# Properties

 Label 3744.2.a.x Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.56155 2.56155
0 0 0 −0.561553 0 0.561553 0 0 0
1.2 0 0 0 3.56155 0 −3.56155 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.x 2
3.b odd 2 1 416.2.a.c 2
4.b odd 2 1 3744.2.a.y 2
8.b even 2 1 7488.2.a.ce 2
8.d odd 2 1 7488.2.a.cf 2
12.b even 2 1 416.2.a.e yes 2
24.f even 2 1 832.2.a.l 2
24.h odd 2 1 832.2.a.o 2
39.d odd 2 1 5408.2.a.p 2
48.i odd 4 2 3328.2.b.ba 4
48.k even 4 2 3328.2.b.u 4
156.h even 2 1 5408.2.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.a.c 2 3.b odd 2 1
416.2.a.e yes 2 12.b even 2 1
832.2.a.l 2 24.f even 2 1
832.2.a.o 2 24.h odd 2 1
3328.2.b.u 4 48.k even 4 2
3328.2.b.ba 4 48.i odd 4 2
3744.2.a.x 2 1.a even 1 1 trivial
3744.2.a.y 2 4.b odd 2 1
5408.2.a.p 2 39.d odd 2 1
5408.2.a.bd 2 156.h even 2 1
7488.2.a.ce 2 8.b even 2 1
7488.2.a.cf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3744))$$:

 $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{7}^{2} + 3 T_{7} - 2$$ $$T_{11} - 2$$ $$T_{29}^{2} - 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 - 3 T + T^{2}$$
$7$ $$-2 + 3 T + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$-2 + 3 T + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$-68 + T^{2}$$
$31$ $$-8 + 6 T + T^{2}$$
$37$ $$-26 + 7 T + T^{2}$$
$41$ $$-8 + 6 T + T^{2}$$
$43$ $$-104 + 3 T + T^{2}$$
$47$ $$-18 + 9 T + T^{2}$$
$53$ $$64 - 18 T + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$-152 - 2 T + T^{2}$$
$67$ $$( 6 + T )^{2}$$
$71$ $$-18 - 9 T + T^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$-128 + 10 T + T^{2}$$
$89$ $$-68 + T^{2}$$
$97$ $$( 6 + T )^{2}$$