# Properties

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

# Related objects

## 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 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} - 18q^{5} + 7q^{7} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} - 18q^{5} + 7q^{7} + 9q^{9} + 72q^{11} + 34q^{13} - 54q^{15} + 6q^{17} - 92q^{19} + 21q^{21} - 180q^{23} + 199q^{25} + 27q^{27} + 114q^{29} + 56q^{31} + 216q^{33} - 126q^{35} + 34q^{37} + 102q^{39} + 6q^{41} - 164q^{43} - 162q^{45} + 168q^{47} + 49q^{49} + 18q^{51} - 654q^{53} - 1296q^{55} - 276q^{57} + 492q^{59} + 250q^{61} + 63q^{63} - 612q^{65} + 124q^{67} - 540q^{69} + 36q^{71} + 1010q^{73} + 597q^{75} + 504q^{77} + 56q^{79} + 81q^{81} - 228q^{83} - 108q^{85} + 342q^{87} + 390q^{89} + 238q^{91} + 168q^{93} + 1656q^{95} - 70q^{97} + 648q^{99} + O(q^{100})$$

## 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 −18.0000 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.o 1
4.b odd 2 1 1344.4.a.a 1
8.b even 2 1 42.4.a.a 1
8.d odd 2 1 336.4.a.l 1
24.f even 2 1 1008.4.a.b 1
24.h odd 2 1 126.4.a.a 1
40.f even 2 1 1050.4.a.g 1
40.i odd 4 2 1050.4.g.a 2
56.e even 2 1 2352.4.a.a 1
56.h odd 2 1 294.4.a.i 1
56.j odd 6 2 294.4.e.b 2
56.p even 6 2 294.4.e.c 2
168.i even 2 1 882.4.a.g 1
168.s odd 6 2 882.4.g.w 2
168.ba even 6 2 882.4.g.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 8.b even 2 1
126.4.a.a 1 24.h odd 2 1
294.4.a.i 1 56.h odd 2 1
294.4.e.b 2 56.j odd 6 2
294.4.e.c 2 56.p even 6 2
336.4.a.l 1 8.d odd 2 1
882.4.a.g 1 168.i even 2 1
882.4.g.o 2 168.ba even 6 2
882.4.g.w 2 168.s odd 6 2
1008.4.a.b 1 24.f even 2 1
1050.4.a.g 1 40.f even 2 1
1050.4.g.a 2 40.i odd 4 2
1344.4.a.a 1 4.b odd 2 1
1344.4.a.o 1 1.a even 1 1 trivial
2352.4.a.a 1 56.e even 2 1

## Hecke kernels

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

 $$T_{5} + 18$$ $$T_{11} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$18 + T$$
$7$ $$-7 + T$$
$11$ $$-72 + T$$
$13$ $$-34 + T$$
$17$ $$-6 + T$$
$19$ $$92 + T$$
$23$ $$180 + T$$
$29$ $$-114 + T$$
$31$ $$-56 + T$$
$37$ $$-34 + T$$
$41$ $$-6 + T$$
$43$ $$164 + T$$
$47$ $$-168 + T$$
$53$ $$654 + T$$
$59$ $$-492 + T$$
$61$ $$-250 + T$$
$67$ $$-124 + T$$
$71$ $$-36 + T$$
$73$ $$-1010 + T$$
$79$ $$-56 + T$$
$83$ $$228 + T$$
$89$ $$-390 + T$$
$97$ $$70 + T$$