# Properties

 Label 63.3.m.b Level $63$ Weight $3$ Character orbit 63.m Analytic conductor $1.717$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 63.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.71662566547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - 4 \zeta_{6} + 4) q^{4} + ( - 3 \zeta_{6} + 8) q^{7}+O(q^{10})$$ q + (-4*z + 4) * q^4 + (-3*z + 8) * q^7 $$q + ( - 4 \zeta_{6} + 4) q^{4} + ( - 3 \zeta_{6} + 8) q^{7} + (14 \zeta_{6} - 7) q^{13} - 16 \zeta_{6} q^{16} + (21 \zeta_{6} - 42) q^{19} + (25 \zeta_{6} - 25) q^{25} + ( - 32 \zeta_{6} + 20) q^{28} + (35 \zeta_{6} + 35) q^{31} - 73 \zeta_{6} q^{37} + 61 q^{43} + ( - 39 \zeta_{6} + 55) q^{49} + (28 \zeta_{6} + 28) q^{52} + (56 \zeta_{6} - 112) q^{61} - 64 q^{64} + ( - 13 \zeta_{6} + 13) q^{67} + ( - 63 \zeta_{6} - 63) q^{73} + (168 \zeta_{6} - 84) q^{76} - 11 \zeta_{6} q^{79} + (91 \zeta_{6} - 14) q^{91} + ( - 224 \zeta_{6} + 112) q^{97} +O(q^{100})$$ q + (-4*z + 4) * q^4 + (-3*z + 8) * q^7 + (14*z - 7) * q^13 - 16*z * q^16 + (21*z - 42) * q^19 + (25*z - 25) * q^25 + (-32*z + 20) * q^28 + (35*z + 35) * q^31 - 73*z * q^37 + 61 * q^43 + (-39*z + 55) * q^49 + (28*z + 28) * q^52 + (56*z - 112) * q^61 - 64 * q^64 + (-13*z + 13) * q^67 + (-63*z - 63) * q^73 + (168*z - 84) * q^76 - 11*z * q^79 + (91*z - 14) * q^91 + (-224*z + 112) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + 13 q^{7}+O(q^{10})$$ 2 * q + 4 * q^4 + 13 * q^7 $$2 q + 4 q^{4} + 13 q^{7} - 16 q^{16} - 63 q^{19} - 25 q^{25} + 8 q^{28} + 105 q^{31} - 73 q^{37} + 122 q^{43} + 71 q^{49} + 84 q^{52} - 168 q^{61} - 128 q^{64} + 13 q^{67} - 189 q^{73} - 11 q^{79} + 63 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + 13 * q^7 - 16 * q^16 - 63 * q^19 - 25 * q^25 + 8 * q^28 + 105 * q^31 - 73 * q^37 + 122 * q^43 + 71 * q^49 + 84 * q^52 - 168 * q^61 - 128 * q^64 + 13 * q^67 - 189 * q^73 - 11 * q^79 + 63 * q^91

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 2.00000 3.46410i 0 0 6.50000 2.59808i 0 0 0
19.1 0 0 2.00000 + 3.46410i 0 0 6.50000 + 2.59808i 0 0 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 63.3.m.b 2
3.b odd 2 1 CM 63.3.m.b 2
4.b odd 2 1 1008.3.cg.d 2
7.b odd 2 1 441.3.m.c 2
7.c even 3 1 441.3.d.c 2
7.c even 3 1 441.3.m.c 2
7.d odd 6 1 inner 63.3.m.b 2
7.d odd 6 1 441.3.d.c 2
12.b even 2 1 1008.3.cg.d 2
21.c even 2 1 441.3.m.c 2
21.g even 6 1 inner 63.3.m.b 2
21.g even 6 1 441.3.d.c 2
21.h odd 6 1 441.3.d.c 2
21.h odd 6 1 441.3.m.c 2
28.f even 6 1 1008.3.cg.d 2
84.j odd 6 1 1008.3.cg.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.m.b 2 1.a even 1 1 trivial
63.3.m.b 2 3.b odd 2 1 CM
63.3.m.b 2 7.d odd 6 1 inner
63.3.m.b 2 21.g even 6 1 inner
441.3.d.c 2 7.c even 3 1
441.3.d.c 2 7.d odd 6 1
441.3.d.c 2 21.g even 6 1
441.3.d.c 2 21.h odd 6 1
441.3.m.c 2 7.b odd 2 1
441.3.m.c 2 7.c even 3 1
441.3.m.c 2 21.c even 2 1
441.3.m.c 2 21.h odd 6 1
1008.3.cg.d 2 4.b odd 2 1
1008.3.cg.d 2 12.b even 2 1
1008.3.cg.d 2 28.f even 6 1
1008.3.cg.d 2 84.j odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 13T + 49$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 147$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 63T + 1323$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 105T + 3675$$
$37$ $$T^{2} + 73T + 5329$$
$41$ $$T^{2}$$
$43$ $$(T - 61)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 168T + 9408$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 189T + 11907$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 37632$$