Properties

 Label 756.2.a.f Level $756$ Weight $2$ Character orbit 756.a Self dual yes Analytic conductor $6.037$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 756.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.03669039281$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + q^{7} + O(q^{10})$$ $$q + 3 q^{5} + q^{7} + 3 q^{11} + 2 q^{13} - 6 q^{17} + 5 q^{19} - 9 q^{23} + 4 q^{25} + 6 q^{29} - q^{31} + 3 q^{35} + 11 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} + q^{49} + 9 q^{55} + 8 q^{61} + 6 q^{65} - 10 q^{67} + 3 q^{71} + 8 q^{73} + 3 q^{77} - 4 q^{79} + 6 q^{83} - 18 q^{85} + 3 q^{89} + 2 q^{91} + 15 q^{95} + 8 q^{97} + O(q^{100})$$

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 1.00000 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 756.2.a.f yes 1
3.b odd 2 1 756.2.a.a 1
4.b odd 2 1 3024.2.a.z 1
7.b odd 2 1 5292.2.a.b 1
9.c even 3 2 2268.2.j.b 2
9.d odd 6 2 2268.2.j.m 2
12.b even 2 1 3024.2.a.e 1
21.c even 2 1 5292.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.a.a 1 3.b odd 2 1
756.2.a.f yes 1 1.a even 1 1 trivial
2268.2.j.b 2 9.c even 3 2
2268.2.j.m 2 9.d odd 6 2
3024.2.a.e 1 12.b even 2 1
3024.2.a.z 1 4.b odd 2 1
5292.2.a.b 1 7.b odd 2 1
5292.2.a.l 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(756))$$:

 $$T_{5} - 3$$ $$T_{11} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$-1 + T$$
$11$ $$-3 + T$$
$13$ $$-2 + T$$
$17$ $$6 + T$$
$19$ $$-5 + T$$
$23$ $$9 + T$$
$29$ $$-6 + T$$
$31$ $$1 + T$$
$37$ $$-11 + T$$
$41$ $$-3 + T$$
$43$ $$4 + T$$
$47$ $$12 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-8 + T$$
$67$ $$10 + T$$
$71$ $$-3 + T$$
$73$ $$-8 + T$$
$79$ $$4 + T$$
$83$ $$-6 + T$$
$89$ $$-3 + T$$
$97$ $$-8 + T$$