# Properties

 Label 1344.4.a.bh Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ Defining polynomial: $$x^{2} - x - 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 2\sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 2 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 2 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 24 - \beta ) q^{11} + ( 26 - 4 \beta ) q^{13} + ( -6 - 3 \beta ) q^{15} + ( -6 + 7 \beta ) q^{17} + ( 80 - 2 \beta ) q^{19} -21 q^{21} + ( -24 - 5 \beta ) q^{23} + ( 27 + 4 \beta ) q^{25} -27 q^{27} + ( 190 - 6 \beta ) q^{29} + ( -92 + 18 \beta ) q^{31} + ( -72 + 3 \beta ) q^{33} + ( 14 + 7 \beta ) q^{35} + ( -42 + 20 \beta ) q^{37} + ( -78 + 12 \beta ) q^{39} + ( 26 - 3 \beta ) q^{41} + ( -192 + 16 \beta ) q^{43} + ( 18 + 9 \beta ) q^{45} + ( -32 - 38 \beta ) q^{47} + 49 q^{49} + ( 18 - 21 \beta ) q^{51} + ( 326 + 36 \beta ) q^{53} + ( -100 + 22 \beta ) q^{55} + ( -240 + 6 \beta ) q^{57} + ( 212 - 22 \beta ) q^{59} + ( 530 - 6 \beta ) q^{61} + 63 q^{63} + ( -540 + 18 \beta ) q^{65} + ( -656 + 30 \beta ) q^{67} + ( 72 + 15 \beta ) q^{69} + ( -336 - 51 \beta ) q^{71} + ( -542 + 2 \beta ) q^{73} + ( -81 - 12 \beta ) q^{75} + ( 168 - 7 \beta ) q^{77} + ( 368 + 38 \beta ) q^{79} + 81 q^{81} + ( -140 + 40 \beta ) q^{83} + ( 1024 + 8 \beta ) q^{85} + ( -570 + 18 \beta ) q^{87} + ( 474 - 87 \beta ) q^{89} + ( 182 - 28 \beta ) q^{91} + ( 276 - 54 \beta ) q^{93} + ( -136 + 76 \beta ) q^{95} + ( 402 - 26 \beta ) q^{97} + ( 216 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 4q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 4q^{5} + 14q^{7} + 18q^{9} + 48q^{11} + 52q^{13} - 12q^{15} - 12q^{17} + 160q^{19} - 42q^{21} - 48q^{23} + 54q^{25} - 54q^{27} + 380q^{29} - 184q^{31} - 144q^{33} + 28q^{35} - 84q^{37} - 156q^{39} + 52q^{41} - 384q^{43} + 36q^{45} - 64q^{47} + 98q^{49} + 36q^{51} + 652q^{53} - 200q^{55} - 480q^{57} + 424q^{59} + 1060q^{61} + 126q^{63} - 1080q^{65} - 1312q^{67} + 144q^{69} - 672q^{71} - 1084q^{73} - 162q^{75} + 336q^{77} + 736q^{79} + 162q^{81} - 280q^{83} + 2048q^{85} - 1140q^{87} + 948q^{89} + 364q^{91} + 552q^{93} - 272q^{95} + 804q^{97} + 432q^{99} + O(q^{100})$$

## 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
 −2.54138 3.54138
0 −3.00000 0 −10.1655 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 14.1655 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.bh 2
4.b odd 2 1 1344.4.a.bp 2
8.b even 2 1 672.4.a.k yes 2
8.d odd 2 1 672.4.a.f 2
24.f even 2 1 2016.4.a.m 2
24.h odd 2 1 2016.4.a.n 2

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

 $$T_{5}^{2} - 4 T_{5} - 144$$ $$T_{11}^{2} - 48 T_{11} + 428$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-144 - 4 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$428 - 48 T + T^{2}$$
$13$ $$-1692 - 52 T + T^{2}$$
$17$ $$-7216 + 12 T + T^{2}$$
$19$ $$5808 - 160 T + T^{2}$$
$23$ $$-3124 + 48 T + T^{2}$$
$29$ $$30772 - 380 T + T^{2}$$
$31$ $$-39488 + 184 T + T^{2}$$
$37$ $$-57436 + 84 T + T^{2}$$
$41$ $$-656 - 52 T + T^{2}$$
$43$ $$-1024 + 384 T + T^{2}$$
$47$ $$-212688 + 64 T + T^{2}$$
$53$ $$-85532 - 652 T + T^{2}$$
$59$ $$-26688 - 424 T + T^{2}$$
$61$ $$275572 - 1060 T + T^{2}$$
$67$ $$297136 + 1312 T + T^{2}$$
$71$ $$-272052 + 672 T + T^{2}$$
$73$ $$293172 + 1084 T + T^{2}$$
$79$ $$-78288 - 736 T + T^{2}$$
$83$ $$-217200 + 280 T + T^{2}$$
$89$ $$-895536 - 948 T + T^{2}$$
$97$ $$61556 - 804 T + T^{2}$$
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