# Properties

 Label 63.3.m.a Level $63$ Weight $3$ Character orbit 63.m Analytic conductor $1.717$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 4) q^{5} - 7 \zeta_{6} q^{7} - 8 q^{8} +O(q^{10})$$ q - 2*z * q^2 + (-2*z + 4) * q^5 - 7*z * q^7 - 8 * q^8 $$q - 2 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 4) q^{5} - 7 \zeta_{6} q^{7} - 8 q^{8} + ( - 4 \zeta_{6} - 4) q^{10} + ( - 10 \zeta_{6} + 10) q^{11} + (14 \zeta_{6} - 7) q^{13} + (14 \zeta_{6} - 14) q^{14} + 16 \zeta_{6} q^{16} + (4 \zeta_{6} + 4) q^{17} + ( - 19 \zeta_{6} + 38) q^{19} - 20 q^{22} + 40 \zeta_{6} q^{23} + (13 \zeta_{6} - 13) q^{25} + ( - 14 \zeta_{6} + 28) q^{26} - 16 q^{29} + (3 \zeta_{6} + 3) q^{31} + ( - 16 \zeta_{6} + 8) q^{34} + ( - 14 \zeta_{6} - 14) q^{35} - 5 \zeta_{6} q^{37} + ( - 38 \zeta_{6} - 38) q^{38} + (16 \zeta_{6} - 32) q^{40} + ( - 28 \zeta_{6} + 14) q^{41} - 19 q^{43} + ( - 80 \zeta_{6} + 80) q^{46} + ( - 30 \zeta_{6} + 60) q^{47} + (49 \zeta_{6} - 49) q^{49} + 26 q^{50} + (32 \zeta_{6} - 32) q^{53} + ( - 40 \zeta_{6} + 20) q^{55} + 56 \zeta_{6} q^{56} + 32 \zeta_{6} q^{58} + ( - 24 \zeta_{6} - 24) q^{59} + ( - 12 \zeta_{6} + 24) q^{61} + ( - 12 \zeta_{6} + 6) q^{62} + 64 q^{64} + 42 \zeta_{6} q^{65} + (59 \zeta_{6} - 59) q^{67} + (56 \zeta_{6} - 28) q^{70} + 26 q^{71} + ( - 11 \zeta_{6} - 11) q^{73} + (10 \zeta_{6} - 10) q^{74} - 70 q^{77} - 47 \zeta_{6} q^{79} + (32 \zeta_{6} + 32) q^{80} + (28 \zeta_{6} - 56) q^{82} + (28 \zeta_{6} - 14) q^{83} + 24 q^{85} + 38 \zeta_{6} q^{86} + (80 \zeta_{6} - 80) q^{88} + (68 \zeta_{6} - 136) q^{89} + ( - 49 \zeta_{6} + 98) q^{91} + ( - 60 \zeta_{6} - 60) q^{94} + ( - 114 \zeta_{6} + 114) q^{95} + ( - 56 \zeta_{6} + 28) q^{97} + 98 q^{98} +O(q^{100})$$ q - 2*z * q^2 + (-2*z + 4) * q^5 - 7*z * q^7 - 8 * q^8 + (-4*z - 4) * q^10 + (-10*z + 10) * q^11 + (14*z - 7) * q^13 + (14*z - 14) * q^14 + 16*z * q^16 + (4*z + 4) * q^17 + (-19*z + 38) * q^19 - 20 * q^22 + 40*z * q^23 + (13*z - 13) * q^25 + (-14*z + 28) * q^26 - 16 * q^29 + (3*z + 3) * q^31 + (-16*z + 8) * q^34 + (-14*z - 14) * q^35 - 5*z * q^37 + (-38*z - 38) * q^38 + (16*z - 32) * q^40 + (-28*z + 14) * q^41 - 19 * q^43 + (-80*z + 80) * q^46 + (-30*z + 60) * q^47 + (49*z - 49) * q^49 + 26 * q^50 + (32*z - 32) * q^53 + (-40*z + 20) * q^55 + 56*z * q^56 + 32*z * q^58 + (-24*z - 24) * q^59 + (-12*z + 24) * q^61 + (-12*z + 6) * q^62 + 64 * q^64 + 42*z * q^65 + (59*z - 59) * q^67 + (56*z - 28) * q^70 + 26 * q^71 + (-11*z - 11) * q^73 + (10*z - 10) * q^74 - 70 * q^77 - 47*z * q^79 + (32*z + 32) * q^80 + (28*z - 56) * q^82 + (28*z - 14) * q^83 + 24 * q^85 + 38*z * q^86 + (80*z - 80) * q^88 + (68*z - 136) * q^89 + (-49*z + 98) * q^91 + (-60*z - 60) * q^94 + (-114*z + 114) * q^95 + (-56*z + 28) * q^97 + 98 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 6 q^{5} - 7 q^{7} - 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 6 * q^5 - 7 * q^7 - 16 * q^8 $$2 q - 2 q^{2} + 6 q^{5} - 7 q^{7} - 16 q^{8} - 12 q^{10} + 10 q^{11} - 14 q^{14} + 16 q^{16} + 12 q^{17} + 57 q^{19} - 40 q^{22} + 40 q^{23} - 13 q^{25} + 42 q^{26} - 32 q^{29} + 9 q^{31} - 42 q^{35} - 5 q^{37} - 114 q^{38} - 48 