# Properties

 Label 1764.2.a.k Level $1764$ Weight $2$ Character orbit 1764.a Self dual yes Analytic conductor $14.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} + O(q^{10})$$ $$q + 4q^{5} - 2q^{11} + 6q^{13} - 4q^{17} + 4q^{19} - 2q^{23} + 11q^{25} + 2q^{29} + 2q^{37} - 4q^{43} + 12q^{47} + 6q^{53} - 8q^{55} - 8q^{59} - 6q^{61} + 24q^{65} - 8q^{67} - 14q^{71} + 2q^{73} + 12q^{79} - 4q^{83} - 16q^{85} + 16q^{95} + 2q^{97} + O(q^{100})$$

## 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 4.00000 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 1764.2.a.k 1
3.b odd 2 1 588.2.a.d 1
4.b odd 2 1 7056.2.a.cd 1
7.b odd 2 1 252.2.a.a 1
7.c even 3 2 1764.2.k.a 2
7.d odd 6 2 1764.2.k.k 2
12.b even 2 1 2352.2.a.a 1
21.c even 2 1 84.2.a.a 1
21.g even 6 2 588.2.i.e 2
21.h odd 6 2 588.2.i.d 2
24.f even 2 1 9408.2.a.df 1
24.h odd 2 1 9408.2.a.bn 1
28.d even 2 1 1008.2.a.a 1
35.c odd 2 1 6300.2.a.w 1
35.f even 4 2 6300.2.k.g 2
56.e even 2 1 4032.2.a.bn 1
56.h odd 2 1 4032.2.a.bm 1
63.l odd 6 2 2268.2.j.n 2
63.o even 6 2 2268.2.j.a 2
84.h odd 2 1 336.2.a.f 1
84.j odd 6 2 2352.2.q.b 2
84.n even 6 2 2352.2.q.z 2
105.g even 2 1 2100.2.a.r 1
105.k odd 4 2 2100.2.k.i 2
168.e odd 2 1 1344.2.a.a 1
168.i even 2 1 1344.2.a.k 1
336.v odd 4 2 5376.2.c.p 2
336.y even 4 2 5376.2.c.q 2
420.o odd 2 1 8400.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a 1 21.c even 2 1
252.2.a.a 1 7.b odd 2 1
336.2.a.f 1 84.h odd 2 1
588.2.a.d 1 3.b odd 2 1
588.2.i.d 2 21.h odd 6 2
588.2.i.e 2 21.g even 6 2
1008.2.a.a 1 28.d even 2 1
1344.2.a.a 1 168.e odd 2 1
1344.2.a.k 1 168.i even 2 1
1764.2.a.k 1 1.a even 1 1 trivial
1764.2.k.a 2 7.c even 3 2
1764.2.k.k 2 7.d odd 6 2
2100.2.a.r 1 105.g even 2 1
2100.2.k.i 2 105.k odd 4 2
2268.2.j.a 2 63.o even 6 2
2268.2.j.n 2 63.l odd 6 2
2352.2.a.a 1 12.b even 2 1
2352.2.q.b 2 84.j odd 6 2
2352.2.q.z 2 84.n even 6 2
4032.2.a.bm 1 56.h odd 2 1
4032.2.a.bn 1 56.e even 2 1
5376.2.c.p 2 336.v odd 4 2
5376.2.c.q 2 336.y even 4 2
6300.2.a.w 1 35.c odd 2 1
6300.2.k.g 2 35.f even 4 2
7056.2.a.cd 1 4.b odd 2 1
8400.2.a.e 1 420.o odd 2 1
9408.2.a.bn 1 24.h odd 2 1
9408.2.a.df 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} - 4$$ $$T_{11} + 2$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$-6 + T$$
$17$ $$4 + T$$
$19$ $$-4 + T$$
$23$ $$2 + T$$
$29$ $$-2 + T$$
$31$ $$T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$4 + T$$
$47$ $$-12 + T$$
$53$ $$-6 + T$$
$59$ $$8 + T$$
$61$ $$6 + T$$
$67$ $$8 + T$$
$71$ $$14 + T$$
$73$ $$-2 + T$$
$79$ $$-12 + T$$
$83$ $$4 + T$$
$89$ $$T$$
$97$ $$-2 + T$$