# Properties

 Label 1344.4.a.bi 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: $$\Q(\sqrt{337})$$ Defining polynomial: $$x^{2} - x - 84$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{337}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 3 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 3 + \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 13 + \beta ) q^{11} + ( -48 - 2 \beta ) q^{13} + ( -9 - 3 \beta ) q^{15} + ( 39 - 3 \beta ) q^{17} + 20 q^{19} -21 q^{21} + ( -11 + 5 \beta ) q^{23} + ( 221 + 6 \beta ) q^{25} -27 q^{27} -102 q^{29} + ( -48 + 16 \beta ) q^{31} + ( -39 - 3 \beta ) q^{33} + ( 21 + 7 \beta ) q^{35} + ( -252 + 2 \beta ) q^{37} + ( 144 + 6 \beta ) q^{39} + ( -51 + 11 \beta ) q^{41} + ( 148 + 16 \beta ) q^{43} + ( 27 + 9 \beta ) q^{45} + ( 390 - 10 \beta ) q^{47} + 49 q^{49} + ( -117 + 9 \beta ) q^{51} + ( -96 - 18 \beta ) q^{53} + ( 376 + 16 \beta ) q^{55} -60 q^{57} + ( 106 + 14 \beta ) q^{59} + ( 50 + 16 \beta ) q^{61} + 63 q^{63} + ( -818 - 54 \beta ) q^{65} + ( 106 - 2 \beta ) q^{67} + ( 33 - 15 \beta ) q^{69} + ( 267 + 11 \beta ) q^{71} + ( 564 + 10 \beta ) q^{73} + ( -663 - 18 \beta ) q^{75} + ( 91 + 7 \beta ) q^{77} + ( -234 - 58 \beta ) q^{79} + 81 q^{81} + ( -412 - 48 \beta ) q^{83} + ( -894 + 30 \beta ) q^{85} + 306 q^{87} + ( -1059 + 27 \beta ) q^{89} + ( -336 - 14 \beta ) q^{91} + ( 144 - 48 \beta ) q^{93} + ( 60 + 20 \beta ) q^{95} + ( 200 + 70 \beta ) q^{97} + ( 117 + 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 6q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 6q^{5} + 14q^{7} + 18q^{9} + 26q^{11} - 96q^{13} - 18q^{15} + 78q^{17} + 40q^{19} - 42q^{21} - 22q^{23} + 442q^{25} - 54q^{27} - 204q^{29} - 96q^{31} - 78q^{33} + 42q^{35} - 504q^{37} + 288q^{39} - 102q^{41} + 296q^{43} + 54q^{45} + 780q^{47} + 98q^{49} - 234q^{51} - 192q^{53} + 752q^{55} - 120q^{57} + 212q^{59} + 100q^{61} + 126q^{63} - 1636q^{65} + 212q^{67} + 66q^{69} + 534q^{71} + 1128q^{73} - 1326q^{75} + 182q^{77} - 468q^{79} + 162q^{81} - 824q^{83} - 1788q^{85} + 612q^{87} - 2118q^{89} - 672q^{91} + 288q^{93} + 120q^{95} + 400q^{97} + 234q^{99} + O(q^{100})$$

## 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
 −8.67878 9.67878
0 −3.00000 0 −15.3576 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 21.3576 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.bi 2
4.b odd 2 1 1344.4.a.bq 2
8.b even 2 1 336.4.a.o 2
8.d odd 2 1 168.4.a.g 2
24.f even 2 1 504.4.a.n 2
24.h odd 2 1 1008.4.a.bg 2
56.e even 2 1 1176.4.a.w 2
56.h odd 2 1 2352.4.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.g 2 8.d odd 2 1
336.4.a.o 2 8.b even 2 1
504.4.a.n 2 24.f even 2 1
1008.4.a.bg 2 24.h odd 2 1
1176.4.a.w 2 56.e even 2 1
1344.4.a.bi 2 1.a even 1 1 trivial
1344.4.a.bq 2 4.b odd 2 1
2352.4.a.br 2 56.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{2} - 6 T_{5} - 328$$ $$T_{11}^{2} - 26 T_{11} - 168$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-328 - 6 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-168 - 26 T + T^{2}$$
$13$ $$956 + 96 T + T^{2}$$
$17$ $$-1512 - 78 T + T^{2}$$
$19$ $$( -20 + T )^{2}$$
$23$ $$-8304 + 22 T + T^{2}$$
$29$ $$( 102 + T )^{2}$$
$31$ $$-83968 + 96 T + T^{2}$$
$37$ $$62156 + 504 T + T^{2}$$
$41$ $$-38176 + 102 T + T^{2}$$
$43$ $$-64368 - 296 T + T^{2}$$
$47$ $$118400 - 780 T + T^{2}$$
$53$ $$-99972 + 192 T + T^{2}$$
$59$ $$-54816 - 212 T + T^{2}$$
$61$ $$-83772 - 100 T + T^{2}$$
$67$ $$9888 - 212 T + T^{2}$$
$71$ $$30512 - 534 T + T^{2}$$
$73$ $$284396 - 1128 T + T^{2}$$
$79$ $$-1078912 + 468 T + T^{2}$$
$83$ $$-606704 + 824 T + T^{2}$$
$89$ $$875808 + 2118 T + T^{2}$$
$97$ $$-1611300 - 400 T + T^{2}$$