Properties

 Label 1764.4.a.c 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 28) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 8q^{5} + O(q^{10})$$ $$q - 8q^{5} + 40q^{11} + 12q^{13} - 58q^{17} - 26q^{19} + 64q^{23} - 61q^{25} + 62q^{29} - 252q^{31} + 26q^{37} + 6q^{41} + 416q^{43} - 396q^{47} + 450q^{53} - 320q^{55} + 274q^{59} + 576q^{61} - 96q^{65} - 476q^{67} + 448q^{71} + 158q^{73} - 936q^{79} + 530q^{83} + 464q^{85} - 390q^{89} + 208q^{95} - 214q^{97} + O(q^{100})$$

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 −8.00000 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 1764.4.a.c 1
3.b odd 2 1 196.4.a.d 1
7.b odd 2 1 252.4.a.d 1
7.c even 3 2 1764.4.k.m 2
7.d odd 6 2 1764.4.k.d 2
12.b even 2 1 784.4.a.a 1
21.c even 2 1 28.4.a.a 1
21.g even 6 2 196.4.e.f 2
21.h odd 6 2 196.4.e.a 2
28.d even 2 1 1008.4.a.o 1
84.h odd 2 1 112.4.a.g 1
105.g even 2 1 700.4.a.n 1
105.k odd 4 2 700.4.e.a 2
168.e odd 2 1 448.4.a.a 1
168.i even 2 1 448.4.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 21.c even 2 1
112.4.a.g 1 84.h odd 2 1
196.4.a.d 1 3.b odd 2 1
196.4.e.a 2 21.h odd 6 2
196.4.e.f 2 21.g even 6 2
252.4.a.d 1 7.b odd 2 1
448.4.a.a 1 168.e odd 2 1
448.4.a.p 1 168.i even 2 1
700.4.a.n 1 105.g even 2 1
700.4.e.a 2 105.k odd 4 2
784.4.a.a 1 12.b even 2 1
1008.4.a.o 1 28.d even 2 1
1764.4.a.c 1 1.a even 1 1 trivial
1764.4.k.d 2 7.d odd 6 2
1764.4.k.m 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} + 8$$ $$T_{11} - 40$$ $$T_{13} - 12$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$8 + T$$
$7$ $$T$$
$11$ $$-40 + T$$
$13$ $$-12 + T$$
$17$ $$58 + T$$
$19$ $$26 + T$$
$23$ $$-64 + T$$
$29$ $$-62 + T$$
$31$ $$252 + T$$
$37$ $$-26 + T$$
$41$ $$-6 + T$$
$43$ $$-416 + T$$
$47$ $$396 + T$$
$53$ $$-450 + T$$
$59$ $$-274 + T$$
$61$ $$-576 + T$$
$67$ $$476 + T$$
$71$ $$-448 + T$$
$73$ $$-158 + T$$
$79$ $$936 + T$$
$83$ $$-530 + T$$
$89$ $$390 + T$$
$97$ $$214 + T$$