Properties

 Label 1764.4.a.k 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

 $$f(q)$$ $$=$$ $$q + 6 q^{5}+O(q^{10})$$ q + 6 * q^5 $$q + 6 q^{5} + 12 q^{11} + 82 q^{13} - 30 q^{17} - 68 q^{19} - 216 q^{23} - 89 q^{25} - 246 q^{29} + 112 q^{31} + 110 q^{37} - 246 q^{41} - 172 q^{43} + 192 q^{47} - 558 q^{53} + 72 q^{55} + 540 q^{59} - 110 q^{61} + 492 q^{65} + 140 q^{67} + 840 q^{71} + 550 q^{73} - 208 q^{79} + 516 q^{83} - 180 q^{85} - 1398 q^{89} - 408 q^{95} - 1586 q^{97}+O(q^{100})$$ q + 6 * q^5 + 12 * q^11 + 82 * q^13 - 30 * q^17 - 68 * q^19 - 216 * q^23 - 89 * q^25 - 246 * q^29 + 112 * q^31 + 110 * q^37 - 246 * q^41 - 172 * q^43 + 192 * q^47 - 558 * q^53 + 72 * q^55 + 540 * q^59 - 110 * q^61 + 492 * q^65 + 140 * q^67 + 840 * q^71 + 550 * q^73 - 208 * q^79 + 516 * q^83 - 180 * q^85 - 1398 * q^89 - 408 * q^95 - 1586 * q^97

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 6.00000 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.k 1
3.b odd 2 1 196.4.a.b 1
7.b odd 2 1 252.4.a.c 1
7.c even 3 2 1764.4.k.e 2
7.d odd 6 2 1764.4.k.k 2
12.b even 2 1 784.4.a.n 1
21.c even 2 1 28.4.a.b 1
21.g even 6 2 196.4.e.c 2
21.h odd 6 2 196.4.e.d 2
28.d even 2 1 1008.4.a.f 1
84.h odd 2 1 112.4.a.c 1
105.g even 2 1 700.4.a.e 1
105.k odd 4 2 700.4.e.f 2
168.e odd 2 1 448.4.a.m 1
168.i even 2 1 448.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 21.c even 2 1
112.4.a.c 1 84.h odd 2 1
196.4.a.b 1 3.b odd 2 1
196.4.e.c 2 21.g even 6 2
196.4.e.d 2 21.h odd 6 2
252.4.a.c 1 7.b odd 2 1
448.4.a.d 1 168.i even 2 1
448.4.a.m 1 168.e odd 2 1
700.4.a.e 1 105.g even 2 1
700.4.e.f 2 105.k odd 4 2
784.4.a.n 1 12.b even 2 1
1008.4.a.f 1 28.d even 2 1
1764.4.a.k 1 1.a even 1 1 trivial
1764.4.k.e 2 7.c even 3 2
1764.4.k.k 2 7.d odd 6 2

Hecke kernels

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

 $$T_{5} - 6$$ T5 - 6 $$T_{11} - 12$$ T11 - 12 $$T_{13} - 82$$ T13 - 82

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 6$$
$7$ $$T$$
$11$ $$T - 12$$
$13$ $$T - 82$$
$17$ $$T + 30$$
$19$ $$T + 68$$
$23$ $$T + 216$$
$29$ $$T + 246$$
$31$ $$T - 112$$
$37$ $$T - 110$$
$41$ $$T + 246$$
$43$ $$T + 172$$
$47$ $$T - 192$$
$53$ $$T + 558$$
$59$ $$T - 540$$
$61$ $$T + 110$$
$67$ $$T - 140$$
$71$ $$T - 840$$
$73$ $$T - 550$$
$79$ $$T + 208$$
$83$ $$T - 516$$
$89$ $$T + 1398$$
$97$ $$T + 1586$$