# Properties

 Label 3744.2.a.w Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1248) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{5} + ( 3 - \beta ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{5} + ( 3 - \beta ) q^{7} + 2 \beta q^{11} - q^{13} + 2 \beta q^{17} + ( 1 + \beta ) q^{19} + ( -2 - 2 \beta ) q^{23} + ( 1 + 2 \beta ) q^{25} + ( 4 - 2 \beta ) q^{29} + ( 7 - \beta ) q^{31} + ( -2 + 2 \beta ) q^{35} + 2 \beta q^{37} + ( 7 - \beta ) q^{41} + ( 2 - 2 \beta ) q^{43} + ( -4 - 2 \beta ) q^{47} + ( 7 - 6 \beta ) q^{49} + ( 4 + 2 \beta ) q^{53} + ( 10 + 2 \beta ) q^{55} + ( -2 - 4 \beta ) q^{59} + ( 8 + 2 \beta ) q^{61} + ( -1 - \beta ) q^{65} + ( -1 + 3 \beta ) q^{67} -10 q^{71} + 2 \beta q^{73} + ( -10 + 6 \beta ) q^{77} -4 \beta q^{79} + ( -6 - 4 \beta ) q^{83} + ( 10 + 2 \beta ) q^{85} + ( 9 - 3 \beta ) q^{89} + ( -3 + \beta ) q^{91} + ( 6 + 2 \beta ) q^{95} + ( 12 - 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 6q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 6q^{7} - 2q^{13} + 2q^{19} - 4q^{23} + 2q^{25} + 8q^{29} + 14q^{31} - 4q^{35} + 14q^{41} + 4q^{43} - 8q^{47} + 14q^{49} + 8q^{53} + 20q^{55} - 4q^{59} + 16q^{61} - 2q^{65} - 2q^{67} - 20q^{71} - 20q^{77} - 12q^{83} + 20q^{85} + 18q^{89} - 6q^{91} + 12q^{95} + 24q^{97} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 0 0 −1.23607 0 5.23607 0 0 0
1.2 0 0 0 3.23607 0 0.763932 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

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 3744.2.a.w 2
3.b odd 2 1 1248.2.a.k 2
4.b odd 2 1 3744.2.a.v 2
8.b even 2 1 7488.2.a.ch 2
8.d odd 2 1 7488.2.a.cg 2
12.b even 2 1 1248.2.a.m yes 2
24.f even 2 1 2496.2.a.bg 2
24.h odd 2 1 2496.2.a.bj 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.k 2 3.b odd 2 1
1248.2.a.m yes 2 12.b even 2 1
2496.2.a.bg 2 24.f even 2 1
2496.2.a.bj 2 24.h odd 2 1
3744.2.a.v 2 4.b odd 2 1
3744.2.a.w 2 1.a even 1 1 trivial
7488.2.a.cg 2 8.d odd 2 1
7488.2.a.ch 2 8.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{7}^{2} - 6 T_{7} + 4$$ $$T_{11}^{2} - 20$$ $$T_{29}^{2} - 8 T_{29} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$4 - 6 T + T^{2}$$
$11$ $$-20 + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$-4 - 2 T + T^{2}$$
$23$ $$-16 + 4 T + T^{2}$$
$29$ $$-4 - 8 T + T^{2}$$
$31$ $$44 - 14 T + T^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$44 - 14 T + T^{2}$$
$43$ $$-16 - 4 T + T^{2}$$
$47$ $$-4 + 8 T + T^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-76 + 4 T + T^{2}$$
$61$ $$44 - 16 T + T^{2}$$
$67$ $$-44 + 2 T + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$-20 + T^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-44 + 12 T + T^{2}$$
$89$ $$36 - 18 T + T^{2}$$
$97$ $$124 - 24 T + T^{2}$$