# Properties

 Label 1764.2.a.a Level 1764 Weight 2 Character orbit 1764.a Self dual yes Analytic conductor 14.086 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.0856109166$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{5} + O(q^{10})$$ $$q - 3q^{5} + 3q^{11} + 2q^{13} - 3q^{17} - q^{19} - 3q^{23} + 4q^{25} + 6q^{29} - 7q^{31} - q^{37} - 6q^{41} - 4q^{43} + 9q^{47} - 3q^{53} - 9q^{55} - 9q^{59} - q^{61} - 6q^{65} - 7q^{67} - q^{73} - 13q^{79} - 12q^{83} + 9q^{85} - 15q^{89} + 3q^{95} - 10q^{97} + O(q^{100})$$

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

## 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.a 1
3.b odd 2 1 196.2.a.b 1
4.b odd 2 1 7056.2.a.f 1
7.b odd 2 1 1764.2.a.j 1
7.c even 3 2 252.2.k.c 2
7.d odd 6 2 1764.2.k.b 2
12.b even 2 1 784.2.a.d 1
15.d odd 2 1 4900.2.a.g 1
15.e even 4 2 4900.2.e.i 2
21.c even 2 1 196.2.a.a 1
21.g even 6 2 196.2.e.a 2
21.h odd 6 2 28.2.e.a 2
24.f even 2 1 3136.2.a.s 1
24.h odd 2 1 3136.2.a.h 1
28.d even 2 1 7056.2.a.bw 1
28.g odd 6 2 1008.2.s.p 2
63.g even 3 2 2268.2.l.a 2
63.h even 3 2 2268.2.i.h 2
63.j odd 6 2 2268.2.i.a 2
63.n odd 6 2 2268.2.l.h 2
84.h odd 2 1 784.2.a.g 1
84.j odd 6 2 784.2.i.d 2
84.n even 6 2 112.2.i.b 2
105.g even 2 1 4900.2.a.n 1
105.k odd 4 2 4900.2.e.h 2
105.o odd 6 2 700.2.i.c 2
105.x even 12 4 700.2.r.b 4
168.e odd 2 1 3136.2.a.k 1
168.i even 2 1 3136.2.a.v 1
168.s odd 6 2 448.2.i.e 2
168.v even 6 2 448.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 21.h odd 6 2
112.2.i.b 2 84.n even 6 2
196.2.a.a 1 21.c even 2 1
196.2.a.b 1 3.b odd 2 1
196.2.e.a 2 21.g even 6 2
252.2.k.c 2 7.c even 3 2
448.2.i.c 2 168.v even 6 2
448.2.i.e 2 168.s odd 6 2
700.2.i.c 2 105.o odd 6 2
700.2.r.b 4 105.x even 12 4
784.2.a.d 1 12.b even 2 1
784.2.a.g 1 84.h odd 2 1
784.2.i.d 2 84.j odd 6 2
1008.2.s.p 2 28.g odd 6 2
1764.2.a.a 1 1.a even 1 1 trivial
1764.2.a.j 1 7.b odd 2 1
1764.2.k.b 2 7.d odd 6 2
2268.2.i.a 2 63.j odd 6 2
2268.2.i.h 2 63.h even 3 2
2268.2.l.a 2 63.g even 3 2
2268.2.l.h 2 63.n odd 6 2
3136.2.a.h 1 24.h odd 2 1
3136.2.a.k 1 168.e odd 2 1
3136.2.a.s 1 24.f even 2 1
3136.2.a.v 1 168.i even 2 1
4900.2.a.g 1 15.d odd 2 1
4900.2.a.n 1 105.g even 2 1
4900.2.e.h 2 105.k odd 4 2
4900.2.e.i 2 15.e even 4 2
7056.2.a.f 1 4.b odd 2 1
7056.2.a.bw 1 28.d even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## Hecke kernels

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

 $$T_{5} + 3$$ $$T_{11} - 3$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 3 T + 5 T^{2}$$
$7$ 1
$11$ $$1 - 3 T + 11 T^{2}$$
$13$ $$1 - 2 T + 13 T^{2}$$
$17$ $$1 + 3 T + 17 T^{2}$$
$19$ $$1 + T + 19 T^{2}$$
$23$ $$1 + 3 T + 23 T^{2}$$
$29$ $$1 - 6 T + 29 T^{2}$$
$31$ $$1 + 7 T + 31 T^{2}$$
$37$ $$1 + T + 37 T^{2}$$
$41$ $$1 + 6 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 - 9 T + 47 T^{2}$$
$53$ $$1 + 3 T + 53 T^{2}$$
$59$ $$1 + 9 T + 59 T^{2}$$
$61$ $$1 + T + 61 T^{2}$$
$67$ $$1 + 7 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 + T + 73 T^{2}$$
$79$ $$1 + 13 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 15 T + 89 T^{2}$$
$97$ $$1 + 10 T + 97 T^{2}$$