# Properties

 Label 1344.4.a.h Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 3q^{3} + 2q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} + 2q^{5} - 7q^{7} + 9q^{9} + 12q^{11} + 66q^{13} - 6q^{15} - 70q^{17} - 92q^{19} + 21q^{21} - 16q^{23} - 121q^{25} - 27q^{27} + 122q^{29} - 64q^{31} - 36q^{33} - 14q^{35} + 306q^{37} - 198q^{39} + 50q^{41} + 20q^{43} + 18q^{45} + 176q^{47} + 49q^{49} + 210q^{51} - 526q^{53} + 24q^{55} + 276q^{57} + 540q^{59} + 818q^{61} - 63q^{63} + 132q^{65} - 228q^{67} + 48q^{69} - 864q^{71} + 106q^{73} + 363q^{75} - 84q^{77} - 736q^{79} + 81q^{81} - 588q^{83} - 140q^{85} - 366q^{87} + 146q^{89} - 462q^{91} + 192q^{93} - 184q^{95} - 1214q^{97} + 108q^{99} + O(q^{100})$$

## 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 2.00000 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.h 1
4.b odd 2 1 1344.4.a.u 1
8.b even 2 1 336.4.a.i 1
8.d odd 2 1 168.4.a.b 1
24.f even 2 1 504.4.a.d 1
24.h odd 2 1 1008.4.a.k 1
56.e even 2 1 1176.4.a.l 1
56.h odd 2 1 2352.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.b 1 8.d odd 2 1
336.4.a.i 1 8.b even 2 1
504.4.a.d 1 24.f even 2 1
1008.4.a.k 1 24.h odd 2 1
1176.4.a.l 1 56.e even 2 1
1344.4.a.h 1 1.a even 1 1 trivial
1344.4.a.u 1 4.b odd 2 1
2352.4.a.k 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} - 2$$ $$T_{11} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$-2 + T$$
$7$ $$7 + T$$
$11$ $$-12 + T$$
$13$ $$-66 + T$$
$17$ $$70 + T$$
$19$ $$92 + T$$
$23$ $$16 + T$$
$29$ $$-122 + T$$
$31$ $$64 + T$$
$37$ $$-306 + T$$
$41$ $$-50 + T$$
$43$ $$-20 + T$$
$47$ $$-176 + T$$
$53$ $$526 + T$$
$59$ $$-540 + T$$
$61$ $$-818 + T$$
$67$ $$228 + T$$
$71$ $$864 + T$$
$73$ $$-106 + T$$
$79$ $$736 + T$$
$83$ $$588 + T$$
$89$ $$-146 + T$$
$97$ $$1214 + T$$
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