# Properties

 Label 1344.4.a.bt Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## 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: 3.3.22700.1 Defining polynomial: $$x^{3} - x^{2} - 28 x + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 3 + \beta_{2} ) q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 3 + \beta_{2} ) q^{5} -7 q^{7} + 9 q^{9} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{11} + ( 16 + \beta_{1} + \beta_{2} ) q^{13} + ( -9 - 3 \beta_{2} ) q^{15} + ( 9 - \beta_{1} + 4 \beta_{2} ) q^{17} + ( -48 - \beta_{1} + 5 \beta_{2} ) q^{19} + 21 q^{21} + ( 17 + 5 \beta_{1} ) q^{23} + ( 105 + 5 \beta_{1} + 5 \beta_{2} ) q^{25} -27 q^{27} + ( -98 - 5 \beta_{1} + \beta_{2} ) q^{29} + ( -24 - 7 \beta_{1} - \beta_{2} ) q^{31} + ( 3 - 3 \beta_{1} - 6 \beta_{2} ) q^{33} + ( -21 - 7 \beta_{2} ) q^{35} + ( -8 + 3 \beta_{1} + 11 \beta_{2} ) q^{37} + ( -48 - 3 \beta_{1} - 3 \beta_{2} ) q^{39} + ( 127 - 5 \beta_{1} + 14 \beta_{2} ) q^{41} + ( -264 + 8 \beta_{2} ) q^{43} + ( 27 + 9 \beta_{2} ) q^{45} + ( 82 + 2 \beta_{2} ) q^{47} + 49 q^{49} + ( -27 + 3 \beta_{1} - 12 \beta_{2} ) q^{51} + ( 11 \beta_{1} - 21 \beta_{2} ) q^{53} + ( 440 + 15 \beta_{1} + 25 \beta_{2} ) q^{55} + ( 144 + 3 \beta_{1} - 15 \beta_{2} ) q^{57} + ( -494 - 4 \beta_{1} - 14 \beta_{2} ) q^{59} + ( 258 - 5 \beta_{1} + 41 \beta_{2} ) q^{61} -63 q^{63} + ( 270 + 10 \beta_{1} + 40 \beta_{2} ) q^{65} + ( -422 - 6 \beta_{2} ) q^{67} + ( -51 - 15 \beta_{1} ) q^{69} + ( 63 - 25 \beta_{1} - 4 \beta_{2} ) q^{71} + ( -384 - 20 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -315 - 15 \beta_{1} - 15 \beta_{2} ) q^{75} + ( 7 - 7 \beta_{1} - 14 \beta_{2} ) q^{77} + ( 278 - 8 \beta_{1} + 14 \beta_{2} ) q^{79} + 81 q^{81} + ( -520 + 10 \beta_{1} - 14 \beta_{2} ) q^{83} + ( 910 + 15 \beta_{1} - 5 \beta_{2} ) q^{85} + ( 294 + 15 \beta_{1} - 3 \beta_{2} ) q^{87} + ( -53 + 5 \beta_{1} - 24 \beta_{2} ) q^{89} + ( -112 - 7 \beta_{1} - 7 \beta_{2} ) q^{91} + ( 72 + 21 \beta_{1} + 3 \beta_{2} ) q^{93} + ( 960 + 20 \beta_{1} - 60 \beta_{2} ) q^{95} + ( -36 + 24 \beta_{1} - 46 \beta_{2} ) q^{97} + ( -9 + 9 \beta_{1} + 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 9q^{3} + 10q^{5} - 21q^{7} + 27q^{9} + O(q^{10})$$ $$3q - 9q^{3} + 10q^{5} - 21q^{7} + 27q^{9} + 50q^{13} - 30q^{15} + 30q^{17} - 140q^{19} + 63q^{21} + 56q^{23} + 325q^{25} - 81q^{27} - 298q^{29} - 80q^{31} - 70q^{35} - 10q^{37} - 150q^{39} + 390q^{41} - 784q^{43} + 90q^{45} + 248q^{47} + 147q^{49} - 90q^{51} - 10q^{53} + 1360q^{55} + 420q^{57} - 1500q^{59} + 810q^{61} - 189q^{63} + 860q^{65} - 1272q^{67} - 168q^{69} + 160q^{71} - 1170q^{73} - 975q^{75} + 840q^{79} + 243q^{81} - 1564q^{83} + 2740q^{85} + 894q^{87} - 178q^{89} - 350q^{91} + 240q^{93} + 2840q^{95} - 130q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 28 x + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{2} + 7 \nu + 17$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - \beta_{1} + 150$$$$)/8$$

## 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
 0.424864 −5.03475 5.60988
0 −3.00000 0 −15.3946 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 4.31394 0 −7.00000 0 9.00000 0
1.3 0 −3.00000 0 21.0807 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.bt 3
4.b odd 2 1 1344.4.a.bv 3
8.b even 2 1 672.4.a.q yes 3
8.d odd 2 1 672.4.a.o 3
24.f even 2 1 2016.4.a.z 3
24.h odd 2 1 2016.4.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.o 3 8.d odd 2 1
672.4.a.q yes 3 8.b even 2 1
1344.4.a.bt 3 1.a even 1 1 trivial
1344.4.a.bv 3 4.b odd 2 1
2016.4.a.y 3 24.h odd 2 1
2016.4.a.z 3 24.f even 2 1

## Hecke kernels

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

 $$T_{5}^{3} - 10 T_{5}^{2} - 300 T_{5} + 1400$$ $$T_{11}^{3} - 2840 T_{11} - 45280$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 3 + T )^{3}$$
$5$ $$1400 - 300 T - 10 T^{2} + T^{3}$$
$7$ $$( 7 + T )^{3}$$
$11$ $$-45280 - 2840 T + T^{3}$$
$13$ $$26920 - 980 T - 50 T^{2} + T^{3}$$
$17$ $$275880 - 6380 T - 30 T^{2} + T^{3}$$
$19$ $$6080 - 3120 T + 140 T^{2} + T^{3}$$
$23$ $$3285792 - 35288 T - 56 T^{2} + T^{3}$$
$29$ $$-5325704 - 6932 T + 298 T^{2} + T^{3}$$
$31$ $$-8700160 - 69600 T + 80 T^{2} + T^{3}$$
$37$ $$-4978760 - 54260 T + 10 T^{2} + T^{3}$$
$41$ $$21358600 - 49100 T - 390 T^{2} + T^{3}$$
$43$ $$12439552 + 183552 T + 784 T^{2} + T^{3}$$
$47$ $$-452096 + 19168 T - 248 T^{2} + T^{3}$$
$53$ $$-32747080 - 316660 T + 10 T^{2} + T^{3}$$
$59$ $$90301120 + 659920 T + 1500 T^{2} + T^{3}$$
$61$ $$276011400 - 372500 T - 810 T^{2} + T^{3}$$
$67$ $$71066624 + 527328 T + 1272 T^{2} + T^{3}$$
$71$ $$-258709600 - 907800 T - 160 T^{2} + T^{3}$$
$73$ $$-337981800 - 125300 T + 1170 T^{2} + T^{3}$$
$79$ $$29596160 + 79840 T - 840 T^{2} + T^{3}$$
$83$ $$33022272 + 608432 T + 1564 T^{2} + T^{3}$$
$89$ $$-53515224 - 214572 T + 178 T^{2} + T^{3}$$
$97$ $$-400247720 - 1507380 T + 130 T^{2} + T^{3}$$
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