# Properties

 Label 63.3.d.b Level $63$ Weight $3$ Character orbit 63.d 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.d (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - 3 q^{4} - \beta q^{5} + ( - \beta + 1) q^{7} + 7 q^{8} +O(q^{10})$$ q - q^2 - 3 * q^4 - b * q^5 + (-b + 1) * q^7 + 7 * q^8 $$q - q^{2} - 3 q^{4} - \beta q^{5} + ( - \beta + 1) q^{7} + 7 q^{8} + \beta q^{10} - 10 q^{11} - \beta q^{13} + (\beta - 1) q^{14} + 5 q^{16} + 3 \beta q^{19} + 3 \beta q^{20} + 10 q^{22} + 14 q^{23} - 23 q^{25} + \beta q^{26} + (3 \beta - 3) q^{28} + 38 q^{29} - 4 \beta q^{31} - 33 q^{32} + ( - \beta - 48) q^{35} + 26 q^{37} - 3 \beta q^{38} - 7 \beta q^{40} - 10 \beta q^{41} + 26 q^{43} + 30 q^{44} - 14 q^{46} + 4 \beta q^{47} + ( - 2 \beta - 47) q^{49} + 23 q^{50} + 3 \beta q^{52} - 10 q^{53} + 10 \beta q^{55} + ( - 7 \beta + 7) q^{56} - 38 q^{58} + 11 \beta q^{59} - 5 \beta q^{61} + 4 \beta q^{62} + 13 q^{64} - 48 q^{65} + 74 q^{67} + (\beta + 48) q^{70} + 62 q^{71} + 6 \beta q^{73} - 26 q^{74} - 9 \beta q^{76} + (10 \beta - 10) q^{77} - 46 q^{79} - 5 \beta q^{80} + 10 \beta q^{82} + 13 \beta q^{83} - 26 q^{86} - 70 q^{88} - 6 \beta q^{89} + ( - \beta - 48) q^{91} - 42 q^{92} - 4 \beta q^{94} + 144 q^{95} - 8 \beta q^{97} + (2 \beta + 47) q^{98} +O(q^{100})$$ q - q^2 - 3 * q^4 - b * q^5 + (-b + 1) * q^7 + 7 * q^8 + b * q^10 - 10 * q^11 - b * q^13 + (b - 1) * q^14 + 5 * q^16 + 3*b * q^19 + 3*b * q^20 + 10 * q^22 + 14 * q^23 - 23 * q^25 + b * q^26 + (3*b - 3) * q^28 + 38 * q^29 - 4*b * q^31 - 33 * q^32 + (-b - 48) * q^35 + 26 * q^37 - 3*b * q^38 - 7*b * q^40 - 10*b * q^41 + 26 * q^43 + 30 * q^44 - 14 * q^46 + 4*b * q^47 + (-2*b - 47) * q^49 + 23 * q^50 + 3*b * q^52 - 10 * q^53 + 10*b * q^55 + (-7*b + 7) * q^56 - 38 * q^58 + 11*b * q^59 - 5*b * q^61 + 4*b * q^62 + 13 * q^64 - 48 * q^65 + 74 * q^67 + (b + 48) * q^70 + 62 * q^71 + 6*b * q^73 - 26 * q^74 - 9*b * q^76 + (10*b - 10) * q^77 - 46 * q^79 - 5*b * q^80 + 10*b * q^82 + 13*b * q^83 - 26 * q^86 - 70 * q^88 - 6*b * q^89 + (-b - 48) * q^91 - 42 * q^92 - 4*b * q^94 + 144 * q^95 - 8*b * q^97 + (2*b + 47) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 6 q^{4} + 2 q^{7} + 14 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 6 * q^4 + 2 * q^7 + 14 * q^8 $$2 q - 2 q^{2} - 6 q^{4} + 2 q^{7} + 14 q^{8} - 20 q^{11} - 2 q^{14} + 10 q^{16} + 20 q^{22} + 28 q^{23} - 46 q^{25} - 6 q^{28} + 76 q^{29} - 66 q^{32} - 96 q^{35} + 52 q^{37} + 52 q^{43} + 60 q^{44} - 28 q^{46} - 94 q^{49} + 46 q^{50} - 20 q^{53} + 14 q^{56} - 76 q^{58} + 26 q^{64} - 96 q^{65} + 148 q^{67} + 96 q^{70} + 124 q^{71} - 52 q^{74} - 20 q^{77} - 92 q^{79} - 52 q^{86} - 140 q^{88} - 96 q^{91} - 84 q^{92} + 288 q^{95} + 94 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 6 * q^4 + 2 * q^7 + 14 * q^8 - 20 * q^11 - 2 * q^14 + 10 * q^16 + 20 * q^22 + 28 * q^23 - 46 * q^25 - 6 * q^28 + 76 * q^29 - 66 * q^32 - 96 * q^35 + 52 * q^37 + 52 * q^43 + 60 * q^44 - 28 * q^46 - 94 * q^49 + 46 * q^50 - 20 * q^53 + 14 * q^56 - 76 * q^58 + 26 * q^64 - 96 * q^65 + 148 * q^67 + 96 * q^70 + 124 * q^71 - 52 * q^74 - 20 * q^77 - 92 * q^79 - 52 * q^86 - 140 * q^88 - 96 * q^91 - 84 * q^92 + 288 * q^95 + 94 * q^98

