# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} - q^{7} +O(q^{10})$$ $$q -\beta q^{5} - q^{7} + \beta q^{11} + 6 q^{13} -2 \beta q^{17} - q^{19} + \beta q^{23} + 8 q^{25} + 2 \beta q^{29} + 9 q^{31} + \beta q^{35} - q^{37} + 3 \beta q^{41} + 8 q^{43} + q^{49} -13 q^{55} -4 \beta q^{59} -6 \beta q^{65} -2 q^{67} + \beta q^{71} + 4 q^{73} -\beta q^{77} -2 \beta q^{83} + 26 q^{85} + 3 \beta q^{89} -6 q^{91} + \beta q^{95} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7} + O(q^{10})$$ $$2 q - 2 q^{7} + 12 q^{13} - 2 q^{19} + 16 q^{25} + 18 q^{31} - 2 q^{37} + 16 q^{43} + 2 q^{49} - 26 q^{55} - 4 q^{67} + 8 q^{73} + 52 q^{85} - 12 q^{91} - 16 q^{97} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 0 0 −3.60555 0 −1.00000 0 0 0
1.2 0 0 0 3.60555 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.a.g 2
3.b odd 2 1 inner 756.2.a.g 2
4.b odd 2 1 3024.2.a.bj 2
7.b odd 2 1 5292.2.a.o 2
9.c even 3 2 2268.2.j.q 4
9.d odd 6 2 2268.2.j.q 4
12.b even 2 1 3024.2.a.bj 2
21.c even 2 1 5292.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.a.g 2 1.a even 1 1 trivial
756.2.a.g 2 3.b odd 2 1 inner
2268.2.j.q 4 9.c even 3 2
2268.2.j.q 4 9.d odd 6 2
3024.2.a.bj 2 4.b odd 2 1
3024.2.a.bj 2 12.b even 2 1
5292.2.a.o 2 7.b odd 2 1
5292.2.a.o 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(756))$$:

 $$T_{5}^{2} - 13$$ $$T_{11}^{2} - 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-13 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-13 + T^{2}$$
$13$ $$( -6 + T )^{2}$$
$17$ $$-52 + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-13 + T^{2}$$
$29$ $$-52 + T^{2}$$
$31$ $$( -9 + T )^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$-117 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$-208 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$-13 + T^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-52 + T^{2}$$
$89$ $$-117 + T^{2}$$
$97$ $$( 8 + T )^{2}$$