# Properties

 Label 63.2.e.b Level $63$ Weight $2$ Character orbit 63.e Analytic conductor $0.503$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 63.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.503057532734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} + ( -3 + \zeta_{6} ) q^{7} + 4 \zeta_{6} q^{10} -2 \zeta_{6} q^{11} + q^{13} + ( -4 + 6 \zeta_{6} ) q^{14} + ( 4 - 4 \zeta_{6} ) q^{16} + ( -1 + \zeta_{6} ) q^{19} + 4 q^{20} -4 q^{22} + \zeta_{6} q^{25} + ( 2 - 2 \zeta_{6} ) q^{26} + ( 2 + 4 \zeta_{6} ) q^{28} -4 q^{29} -9 \zeta_{6} q^{31} -8 \zeta_{6} q^{32} + ( 4 - 6 \zeta_{6} ) q^{35} + ( -3 + 3 \zeta_{6} ) q^{37} + 2 \zeta_{6} q^{38} + 10 q^{41} + 5 q^{43} + ( -4 + 4 \zeta_{6} ) q^{44} + ( -6 + 6 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + 2 q^{50} -2 \zeta_{6} q^{52} + 12 \zeta_{6} q^{53} + 4 q^{55} + ( -8 + 8 \zeta_{6} ) q^{58} -12 \zeta_{6} q^{59} + ( -10 + 10 \zeta_{6} ) q^{61} -18 q^{62} -8 q^{64} + ( -2 + 2 \zeta_{6} ) q^{65} + 5 \zeta_{6} q^{67} + ( -4 - 8 \zeta_{6} ) q^{70} + 6 q^{71} + 3 \zeta_{6} q^{73} + 6 \zeta_{6} q^{74} + 2 q^{76} + ( 2 + 4 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} + 8 \zeta_{6} q^{80} + ( 20 - 20 \zeta_{6} ) q^{82} -6 q^{83} + ( 10 - 10 \zeta_{6} ) q^{86} + ( 16 - 16 \zeta_{6} ) q^{89} + ( -3 + \zeta_{6} ) q^{91} + 12 \zeta_{6} q^{94} -2 \zeta_{6} q^{95} -6 q^{97} + ( 6 - 16 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} - 2q^{5} - 5q^{7} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} - 2q^{5} - 5q^{7} + 4q^{10} - 2q^{11} + 2q^{13} - 2q^{14} + 4q^{16} - q^{19} + 8q^{20} - 8q^{22} + q^{25} + 2q^{26} + 8q^{28} - 8q^{29} - 9q^{31} - 8q^{32} + 2q^{35} - 3q^{37} + 2q^{38} + 20q^{41} + 10q^{43} - 4q^{44} - 6q^{47} + 11q^{49} + 4q^{50} - 2q^{52} + 12q^{53} + 8q^{55} - 8q^{58} - 12q^{59} - 10q^{61} - 36q^{62} - 16q^{64} - 2q^{65} + 5q^{67} - 16q^{70} + 12q^{71} + 3q^{73} + 6q^{74} + 4q^{76} + 8q^{77} + q^{79} + 8q^{80} + 20q^{82} - 12q^{83} + 10q^{86} + 16q^{89} - 5q^{91} + 12q^{94} - 2q^{95} - 12q^{97} - 4q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0 −1.00000 1.73205i −1.00000 + 1.73205i 0 −2.50000 + 0.866025i 0 0 2.00000 + 3.46410i
46.1 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −1.00000 1.73205i 0 −2.50000 0.866025i 0 0 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 63.2.e.b 2
3.b odd 2 1 21.2.e.a 2