## Character values

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

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 0 −3.00000 6.92820i 0 1.00000 6.92820i 7.00000 0 6.92820i
55.2 −1.00000 0 −3.00000 6.92820i 0 1.00000 + 6.92820i 7.00000 0 6.92820i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.d.b 2
3.b odd 2 1 21.3.d.a 2
4.b odd 2 1 1008.3.f.d 2
7.b odd 2 1 inner 63.3.d.b 2
7.c even 3 1 441.3.m.d 2
7.c even 3 1 441.3.m.f 2
7.d odd 6 1 441.3.m.d 2
7.d odd 6 1 441.3.m.f 2
12.b even 2 1 336.3.f.a 2
15.d odd 2 1 525.3.h.a 2
15.e even 4 2 525.3.e.a 4
21.c even 2 1 21.3.d.a 2
21.g even 6 1 147.3.f.b 2
21.g even 6 1 147.3.f.d 2
21.h odd 6 1 147.3.f.b 2
21.h odd 6 1 147.3.f.d 2
24.f even 2 1 1344.3.f.b 2
24.h odd 2 1 1344.3.f.c 2
28.d even 2 1 1008.3.f.d 2
84.h odd 2 1 336.3.f.a 2
105.g even 2 1 525.3.h.a 2
105.k odd 4 2 525.3.e.a 4
168.e odd 2 1 1344.3.f.b 2
168.i even 2 1 1344.3.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 3.b odd 2 1
21.3.d.a 2 21.c even 2 1
63.3.d.b 2 1.a even 1 1 trivial
63.3.d.b 2 7.b odd 2 1 inner
147.3.f.b 2 21.g even 6 1
147.3.f.b 2 21.h odd 6 1
147.3.f.d 2 21.g even 6 1
147.3.f.d 2 21.h odd 6 1
336.3.f.a 2 12.b even 2 1
336.3.f.a 2 84.h odd 2 1
441.3.m.d 2 7.c even 3 1
441.3.m.d 2 7.d odd 6 1
441.3.m.f 2 7.c even 3 1
441.3.m.f 2 7.d odd 6 1
525.3.e.a 4 15.e even 4 2
525.3.e.a 4 105.k odd 4 2
525.3.h.a 2 15.d odd 2 1
525.3.h.a 2 105.g even 2 1
1008.3.f.d 2 4.b odd 2 1
1008.3.f.d 2 28.d even 2 1
1344.3.f.b 2 24.f even 2 1
1344.3.f.b 2 168.e odd 2 1
1344.3.f.c 2 24.h odd 2 1
1344.3.f.c 2 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 48$$
$7$ $$T^{2} - 2T + 49$$
$11$ $$(T + 10)^{2}$$
$13$ $$T^{2} + 48$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 432$$
$23$ $$(T - 14)^{2}$$
$29$ $$(T - 38)^{2}$$
$31$ $$T^{2} + 768$$
$37$ $$(T - 26)^{2}$$
$41$ $$T^{2} + 4800$$
$43$ $$(T - 26)^{2}$$
$47$ $$T^{2} + 768$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 5808$$
$61$ $$T^{2} + 1200$$
$67$ $$(T - 74)^{2}$$
$71$ $$(T - 62)^{2}$$
$73$ $$T^{2} + 1728$$
$79$ $$(T + 46)^{2}$$
$83$ $$T^{2} + 8112$$
$89$ $$T^{2} + 1728$$
$97$ $$T^{2} + 3072$$