# Properties

 Label 1764.4.a.a Level $1764$ Weight $4$ Character orbit 1764.a Self dual yes Analytic conductor $104.079$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$104.079369250$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 20q^{5} + O(q^{10})$$ $$q - 20q^{5} - 44q^{11} + 44q^{13} + 72q^{17} - 100q^{19} + 120q^{23} + 275q^{25} - 218q^{29} + 280q^{31} - 30q^{37} + 120q^{41} + 220q^{43} + 88q^{47} - 110q^{53} + 880q^{55} + 580q^{59} - 380q^{61} - 880q^{65} - 980q^{67} + 112q^{71} + 640q^{73} - 488q^{79} + 660q^{83} - 1440q^{85} + 320q^{89} + 2000q^{95} - 248q^{97} + O(q^{100})$$

## 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
0 0 0 −20.0000 0 0 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 1764.4.a.a 1
3.b odd 2 1 196.4.a.c yes 1
7.b odd 2 1 1764.4.a.m 1
7.c even 3 2 1764.4.k.p 2
7.d odd 6 2 1764.4.k.a 2
12.b even 2 1 784.4.a.f 1
21.c even 2 1 196.4.a.a 1
21.g even 6 2 196.4.e.e 2
21.h odd 6 2 196.4.e.b 2
84.h odd 2 1 784.4.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.4.a.a 1 21.c even 2 1
196.4.a.c yes 1 3.b odd 2 1
196.4.e.b 2 21.h odd 6 2
196.4.e.e 2 21.g even 6 2
784.4.a.f 1 12.b even 2 1
784.4.a.m 1 84.h odd 2 1
1764.4.a.a 1 1.a even 1 1 trivial
1764.4.a.m 1 7.b odd 2 1
1764.4.k.a 2 7.d odd 6 2
1764.4.k.p 2 7.c even 3 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} + 20$$ $$T_{11} + 44$$ $$T_{13} - 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$20 + T$$
$7$ $$T$$
$11$ $$44 + T$$
$13$ $$-44 + T$$
$17$ $$-72 + T$$
$19$ $$100 + T$$
$23$ $$-120 + T$$
$29$ $$218 + T$$
$31$ $$-280 + T$$
$37$ $$30 + T$$
$41$ $$-120 + T$$
$43$ $$-220 + T$$
$47$ $$-88 + T$$
$53$ $$110 + T$$
$59$ $$-580 + T$$
$61$ $$380 + T$$
$67$ $$980 + T$$
$71$ $$-112 + T$$
$73$ $$-640 + T$$
$79$ $$488 + T$$
$83$ $$-660 + T$$
$89$ $$-320 + T$$
$97$ $$248 + T$$