q^{40} - 38 q^{43} + 80 q^{46} + 90 q^{47} - 49 q^{49} + 52 q^{50} - 32 q^{53} + 56 q^{56} + 32 q^{58} - 72 q^{59} + 36 q^{61} + 128 q^{64} + 42 q^{65} - 59 q^{67} + 52 q^{71} - 33 q^{73} - 10 q^{74} - 140 q^{77} - 47 q^{79} + 96 q^{80} - 84 q^{82} + 48 q^{85} + 38 q^{86} - 80 q^{88} - 204 q^{89} + 147 q^{91} - 180 q^{94} + 114 q^{95} + 196 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 6 * q^5 - 7 * q^7 - 16 * q^8 - 12 * q^10 + 10 * q^11 - 14 * q^14 + 16 * q^16 + 12 * q^17 + 57 * q^19 - 40 * q^22 + 40 * q^23 - 13 * q^25 + 42 * q^26 - 32 * q^29 + 9 * q^31 - 42 * q^35 - 5 * q^37 - 114 * q^38 - 48 * q^40 - 38 * q^43 + 80 * q^46 + 90 * q^47 - 49 * q^49 + 52 * q^50 - 32 * q^53 + 56 * q^56 + 32 * q^58 - 72 * q^59 + 36 * q^61 + 128 * q^64 + 42 * q^65 - 59 * q^67 + 52 * q^71 - 33 * q^73 - 10 * q^74 - 140 * q^77 - 47 * q^79 + 96 * q^80 - 84 * q^82 + 48 * q^85 + 38 * q^86 - 80 * q^88 - 204 * q^89 + 147 * q^91 - 180 * q^94 + 114 * q^95 + 196 * q^98

## 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
−1.00000 1.73205i 0 0 3.00000 1.73205i 0 −3.50000 6.06218i −8.00000 0 −6.00000 3.46410i
19.1 −1.00000 + 1.73205i 0 0 3.00000 + 1.73205i 0 −3.50000 + 6.06218i −8.00000 0 −6.00000 + 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.d odd 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.a 2
3.b odd 2 1 21.3.f.c 2
4.b odd 2 1 1008.3.cg.f 2
7.b odd 2 1 441.3.m.b 2
7.c even 3 1 441.3.d.d 2
7.c even 3 1 441.3.m.b 2
7.d odd 6 1 inner 63.3.m.a 2
7.d odd 6 1 441.3.d.d 2
12.b even 2 1 336.3.bh.c 2
15.d odd 2 1 525.3.o.b 2
15.e even 4 2 525.3.s.d 4
21.c even 2 1 147.3.f.e 2
21.g even 6 1 21.3.f.c 2
21.g even 6 1 147.3.d.a 2
21.h odd 6 1 147.3.d.a 2
21.h odd 6 1 147.3.f.e 2
28.f even 6 1 1008.3.cg.f 2
84.j odd 6 1 336.3.bh.c 2
84.j odd 6 1 2352.3.f.b 2
84.n even 6 1 2352.3.f.b 2
105.p even 6 1 525.3.o.b 2
105.w odd 12 2 525.3.s.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.c 2 3.b odd 2 1
21.3.f.c 2 21.g even 6 1
63.3.m.a 2 1.a even 1 1 trivial
63.3.m.a 2 7.d odd 6 1 inner
147.3.d.a 2 21.g even 6 1
147.3.d.a 2 21.h odd 6 1
147.3.f.e 2 21.c even 2 1
147.3.f.e 2 21.h odd 6 1
336.3.bh.c 2 12.b even 2 1
336.3.bh.c 2 84.j odd 6 1
441.3.d.d 2 7.c even 3 1
441.3.d.d 2 7.d odd 6 1
441.3.m.b 2 7.b odd 2 1
441.3.m.b 2 7.c even 3 1
525.3.o.b 2 15.d odd 2 1
525.3.o.b 2 105.p even 6 1
525.3.s.d 4 15.e even 4 2
525.3.s.d 4 105.w odd 12 2
1008.3.cg.f 2 4.b odd 2 1
1008.3.cg.f 2 28.f even 6 1
2352.3.f.b 2 84.j odd 6 1
2352.3.f.b 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} + 4$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 12$$
$7$ $$T^{2} + 7T + 49$$
$11$ $$T^{2} - 10T + 100$$
$13$ $$T^{2} + 147$$
$17$ $$T^{2} - 12T + 48$$
$19$ $$T^{2} - 57T + 1083$$
$23$ $$T^{2} - 40T + 1600$$
$29$ $$(T + 16)^{2}$$
$31$ $$T^{2} - 9T + 27$$
$37$ $$T^{2} + 5T + 25$$
$41$ $$T^{2} + 588$$
$43$ $$(T + 19)^{2}$$
$47$ $$T^{2} - 90T + 2700$$
$53$ $$T^{2} + 32T + 1024$$
$59$ $$T^{2} + 72T + 1728$$
$61$ $$T^{2} - 36T + 432$$
$67$ $$T^{2} + 59T + 3481$$
$71$ $$(T - 26)^{2}$$
$73$ $$T^{2} + 33T + 363$$
$79$ $$T^{2} + 47T + 2209$$
$83$ $$T^{2} + 588$$
$89$ $$T^{2} + 204T + 13872$$
$97$ $$T^{2} + 2352$$