q^{40} + 2 q^{41} -4 q^{43} + 4 \zeta_{6} q^{44} + ( 2 - 2 \zeta_{6} ) q^{45} - q^{48} + q^{50} + 6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} -8 q^{55} + 4 q^{57} -2 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + ( -2 + 2 \zeta_{6} ) q^{60} + 2 \zeta_{6} q^{61} + 7 q^{64} -4 \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{66} + ( -4 + 4 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} -3 \zeta_{6} q^{72} + ( 6 - 6 \zeta_{6} ) q^{73} + ( 6 - 6 \zeta_{6} ) q^{74} + \zeta_{6} q^{75} -4 q^{76} + 2 q^{78} + 16 \zeta_{6} q^{79} + ( -2 + 2 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 2 \zeta_{6} q^{82} -12 q^{83} + 12 q^{85} -4 \zeta_{6} q^{86} + ( 2 - 2 \zeta_{6} ) q^{87} + ( -12 + 12 \zeta_{6} ) q^{88} + 14 \zeta_{6} q^{89} + 2 q^{90} + ( 8 - 8 \zeta_{6} ) q^{95} + 5 \zeta_{6} q^{96} + 18 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} + q^{4} + 2q^{5} - 2q^{6} + 6q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} + q^{4} + 2q^{5} - 2q^{6} + 6q^{8} - q^{9} - 2q^{10} - 4q^{11} + q^{12} - 4q^{13} - 4q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + 4q^{20} - 8q^{22} - 3q^{24} + q^{25} - 2q^{26} + 2q^{27} - 4q^{29} - 2q^{30} + 5q^{32} - 4q^{33} + 12q^{34} - 2q^{36} - 6q^{37} + 4q^{38} + 2q^{39} + 6q^{40} + 4q^{41} - 8q^{43} + 4q^{44} + 2q^{45} - 2q^{48} + 2q^{50} + 6q^{51} - 2q^{52} - 6q^{53} + q^{54} - 16q^{55} + 8q^{57} - 2q^{58} - 12q^{59} - 2q^{60} + 2q^{61} + 14q^{64} - 4q^{65} + 4q^{66} - 4q^{67} - 6q^{68} - 3q^{72} + 6q^{73} + 6q^{74} + q^{75} - 8q^{76} + 4q^{78} + 16q^{79} - 2q^{80} - q^{81} + 2q^{82} - 24q^{83} + 24q^{85} - 4q^{86} + 2q^{87} - 12q^{88} + 14q^{89} + 4q^{90} + 8q^{95} + 5q^{96} + 36q^{97} + 8q^{99} + O(q^{100})$$

## 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
0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i −1.00000 0 3.00000 −0.500000 0.866025i −1.00000 + 1.73205i
79.1 0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i −1.00000 0 3.00000 −0.500000 + 0.866025i −1.00000 1.73205i
 $$n$$: 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.b 2
3.b odd 2 1 441.2.e.a 2
4.b odd 2 1 2352.2.q.x 2
7.b odd 2 1 147.2.e.c 2
7.c even 3 1 21.2.a.a 1
7.c even 3 1 inner 147.2.e.b 2
7.d odd 6 1 147.2.a.a 1
7.d odd 6 1 147.2.e.c 2
21.c even 2 1 441.2.e.b 2
21.g even 6 1 441.2.a.f 1
21.g even 6 1 441.2.e.b 2
21.h odd 6 1 63.2.a.a 1
21.h odd 6 1 441.2.e.a 2
28.d even 2 1 2352.2.q.e 2
28.f even 6 1 2352.2.a.v 1
28.f even 6 1 2352.2.q.e 2
28.g odd 6 1 336.2.a.a 1
28.g odd 6 1 2352.2.q.x 2
35.i odd 6 1 3675.2.a.n 1
35.j even 6 1 525.2.a.d 1
35.l odd 12 2 525.2.d.a 2
56.j odd 6 1 9408.2.a.bv 1
56.k odd 6 1 1344.2.a.s 1
56.m even 6 1 9408.2.a.m 1
56.p even 6 1 1344.2.a.g 1
63.g even 3 1 567.2.f.g 2
63.h even 3 1 567.2.f.g 2
63.j odd 6 1 567.2.f.b 2
63.n odd 6 1 567.2.f.b 2
77.h odd 6 1 2541.2.a.j 1
84.j odd 6 1 7056.2.a.p 1
84.n even 6 1 1008.2.a.l 1
91.r even 6 1 3549.2.a.c 1
105.o odd 6 1 1575.2.a.c 1
105.x even 12 2 1575.2.d.a 2
112.u odd 12 2 5376.2.c.l 2
112.w even 12 2 5376.2.c.r 2
119.j even 6 1 6069.2.a.b 1
133.r odd 6 1 7581.2.a.d 1
140.p odd 6 1 8400.2.a.bn 1
168.s odd 6 1 4032.2.a.h 1
168.v even 6 1 4032.2.a.k 1
231.l even 6 1 7623.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 7.c even 3 1
63.2.a.a 1 21.h odd 6 1
147.2.a.a 1 7.d odd 6 1
147.2.e.b 2 1.a even 1 1 trivial
147.2.e.b 2 7.c even 3 1 inner
147.2.e.c 2 7.b odd 2 1
147.2.e.c 2 7.d odd 6 1
336.2.a.a 1 28.g odd 6 1
441.2.a.f 1 21.g even 6 1
441.2.e.a 2 3.b odd 2 1
441.2.e.a 2 21.h odd 6 1
441.2.e.b 2 21.c even 2 1
441.2.e.b 2 21.g even 6 1
525.2.a.d 1 35.j even 6 1
525.2.d.a 2 35.l odd 12 2
567.2.f.b 2 63.j odd 6 1
567.2.f.b 2 63.n odd 6 1
567.2.f.g 2 63.g even 3 1
567.2.f.g 2 63.h even 3 1
1008.2.a.l 1 84.n even 6 1
1344.2.a.g 1 56.p even 6 1
1344.2.a.s 1 56.k odd 6 1
1575.2.a.c 1 105.o odd 6 1
1575.2.d.a 2 105.x even 12 2
2352.2.a.v 1 28.f even 6 1
2352.2.q.e 2 28.d even 2 1
2352.2.q.e 2 28.f even 6 1
2352.2.q.x 2 4.b odd 2 1
2352.2.q.x 2 28.g odd 6 1
2541.2.a.j 1 77.h odd 6 1
3549.2.a.c 1 91.r even 6 1
3675.2.a.n 1 35.i odd 6 1
4032.2.a.h 1 168.s odd 6 1
4032.2.a.k 1 168.v even 6 1
5376.2.c.l 2 112.u odd 12 2
5376.2.c.r 2 112.w even 12 2
6069.2.a.b 1 119.j even 6 1
7056.2.a.p 1 84.j odd 6 1
7581.2.a.d 1 133.r odd 6 1
7623.2.a.g 1 231.l even 6 1
8400.2.a.bn 1 140.p odd 6 1
9408.2.a.m 1 56.m even 6 1
9408.2.a.bv 1 56.j odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5}^{2} - 2 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$196 - 14 T + T^{2}$$
$97$ $$( -18 + T )^{2}$$