# Properties

 Label 1152.4.a.t Level $1152$ Weight $4$ Character orbit 1152.a Self dual yes Analytic conductor $67.970$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1152.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.9702003266$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 128) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \beta ) q^{5} + ( 4 + 2 \beta ) q^{7} +O(q^{10})$$ $$q + ( 2 + 2 \beta ) q^{5} + ( 4 + 2 \beta ) q^{7} + ( -46 - \beta ) q^{11} + ( -50 + 6 \beta ) q^{13} + ( -46 - 12 \beta ) q^{17} + ( 2 + 7 \beta ) q^{19} + ( 4 + 18 \beta ) q^{23} + ( 71 + 8 \beta ) q^{25} + ( 42 + 10 \beta ) q^{29} + ( 192 + 16 \beta ) q^{31} + ( 200 + 12 \beta ) q^{35} + ( 86 + 14 \beta ) q^{37} + ( 150 - 8 \beta ) q^{41} + ( 150 + 5 \beta ) q^{43} + ( -8 + 44 \beta ) q^{47} + ( -135 + 16 \beta ) q^{49} + ( -6 - 14 \beta ) q^{53} + ( -188 - 94 \beta ) q^{55} + ( 322 - 33 \beta ) q^{59} + ( -146 + 70 \beta ) q^{61} + ( 476 - 88 \beta ) q^{65} + ( -86 + 83 \beta ) q^{67} + ( 204 - 42 \beta ) q^{71} + ( 206 - 52 \beta ) q^{73} + ( -280 - 96 \beta ) q^{77} + ( 200 + 36 \beta ) q^{79} + ( -474 + 13 \beta ) q^{83} + ( -1244 - 116 \beta ) q^{85} + ( -286 + 116 \beta ) q^{89} + ( 376 - 76 \beta ) q^{91} + ( 676 + 18 \beta ) q^{95} + ( 1102 + 92 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + 8q^{7} + O(q^{10})$$ $$2q + 4q^{5} + 8q^{7} - 92q^{11} - 100q^{13} - 92q^{17} + 4q^{19} + 8q^{23} + 142q^{25} + 84q^{29} + 384q^{31} + 400q^{35} + 172q^{37} + 300q^{41} + 300q^{43} - 16q^{47} - 270q^{49} - 12q^{53} - 376q^{55} + 644q^{59} - 292q^{61} + 952q^{65} - 172q^{67} + 408q^{71} + 412q^{73} - 560q^{77} + 400q^{79} - 948q^{83} - 2488q^{85} - 572q^{89} + 752q^{91} + 1352q^{95} + 2204q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 0 0 −11.8564 0 −9.85641 0 0 0
1.2 0 0 0 15.8564 0 17.8564 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$

## 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 1152.4.a.t 2
3.b odd 2 1 128.4.a.g yes 2
4.b odd 2 1 1152.4.a.s 2
8.b even 2 1 1152.4.a.r 2
8.d odd 2 1 1152.4.a.q 2
12.b even 2 1 128.4.a.e 2
24.f even 2 1 128.4.a.h yes 2
24.h odd 2 1 128.4.a.f yes 2
48.i odd 4 2 256.4.b.h 4
48.k even 4 2 256.4.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 12.b even 2 1
128.4.a.f yes 2 24.h odd 2 1
128.4.a.g yes 2 3.b odd 2 1
128.4.a.h yes 2 24.f even 2 1
256.4.b.h 4 48.i odd 4 2
256.4.b.i 4 48.k even 4 2
1152.4.a.q 2 8.d odd 2 1
1152.4.a.r 2 8.b even 2 1
1152.4.a.s 2 4.b odd 2 1
1152.4.a.t 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1152))$$:

 $$T_{5}^{2} - 4 T_{5} - 188$$ $$T_{7}^{2} - 8 T_{7} - 176$$ $$T_{13}^{2} + 100 T_{13} + 772$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-188 - 4 T + T^{2}$$
$7$ $$-176 - 8 T + T^{2}$$
$11$ $$2068 + 92 T + T^{2}$$
$13$ $$772 + 100 T + T^{2}$$
$17$ $$-4796 + 92 T + T^{2}$$
$19$ $$-2348 - 4 T + T^{2}$$
$23$ $$-15536 - 8 T + T^{2}$$
$29$ $$-3036 - 84 T + T^{2}$$
$31$ $$24576 - 384 T + T^{2}$$
$37$ $$-2012 - 172 T + T^{2}$$
$41$ $$19428 - 300 T + T^{2}$$
$43$ $$21300 - 300 T + T^{2}$$
$47$ $$-92864 + 16 T + T^{2}$$
$53$ $$-9372 + 12 T + T^{2}$$
$59$ $$51412 - 644 T + T^{2}$$
$61$ $$-213884 + 292 T + T^{2}$$
$67$ $$-323276 + 172 T + T^{2}$$
$71$ $$-43056 - 408 T + T^{2}$$
$73$ $$-87356 - 412 T + T^{2}$$
$79$ $$-22208 - 400 T + T^{2}$$
$83$ $$216564 + 948 T + T^{2}$$
$89$ $$-564092 + 572 T + T^{2}$$
$97$ $$808132 - 2204 T + T^{2}$$