# Properties

 Label 1584.4.a.bc Level $1584$ Weight $4$ Character orbit 1584.a Self dual yes Analytic conductor $93.459$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1584.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$93.4590254491$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta ) q^{5} + ( -10 - \beta ) q^{7} +O(q^{10})$$ $$q + ( -1 + 2 \beta ) q^{5} + ( -10 - \beta ) q^{7} -11 q^{11} + ( 40 + 5 \beta ) q^{13} + ( 62 + 3 \beta ) q^{17} + ( -36 + 15 \beta ) q^{19} + ( -49 + 9 \beta ) q^{23} + ( 68 - 4 \beta ) q^{25} + ( -72 - 14 \beta ) q^{29} + ( 17 + 7 \beta ) q^{31} + ( -86 - 19 \beta ) q^{35} + ( 27 + 2 \beta ) q^{37} + ( -268 - \beta ) q^{41} + ( 30 - 4 \beta ) q^{43} + ( -136 + 30 \beta ) q^{47} + ( -195 + 20 \beta ) q^{49} + ( 246 - 14 \beta ) q^{53} + ( 11 - 22 \beta ) q^{55} + ( 317 + 33 \beta ) q^{59} + ( 420 - 46 \beta ) q^{61} + ( 440 + 75 \beta ) q^{65} + ( -377 - 5 \beta ) q^{67} + ( -339 - 19 \beta ) q^{71} + ( -200 + 117 \beta ) q^{73} + ( 110 + 11 \beta ) q^{77} + ( -158 + 164 \beta ) q^{79} + ( 234 - 30 \beta ) q^{83} + ( 226 + 121 \beta ) q^{85} + ( 921 - 82 \beta ) q^{89} + ( -640 - 90 \beta ) q^{91} + ( 1476 - 87 \beta ) q^{95} + ( 1097 - 36 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 20q^{7} + O(q^{10})$$ $$2q - 2q^{5} - 20q^{7} - 22q^{11} + 80q^{13} + 124q^{17} - 72q^{19} - 98q^{23} + 136q^{25} - 144q^{29} + 34q^{31} - 172q^{35} + 54q^{37} - 536q^{41} + 60q^{43} - 272q^{47} - 390q^{49} + 492q^{53} + 22q^{55} + 634q^{59} + 840q^{61} + 880q^{65} - 754q^{67} - 678q^{71} - 400q^{73} + 220q^{77} - 316q^{79} + 468q^{83} + 452q^{85} + 1842q^{89} - 1280q^{91} + 2952q^{95} + 2194q^{97} + O(q^{100})$$

## 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 −14.8564 0 −3.07180 0 0 0
1.2 0 0 0 12.8564 0 −16.9282 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 1584.4.a.bc 2
3.b odd 2 1 176.4.a.i 2
4.b odd 2 1 99.4.a.c 2
12.b even 2 1 11.4.a.a 2
20.d odd 2 1 2475.4.a.q 2
24.f even 2 1 704.4.a.p 2
24.h odd 2 1 704.4.a.n 2
33.d even 2 1 1936.4.a.w 2
44.c even 2 1 1089.4.a.v 2
60.h even 2 1 275.4.a.b 2
60.l odd 4 2 275.4.b.c 4
84.h odd 2 1 539.4.a.e 2
132.d odd 2 1 121.4.a.c 2
132.n odd 10 4 121.4.c.f 8
132.o even 10 4 121.4.c.c 8
156.h even 2 1 1859.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 12.b even 2 1
99.4.a.c 2 4.b odd 2 1
121.4.a.c 2 132.d odd 2 1
121.4.c.c 8 132.o even 10 4
121.4.c.f 8 132.n odd 10 4
176.4.a.i 2 3.b odd 2 1
275.4.a.b 2 60.h even 2 1
275.4.b.c 4 60.l odd 4 2
539.4.a.e 2 84.h odd 2 1
704.4.a.n 2 24.h odd 2 1
704.4.a.p 2 24.f even 2 1
1089.4.a.v 2 44.c even 2 1
1584.4.a.bc 2 1.a even 1 1 trivial
1859.4.a.a 2 156.h even 2 1
1936.4.a.w 2 33.d even 2 1
2475.4.a.q 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1584))$$:

 $$T_{5}^{2} + 2 T_{5} - 191$$ $$T_{7}^{2} + 20 T_{7} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-191 + 2 T + T^{2}$$
$7$ $$52 + 20 T + T^{2}$$
$11$ $$( 11 + T )^{2}$$
$13$ $$400 - 80 T + T^{2}$$
$17$ $$3412 - 124 T + T^{2}$$
$19$ $$-9504 + 72 T + T^{2}$$
$23$ $$-1487 + 98 T + T^{2}$$
$29$ $$-4224 + 144 T + T^{2}$$
$31$ $$-2063 - 34 T + T^{2}$$
$37$ $$537 - 54 T + T^{2}$$
$41$ $$71776 + 536 T + T^{2}$$
$43$ $$132 - 60 T + T^{2}$$
$47$ $$-24704 + 272 T + T^{2}$$
$53$ $$51108 - 492 T + T^{2}$$
$59$ $$48217 - 634 T + T^{2}$$
$61$ $$74832 - 840 T + T^{2}$$
$67$ $$140929 + 754 T + T^{2}$$
$71$ $$97593 + 678 T + T^{2}$$
$73$ $$-617072 + 400 T + T^{2}$$
$79$ $$-1266044 + 316 T + T^{2}$$
$83$ $$11556 - 468 T + T^{2}$$
$89$ $$525489 - 1842 T + T^{2}$$
$97$ $$1141201 - 2194 T + T^{2}$$