# Properties

 Label 1344.4.a.be 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: $$\Q(\sqrt{137})$$ Defining polynomial: $$x^{2} - x - 34$$ 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 $$\beta = \sqrt{137}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( -11 - \beta ) q^{11} + ( -12 + 2 \beta ) q^{13} + ( 15 + 3 \beta ) q^{15} + ( 15 - \beta ) q^{17} + ( -16 - 4 \beta ) q^{19} -21 q^{21} + ( 41 + 3 \beta ) q^{23} + ( 37 + 10 \beta ) q^{25} -27 q^{27} + ( -18 + 20 \beta ) q^{29} + ( 56 + 8 \beta ) q^{31} + ( 33 + 3 \beta ) q^{33} + ( -35 - 7 \beta ) q^{35} + ( -24 - 14 \beta ) q^{37} + ( 36 - 6 \beta ) q^{39} + ( -35 + \beta ) q^{41} + ( -20 + 20 \beta ) q^{43} + ( -45 - 9 \beta ) q^{45} + ( 210 - 34 \beta ) q^{47} + 49 q^{49} + ( -45 + 3 \beta ) q^{51} + ( -88 + 18 \beta ) q^{53} + ( 192 + 16 \beta ) q^{55} + ( 48 + 12 \beta ) q^{57} + ( -202 - 2 \beta ) q^{59} + ( 78 + 40 \beta ) q^{61} + 63 q^{63} + ( -214 + 2 \beta ) q^{65} + ( 2 + 30 \beta ) q^{67} + ( -123 - 9 \beta ) q^{69} + ( 407 + 5 \beta ) q^{71} + ( -108 + 46 \beta ) q^{73} + ( -111 - 30 \beta ) q^{75} + ( -77 - 7 \beta ) q^{77} + ( 790 - 22 \beta ) q^{79} + 81 q^{81} + ( -232 + 4 \beta ) q^{83} + ( 62 - 10 \beta ) q^{85} + ( 54 - 60 \beta ) q^{87} + ( -579 - 87 \beta ) q^{89} + ( -84 + 14 \beta ) q^{91} + ( -168 - 24 \beta ) q^{93} + ( 628 + 36 \beta ) q^{95} + ( -880 - 62 \beta ) q^{97} + ( -99 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 10q^{5} + 14q^{7} + 18q^{9} - 22q^{11} - 24q^{13} + 30q^{15} + 30q^{17} - 32q^{19} - 42q^{21} + 82q^{23} + 74q^{25} - 54q^{27} - 36q^{29} + 112q^{31} + 66q^{33} - 70q^{35} - 48q^{37} + 72q^{39} - 70q^{41} - 40q^{43} - 90q^{45} + 420q^{47} + 98q^{49} - 90q^{51} - 176q^{53} + 384q^{55} + 96q^{57} - 404q^{59} + 156q^{61} + 126q^{63} - 428q^{65} + 4q^{67} - 246q^{69} + 814q^{71} - 216q^{73} - 222q^{75} - 154q^{77} + 1580q^{79} + 162q^{81} - 464q^{83} + 124q^{85} + 108q^{87} - 1158q^{89} - 168q^{91} - 336q^{93} + 1256q^{95} - 1760q^{97} - 198q^{99} + O(q^{100})$$

## 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
 6.35235 −5.35235
0 −3.00000 0 −16.7047 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 6.70470 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.be 2
4.b odd 2 1 1344.4.a.bm 2
8.b even 2 1 672.4.a.m yes 2
8.d odd 2 1 672.4.a.h 2
24.f even 2 1 2016.4.a.i 2
24.h odd 2 1 2016.4.a.j 2

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

 $$T_{5}^{2} + 10 T_{5} - 112$$ $$T_{11}^{2} + 22 T_{11} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-112 + 10 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-16 + 22 T + T^{2}$$
$13$ $$-404 + 24 T + T^{2}$$
$17$ $$88 - 30 T + T^{2}$$
$19$ $$-1936 + 32 T + T^{2}$$
$23$ $$448 - 82 T + T^{2}$$
$29$ $$-54476 + 36 T + T^{2}$$
$31$ $$-5632 - 112 T + T^{2}$$
$37$ $$-26276 + 48 T + T^{2}$$
$41$ $$1088 + 70 T + T^{2}$$
$43$ $$-54400 + 40 T + T^{2}$$
$47$ $$-114272 - 420 T + T^{2}$$
$53$ $$-36644 + 176 T + T^{2}$$
$59$ $$40256 + 404 T + T^{2}$$
$61$ $$-213116 - 156 T + T^{2}$$
$67$ $$-123296 - 4 T + T^{2}$$
$71$ $$162224 - 814 T + T^{2}$$
$73$ $$-278228 + 216 T + T^{2}$$
$79$ $$557792 - 1580 T + T^{2}$$
$83$ $$51632 + 464 T + T^{2}$$
$89$ $$-701712 + 1158 T + T^{2}$$
$97$ $$247772 + 1760 T + T^{2}$$