# Properties

 Label 588.4.a.f Level $588$ Weight $4$ Character orbit 588.a Self dual yes Analytic conductor $34.693$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ Defining polynomial: $$x^{2} - x - 48$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{193})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( -5 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( -5 - \beta ) q^{5} + 9 q^{9} + ( 1 - 7 \beta ) q^{11} -5 \beta q^{13} + ( 15 + 3 \beta ) q^{15} + ( -52 + 4 \beta ) q^{17} + ( -32 - 3 \beta ) q^{19} + ( 28 + 20 \beta ) q^{23} + ( -52 + 11 \beta ) q^{25} -27 q^{27} + ( 143 - 11 \beta ) q^{29} + ( -171 - 20 \beta ) q^{31} + ( -3 + 21 \beta ) q^{33} + ( 20 - 45 \beta ) q^{37} + 15 \beta q^{39} + ( 72 + 18 \beta ) q^{41} + ( 362 - 3 \beta ) q^{43} + ( -45 - 9 \beta ) q^{45} + ( 126 - 36 \beta ) q^{47} + ( 156 - 12 \beta ) q^{51} + ( 243 + 9 \beta ) q^{53} + ( 331 + 41 \beta ) q^{55} + ( 96 + 9 \beta ) q^{57} + ( -113 + 53 \beta ) q^{59} + ( 246 + 40 \beta ) q^{61} + ( 240 + 30 \beta ) q^{65} + ( 94 - 77 \beta ) q^{67} + ( -84 - 60 \beta ) q^{69} + ( 778 + 44 \beta ) q^{71} + ( -634 + 53 \beta ) q^{73} + ( 156 - 33 \beta ) q^{75} + ( 699 + 62 \beta ) q^{79} + 81 q^{81} + ( 755 - 101 \beta ) q^{83} + ( 68 + 28 \beta ) q^{85} + ( -429 + 33 \beta ) q^{87} + ( 966 + 42 \beta ) q^{89} + ( 513 + 60 \beta ) q^{93} + ( 304 + 50 \beta ) q^{95} + ( -295 + 29 \beta ) q^{97} + ( 9 - 63 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} - 11q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} - 11q^{5} + 18q^{9} - 5q^{11} - 5q^{13} + 33q^{15} - 100q^{17} - 67q^{19} + 76q^{23} - 93q^{25} - 54q^{27} + 275q^{29} - 362q^{31} + 15q^{33} - 5q^{37} + 15q^{39} + 162q^{41} + 721q^{43} - 99q^{45} + 216q^{47} + 300q^{51} + 495q^{53} + 703q^{55} + 201q^{57} - 173q^{59} + 532q^{61} + 510q^{65} + 111q^{67} - 228q^{69} + 1600q^{71} - 1215q^{73} + 279q^{75} + 1460q^{79} + 162q^{81} + 1409q^{83} + 164q^{85} - 825q^{87} + 1974q^{89} + 1086q^{93} + 658q^{95} - 561q^{97} - 45q^{99} + O(q^{100})$$

## 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
 7.44622 −6.44622
0 −3.00000 0 −12.4462 0 0 0 9.00000 0
1.2 0 −3.00000 0 1.44622 0 0 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 588.4.a.f 2
3.b odd 2 1 1764.4.a.y 2
4.b odd 2 1 2352.4.a.bx 2
7.b odd 2 1 588.4.a.i 2
7.c even 3 2 588.4.i.j 4
7.d odd 6 2 84.4.i.a 4
21.c even 2 1 1764.4.a.o 2
21.g even 6 2 252.4.k.f 4
21.h odd 6 2 1764.4.k.q 4
28.d even 2 1 2352.4.a.bt 2
28.f even 6 2 336.4.q.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 7.d odd 6 2
252.4.k.f 4 21.g even 6 2
336.4.q.i 4 28.f even 6 2
588.4.a.f 2 1.a even 1 1 trivial
588.4.a.i 2 7.b odd 2 1
588.4.i.j 4 7.c even 3 2
1764.4.a.o 2 21.c even 2 1
1764.4.a.y 2 3.b odd 2 1
1764.4.k.q 4 21.h odd 6 2
2352.4.a.bt 2 28.d even 2 1
2352.4.a.bx 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 11 T_{5} - 18$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-18 + 11 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2358 + 5 T + T^{2}$$
$13$ $$-1200 + 5 T + T^{2}$$
$17$ $$1728 + 100 T + T^{2}$$
$19$ $$688 + 67 T + T^{2}$$
$23$ $$-17856 - 76 T + T^{2}$$
$29$ $$13068 - 275 T + T^{2}$$
$31$ $$13461 + 362 T + T^{2}$$
$37$ $$-97700 + 5 T + T^{2}$$
$41$ $$-9072 - 162 T + T^{2}$$
$43$ $$129526 - 721 T + T^{2}$$
$47$ $$-50868 - 216 T + T^{2}$$
$53$ $$57348 - 495 T + T^{2}$$
$59$ $$-128052 + 173 T + T^{2}$$
$61$ $$-6444 - 532 T + T^{2}$$
$67$ $$-282994 - 111 T + T^{2}$$
$71$ $$546588 - 1600 T + T^{2}$$
$73$ $$233522 + 1215 T + T^{2}$$
$79$ $$347427 - 1460 T + T^{2}$$
$83$ $$4122 - 1409 T + T^{2}$$
$89$ $$889056 - 1974 T + T^{2}$$
$97$ $$38102 + 561 T + T^{2}$$