# Properties

 Label 63.3.m.d 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{4} + (3 \zeta_{6} - 6) q^{5} + (3 \zeta_{6} + 5) q^{7} - 3 q^{8}+O(q^{10})$$ q + 3*z * q^2 + (5*z - 5) * q^4 + (3*z - 6) * q^5 + (3*z + 5) * q^7 - 3 * q^8 $$q + 3 \zeta_{6} q^{2} + (5 \zeta_{6} - 5) q^{4} + (3 \zeta_{6} - 6) q^{5} + (3 \zeta_{6} + 5) q^{7} - 3 q^{8} + ( - 9 \zeta_{6} - 9) q^{10} + ( - 15 \zeta_{6} + 15) q^{11} + ( - 16 \zeta_{6} + 8) q^{13} + (24 \zeta_{6} - 9) q^{14} + 11 \zeta_{6} q^{16} + ( - 6 \zeta_{6} - 6) q^{17} + (6 \zeta_{6} - 12) q^{19} + ( - 30 \zeta_{6} + 15) q^{20} + 45 q^{22} + ( - 2 \zeta_{6} + 2) q^{25} + ( - 24 \zeta_{6} + 48) q^{26} + (25 \zeta_{6} - 40) q^{28} + 9 q^{29} + ( - 7 \zeta_{6} - 7) q^{31} + (45 \zeta_{6} - 45) q^{32} + ( - 36 \zeta_{6} + 18) q^{34} + (6 \zeta_{6} - 39) q^{35} - 10 \zeta_{6} q^{37} + ( - 18 \zeta_{6} - 18) q^{38} + ( - 9 \zeta_{6} + 18) q^{40} + (12 \zeta_{6} - 6) q^{41} - 74 q^{43} + 75 \zeta_{6} q^{44} + (39 \zeta_{6} + 16) q^{49} + 6 q^{50} + (40 \zeta_{6} + 40) q^{52} + ( - 33 \zeta_{6} + 33) q^{53} + (90 \zeta_{6} - 45) q^{55} + ( - 9 \zeta_{6} - 15) q^{56} + 27 \zeta_{6} q^{58} + ( - 9 \zeta_{6} - 9) q^{59} + ( - 52 \zeta_{6} + 104) q^{61} + ( - 42 \zeta_{6} + 21) q^{62} - 91 q^{64} + 72 \zeta_{6} q^{65} + ( - 76 \zeta_{6} + 76) q^{67} + ( - 30 \zeta_{6} + 60) q^{68} + ( - 99 \zeta_{6} - 18) q^{70} - 84 q^{71} + ( - 36 \zeta_{6} - 36) q^{73} + ( - 30 \zeta_{6} + 30) q^{74} + ( - 60 \zeta_{6} + 30) q^{76} + ( - 75 \zeta_{6} + 120) q^{77} + 43 \zeta_{6} q^{79} + ( - 33 \zeta_{6} - 33) q^{80} + (18 \zeta_{6} - 36) q^{82} + (138 \zeta_{6} - 69) q^{83} + 54 q^{85} - 222 \zeta_{6} q^{86} + (45 \zeta_{6} - 45) q^{88} + ( - 42 \zeta_{6} + 84) q^{89} + ( - 104 \zeta_{6} + 88) q^{91} + ( - 54 \zeta_{6} + 54) q^{95} + (214 \zeta_{6} - 107) q^{97} + (165 \zeta_{6} - 117) q^{98} +O(q^{100})$$ q + 3*z * q^2 + (5*z - 5) * q^4 + (3*z - 6) * q^5 + (3*z + 5) * q^7 - 3 * q^8 + (-9*z - 9) * q^10 + (-15*z + 15) * q^11 + (-16*z + 8) * q^13 + (24*z - 9) * q^14 + 11*z * q^16 + (-6*z - 6) * q^17 + (6*z - 12) * q^19 + (-30*z + 15) * q^20 + 45 * q^22 + (-2*z + 2) * q^25 + (-24*z + 48) * q^26 + (25*z - 40) * q^28 + 9 * q^29 + (-7*z - 7) * q^31 + (45*z - 45) * q^32 + (-36*z + 18) * q^34 + (6*z - 39) * q^35 - 10*z * q^37 + (-18*z - 18) * q^38 + (-9*z + 18) * q^40 + (12*z - 6) * q^41 - 74 * q^43 + 75*z * q^44 + (39*z + 16) * q^49 + 6 * q^50 + (40*z + 40) * q^52 + (-33*z + 33) * q^53 + (90*z - 45) * q^55 + (-9*z - 15) * q^56 + 27*z * q^58 + (-9*z - 9) * q^59 + (-52*z + 104) * q^61 + (-42*z + 21) * q^62 - 91 * q^64 + 72*z * q^65 + (-76*z + 76) * q^67 + (-30*z + 60) * q^68 + (-99*z - 18) * q^70 - 84 * q^71 + (-36*z - 36) * q^73 + (-30*z + 30) * q^74 + (-60*z + 30) * q^76 + (-75*z + 120) * q^77 + 43*z * q^79 + (-33*z - 33) * q^80 + (18*z - 36) * q^82 + (138*z - 69) * q^83 + 54 * q^85 - 222*z * q^86 + (45*z - 45) * q^88 + (-42*z + 84) * q^89 + (-104*z + 88) * q^91 + (-54*z + 54) * q^95 + (214*z - 