# Properties

 Label 1344.4.a.bo 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: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{57}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( -3 - \beta ) q^{5} -7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( -3 - \beta ) q^{5} -7 q^{7} + 9 q^{9} + ( -3 + 5 \beta ) q^{11} + ( -8 + 6 \beta ) q^{13} + ( -9 - 3 \beta ) q^{15} + ( -3 - \beta ) q^{17} + ( 32 - 12 \beta ) q^{19} -21 q^{21} + ( -3 + 17 \beta ) q^{23} + ( -59 + 6 \beta ) q^{25} + 27 q^{27} + ( 126 - 12 \beta ) q^{29} + ( -20 - 36 \beta ) q^{31} + ( -9 + 15 \beta ) q^{33} + ( 21 + 7 \beta ) q^{35} + ( 124 + 18 \beta ) q^{37} + ( -24 + 18 \beta ) q^{39} + ( -225 - 15 \beta ) q^{41} + ( 188 - 24 \beta ) q^{43} + ( -27 - 9 \beta ) q^{45} + ( 6 - 34 \beta ) q^{47} + 49 q^{49} + ( -9 - 3 \beta ) q^{51} + ( 552 - 2 \beta ) q^{53} + ( -276 - 12 \beta ) q^{55} + ( 96 - 36 \beta ) q^{57} + ( 402 - 58 \beta ) q^{59} + ( 214 - 36 \beta ) q^{61} -63 q^{63} + ( -318 - 10 \beta ) q^{65} + ( 74 + 54 \beta ) q^{67} + ( -9 + 51 \beta ) q^{69} + ( -477 + 15 \beta ) q^{71} + ( 536 + 6 \beta ) q^{73} + ( -177 + 18 \beta ) q^{75} + ( 21 - 35 \beta ) q^{77} + ( 286 + 54 \beta ) q^{79} + 81 q^{81} + ( 972 + 48 \beta ) q^{83} + ( 66 + 6 \beta ) q^{85} + ( 378 - 36 \beta ) q^{87} + ( 183 - 71 \beta ) q^{89} + ( 56 - 42 \beta ) q^{91} + ( -60 - 108 \beta ) q^{93} + ( 588 + 4 \beta ) q^{95} + ( 404 + 138 \beta ) q^{97} + ( -27 + 45 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} - 6q^{5} - 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} - 6q^{5} - 14q^{7} + 18q^{9} - 6q^{11} - 16q^{13} - 18q^{15} - 6q^{17} + 64q^{19} - 42q^{21} - 6q^{23} - 118q^{25} + 54q^{27} + 252q^{29} - 40q^{31} - 18q^{33} + 42q^{35} + 248q^{37} - 48q^{39} - 450q^{41} + 376q^{43} - 54q^{45} + 12q^{47} + 98q^{49} - 18q^{51} + 1104q^{53} - 552q^{55} + 192q^{57} + 804q^{59} + 428q^{61} - 126q^{63} - 636q^{65} + 148q^{67} - 18q^{69} - 954q^{71} + 1072q^{73} - 354q^{75} + 42q^{77} + 572q^{79} + 162q^{81} + 1944q^{83} + 132q^{85} + 756q^{87} + 366q^{89} + 112q^{91} - 120q^{93} + 1176q^{95} + 808q^{97} - 54q^{99} + O(q^{100})$$

## 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
 4.27492 −3.27492
0 3.00000 0 −10.5498 0 −7.00000 0 9.00000 0
1.2 0 3.00000 0 4.54983 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.bo 2
4.b odd 2 1 1344.4.a.bg 2
8.b even 2 1 336.4.a.m 2
8.d odd 2 1 21.4.a.c 2
24.f even 2 1 63.4.a.e 2
24.h odd 2 1 1008.4.a.ba 2
40.e odd 2 1 525.4.a.n 2
40.k even 4 2 525.4.d.g 4
56.e even 2 1 147.4.a.i 2
56.h odd 2 1 2352.4.a.bz 2
56.k odd 6 2 147.4.e.l 4
56.m even 6 2 147.4.e.m 4
120.m even 2 1 1575.4.a.p 2
168.e odd 2 1 441.4.a.r 2
168.v even 6 2 441.4.e.q 4
168.be odd 6 2 441.4.e.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 8.d odd 2 1
63.4.a.e 2 24.f even 2 1
147.4.a.i 2 56.e even 2 1
147.4.e.l 4 56.k odd 6 2
147.4.e.m 4 56.m even 6 2
336.4.a.m 2 8.b even 2 1
441.4.a.r 2 168.e odd 2 1
441.4.e.p 4 168.be odd 6 2
441.4.e.q 4 168.v even 6 2
525.4.a.n 2 40.e odd 2 1
525.4.d.g 4 40.k even 4 2
1008.4.a.ba 2 24.h odd 2 1
1344.4.a.bg 2 4.b odd 2 1
1344.4.a.bo 2 1.a even 1 1 trivial
1575.4.a.p 2 120.m even 2 1
2352.4.a.bz 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{2} + 6 T_{5} - 48$$ $$T_{11}^{2} + 6 T_{11} - 1416$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-48 + 6 T + T^{2}$$
$7$ $$( 7 + T )^{2}$$
$11$ $$-1416 + 6 T + T^{2}$$
$13$ $$-1988 + 16 T + T^{2}$$
$17$ $$-48 + 6 T + T^{2}$$
$19$ $$-7184 - 64 T + T^{2}$$
$23$ $$-16464 + 6 T + T^{2}$$
$29$ $$7668 - 252 T + T^{2}$$
$31$ $$-73472 + 40 T + T^{2}$$
$37$ $$-3092 - 248 T + T^{2}$$
$41$ $$37800 + 450 T + T^{2}$$
$43$ $$2512 - 376 T + T^{2}$$
$47$ $$-65856 - 12 T + T^{2}$$
$53$ $$304476 - 1104 T + T^{2}$$
$59$ $$-30144 - 804 T + T^{2}$$
$61$ $$-28076 - 428 T + T^{2}$$
$67$ $$-160736 - 148 T + T^{2}$$
$71$ $$214704 + 954 T + T^{2}$$
$73$ $$285244 - 1072 T + T^{2}$$
$79$ $$-84416 - 572 T + T^{2}$$
$83$ $$813456 - 1944 T + T^{2}$$
$89$ $$-253848 - 366 T + T^{2}$$
$97$ $$-922292 - 808 T + T^{2}$$