# Properties

 Label 3744.2.a.bb Level $3744$ Weight $2$ Character orbit 3744.a Self dual yes Analytic conductor $29.896$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5 x^{2} + 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{3} ) q^{5} -\beta_{2} q^{7} + ( \beta_{1} + \beta_{2} ) q^{11} + q^{13} + ( -2 - 2 \beta_{3} ) q^{17} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( 1 - 2 \beta_{3} ) q^{25} -4 q^{29} -\beta_{2} q^{31} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -2 - 4 \beta_{3} ) q^{37} + ( -3 - \beta_{3} ) q^{41} + 2 \beta_{1} q^{43} + ( -3 \beta_{1} + \beta_{2} ) q^{47} + ( 3 - 2 \beta_{3} ) q^{49} + ( -4 + 4 \beta_{3} ) q^{53} + 2 \beta_{1} q^{55} + ( 3 \beta_{1} - \beta_{2} ) q^{59} + 2 \beta_{3} q^{61} + ( -1 + \beta_{3} ) q^{65} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{2} ) q^{71} + 6 q^{73} + ( -10 - 2 \beta_{3} ) q^{77} -4 \beta_{1} q^{79} + ( \beta_{1} - 3 \beta_{2} ) q^{83} -8 q^{85} + ( -9 - 3 \beta_{3} ) q^{89} -\beta_{2} q^{91} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{95} + ( 2 + 4 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + O(q^{10})$$ $$4q - 4q^{5} + 4q^{13} - 8q^{17} + 4q^{25} - 16q^{29} - 8q^{37} - 12q^{41} + 12q^{49} - 16q^{53} - 4q^{65} + 24q^{73} - 40q^{77} - 32q^{85} - 36q^{89} + 8q^{97} + O(q^{100})$$

## 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
 −1.17557 1.17557 1.90211 −1.90211
0 0 0 −3.23607 0 −3.80423 0 0 0
1.2 0 0 0 −3.23607 0 3.80423 0 0 0
1.3 0 0 0 1.23607 0 −2.35114 0 0 0
1.4 0 0 0 1.23607 0 2.35114 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.bb 4
3.b odd 2 1 3744.2.a.bf yes 4
4.b odd 2 1 inner 3744.2.a.bb 4
8.b even 2 1 7488.2.a.dd 4
8.d odd 2 1 7488.2.a.dd 4
12.b even 2 1 3744.2.a.bf yes 4
24.f even 2 1 7488.2.a.cz 4
24.h odd 2 1 7488.2.a.cz 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3744.2.a.bb 4 1.a even 1 1 trivial
3744.2.a.bb 4 4.b odd 2 1 inner
3744.2.a.bf yes 4 3.b odd 2 1
3744.2.a.bf yes 4 12.b even 2 1
7488.2.a.cz 4 24.f even 2 1
7488.2.a.cz 4 24.h odd 2 1
7488.2.a.dd 4 8.b even 2 1
7488.2.a.dd 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3744))$$:

 $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{7}^{4} - 20 T_{7}^{2} + 80$$ $$T_{11}^{4} - 40 T_{11}^{2} + 80$$ $$T_{29} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -4 + 2 T + T^{2} )^{2}$$
$7$ $$80 - 20 T^{2} + T^{4}$$
$11$ $$80 - 40 T^{2} + T^{4}$$
$13$ $$( -1 + T )^{4}$$
$17$ $$( -16 + 4 T + T^{2} )^{2}$$
$19$ $$2000 - 100 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 4 + T )^{4}$$
$31$ $$80 - 20 T^{2} + T^{4}$$
$37$ $$( -76 + 4 T + T^{2} )^{2}$$
$41$ $$( 4 + 6 T + T^{2} )^{2}$$
$43$ $$1280 - 80 T^{2} + T^{4}$$
$47$ $$9680 - 200 T^{2} + T^{4}$$
$53$ $$( -64 + 8 T + T^{2} )^{2}$$
$59$ $$9680 - 200 T^{2} + T^{4}$$
$61$ $$( -20 + T^{2} )^{2}$$
$67$ $$9680 - 260 T^{2} + T^{4}$$
$71$ $$80 - 40 T^{2} + T^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$20480 - 320 T^{2} + T^{4}$$
$83$ $$2000 - 200 T^{2} + T^{4}$$
$89$ $$( 36 + 18 T + T^{2} )^{2}$$
$97$ $$( -76 - 4 T + T^{2} )^{2}$$