Properties

 Label 63.2.a.b Level $63$ Weight $2$ Character orbit 63.a Self dual yes 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.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$0.503057532734$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} - 2 \beta q^{5} + q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 - 2*b * q^5 + q^7 - b * q^8 $$q + \beta q^{2} + q^{4} - 2 \beta q^{5} + q^{7} - \beta q^{8} - 6 q^{10} + 2 \beta q^{11} + 2 q^{13} + \beta q^{14} - 5 q^{16} + 2 \beta q^{17} - 4 q^{19} - 2 \beta q^{20} + 6 q^{22} - 2 \beta q^{23} + 7 q^{25} + 2 \beta q^{26} + q^{28} - 4 q^{31} - 3 \beta q^{32} + 6 q^{34} - 2 \beta q^{35} + 2 q^{37} - 4 \beta q^{38} + 6 q^{40} + 6 \beta q^{41} - 4 q^{43} + 2 \beta q^{44} - 6 q^{46} + 4 \beta q^{47} + q^{49} + 7 \beta q^{50} + 2 q^{52} - 4 \beta q^{53} - 12 q^{55} - \beta q^{56} - 4 \beta q^{59} - 10 q^{61} - 4 \beta q^{62} + q^{64} - 4 \beta q^{65} - 4 q^{67} + 2 \beta q^{68} - 6 q^{70} - 6 \beta q^{71} + 14 q^{73} + 2 \beta q^{74} - 4 q^{76} + 2 \beta q^{77} + 8 q^{79} + 10 \beta q^{80} + 18 q^{82} - 12 q^{85} - 4 \beta q^{86} - 6 q^{88} - 2 \beta q^{89} + 2 q^{91} - 2 \beta q^{92} + 12 q^{94} + 8 \beta q^{95} + 14 q^{97} + \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 - 2*b * q^5 + q^7 - b * q^8 - 6 * q^10 + 2*b * q^11 + 2 * q^13 + b * q^14 - 5 * q^16 + 2*b * q^17 - 4 * q^19 - 2*b * q^20 + 6 * q^22 - 2*b * q^23 + 7 * q^25 + 2*b * q^26 + q^28 - 4 * q^31 - 3*b * q^32 + 6 * q^34 - 2*b * q^35 + 2 * q^37 - 4*b * q^38 + 6 * q^40 + 6*b * q^41 - 4 * q^43 + 2*b * q^44 - 6 * q^46 + 4*b * q^47 + q^49 + 7*b * q^50 + 2 * q^52 - 4*b * q^53 - 12 * q^55 - b * q^56 - 4*b * q^59 - 10 * q^61 - 4*b * q^62 + q^64 - 4*b * q^65 - 4 * q^67 + 2*b * q^68 - 6 * q^70 - 6*b * q^71 + 14 * q^73 + 2*b * q^74 - 4 * q^76 + 2*b * q^77 + 8 * q^79 + 10*b * q^80 + 18 * q^82 - 12 * q^85 - 4*b * q^86 - 6 * q^88 - 2*b * q^89 + 2 * q^91 - 2*b * q^92 + 12 * q^94 + 8*b * q^95 + 14 * q^97 + b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^7 $$2 q + 2 q^{4} + 2 q^{7} - 12 q^{10} + 4 q^{13} - 10 q^{16} - 8 q^{19} + 12 q^{22} + 14 q^{25} + 2 q^{28} - 8 q^{31} + 12 q^{34} + 4 q^{37} + 12 q^{40} - 8 q^{43} - 12 q^{46} + 2 q^{49} + 4 q^{52} - 24 q^{55} - 20 q^{61} + 2 q^{64} - 8 q^{67} - 12 q^{70} + 28 q^{73} - 8 q^{76} + 16 q^{79} + 36 q^{82} - 24 q^{85} - 12 q^{88} + 4 q^{91} + 24 q^{94} + 28 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^7 - 12 * q^10 + 4 * q^13 - 10 * q^16 - 8 * q^19 + 12 * q^22 + 14 * q^25 + 2 * q^28 - 8 * q^31 + 12 * q^34 + 4 * q^37 + 12 * q^40 - 8 * q^43 - 12 * q^46 + 2 * q^49 + 4 * q^52 - 24 * q^55 - 20 * q^61 + 2 * q^64 - 8 * q^67 - 12 * q^70 + 28 * q^73 - 8 * q^76 + 16 * q^79 + 36 * q^82 - 24 * q^85 - 12 * q^88 + 4 * q^91 + 24 * q^94 + 28 * q^97

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 0 1.00000 3.46410 0 1.00000 1.73205 0 −6.00000
1.2 1.73205 0 1.00000 −3.46410 0 1.00000 −1.73205 0 −6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.a.b 2
3.b odd 2 1 inner 63.2.a.b 2
4.b odd 2 1 1008.2.a.n 2
5.b even 2 1 1575.2.a.q 2
5.c odd 4 2 1575.2.d.i 4
7.b odd 2 1 441.2.a.g 2
7.c even 3 2 441.2.e.j 4
7.d odd 6 2 441.2.e.i 4
8.b even 2 1 4032.2.a.bt 2
8.d odd 2 1 4032.2.a.bq 2
9.c even 3 2 567.2.f.j 4
9.d odd 6 2 567.2.f.j 4
11.b odd 2 1 7623.2.a.bi 2
12.b even 2 1 1008.2.a.n 2
15.d odd 2 1 1575.2.a.q 2
15.e even 4 2 1575.2.d.i 4
21.c even 2 1 441.2.a.g 2
21.g even 6 2 441.2.e.i 4
21.h odd 6 2 441.2.e.j 4
24.f even 2 1 4032.2.a.bq 2
24.h odd 2 1 4032.2.a.bt 2
28.d even 2 1 7056.2.a.cm 2
33.d even 2 1 7623.2.a.bi 2
84.h odd 2 1 7056.2.a.cm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 1.a even 1 1 trivial
63.2.a.b 2 3.b odd 2 1 inner
441.2.a.g 2 7.b odd 2 1
441.2.a.g 2 21.c even 2 1
441.2.e.i 4 7.d odd 6 2
441.2.e.i 4 21.g even 6 2
441.2.e.j 4 7.c even 3 2
441.2.e.j 4 21.h odd 6 2
567.2.f.j 4 9.c even 3 2
567.2.f.j 4 9.d odd 6 2
1008.2.a.n 2 4.b odd 2 1
1008.2.a.n 2 12.b even 2 1
1575.2.a.q 2 5.b even 2 1
1575.2.a.q 2 15.d odd 2 1
1575.2.d.i 4 5.c odd 4 2
1575.2.d.i 4 15.e even 4 2
4032.2.a.bq 2 8.d odd 2 1
4032.2.a.bq 2 24.f even 2 1
4032.2.a.bt 2 8.b even 2 1
4032.2.a.bt 2 24.h odd 2 1
7056.2.a.cm 2 28.d even 2 1
7056.2.a.cm 2 84.h odd 2 1
7623.2.a.bi 2 11.b odd 2 1
7623.2.a.bi 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(63))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 12$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 12$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 12$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 108$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} - 48$$
$59$ $$T^{2} - 48$$
$61$ $$(T + 10)^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 108$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 12$$
$97$ $$(T - 14)^{2}$$