4.b odd 2 1 1008.2.s.d 2
7.b odd 2 1 441.2.e.e 2
7.c even 3 1 inner 63.2.e.b 2
7.c even 3 1 441.2.a.b 1
7.d odd 6 1 441.2.a.a 1
7.d odd 6 1 441.2.e.e 2
9.c even 3 1 567.2.g.f 2
9.c even 3 1 567.2.h.a 2
9.d odd 6 1 567.2.g.a 2
9.d odd 6 1 567.2.h.f 2
12.b even 2 1 336.2.q.f 2
15.d odd 2 1 525.2.i.e 2
15.e even 4 2 525.2.r.e 4
21.c even 2 1 147.2.e.a 2
21.g even 6 1 147.2.a.b 1
21.g even 6 1 147.2.e.a 2
21.h odd 6 1 21.2.e.a 2
21.h odd 6 1 147.2.a.c 1
24.f even 2 1 1344.2.q.c 2
24.h odd 2 1 1344.2.q.m 2
28.f even 6 1 7056.2.a.m 1
28.g odd 6 1 1008.2.s.d 2
28.g odd 6 1 7056.2.a.bp 1
63.g even 3 1 567.2.h.a 2
63.h even 3 1 567.2.g.f 2
63.j odd 6 1 567.2.g.a 2
63.n odd 6 1 567.2.h.f 2
84.h odd 2 1 2352.2.q.c 2
84.j odd 6 1 2352.2.a.w 1
84.j odd 6 1 2352.2.q.c 2
84.n even 6 1 336.2.q.f 2
84.n even 6 1 2352.2.a.d 1
105.o odd 6 1 525.2.i.e 2
105.o odd 6 1 3675.2.a.a 1
105.p even 6 1 3675.2.a.c 1
105.x even 12 2 525.2.r.e 4
168.s odd 6 1 1344.2.q.m 2
168.s odd 6 1 9408.2.a.bg 1
168.v even 6 1 1344.2.q.c 2
168.v even 6 1 9408.2.a.cv 1
168.ba even 6 1 9408.2.a.bz 1
168.be odd 6 1 9408.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 3.b odd 2 1
21.2.e.a 2 21.h odd 6 1
63.2.e.b 2 1.a even 1 1 trivial
63.2.e.b 2 7.c even 3 1 inner
147.2.a.b 1 21.g even 6 1
147.2.a.c 1 21.h odd 6 1
147.2.e.a 2 21.c even 2 1
147.2.e.a 2 21.g even 6 1
336.2.q.f 2 12.b even 2 1
336.2.q.f 2 84.n even 6 1
441.2.a.a 1 7.d odd 6 1
441.2.a.b 1 7.c even 3 1
441.2.e.e 2 7.b odd 2 1
441.2.e.e 2 7.d odd 6 1
525.2.i.e 2 15.d odd 2 1
525.2.i.e 2 105.o odd 6 1
525.2.r.e 4 15.e even 4 2
525.2.r.e 4 105.x even 12 2
567.2.g.a 2 9.d odd 6 1
567.2.g.a 2 63.j odd 6 1
567.2.g.f 2 9.c even 3 1
567.2.g.f 2 63.h even 3 1
567.2.h.a 2 9.c even 3 1
567.2.h.a 2 63.g even 3 1
567.2.h.f 2 9.d odd 6 1
567.2.h.f 2 63.n odd 6 1
1008.2.s.d 2 4.b odd 2 1
1008.2.s.d 2 28.g odd 6 1
1344.2.q.c 2 24.f even 2 1
1344.2.q.c 2 168.v even 6 1
1344.2.q.m 2 24.h odd 2 1
1344.2.q.m 2 168.s odd 6 1
2352.2.a.d 1 84.n even 6 1
2352.2.a.w 1 84.j odd 6 1
2352.2.q.c 2 84.h odd 2 1
2352.2.q.c 2 84.j odd 6 1
3675.2.a.a 1 105.o odd 6 1
3675.2.a.c 1 105.p even 6 1
7056.2.a.m 1 28.f even 6 1
7056.2.a.bp 1 28.g odd 6 1
9408.2.a.k 1 168.be odd 6 1
9408.2.a.bg 1 168.s odd 6 1
9408.2.a.bz 1 168.ba even 6 1
9408.2.a.cv 1 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

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