# Properties

 Label 1344.4.a.bs Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.37341.1 Defining polynomial: $$x^{3} - 57 x - 148$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -2 + \beta_{1} ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -2 + \beta_{1} ) q^{5} + 7 q^{7} + 9 q^{9} + ( -16 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{13} + ( 6 - 3 \beta_{1} ) q^{15} + ( 18 - \beta_{1} + \beta_{2} ) q^{17} + ( -28 - 2 \beta_{1} - \beta_{2} ) q^{19} -21 q^{21} + ( 16 + 3 \beta_{1} + \beta_{2} ) q^{23} + ( 31 + 4 \beta_{1} + \beta_{2} ) q^{25} -27 q^{27} + ( -6 + 10 \beta_{1} - 3 \beta_{2} ) q^{29} + ( 24 + 2 \beta_{1} + 3 \beta_{2} ) q^{31} + ( 48 + 3 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -14 + 7 \beta_{1} ) q^{35} + ( -70 + 4 \beta_{1} + \beta_{2} ) q^{37} + ( 6 - 12 \beta_{1} - 3 \beta_{2} ) q^{39} + ( 138 + 5 \beta_{1} - 3 \beta_{2} ) q^{41} + ( -56 + 8 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -18 + 9 \beta_{1} ) q^{45} + ( 24 + 26 \beta_{1} ) q^{47} + 49 q^{49} + ( -54 + 3 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -134 - 12 \beta_{1} - 3 \beta_{2} ) q^{53} + ( -152 - 10 \beta_{1} - 11 \beta_{2} ) q^{55} + ( 84 + 6 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -180 - 38 \beta_{1} ) q^{59} + ( -266 - 30 \beta_{1} + 7 \beta_{2} ) q^{61} + 63 q^{63} + ( 580 + 34 \beta_{1} - 6 \beta_{2} ) q^{65} + ( -16 + 38 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -48 - 9 \beta_{1} - 3 \beta_{2} ) q^{69} + ( -152 + 21 \beta_{1} - \beta_{2} ) q^{71} + ( 410 + 18 \beta_{1} + 16 \beta_{2} ) q^{73} + ( -93 - 12 \beta_{1} - 3 \beta_{2} ) q^{75} + ( -112 - 7 \beta_{1} + 7 \beta_{2} ) q^{77} + ( -456 - 26 \beta_{1} + 8 \beta_{2} ) q^{79} + 81 q^{81} + ( -20 - 40 \beta_{1} - 10 \beta_{2} ) q^{83} + ( -220 + 24 \beta_{1} - 11 \beta_{2} ) q^{85} + ( 18 - 30 \beta_{1} + 9 \beta_{2} ) q^{87} + ( 914 - 31 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -14 + 28 \beta_{1} + 7 \beta_{2} ) q^{91} + ( -72 - 6 \beta_{1} - 9 \beta_{2} ) q^{93} + ( -216 - 52 \beta_{1} + 8 \beta_{2} ) q^{95} + ( 650 - 26 \beta_{1} - 12 \beta_{2} ) q^{97} + ( -144 - 9 \beta_{1} + 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 9q^{3} - 6q^{5} + 21q^{7} + 27q^{9} + O(q^{10})$$ $$3q - 9q^{3} - 6q^{5} + 21q^{7} + 27q^{9} - 48q^{11} - 6q^{13} + 18q^{15} + 54q^{17} - 84q^{19} - 63q^{21} + 48q^{23} + 93q^{25} - 81q^{27} - 18q^{29} + 72q^{31} + 144q^{33} - 42q^{35} - 210q^{37} + 18q^{39} + 414q^{41} - 168q^{43} - 54q^{45} + 72q^{47} + 147q^{49} - 162q^{51} - 402q^{53} - 456q^{55} + 252q^{57} - 540q^{59} - 798q^{61} + 189q^{63} + 1740q^{65} - 48q^{67} - 144q^{69} - 456q^{71} + 1230q^{73} - 279q^{75} - 336q^{77} - 1368q^{79} + 243q^{81} - 60q^{83} - 660q^{85} + 54q^{87} + 2742q^{89} - 42q^{91} - 216q^{93} - 648q^{95} + 1950q^{97} - 432q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 57 x - 148$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} - 16 \nu - 152$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 8 \beta_{1} + 152$$$$)/4$$

## 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
 −5.47374 −3.13924 8.61298
0 −3.00000 0 −12.9475 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 −8.27848 0 7.00000 0 9.00000 0
1.3 0 −3.00000 0 15.2260 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.bs 3
4.b odd 2 1 1344.4.a.bu 3
8.b even 2 1 672.4.a.r yes 3
8.d odd 2 1 672.4.a.p 3
24.f even 2 1 2016.4.a.u 3
24.h odd 2 1 2016.4.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.p 3 8.d odd 2 1
672.4.a.r yes 3 8.b even 2 1
1344.4.a.bs 3 1.a even 1 1 trivial
1344.4.a.bu 3 4.b odd 2 1
2016.4.a.u 3 24.f even 2 1
2016.4.a.v 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{3} + 6 T_{5}^{2} - 216 T_{5} - 1632$$ $$T_{11}^{3} + 48 T_{11}^{2} - 3060 T_{11} - 95488$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$-1632 - 216 T + 6 T^{2} + T^{3}$$
$7$ $$( -7 + T )^{3}$$
$11$ $$-95488 - 3060 T + 48 T^{2} + T^{3}$$
$13$ $$63656 - 6756 T + 6 T^{2} + T^{3}$$
$17$ $$24736 - 2856 T - 54 T^{2} + T^{3}$$
$19$ $$-200256 - 1872 T + 84 T^{2} + T^{3}$$
$23$ $$187648 - 4500 T - 48 T^{2} + T^{3}$$
$29$ $$4848088 - 57108 T + 18 T^{2} + T^{3}$$
$31$ $$2342912 - 30144 T - 72 T^{2} + T^{3}$$
$37$ $$-53576 + 7932 T + 210 T^{2} + T^{3}$$
$41$ $$4957664 + 18456 T - 414 T^{2} + T^{3}$$
$43$ $$-722944 - 17664 T + 168 T^{2} + T^{3}$$
$47$ $$-17124736 - 152400 T - 72 T^{2} + T^{3}$$
$53$ $$-7840072 - 7044 T + 402 T^{2} + T^{3}$$
$59$ $$11538688 - 232032 T + 540 T^{2} + T^{3}$$
$61$ $$-170037272 - 184788 T + 798 T^{2} + T^{3}$$
$67$ $$-44046592 - 349776 T + 48 T^{2} + T^{3}$$
$71$ $$-19646112 - 36756 T + 456 T^{2} + T^{3}$$
$73$ $$640250616 - 438948 T - 1230 T^{2} + T^{3}$$
$79$ $$-182717824 + 225456 T + 1368 T^{2} + T^{3}$$
$83$ $$-90712000 - 675600 T + 60 T^{2} + T^{3}$$
$89$ $$-524337504 + 2246616 T - 2742 T^{2} + T^{3}$$
$97$ $$-50085448 + 638748 T - 1950 T^{2} + T^{3}$$