# Properties

 Label 3744.2.a.q Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3744 = 2^{5} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3744.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.8959905168$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 416) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{5} -\beta q^{7} +O(q^{10})$$ $$q -3 q^{5} -\beta q^{7} -2 \beta q^{11} - q^{13} + 3 q^{17} + 2 \beta q^{19} -4 \beta q^{23} + 4 q^{25} -10 q^{29} + 3 \beta q^{35} + 3 q^{37} -3 \beta q^{43} + \beta q^{47} -2 q^{49} -4 q^{53} + 6 \beta q^{55} -2 \beta q^{59} + 3 q^{65} -6 \beta q^{67} + 3 \beta q^{71} + 14 q^{73} + 10 q^{77} + 4 \beta q^{79} + 8 \beta q^{83} -9 q^{85} + 10 q^{89} + \beta q^{91} -6 \beta q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{5} + O(q^{10})$$ $$2q - 6q^{5} - 2q^{13} + 6q^{17} + 8q^{25} - 20q^{29} + 6q^{37} - 4q^{49} - 8q^{53} + 6q^{65} + 28q^{73} + 20q^{77} - 18q^{85} + 20q^{89} - 4q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 0 0 −3.00000 0 −2.23607 0 0 0
1.2 0 0 0 −3.00000 0 2.23607 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.a.q 2
3.b odd 2 1 416.2.a.d 2
4.b odd 2 1 inner 3744.2.a.q 2
8.b even 2 1 7488.2.a.cw 2
8.d odd 2 1 7488.2.a.cw 2
12.b even 2 1 416.2.a.d 2
24.f even 2 1 832.2.a.m 2
24.h odd 2 1 832.2.a.m 2
39.d odd 2 1 5408.2.a.q 2
48.i odd 4 2 3328.2.b.x 4
48.k even 4 2 3328.2.b.x 4
156.h even 2 1 5408.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.d 2 3.b odd 2 1
416.2.a.d 2 12.b even 2 1
832.2.a.m 2 24.f even 2 1
832.2.a.m 2 24.h odd 2 1
3328.2.b.x 4 48.i odd 4 2
3328.2.b.x 4 48.k even 4 2
3744.2.a.q 2 1.a even 1 1 trivial
3744.2.a.q 2 4.b odd 2 1 inner
5408.2.a.q 2 39.d odd 2 1
5408.2.a.q 2 156.h even 2 1
7488.2.a.cw 2 8.b even 2 1
7488.2.a.cw 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3744))$$:

 $$T_{5} + 3$$ $$T_{7}^{2} - 5$$ $$T_{11}^{2} - 20$$ $$T_{29} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 3 + T )^{2}$$
$7$ $$-5 + T^{2}$$
$11$ $$-20 + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$-80 + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( -3 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$-45 + T^{2}$$
$47$ $$-5 + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$-20 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$-180 + T^{2}$$
$71$ $$-45 + T^{2}$$
$73$ $$( -14 + T )^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-320 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$