# Properties

 Label 1344.4.a.b 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 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 14q^{5} - 7q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 14q^{5} - 7q^{7} + 9q^{9} - 4q^{11} - 54q^{13} + 42q^{15} - 14q^{17} - 92q^{19} + 21q^{21} - 152q^{23} + 71q^{25} - 27q^{27} + 106q^{29} - 144q^{31} + 12q^{33} + 98q^{35} - 158q^{37} + 162q^{39} - 390q^{41} + 508q^{43} - 126q^{45} - 528q^{47} + 49q^{49} + 42q^{51} - 606q^{53} + 56q^{55} + 276q^{57} + 364q^{59} - 678q^{61} - 63q^{63} + 756q^{65} - 844q^{67} + 456q^{69} - 8q^{71} - 422q^{73} - 213q^{75} + 28q^{77} + 384q^{79} + 81q^{81} + 548q^{83} + 196q^{85} - 318q^{87} + 1194q^{89} + 378q^{91} + 432q^{93} + 1288q^{95} - 1502q^{97} - 36q^{99} + O(q^{100})$$

## 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 −14.0000 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.b 1
4.b odd 2 1 1344.4.a.p 1
8.b even 2 1 84.4.a.b 1
8.d odd 2 1 336.4.a.e 1
24.f even 2 1 1008.4.a.d 1
24.h odd 2 1 252.4.a.a 1
40.f even 2 1 2100.4.a.g 1
40.i odd 4 2 2100.4.k.g 2
56.e even 2 1 2352.4.a.v 1
56.h odd 2 1 588.4.a.a 1
56.j odd 6 2 588.4.i.h 2
56.p even 6 2 588.4.i.a 2
168.i even 2 1 1764.4.a.l 1
168.s odd 6 2 1764.4.k.n 2
168.ba even 6 2 1764.4.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 8.b even 2 1
252.4.a.a 1 24.h odd 2 1
336.4.a.e 1 8.d odd 2 1
588.4.a.a 1 56.h odd 2 1
588.4.i.a 2 56.p even 6 2
588.4.i.h 2 56.j odd 6 2
1008.4.a.d 1 24.f even 2 1
1344.4.a.b 1 1.a even 1 1 trivial
1344.4.a.p 1 4.b odd 2 1
1764.4.a.l 1 168.i even 2 1
1764.4.k.c 2 168.ba even 6 2
1764.4.k.n 2 168.s odd 6 2
2100.4.a.g 1 40.f even 2 1
2100.4.k.g 2 40.i odd 4 2
2352.4.a.v 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} + 14$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$14 + T$$
$7$ $$7 + T$$
$11$ $$4 + T$$
$13$ $$54 + T$$
$17$ $$14 + T$$
$19$ $$92 + T$$
$23$ $$152 + T$$
$29$ $$-106 + T$$
$31$ $$144 + T$$
$37$ $$158 + T$$
$41$ $$390 + T$$
$43$ $$-508 + T$$
$47$ $$528 + T$$
$53$ $$606 + T$$
$59$ $$-364 + T$$
$61$ $$678 + T$$
$67$ $$844 + T$$
$71$ $$8 + T$$
$73$ $$422 + T$$
$79$ $$-384 + T$$
$83$ $$-548 + T$$
$89$ $$-1194 + T$$
$97$ $$1502 + T$$