# Properties

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

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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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1248) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{7} + O(q^{10})$$ $$q + 2q^{7} + q^{13} - 2q^{17} - 2q^{19} - 8q^{23} - 5q^{25} - 6q^{29} + 2q^{31} - 6q^{37} + 4q^{43} - 8q^{47} - 3q^{49} + 6q^{53} - 4q^{59} + 2q^{61} + 2q^{67} - 4q^{71} - 2q^{73} + 12q^{79} + 12q^{83} - 12q^{89} + 2q^{91} - 18q^{97} + O(q^{100})$$

## 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
0 0 0 0 0 2.00000 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.j 1
3.b odd 2 1 1248.2.a.i yes 1
4.b odd 2 1 3744.2.a.g 1
8.b even 2 1 7488.2.a.be 1
8.d odd 2 1 7488.2.a.ba 1
12.b even 2 1 1248.2.a.c 1
24.f even 2 1 2496.2.a.w 1
24.h odd 2 1 2496.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.c 1 12.b even 2 1
1248.2.a.i yes 1 3.b odd 2 1
2496.2.a.i 1 24.h odd 2 1
2496.2.a.w 1 24.f even 2 1
3744.2.a.g 1 4.b odd 2 1
3744.2.a.j 1 1.a even 1 1 trivial
7488.2.a.ba 1 8.d odd 2 1
7488.2.a.be 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3744))$$:

 $$T_{5}$$ $$T_{7} - 2$$ $$T_{11}$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$T$$
$13$ $$-1 + T$$
$17$ $$2 + T$$
$19$ $$2 + T$$
$23$ $$8 + T$$
$29$ $$6 + T$$
$31$ $$-2 + T$$
$37$ $$6 + T$$
$41$ $$T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$-6 + T$$
$59$ $$4 + T$$
$61$ $$-2 + T$$
$67$ $$-2 + T$$
$71$ $$4 + T$$
$73$ $$2 + T$$
$79$ $$-12 + T$$
$83$ $$-12 + T$$
$89$ $$12 + T$$
$97$ $$18 + T$$
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