# Properties

 Label 147.4.g.a Level $147$ Weight $4$ Character orbit 147.g Analytic conductor $8.673$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -3 - 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 27 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 - 3 \zeta_{6} ) q^{3} + ( -8 + 8 \zeta_{6} ) q^{4} + 27 \zeta_{6} q^{9} + ( 48 - 24 \zeta_{6} ) q^{12} + ( 36 - 72 \zeta_{6} ) q^{13} -64 \zeta_{6} q^{16} + ( 180 - 90 \zeta_{6} ) q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} + ( 81 - 162 \zeta_{6} ) q^{27} + ( -90 - 90 \zeta_{6} ) q^{31} -216 q^{36} + 110 \zeta_{6} q^{37} + ( -324 + 324 \zeta_{6} ) q^{39} + 520 q^{43} + ( -192 + 384 \zeta_{6} ) q^{48} + ( 288 + 288 \zeta_{6} ) q^{52} -810 q^{57} + ( -1080 + 540 \zeta_{6} ) q^{61} + 512 q^{64} + ( 880 - 880 \zeta_{6} ) q^{67} + ( -216 - 216 \zeta_{6} ) q^{73} + ( -750 + 375 \zeta_{6} ) q^{75} + ( -720 + 1440 \zeta_{6} ) q^{76} -884 \zeta_{6} q^{79} + ( -729 + 729 \zeta_{6} ) q^{81} + 810 \zeta_{6} q^{93} + ( 792 - 1584 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + O(q^{10})$$ $$2 q - 9 q^{3} - 8 q^{4} + 27 q^{9} + 72 q^{12} - 64 q^{16} + 270 q^{19} + 125 q^{25} - 270 q^{31} - 432 q^{36} + 110 q^{37} - 324 q^{39} + 1040 q^{43} + 864 q^{52} - 1620 q^{57} - 1620 q^{61} + 1024 q^{64} + 880 q^{67} - 648 q^{73} - 1125 q^{75} - 884 q^{79} - 729 q^{81} + 810 q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
68.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −4.50000 + 2.59808i −4.00000 6.92820i 0 0 0 0 13.5000 23.3827i 0
80.1 0 −4.50000 2.59808i −4.00000 + 6.92820i 0 0 0 0 13.5000 + 23.3827i 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.g.a 2
3.b odd 2 1 CM 147.4.g.a 2
7.b odd 2 1 147.4.g.b 2
7.c even 3 1 21.4.c.a 2
7.c even 3 1 147.4.g.b 2
7.d odd 6 1 21.4.c.a 2
7.d odd 6 1 inner 147.4.g.a 2
21.c even 2 1 147.4.g.b 2
21.g even 6 1 21.4.c.a 2
21.g even 6 1 inner 147.4.g.a 2
21.h odd 6 1 21.4.c.a 2
21.h odd 6 1 147.4.g.b 2
28.f even 6 1 336.4.k.a 2
28.g odd 6 1 336.4.k.a 2
84.j odd 6 1 336.4.k.a 2
84.n even 6 1 336.4.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 7.c even 3 1
21.4.c.a 2 7.d odd 6 1
21.4.c.a 2 21.g even 6 1
21.4.c.a 2 21.h odd 6 1
147.4.g.a 2 1.a even 1 1 trivial
147.4.g.a 2 3.b odd 2 1 CM
147.4.g.a 2 7.d odd 6 1 inner
147.4.g.a 2 21.g even 6 1 inner
147.4.g.b 2 7.b odd 2 1
147.4.g.b 2 7.c even 3 1
147.4.g.b 2 21.c even 2 1
147.4.g.b 2 21.h odd 6 1
336.4.k.a 2 28.f even 6 1
336.4.k.a 2 28.g odd 6 1
336.4.k.a 2 84.j odd 6 1
336.4.k.a 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$:

 $$T_{2}$$ $$T_{19}^{2} - 270 T_{19} + 24300$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$27 + 9 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$3888 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$24300 - 270 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$24300 + 270 T + T^{2}$$
$37$ $$12100 - 110 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -520 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$874800 + 1620 T + T^{2}$$
$67$ $$774400 - 880 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$139968 + 648 T + T^{2}$$
$79$ $$781456 + 884 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$1881792 + T^{2}$$