# Properties

 Label 1344.4.a.bf Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ 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 $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -4 + \beta ) q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -4 + \beta ) q^{5} -7 q^{7} + 9 q^{9} + ( 22 - 7 \beta ) q^{11} + ( 30 + 8 \beta ) q^{13} + ( 12 - 3 \beta ) q^{15} + ( 24 + 13 \beta ) q^{17} + ( 40 - 2 \beta ) q^{19} + 21 q^{21} + ( -70 + 3 \beta ) q^{23} + ( -65 - 8 \beta ) q^{25} -27 q^{27} + ( 30 - 2 \beta ) q^{29} + ( -116 - 2 \beta ) q^{31} + ( -66 + 21 \beta ) q^{33} + ( 28 - 7 \beta ) q^{35} + ( 90 - 32 \beta ) q^{37} + ( -90 - 24 \beta ) q^{39} + ( -172 - 37 \beta ) q^{41} + ( -112 + 40 \beta ) q^{43} + ( -36 + 9 \beta ) q^{45} + ( -156 + 50 \beta ) q^{47} + 49 q^{49} + ( -72 - 39 \beta ) q^{51} + ( 10 + 72 \beta ) q^{53} + ( -396 + 50 \beta ) q^{55} + ( -120 + 6 \beta ) q^{57} + ( 32 + 70 \beta ) q^{59} + ( -102 - 26 \beta ) q^{61} -63 q^{63} + ( 232 - 2 \beta ) q^{65} + ( 436 - 6 \beta ) q^{67} + ( 210 - 9 \beta ) q^{69} + ( -82 - 115 \beta ) q^{71} + ( 750 + 22 \beta ) q^{73} + ( 195 + 24 \beta ) q^{75} + ( -154 + 49 \beta ) q^{77} + ( -820 - 74 \beta ) q^{79} + 81 q^{81} + ( 1100 + 40 \beta ) q^{83} + ( 476 - 28 \beta ) q^{85} + ( -90 + 6 \beta ) q^{87} + ( -132 + 123 \beta ) q^{89} + ( -210 - 56 \beta ) q^{91} + ( 348 + 6 \beta ) q^{93} + ( -248 + 48 \beta ) q^{95} + ( 1046 + 58 \beta ) q^{97} + ( 198 - 63 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 8q^{5} - 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 8q^{5} - 14q^{7} + 18q^{9} + 44q^{11} + 60q^{13} + 24q^{15} + 48q^{17} + 80q^{19} + 42q^{21} - 140q^{23} - 130q^{25} - 54q^{27} + 60q^{29} - 232q^{31} - 132q^{33} + 56q^{35} + 180q^{37} - 180q^{39} - 344q^{41} - 224q^{43} - 72q^{45} - 312q^{47} + 98q^{49} - 144q^{51} + 20q^{53} - 792q^{55} - 240q^{57} + 64q^{59} - 204q^{61} - 126q^{63} + 464q^{65} + 872q^{67} + 420q^{69} - 164q^{71} + 1500q^{73} + 390q^{75} - 308q^{77} - 1640q^{79} + 162q^{81} + 2200q^{83} + 952q^{85} - 180q^{87} - 264q^{89} - 420q^{91} + 696q^{93} - 496q^{95} + 2092q^{97} + 396q^{99} + O(q^{100})$$

## 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
 −3.31662 3.31662
0 −3.00000 0 −10.6332 0 −7.00000 0 9.00000 0
1.2 0 −3.00000 0 2.63325 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.bf 2
4.b odd 2 1 1344.4.a.bn 2
8.b even 2 1 672.4.a.l yes 2
8.d odd 2 1 672.4.a.g 2
24.f even 2 1 2016.4.a.l 2
24.h odd 2 1 2016.4.a.k 2

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

 $$T_{5}^{2} + 8 T_{5} - 28$$ $$T_{11}^{2} - 44 T_{11} - 1672$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-28 + 8 T + T^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$-1672 - 44 T + T^{2}$$
$13$ $$-1916 - 60 T + T^{2}$$
$17$ $$-6860 - 48 T + T^{2}$$
$19$ $$1424 - 80 T + T^{2}$$
$23$ $$4504 + 140 T + T^{2}$$
$29$ $$724 - 60 T + T^{2}$$
$31$ $$13280 + 232 T + T^{2}$$
$37$ $$-36956 - 180 T + T^{2}$$
$41$ $$-30652 + 344 T + T^{2}$$
$43$ $$-57856 + 224 T + T^{2}$$
$47$ $$-85664 + 312 T + T^{2}$$
$53$ $$-227996 - 20 T + T^{2}$$
$59$ $$-214576 - 64 T + T^{2}$$
$61$ $$-19340 + 204 T + T^{2}$$
$67$ $$188512 - 872 T + T^{2}$$
$71$ $$-575176 + 164 T + T^{2}$$
$73$ $$541204 - 1500 T + T^{2}$$
$79$ $$431456 + 1640 T + T^{2}$$
$83$ $$1139600 - 2200 T + T^{2}$$
$89$ $$-648252 + 264 T + T^{2}$$
$97$ $$946100 - 2092 T + T^{2}$$