# Properties

 Label 63.6.e.b Level $63$ Weight $6$ Character orbit 63.e Analytic conductor $10.104$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.1041806482$$ 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: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 32 \zeta_{6} q^{4} + (87 \zeta_{6} - 149) q^{7}+O(q^{10})$$ q + 32*z * q^4 + (87*z - 149) * q^7 $$q + 32 \zeta_{6} q^{4} + (87 \zeta_{6} - 149) q^{7} - 427 q^{13} + (1024 \zeta_{6} - 1024) q^{16} + (3143 \zeta_{6} - 3143) q^{19} + 3125 \zeta_{6} q^{25} + ( - 1984 \zeta_{6} - 2784) q^{28} - 2723 \zeta_{6} q^{31} + ( - 6661 \zeta_{6} + 6661) q^{37} + 22475 q^{43} + ( - 18357 \zeta_{6} + 14632) q^{49} - 13664 \zeta_{6} q^{52} + ( - 38626 \zeta_{6} + 38626) q^{61} - 32768 q^{64} + 37939 \zeta_{6} q^{67} + 78127 \zeta_{6} q^{73} - 100576 q^{76} + (90857 \zeta_{6} - 90857) q^{79} + ( - 37149 \zeta_{6} + 63623) q^{91} - 134386 q^{97} +O(q^{100})$$ q + 32*z * q^4 + (87*z - 149) * q^7 - 427 * q^13 + (1024*z - 1024) * q^16 + (3143*z - 3143) * q^19 + 3125*z * q^25 + (-1984*z - 2784) * q^28 - 2723*z * q^31 + (-6661*z + 6661) * q^37 + 22475 * q^43 + (-18357*z + 14632) * q^49 - 13664*z * q^52 + (-38626*z + 38626) * q^61 - 32768 * q^64 + 37939*z * q^67 + 78127*z * q^73 - 100576 * q^76 + (90857*z - 90857) * q^79 + (-37149*z + 63623) * q^91 - 134386 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{4} - 211 q^{7}+O(q^{10})$$ 2 * q + 32 * q^4 - 211 * q^7 $$2 q + 32 q^{4} - 211 q^{7} - 854 q^{13} - 1024 q^{16} - 3143 q^{19} + 3125 q^{25} - 7552 q^{28} - 2723 q^{31} + 6661 q^{37} + 44950 q^{43} + 10907 q^{49} - 13664 q^{52} + 38626 q^{61} - 65536 q^{64} + 37939 q^{67} + 78127 q^{73} - 201152 q^{76} - 90857 q^{79} + 90097 q^{91} - 268772 q^{97}+O(q^{100})$$ 2 * q + 32 * q^4 - 211 * q^7 - 854 * q^13 - 1024 * q^16 - 3143 * q^19 + 3125 * q^25 - 7552 * q^28 - 2723 * q^31 + 6661 * q^37 + 44950 * q^43 + 10907 * q^49 - 13664 * q^52 + 38626 * q^61 - 65536 * q^64 + 37939 * q^67 + 78127 * q^73 - 201152 * q^76 - 90857 * q^79 + 90097 * q^91 - 268772 * q^97

## 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
0 0 16.0000 + 27.7128i 0 0 −105.500 + 75.3442i 0 0 0
46.1 0 0 16.0000 27.7128i 0 0 −105.500 75.3442i 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.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.b 2
3.b odd 2 1 CM 63.6.e.b 2
7.c even 3 1 inner 63.6.e.b 2
7.c even 3 1 441.6.a.e 1
7.d odd 6 1 441.6.a.f 1
21.g even 6 1 441.6.a.f 1
21.h odd 6 1 inner 63.6.e.b 2
21.h odd 6 1 441.6.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.e.b 2 1.a even 1 1 trivial
63.6.e.b 2 3.b odd 2 1 CM
63.6.e.b 2 7.c even 3 1 inner
63.6.e.b 2 21.h odd 6 1 inner
441.6.a.e 1 7.c even 3 1
441.6.a.e 1 21.h odd 6 1
441.6.a.f 1 7.d odd 6 1
441.6.a.f 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{6}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 211T + 16807$$
$11$ $$T^{2}$$
$13$ $$(T + 427)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 3143 T + 9878449$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 2723 T + 7414729$$
$37$ $$T^{2} - 6661 T + 44368921$$
$41$ $$T^{2}$$
$43$ $$(T - 22475)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 38626 T + 1491967876$$
$67$ $$T^{2} - 37939 T + 1439367721$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 78127 T + 6103828129$$
$79$ $$T^{2} + 90857 T + 8254994449$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 134386)^{2}$$