# Properties

 Label 147.2.e.d Level $147$ Weight $2$ Character orbit 147.e Analytic conductor $1.174$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + \beta_{2} q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{4} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( -3 + \beta_{3} ) q^{8} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} + \beta_{2} ) q^{2} + \beta_{2} q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{4} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 + \beta_{3} ) q^{6} + ( -3 + \beta_{3} ) q^{8} + ( -1 - \beta_{2} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{10} -2 \beta_{2} q^{11} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{12} + ( 4 - \beta_{3} ) q^{13} + ( 2 + \beta_{3} ) q^{15} + ( -3 - 3 \beta_{2} ) q^{16} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{18} -2 \beta_{1} q^{19} + ( -2 - 3 \beta_{3} ) q^{20} + ( 2 - 2 \beta_{3} ) q^{22} + ( 2 - 4 \beta_{1} + 2 \beta_{2} ) q^{23} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{24} + ( -4 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{25} + ( 6 + 5 \beta_{1} + 6 \beta_{2} ) q^{26} + q^{27} + ( -4 - 2 \beta_{3} ) q^{29} + \beta_{1} q^{30} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{32} + ( 2 + 2 \beta_{2} ) q^{33} + ( -8 + 5 \beta_{3} ) q^{34} + ( 1 - 2 \beta_{3} ) q^{36} + ( 4 + 4 \beta_{2} ) q^{37} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{38} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{39} + ( 4 - \beta_{1} + 4 \beta_{2} ) q^{40} + ( 2 + 3 \beta_{3} ) q^{41} + 4 \beta_{3} q^{43} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{45} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{46} -2 \beta_{1} q^{47} + 3 q^{48} + ( 7 - 3 \beta_{3} ) q^{50} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 9 \beta_{1} + 8 \beta_{2} + 9 \beta_{3} ) q^{52} -2 \beta_{2} q^{53} + ( 1 + \beta_{1} + \beta_{2} ) q^{54} + ( -4 - 2 \beta_{3} ) q^{55} -2 \beta_{3} q^{57} -2 \beta_{1} q^{58} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{60} + ( -8 + 3 \beta_{1} - 8 \beta_{2} ) q^{61} + ( 8 - 6 \beta_{3} ) q^{62} + ( -7 + 2 \beta_{3} ) q^{64} + ( -6 + 2 \beta_{1} - 6 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{67} + ( -14 - 7 \beta_{1} - 14 \beta_{2} ) q^{68} + ( -2 - 4 \beta_{3} ) q^{69} + ( -2 + 8 \beta_{3} ) q^{71} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{72} + ( -7 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} ) q^{73} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{74} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{75} + ( 8 - 2 \beta_{3} ) q^{76} + ( -6 + 5 \beta_{3} ) q^{78} + ( -8 + 4 \beta_{1} - 8 \beta_{2} ) q^{79} + ( -3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{80} + \beta_{2} q^{81} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{82} + ( -4 + 8 \beta_{3} ) q^{83} + ( -2 - 4 \beta_{3} ) q^{85} + ( -8 - 4 \beta_{1} - 8 \beta_{2} ) q^{86} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{88} + ( 10 - 3 \beta_{1} + 10 \beta_{2} ) q^{89} + \beta_{3} q^{90} + 14 q^{92} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{94} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{95} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{96} + ( 4 - \beta_{3} ) q^{97} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{3} - 2q^{4} - 4q^{5} - 4q^{6} - 12q^{8} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{3} - 2q^{4} - 4q^{5} - 4q^{6} - 12q^{8} - 2q^{9} + 4q^{11} - 2q^{12} + 16q^{13} + 8q^{15} - 6q^{16} - 4q^{17} + 2q^{18} - 8q^{20} + 8q^{22} + 4q^{23} + 6q^{24} - 2q^{25} + 12q^{26} + 4q^{27} - 16q^{29} + 8q^{31} - 6q^{32} + 4q^{33} - 32q^{34} + 4q^{36} + 8q^{37} + 8q^{38} - 8q^{39} + 8q^{40} + 8q^{41} + 4q^{44} - 4q^{45} + 12q^{46} + 12q^{48} + 28q^{50} - 4q^{51} - 16q^{52} + 4q^{53} + 2q^{54} - 16q^{55} + 8q^{59} + 4q^{60} - 16q^{61} + 32q^{62} - 28q^{64} - 12q^{65} - 4q^{66} - 28q^{68} - 8q^{69} - 8q^{71} + 6q^{72} - 8q^{73} - 8q^{74} - 2q^{75} + 32q^{76} - 24q^{78} - 16q^{79} - 12q^{80} - 2q^{81} - 8q^{82} - 16q^{83} - 8q^{85} - 16q^{86} + 8q^{87} - 12q^{88} + 20q^{89} + 56q^{92} + 8q^{93} + 8q^{94} + 8q^{95} - 6q^{96} + 16q^{97} - 8q^{99} + O(q^{100})$$

