# Properties

 Label 1344.4.a.z Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 16q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} + 16q^{5} - 7q^{7} + 9q^{9} - 18q^{11} + 54q^{13} + 48q^{15} - 128q^{17} + 52q^{19} - 21q^{21} + 202q^{23} + 131q^{25} + 27q^{27} - 302q^{29} + 200q^{31} - 54q^{33} - 112q^{35} + 150q^{37} + 162q^{39} + 172q^{41} + 164q^{43} + 144q^{45} + 460q^{47} + 49q^{49} - 384q^{51} + 190q^{53} - 288q^{55} + 156q^{57} + 96q^{59} - 622q^{61} - 63q^{63} + 864q^{65} + 744q^{67} + 606q^{69} + 54q^{71} + 742q^{73} + 393q^{75} + 126q^{77} + 92q^{79} + 81q^{81} - 228q^{83} - 2048q^{85} - 906q^{87} - 116q^{89} - 378q^{91} + 600q^{93} + 832q^{95} - 554q^{97} - 162q^{99} + O(q^{100})$$

## 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 3.00000 0 16.0000 0 −7.00000 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 1344.4.a.z 1
4.b odd 2 1 1344.4.a.l 1
8.b even 2 1 336.4.a.a 1
8.d odd 2 1 168.4.a.d 1
24.f even 2 1 504.4.a.h 1
24.h odd 2 1 1008.4.a.t 1
56.e even 2 1 1176.4.a.h 1
56.h odd 2 1 2352.4.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.d 1 8.d odd 2 1
336.4.a.a 1 8.b even 2 1
504.4.a.h 1 24.f even 2 1
1008.4.a.t 1 24.h odd 2 1
1176.4.a.h 1 56.e even 2 1
1344.4.a.l 1 4.b odd 2 1
1344.4.a.z 1 1.a even 1 1 trivial
2352.4.a.bj 1 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} - 16$$ $$T_{11} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$-16 + T$$
$7$ $$7 + T$$
$11$ $$18 + T$$
$13$ $$-54 + T$$
$17$ $$128 + T$$
$19$ $$-52 + T$$
$23$ $$-202 + T$$
$29$ $$302 + T$$
$31$ $$-200 + T$$
$37$ $$-150 + T$$
$41$ $$-172 + T$$
$43$ $$-164 + T$$
$47$ $$-460 + T$$
$53$ $$-190 + T$$
$59$ $$-96 + T$$
$61$ $$622 + T$$
$67$ $$-744 + T$$
$71$ $$-54 + T$$
$73$ $$-742 + T$$
$79$ $$-92 + T$$
$83$ $$228 + T$$
$89$ $$116 + T$$
$97$ $$554 + T$$
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