# Properties

 Label 1344.4.a.bk Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{43})$$ Defining polynomial: $$x^{2} - 43$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{43}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( -8 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( -8 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( -2 - \beta ) q^{11} -26 q^{13} + ( -24 + 3 \beta ) q^{15} + ( 28 - 3 \beta ) q^{17} + ( -16 + 2 \beta ) q^{19} + 21 q^{21} + ( -22 - 11 \beta ) q^{23} + ( 111 - 16 \beta ) q^{25} + 27 q^{27} + ( -10 - 2 \beta ) q^{29} + ( -148 + 10 \beta ) q^{31} + ( -6 - 3 \beta ) q^{33} + ( -56 + 7 \beta ) q^{35} + ( -86 - 16 \beta ) q^{37} -78 q^{39} + ( 112 + 11 \beta ) q^{41} + ( 72 - 16 \beta ) q^{43} + ( -72 + 9 \beta ) q^{45} + ( -172 + 6 \beta ) q^{47} + 49 q^{49} + ( 84 - 9 \beta ) q^{51} + ( -182 - 16 \beta ) q^{53} + ( -156 + 6 \beta ) q^{55} + ( -48 + 6 \beta ) q^{57} + ( -232 + 34 \beta ) q^{59} + ( -310 - 18 \beta ) q^{61} + 63 q^{63} + ( 208 - 26 \beta ) q^{65} + ( -452 + 30 \beta ) q^{67} + ( -66 - 33 \beta ) q^{69} + ( 254 - 37 \beta ) q^{71} + ( 6 + 54 \beta ) q^{73} + ( 333 - 48 \beta ) q^{75} + ( -14 - 7 \beta ) q^{77} + ( 172 - 6 \beta ) q^{79} + 81 q^{81} + ( -140 - 64 \beta ) q^{83} + ( -740 + 52 \beta ) q^{85} + ( -30 - 6 \beta ) q^{87} + ( 424 - 5 \beta ) q^{89} -182 q^{91} + ( -444 + 30 \beta ) q^{93} + ( 472 - 32 \beta ) q^{95} + ( 686 - 38 \beta ) q^{97} + ( -18 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 16q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 16q^{5} + 14q^{7} + 18q^{9} - 4q^{11} - 52q^{13} - 48q^{15} + 56q^{17} - 32q^{19} + 42q^{21} - 44q^{23} + 222q^{25} + 54q^{27} - 20q^{29} - 296q^{31} - 12q^{33} - 112q^{35} - 172q^{37} - 156q^{39} + 224q^{41} + 144q^{43} - 144q^{45} - 344q^{47} + 98q^{49} + 168q^{51} - 364q^{53} - 312q^{55} - 96q^{57} - 464q^{59} - 620q^{61} + 126q^{63} + 416q^{65} - 904q^{67} - 132q^{69} + 508q^{71} + 12q^{73} + 666q^{75} - 28q^{77} + 344q^{79} + 162q^{81} - 280q^{83} - 1480q^{85} - 60q^{87} + 848q^{89} - 364q^{91} - 888q^{93} + 944q^{95} + 1372q^{97} - 36q^{99} + O(q^{100})$$

## 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
 −6.55744 6.55744
0 3.00000 0 −21.1149 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 5.11488 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.bk 2
4.b odd 2 1 1344.4.a.bc 2
8.b even 2 1 672.4.a.i 2
8.d odd 2 1 672.4.a.n yes 2
24.f even 2 1 2016.4.a.g 2
24.h odd 2 1 2016.4.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.i 2 8.b even 2 1
672.4.a.n yes 2 8.d odd 2 1
1344.4.a.bc 2 4.b odd 2 1
1344.4.a.bk 2 1.a even 1 1 trivial
2016.4.a.g 2 24.f even 2 1
2016.4.a.h 2 24.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{2} + 16 T_{5} - 108$$ $$T_{11}^{2} + 4 T_{11} - 168$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-108 + 16 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-168 + 4 T + T^{2}$$
$13$ $$( 26 + T )^{2}$$
$17$ $$-764 - 56 T + T^{2}$$
$19$ $$-432 + 32 T + T^{2}$$
$23$ $$-20328 + 44 T + T^{2}$$
$29$ $$-588 + 20 T + T^{2}$$
$31$ $$4704 + 296 T + T^{2}$$
$37$ $$-36636 + 172 T + T^{2}$$
$41$ $$-8268 - 224 T + T^{2}$$
$43$ $$-38848 - 144 T + T^{2}$$
$47$ $$23392 + 344 T + T^{2}$$
$53$ $$-10908 + 364 T + T^{2}$$
$59$ $$-145008 + 464 T + T^{2}$$
$61$ $$40372 + 620 T + T^{2}$$
$67$ $$49504 + 904 T + T^{2}$$
$71$ $$-170952 - 508 T + T^{2}$$
$73$ $$-501516 - 12 T + T^{2}$$
$79$ $$23392 - 344 T + T^{2}$$
$83$ $$-684912 + 280 T + T^{2}$$
$89$ $$175476 - 848 T + T^{2}$$
$97$ $$222228 - 1372 T + T^{2}$$