# Properties

 Label 64.2.a.a Level $64$ Weight $2$ Character orbit 64.a Self dual yes Analytic conductor $0.511$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 64.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.511042572936$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{5} - 3q^{9} + O(q^{10})$$ $$q + 2q^{5} - 3q^{9} - 6q^{13} + 2q^{17} - q^{25} + 10q^{29} + 2q^{37} + 10q^{41} - 6q^{45} - 7q^{49} - 14q^{53} + 10q^{61} - 12q^{65} - 6q^{73} + 9q^{81} + 4q^{85} + 10q^{89} + 18q^{97} + O(q^{100})$$

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(8z)^{8}}{\eta(4z)^{2}\eta(16z)^{2}}=q\prod_{n=1}^\infty(1 - q^{4n})^{-2}(1 - q^{8n})^{8}(1 - q^{16n})^{-2}$$

## 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
0 0 0 2.00000 0 0 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.2.a.a 1
3.b odd 2 1 576.2.a.c 1
4.b odd 2 1 CM 64.2.a.a 1
5.b even 2 1 1600.2.a.n 1
5.c odd 4 2 1600.2.c.l 2
7.b odd 2 1 3136.2.a.m 1
8.b even 2 1 32.2.a.a 1
8.d odd 2 1 32.2.a.a 1
11.b odd 2 1 7744.2.a.v 1
12.b even 2 1 576.2.a.c 1
16.e even 4 2 256.2.b.b 2
16.f odd 4 2 256.2.b.b 2
20.d odd 2 1 1600.2.a.n 1
20.e even 4 2 1600.2.c.l 2
24.f even 2 1 288.2.a.d 1
24.h odd 2 1 288.2.a.d 1
28.d even 2 1 3136.2.a.m 1
32.g even 8 4 1024.2.e.j 4
32.h odd 8 4 1024.2.e.j 4
40.e odd 2 1 800.2.a.d 1
40.f even 2 1 800.2.a.d 1
40.i odd 4 2 800.2.c.e 2
40.k even 4 2 800.2.c.e 2
44.c even 2 1 7744.2.a.v 1
48.i odd 4 2 2304.2.d.j 2
48.k even 4 2 2304.2.d.j 2
56.e even 2 1 1568.2.a.e 1
56.h odd 2 1 1568.2.a.e 1
56.j odd 6 2 1568.2.i.f 2
56.k odd 6 2 1568.2.i.g 2
56.m even 6 2 1568.2.i.f 2
56.p even 6 2 1568.2.i.g 2
72.j odd 6 2 2592.2.i.e 2
72.l even 6 2 2592.2.i.e 2
72.n even 6 2 2592.2.i.t 2
72.p odd 6 2 2592.2.i.t 2
88.b odd 2 1 3872.2.a.f 1
88.g even 2 1 3872.2.a.f 1
104.e even 2 1 5408.2.a.g 1
104.h odd 2 1 5408.2.a.g 1
120.i odd 2 1 7200.2.a.v 1
120.m even 2 1 7200.2.a.v 1
120.q odd 4 2 7200.2.f.m 2
120.w even 4 2 7200.2.f.m 2
136.e odd 2 1 9248.2.a.f 1
136.h even 2 1 9248.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 8.b even 2 1
32.2.a.a 1 8.d odd 2 1
64.2.a.a 1 1.a even 1 1 trivial
64.2.a.a 1 4.b odd 2 1 CM
256.2.b.b 2 16.e even 4 2
256.2.b.b 2 16.f odd 4 2
288.2.a.d 1 24.f even 2 1
288.2.a.d 1 24.h odd 2 1
576.2.a.c 1 3.b odd 2 1
576.2.a.c 1 12.b even 2 1
800.2.a.d 1 40.e odd 2 1
800.2.a.d 1 40.f even 2 1
800.2.c.e 2 40.i odd 4 2
800.2.c.e 2 40.k even 4 2
1024.2.e.j 4 32.g even 8 4
1024.2.e.j 4 32.h odd 8 4
1568.2.a.e 1 56.e even 2 1
1568.2.a.e 1 56.h odd 2 1
1568.2.i.f 2 56.j odd 6 2
1568.2.i.f 2 56.m even 6 2
1568.2.i.g 2 56.k odd 6 2
1568.2.i.g 2 56.p even 6 2
1600.2.a.n 1 5.b even 2 1
1600.2.a.n 1 20.d odd 2 1
1600.2.c.l 2 5.c odd 4 2
1600.2.c.l 2 20.e even 4 2
2304.2.d.j 2 48.i odd 4 2
2304.2.d.j 2 48.k even 4 2
2592.2.i.e 2 72.j odd 6 2
2592.2.i.e 2 72.l even 6 2
2592.2.i.t 2 72.n even 6 2
2592.2.i.t 2 72.p odd 6 2
3136.2.a.m 1 7.b odd 2 1
3136.2.a.m 1 28.d even 2 1
3872.2.a.f 1 88.b odd 2 1
3872.2.a.f 1 88.g even 2 1
5408.2.a.g 1 104.e even 2 1
5408.2.a.g 1 104.h odd 2 1
7200.2.a.v 1 120.i odd 2 1
7200.2.a.v 1 120.m even 2 1
7200.2.f.m 2 120.q odd 4 2
7200.2.f.m 2 120.w even 4 2
7744.2.a.v 1 11.b odd 2 1
7744.2.a.v 1 44.c even 2 1
9248.2.a.f 1 136.e odd 2 1
9248.2.a.f 1 136.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$6 + T$$
$17$ $$-2 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-10 + T$$
$31$ $$T$$
$37$ $$-2 + T$$
$41$ $$-10 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$14 + T$$
$59$ $$T$$
$61$ $$-10 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$6 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$-10 + T$$
$97$ $$-18 + T$$