# Properties

 Label 63.6.a.c Level $63$ Weight $6$ Character orbit 63.a Self dual yes Analytic conductor $10.104$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.1041806482$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + 63q^{8} + O(q^{10})$$ $$q - q^{2} - 31q^{4} + 34q^{5} - 49q^{7} + 63q^{8} - 34q^{10} + 340q^{11} + 454q^{13} + 49q^{14} + 929q^{16} + 798q^{17} + 892q^{19} - 1054q^{20} - 340q^{22} + 3192q^{23} - 1969q^{25} - 454q^{26} + 1519q^{28} + 8242q^{29} - 2496q^{31} - 2945q^{32} - 798q^{34} - 1666q^{35} + 9798q^{37} - 892q^{38} + 2142q^{40} - 19834q^{41} - 17236q^{43} - 10540q^{44} - 3192q^{46} - 8928q^{47} + 2401q^{49} + 1969q^{50} - 14074q^{52} - 150q^{53} + 11560q^{55} - 3087q^{56} - 8242q^{58} + 42396q^{59} + 14758q^{61} + 2496q^{62} - 26783q^{64} + 15436q^{65} - 1676q^{67} - 24738q^{68} + 1666q^{70} - 14568q^{71} + 78378q^{73} - 9798q^{74} - 27652q^{76} - 16660q^{77} - 2272q^{79} + 31586q^{80} + 19834q^{82} + 37764q^{83} + 27132q^{85} + 17236q^{86} + 21420q^{88} + 117286q^{89} - 22246q^{91} - 98952q^{92} + 8928q^{94} + 30328q^{95} + 10002q^{97} - 2401q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 −31.0000 34.0000 0 −49.0000 63.0000 0 −34.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.a.c 1
3.b odd 2 1 21.6.a.b 1
4.b odd 2 1 1008.6.a.t 1
7.b odd 2 1 441.6.a.d 1
12.b even 2 1 336.6.a.l 1
15.d odd 2 1 525.6.a.c 1
15.e even 4 2 525.6.d.d 2
21.c even 2 1 147.6.a.e 1
21.g even 6 2 147.6.e.e 2
21.h odd 6 2 147.6.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.b 1 3.b odd 2 1
63.6.a.c 1 1.a even 1 1 trivial
147.6.a.e 1 21.c even 2 1
147.6.e.e 2 21.g even 6 2
147.6.e.f 2 21.h odd 6 2
336.6.a.l 1 12.b even 2 1
441.6.a.d 1 7.b odd 2 1
525.6.a.c 1 15.d odd 2 1
525.6.d.d 2 15.e even 4 2
1008.6.a.t 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$-34 + T$$
$7$ $$49 + T$$
$11$ $$-340 + T$$
$13$ $$-454 + T$$
$17$ $$-798 + T$$
$19$ $$-892 + T$$
$23$ $$-3192 + T$$
$29$ $$-8242 + T$$
$31$ $$2496 + T$$
$37$ $$-9798 + T$$
$41$ $$19834 + T$$
$43$ $$17236 + T$$
$47$ $$8928 + T$$
$53$ $$150 + T$$
$59$ $$-42396 + T$$
$61$ $$-14758 + T$$
$67$ $$1676 + T$$
$71$ $$14568 + T$$
$73$ $$-78378 + T$$
$79$ $$2272 + T$$
$83$ $$-37764 + T$$
$89$ $$-117286 + T$$
$97$ $$-10002 + T$$