# Properties

 Label 63.4.a.a Level $63$ Weight $4$ Character orbit 63.a Self dual yes Analytic conductor $3.717$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 63.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.71712033036$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 8q^{4} + 4q^{5} - 7q^{7} + O(q^{10})$$ $$q - 4q^{2} + 8q^{4} + 4q^{5} - 7q^{7} - 16q^{10} - 62q^{11} - 62q^{13} + 28q^{14} - 64q^{16} - 84q^{17} + 100q^{19} + 32q^{20} + 248q^{22} + 42q^{23} - 109q^{25} + 248q^{26} - 56q^{28} + 10q^{29} - 48q^{31} + 256q^{32} + 336q^{34} - 28q^{35} - 246q^{37} - 400q^{38} + 248q^{41} + 68q^{43} - 496q^{44} - 168q^{46} - 324q^{47} + 49q^{49} + 436q^{50} - 496q^{52} - 258q^{53} - 248q^{55} - 40q^{58} - 120q^{59} + 622q^{61} + 192q^{62} - 512q^{64} - 248q^{65} + 904q^{67} - 672q^{68} + 112q^{70} + 678q^{71} - 642q^{73} + 984q^{74} + 800q^{76} + 434q^{77} + 740q^{79} - 256q^{80} - 992q^{82} - 468q^{83} - 336q^{85} - 272q^{86} - 200q^{89} + 434q^{91} + 336q^{92} + 1296q^{94} + 400q^{95} - 1266q^{97} - 196q^{98} + O(q^{100})$$

## 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
−4.00000 0 8.00000 4.00000 0 −7.00000 0 0 −16.0000
 $$n$$: 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 63.4.a.a 1
3.b odd 2 1 21.4.a.b 1
4.b odd 2 1 1008.4.a.m 1
5.b even 2 1 1575.4.a.k 1
7.b odd 2 1 441.4.a.b 1
7.c even 3 2 441.4.e.m 2
7.d odd 6 2 441.4.e.n 2
12.b even 2 1 336.4.a.h 1
15.d odd 2 1 525.4.a.b 1
15.e even 4 2 525.4.d.b 2
21.c even 2 1 147.4.a.g 1
21.g even 6 2 147.4.e.b 2
21.h odd 6 2 147.4.e.c 2
24.f even 2 1 1344.4.a.i 1
24.h odd 2 1 1344.4.a.w 1
84.h odd 2 1 2352.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 3.b odd 2 1
63.4.a.a 1 1.a even 1 1 trivial
147.4.a.g 1 21.c even 2 1
147.4.e.b 2 21.g even 6 2
147.4.e.c 2 21.h odd 6 2
336.4.a.h 1 12.b even 2 1
441.4.a.b 1 7.b odd 2 1
441.4.e.m 2 7.c even 3 2
441.4.e.n 2 7.d odd 6 2
525.4.a.b 1 15.d odd 2 1
525.4.d.b 2 15.e even 4 2
1008.4.a.m 1 4.b odd 2 1
1344.4.a.i 1 24.f even 2 1
1344.4.a.w 1 24.h odd 2 1
1575.4.a.k 1 5.b even 2 1
2352.4.a.l 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(63))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$7 + T$$
$11$ $$62 + T$$
$13$ $$62 + T$$
$17$ $$84 + T$$
$19$ $$-100 + T$$
$23$ $$-42 + T$$
$29$ $$-10 + T$$
$31$ $$48 + T$$
$37$ $$246 + T$$
$41$ $$-248 + T$$
$43$ $$-68 + T$$
$47$ $$324 + T$$
$53$ $$258 + T$$
$59$ $$120 + T$$
$61$ $$-622 + T$$
$67$ $$-904 + T$$
$71$ $$-678 + T$$
$73$ $$642 + T$$
$79$ $$-740 + T$$
$83$ $$468 + T$$
$89$ $$200 + T$$
$97$ $$1266 + T$$