# Properties

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

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

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