107) * q^97 + (165*z - 117) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 5 q^{4} - 9 q^{5} + 13 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 - 5 * q^4 - 9 * q^5 + 13 * q^7 - 6 * q^8 $$2 q + 3 q^{2} - 5 q^{4} - 9 q^{5} + 13 q^{7} - 6 q^{8} - 27 q^{10} + 15 q^{11} + 6 q^{14} + 11 q^{16} - 18 q^{17} - 18 q^{19} + 90 q^{22} + 2 q^{25} + 72 q^{26} - 55 q^{28} + 18 q^{29} - 21 q^{31} - 45 q^{32} - 72 q^{35} - 10 q^{37} - 54 q^{38} + 27 q^{40} - 148 q^{43} + 75 q^{44} + 71 q^{49} + 12 q^{50} + 120 q^{52} + 33 q^{53} - 39 q^{56} + 27 q^{58} - 27 q^{59} + 156 q^{61} - 182 q^{64} + 72 q^{65} + 76 q^{67} + 90 q^{68} - 135 q^{70} - 168 q^{71} - 108 q^{73} + 30 q^{74} + 165 q^{77} + 43 q^{79} - 99 q^{80} - 54 q^{82} + 108 q^{85} - 222 q^{86} - 45 q^{88} + 126 q^{89} + 72 q^{91} + 54 q^{95} - 69 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 - 5 * q^4 - 9 * q^5 + 13 * q^7 - 6 * q^8 - 27 * q^10 + 15 * q^11 + 6 * q^14 + 11 * q^16 - 18 * q^17 - 18 * q^19 + 90 * q^22 + 2 * q^25 + 72 * q^26 - 55 * q^28 + 18 * q^29 - 21 * q^31 - 45 * q^32 - 72 * q^35 - 10 * q^37 - 54 * q^38 + 27 * q^40 - 148 * q^43 + 75 * q^44 + 71 * q^49 + 12 * q^50 + 120 * q^52 + 33 * q^53 - 39 * q^56 + 27 * q^58 - 27 * q^59 + 156 * q^61 - 182 * q^64 + 72 * q^65 + 76 * q^67 + 90 * q^68 - 135 * q^70 - 168 * q^71 - 108 * q^73 + 30 * q^74 + 165 * q^77 + 43 * q^79 - 99 * q^80 - 54 * q^82 + 108 * q^85 - 222 * q^86 - 45 * q^88 + 126 * q^89 + 72 * q^91 + 54 * q^95 - 69 * 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
1.50000 + 2.59808i 0 −2.50000 + 4.33013i −4.50000 + 2.59808i 0 6.50000 + 2.59808i −3.00000 0 −13.5000 7.79423i
19.1 1.50000 2.59808i 0 −2.50000 4.33013i −4.50000 2.59808i 0 6.50000 2.59808i −3.00000 0 −13.5000 + 7.79423i
 $$n$$: 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.d 2
3.b odd 2 1 21.3.f.a 2
4.b odd 2 1 1008.3.cg.a 2
7.b odd 2 1 441.3.m.g 2
7.c even 3 1 441.3.d.a 2
7.c even 3 1 441.3.m.g 2
7.d odd 6 1 inner 63.3.m.d 2
7.d odd 6 1 441.3.d.a 2
12.b even 2 1 336.3.bh.d 2
15.d odd 2 1 525.3.o.h 2
15.e even 4 2 525.3.s.e 4
21.c even 2 1 147.3.f.a 2
21.g even 6 1 21.3.f.a 2
21.g even 6 1 147.3.d.c 2
21.h odd 6 1 147.3.d.c 2
21.h odd 6 1 147.3.f.a 2
28.f even 6 1 1008.3.cg.a 2
84.j odd 6 1 336.3.bh.d 2
84.j odd 6 1 2352.3.f.a 2
84.n even 6 1 2352.3.f.a 2
105.p even 6 1 525.3.o.h 2
105.w odd 12 2 525.3.s.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9T + 27$$
$7$ $$T^{2} - 13T + 49$$
$11$ $$T^{2} - 15T + 225$$
$13$ $$T^{2} + 192$$
$17$ $$T^{2} + 18T + 108$$
$19$ $$T^{2} + 18T + 108$$
$23$ $$T^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 21T + 147$$
$37$ $$T^{2} + 10T + 100$$
$41$ $$T^{2} + 108$$
$43$ $$(T + 74)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 33T + 1089$$
$59$ $$T^{2} + 27T + 243$$
$61$ $$T^{2} - 156T + 8112$$
$67$ $$T^{2} - 76T + 5776$$
$71$ $$(T + 84)^{2}$$
$73$ $$T^{2} + 108T + 3888$$
$79$ $$T^{2} - 43T + 1849$$
$83$ $$T^{2} + 14283$$
$89$ $$T^{2} - 126T + 5292$$
$97$ $$T^{2} + 34347$$