# Properties

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

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## 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
0 0 0 0 0 −2.82843 0 0 0
1.2 0 0 0 0 0 2.82843 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.t 2
3.b odd 2 1 3744.2.a.u yes 2
4.b odd 2 1 inner 3744.2.a.t 2
8.b even 2 1 7488.2.a.cn 2
8.d odd 2 1 7488.2.a.cn 2
12.b even 2 1 3744.2.a.u yes 2
24.f even 2 1 7488.2.a.cm 2
24.h odd 2 1 7488.2.a.cm 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-32 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$-200 + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-72 + T^{2}$$
$71$ $$-72 + T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$( -4 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$
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