# Properties

 Label 588.4.a.h Level $588$ Weight $4$ Character orbit 588.a Self dual yes Analytic conductor $34.693$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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 $$\beta = \frac{1}{2}(-1 + 3\sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 2 + \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 2 + \beta ) q^{5} + 9 q^{9} + ( -26 - \beta ) q^{11} + ( -33 - 5 \beta ) q^{13} + ( 6 + 3 \beta ) q^{15} + ( -16 - 8 \beta ) q^{17} + ( -85 - \beta ) q^{19} -96 q^{23} + ( 7 + 3 \beta ) q^{25} + 27 q^{27} + ( -28 - 17 \beta ) q^{29} + ( 51 + 10 \beta ) q^{31} + ( -78 - 3 \beta ) q^{33} + ( -77 + 19 \beta ) q^{37} + ( -99 - 15 \beta ) q^{39} + ( -70 + 34 \beta ) q^{41} + ( -239 + 19 \beta ) q^{43} + ( 18 + 9 \beta ) q^{45} + ( -98 - 16 \beta ) q^{47} + ( -48 - 24 \beta ) q^{51} + ( 156 + 27 \beta ) q^{53} + ( -180 - 27 \beta ) q^{55} + ( -255 - 3 \beta ) q^{57} + ( 636 + 3 \beta ) q^{59} + ( -146 + 36 \beta ) q^{61} + ( -706 - 38 \beta ) q^{65} + ( -433 + 9 \beta ) q^{67} -288 q^{69} + ( -686 + 32 \beta ) q^{71} + ( 691 + 21 \beta ) q^{73} + ( 21 + 9 \beta ) q^{75} + ( -93 - 4 \beta ) q^{79} + 81 q^{81} + ( 178 - 43 \beta ) q^{83} + ( -1056 - 24 \beta ) q^{85} + ( -84 - 51 \beta ) q^{87} + ( -400 + 22 \beta ) q^{89} + ( 153 + 30 \beta ) q^{93} + ( -298 - 86 \beta ) q^{95} + ( -410 + 21 \beta ) q^{97} + ( -234 - 9 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 3q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 3q^{5} + 18q^{9} - 51q^{11} - 61q^{13} + 9q^{15} - 24q^{17} - 169q^{19} - 192q^{23} + 11q^{25} + 54q^{27} - 39q^{29} + 92q^{31} - 153q^{33} - 173q^{37} - 183q^{39} - 174q^{41} - 497q^{43} + 27q^{45} - 180q^{47} - 72q^{51} + 285q^{53} - 333q^{55} - 507q^{57} + 1269q^{59} - 328q^{61} - 1374q^{65} - 875q^{67} - 576q^{69} - 1404q^{71} + 1361q^{73} + 33q^{75} - 182q^{79} + 162q^{81} + 399q^{83} - 2088q^{85} - 117q^{87} - 822q^{89} + 276q^{93} - 510q^{95} - 841q^{97} - 459q^{99} + O(q^{100})$$

## 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
 −3.27492 4.27492
0 3.00000 0 −9.82475 0 0 0 9.00000 0
1.2 0 3.00000 0 12.8248 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.h 2
3.b odd 2 1 1764.4.a.p 2
4.b odd 2 1 2352.4.a.bp 2
7.b odd 2 1 588.4.a.g 2
7.c even 3 2 588.4.i.i 4
7.d odd 6 2 84.4.i.b 4
21.c even 2 1 1764.4.a.x 2
21.g even 6 2 252.4.k.d 4
21.h odd 6 2 1764.4.k.z 4
28.d even 2 1 2352.4.a.cb 2
28.f even 6 2 336.4.q.h 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3 T_{5} - 126$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-126 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$522 + 51 T + T^{2}$$
$13$ $$-2276 + 61 T + T^{2}$$
$17$ $$-8064 + 24 T + T^{2}$$
$19$ $$7012 + 169 T + T^{2}$$
$23$ $$( 96 + T )^{2}$$
$29$ $$-36684 + 39 T + T^{2}$$
$31$ $$-10709 - 92 T + T^{2}$$
$37$ $$-38816 + 173 T + T^{2}$$
$41$ $$-140688 + 174 T + T^{2}$$
$43$ $$15454 + 497 T + T^{2}$$
$47$ $$-24732 + 180 T + T^{2}$$
$53$ $$-73188 - 285 T + T^{2}$$
$59$ $$401436 - 1269 T + T^{2}$$
$61$ $$-139316 + 328 T + T^{2}$$
$67$ $$181018 + 875 T + T^{2}$$
$71$ $$361476 + 1404 T + T^{2}$$
$73$ $$406522 - 1361 T + T^{2}$$
$79$ $$6229 + 182 T + T^{2}$$
$83$ $$-197334 - 399 T + T^{2}$$
$89$ $$106848 + 822 T + T^{2}$$
$97$ $$120262 + 841 T + T^{2}$$