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 $$\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$$.

 $$n$$ $$50$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.207107 0.358719i −0.500000 + 0.866025i 0.914214 1.58346i −1.70711 2.95680i 0.414214 0 −1.58579 −0.500000 0.866025i −0.707107 + 1.22474i
67.2 1.20711 + 2.09077i −0.500000 + 0.866025i −1.91421 + 3.31552i −0.292893 0.507306i −2.41421 0 −4.41421 −0.500000 0.866025i 0.707107 1.22474i
79.1 −0.207107 + 0.358719i −0.500000 0.866025i 0.914214 + 1.58346i −1.70711 + 2.95680i 0.414214 0 −1.58579 −0.500000 + 0.866025i −0.707107 1.22474i
79.2 1.20711 2.09077i −0.500000 0.866025i −1.91421 3.31552i −0.292893 + 0.507306i −2.41421 0 −4.41421 −0.500000 + 0.866025i 0.707107 + 1.22474i
 $$n$$: 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.d 4
3.b odd 2 1 441.2.e.g 4
4.b odd 2 1 2352.2.q.bd 4
7.b odd 2 1 147.2.e.e 4
7.c even 3 1 147.2.a.e yes 2
7.c even 3 1 inner 147.2.e.d 4
7.d odd 6 1 147.2.a.d 2
7.d odd 6 1 147.2.e.e 4
21.c even 2 1 441.2.e.f 4
21.g even 6 1 441.2.a.j 2
21.g even 6 1 441.2.e.f 4
21.h odd 6 1 441.2.a.i 2
21.h odd 6 1 441.2.e.g 4
28.d even 2 1 2352.2.q.bb 4
28.f even 6 1 2352.2.a.be 2
28.f even 6 1 2352.2.q.bb 4
28.g odd 6 1 2352.2.a.bc 2
28.g odd 6 1 2352.2.q.bd 4
35.i odd 6 1 3675.2.a.bf 2
35.j even 6 1 3675.2.a.bd 2
56.j odd 6 1 9408.2.a.ef 2
56.k odd 6 1 9408.2.a.dt 2
56.m even 6 1 9408.2.a.dq 2
56.p even 6 1 9408.2.a.di 2
84.j odd 6 1 7056.2.a.cv 2
84.n even 6 1 7056.2.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 7.d odd 6 1
147.2.a.e yes 2 7.c even 3 1
147.2.e.d 4 1.a even 1 1 trivial
147.2.e.d 4 7.c even 3 1 inner
147.2.e.e 4 7.b odd 2 1
147.2.e.e 4 7.d odd 6 1
441.2.a.i 2 21.h odd 6 1
441.2.a.j 2 21.g even 6 1
441.2.e.f 4 21.c even 2 1
441.2.e.f 4 21.g even 6 1
441.2.e.g 4 3.b odd 2 1
441.2.e.g 4 21.h odd 6 1
2352.2.a.bc 2 28.g odd 6 1
2352.2.a.be 2 28.f even 6 1
2352.2.q.bb 4 28.d even 2 1
2352.2.q.bb 4 28.f even 6 1
2352.2.q.bd 4 4.b odd 2 1
2352.2.q.bd 4 28.g odd 6 1
3675.2.a.bd 2 35.j even 6 1
3675.2.a.bf 2 35.i odd 6 1
7056.2.a.cf 2 84.n even 6 1
7056.2.a.cv 2 84.j odd 6 1
9408.2.a.di 2 56.p even 6 1
9408.2.a.dq 2 56.m even 6 1
9408.2.a.dt 2 56.k odd 6 1
9408.2.a.ef 2 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 $$S_{2}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} + 2 T_{2} + 1$$ $$T_{5}^{4} + 4 T_{5}^{3} + 14 T_{5}^{2} + 8 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 - 2 T + T^{2} )^{2}$$
$13$ $$( 14 - 8 T + T^{2} )^{2}$$
$17$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$784 + 112 T + 44 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( 8 + 8 T + T^{2} )^{2}$$
$31$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$( 16 - 4 T + T^{2} )^{2}$$
$41$ $$( -14 - 4 T + T^{2} )^{2}$$
$43$ $$( -32 + T^{2} )^{2}$$
$47$ $$64 + 8 T^{2} + T^{4}$$
$53$ $$( 4 - 2 T + T^{2} )^{2}$$
$59$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$61$ $$2116 + 736 T + 210 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$1024 + 32 T^{2} + T^{4}$$
$71$ $$( -124 + 4 T + T^{2} )^{2}$$
$73$ $$6724 - 656 T + 146 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$1024 + 512 T + 224 T^{2} + 16 T^{3} + T^{4}$$
$83$ $$( -112 + 8 T + T^{2} )^{2}$$
$89$ $$6724 - 1640 T + 318 T^{2} - 20 T^{3} + T^{4}$$
$97$ $$( 14 - 8 T + T^{2} )^{2}$$