# Properties

 Label 63.3.d.a Level $63$ Weight $3$ Character orbit 63.d Self dual yes Analytic conductor $1.717$ Analytic rank $0$ Dimension $1$ CM discriminant -7 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: yes Analytic conductor: $$1.71662566547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 5 q^{4} - 7 q^{7} + 3 q^{8}+O(q^{10})$$ q + 3 * q^2 + 5 * q^4 - 7 * q^7 + 3 * q^8 $$q + 3 q^{2} + 5 q^{4} - 7 q^{7} + 3 q^{8} + 6 q^{11} - 21 q^{14} - 11 q^{16} + 18 q^{22} - 18 q^{23} + 25 q^{25} - 35 q^{28} + 54 q^{29} - 45 q^{32} - 38 q^{37} + 58 q^{43} + 30 q^{44} - 54 q^{46} + 49 q^{49} + 75 q^{50} + 6 q^{53} - 21 q^{56} + 162 q^{58} - 91 q^{64} - 118 q^{67} - 114 q^{71} - 114 q^{74} - 42 q^{77} - 94 q^{79} + 174 q^{86} + 18 q^{88} - 90 q^{92} + 147 q^{98}+O(q^{100})$$ q + 3 * q^2 + 5 * q^4 - 7 * q^7 + 3 * q^8 + 6 * q^11 - 21 * q^14 - 11 * q^16 + 18 * q^22 - 18 * q^23 + 25 * q^25 - 35 * q^28 + 54 * q^29 - 45 * q^32 - 38 * q^37 + 58 * q^43 + 30 * q^44 - 54 * q^46 + 49 * q^49 + 75 * q^50 + 6 * q^53 - 21 * q^56 + 162 * q^58 - 91 * q^64 - 118 * q^67 - 114 * q^71 - 114 * q^74 - 42 * q^77 - 94 * q^79 + 174 * q^86 + 18 * q^88 - 90 * q^92 + 147 * 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
3.00000 0 5.00000 0 0 −7.00000 3.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.d.a 1
3.b odd 2 1 7.3.b.a 1
4.b odd 2 1 1008.3.f.a 1
7.b odd 2 1 CM 63.3.d.a 1
7.c even 3 2 441.3.m.a 2
7.d odd 6 2 441.3.m.a 2
12.b even 2 1 112.3.c.a 1
15.d odd 2 1 175.3.d.a 1
15.e even 4 2 175.3.c.a 2
21.c even 2 1 7.3.b.a 1
21.g even 6 2 49.3.d.a 2
21.h odd 6 2 49.3.d.a 2
24.f even 2 1 448.3.c.b 1
24.h odd 2 1 448.3.c.a 1
28.d even 2 1 1008.3.f.a 1
84.h odd 2 1 112.3.c.a 1
84.j odd 6 2 784.3.s.a 2
84.n even 6 2 784.3.s.a 2
105.g even 2 1 175.3.d.a 1
105.k odd 4 2 175.3.c.a 2
168.e odd 2 1 448.3.c.b 1
168.i even 2 1 448.3.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 3.b odd 2 1
7.3.b.a 1 21.c even 2 1
49.3.d.a 2 21.g even 6 2
49.3.d.a 2 21.h odd 6 2
63.3.d.a 1 1.a even 1 1 trivial
63.3.d.a 1 7.b odd 2 1 CM
112.3.c.a 1 12.b even 2 1
112.3.c.a 1 84.h odd 2 1
175.3.c.a 2 15.e even 4 2
175.3.c.a 2 105.k odd 4 2
175.3.d.a 1 15.d odd 2 1
175.3.d.a 1 105.g even 2 1
441.3.m.a 2 7.c even 3 2
441.3.m.a 2 7.d odd 6 2
448.3.c.a 1 24.h odd 2 1
448.3.c.a 1 168.i even 2 1
448.3.c.b 1 24.f even 2 1
448.3.c.b 1 168.e odd 2 1
784.3.s.a 2 84.j odd 6 2
784.3.s.a 2 84.n even 6 2
1008.3.f.a 1 4.b odd 2 1
1008.3.f.a 1 28.d even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 7$$
$11$ $$T - 6$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 18$$
$29$ $$T - 54$$
$31$ $$T$$
$37$ $$T + 38$$
$41$ $$T$$
$43$ $$T - 58$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 118$$
$71$ $$T + 114$$
$73$ $$T$$
$79$ $$T + 94$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$